Mathematical Concepts of Quantum Mechanics [3 ed.] 9783030595616, 9783030595623

Table of contents :
Preface
Preface to the second edition
Preface to the enlarged second printing
From the preface to the first edition
Contents
1 Physical Background
1.1 The Double-Slit Experiment
1.2 Wave Functions
1.3 State Space
1.4 The Schrӧdinger Equation
1.5 Classical Limit
2 Dynamics
2.1 Conservation of Probability
2.2 Self-adjointness
2.3 Existence of Dynamics
2.4 The Free Propagator
2.5 Semi-classical Approximation
3 Observables
3.1 The Position and Momentum Operators
3.2 General Observables
3.3 The Heisenberg Representation
3.4 Conservation Laws
3.5 Conserved Currents
4 Quantization
4.1 Quantization
4.2 Quantization and Correspondence Principle
4.3 A Particle in an External Electro-magnetic Field
4.4 Spin
4.5 Many-particle Systems
4.6 Identical Particles
4.7 Supplement: Hamiltonian Formulation of Classical Mechanics
5 Uncertainty Principle and Stability of Atoms
and Molecules
5.1 The Heisenberg Uncertainty Principle
5.2 A Refined Uncertainty Principle
5.3 Application: Stability of Atoms and Molecules
6 Spectrum and Dynamics
6.1 The Spectrum of an Operator
6.2 Spectrum and Measurement Outcomes
6.3 Classification of Spectra
6.4 Bound and Decaying States
6.5 Spectra of Schrӧdinger Operators
6.6 Particle in a Periodic Potential
6.7 Angular Momentum
7
Special Cases and Extensions
7.1 The Square Well and Torus
7.2 Motion in a Spherically Symmetric Potential
7.3 The Hydrogen Atom
7.4 The Harmonic Oscillator
7.5 A Particle in a Constant Magnetic Field
7.6 Aharonov-Bohm Effect
7.7 Linearized Ginzburg-Landau Equations of Superconductivity
7.8 Ideal Quantum Gas and Ground States of Atoms
7.9 Supplement: L−equivariant functions
8 Bound States and Variational Principle
8.1 Variational Characterization of Eigenvalues
8.2 Exponential Decay of Bound States
8.3 Number of Bound States
9 Scattering States
9.1 Short-range Interactions: μ > 1
9.2 Long-range Interactions: μ ≤ 1
9.3 Wave Operators
9.4 Appendix: The Potential Step and Square Well
10 Existence of Atoms and Molecules
10.1 Essential Spectra of Atoms and Molecules
10.2 Bound States of Atoms and BO Molecules
10.3 Open Problems
11 Perturbation Theory: Feshbach-Schur Method
11.1 The Feshbach-Schur Method
11.2 The Zeeman Effect
11.3 Time-Dependent Perturbations
11.4 Appendix: Proof of Theorem 11.1
12 Born-Oppenheimer Approximation and Adiabatic Dynamics
12.1 Problem and Heuristics
12.2 Stationary Born-Oppenheimer Approximation
12.3 Complex ψy and Gauge Fields
12.4 Time-dependent Born-Oppenheimer Approximation
12.5 Adiabatic Motion
12.6 Geometrical Phases
12.7 Appendix: Projecting-out Procedure
12.8 Appendix: Proof of Theorem 12.11
13
General Theory of Many-particle Systems
13.1 Many-particle Schrӧdinger Operators
13.2 Separation of the Centre-of-mass Motion
13.3 Break-ups
13.4 The HVZ Theorem
13.5 Intra- vs. Inter-cluster Motion
13.6 Exponential Decay of Bound States
13.7 Remarks on Discrete Spectrum
13.8 Scattering States
14 Self-consistent Approximations
14.1 Hartree, Hartree-Fock and Gross-Pitaevski equations
14.2 Appendix: BEC at T=0
15 The Feynman Path Integral
15.1 The Feynman Path Integral
15.2 Generalizations of the Path Integral
15.3 Mathematical Supplement: the Trotter Product Formula
16 Semi-classical Analysis
16.1 Semi-classical Asymptotics of the Propagator
16.2 Semi-classical Asymptotics of Green’s Function
16.2.1 Appendix
16.3 Bohr-Sommerfeld Semi-classical Quantization
16.4 Semi-classical Asymptotics for the Ground State Energy
16.5 Mathematical Supplement: The Action of the Critical Path
16.6 Appendix: Connection to Geodesics
17 Resonances
17.1 Complex Deformation and Resonances
17.2 Tunneling and Resonances
17.3 The Free Resonance Energy
17.4 Instantons
17.5 Positive Temperatures
17.6 Pre-exponential Factor for the Bounce
17.7 Contribution of the Zero-mode
17.8 Bohr-Sommerfeld Quantization for Resonances
18 Quantum Statistics
18.1 Density Matrices
18.2 Quantum Statistics: General Framework
18.3 Stationary States
18.4 Hilbert Space Approach
18.5 Semi-classical Limit
18.6 Generalized Hartree-Fock and Kohn-Sham Equations
19 Open Quantum Systems
19.1 Information Reduction
19.2 Reduced dynamics
19.3 Some Proofs
19.4 Communication Channels
19.5 Quantum Dynamical Semigroups
19.6 Irreversibility
19.7 Decoherence and Thermalization
20 The Second Quantization
20.1 Fock Space and Creation and Annihilation Operators
20.2 Many-body Hamiltonian
20.3 Evolution of Quantum Fields
20.4 Relation to Quantum Harmonic Oscillator
20.5 Scalar Fermions
20.6 Mean Field Regime
20.7 Appendix: the Ideal Bose Gas
20.7.1 Bose-Einstein Condensation
21 Quantum Electro-Magnetic Field - Photons
21.1 Klein-Gordon Classical Field Theory
21.1.1 Principle of minimum action
21.1.2 Hamiltonians
21.1.3 Hamiltonian System
21.1.4 Complexification of the Klein-Gordon Equation
21.2 Quantization of the Klein-Gordon Equation
21.3 The Gaussian Spaces
21.4 Wick Quantization
21.5 Fock Space
21.6 Quantization of Maxwell’s Equations
22 Standard Model of Non-relativistic Matter and Radiation
22.1 Classical Particle System Interacting with an Electro-magnetic Field
22.2 Quantum Hamiltonian of Non-relativistic QED
22.2.1 Translation invariance
22.2.2 Fiber decomposition with respect to total momentum
22.3 Rescaling and decoupling scalar and vector
potentials
22.3.1 Self-adjointness of H(ε)
22.4 Mass Renormalization
22.5 Appendix: Relative bound on I(ε) and Pull-through Formulae
23
Theory of Radiation
23.1 Spectrum of the Uncoupled System
23.2 Complex Deformations and Resonances
23.3 Results
23.4 Idea of the proof of Theorem 23.1
23.5 Generalized Pauli-Fierz Transformation
23.6 Elimination of Particle and High Photon Energy Degrees of Freedom
23.7 The Hamiltonian H0(ε, z)
23.8 Estimates on the operator H0(ε, z)
23.9 Ground state of H(ε)
23.10 Appendix: Estimates on Iε and HPρˆ(ε)
23.11 Appendix: Key Bound
24 Renormalization Group
24.1 Main Result
24.2 A Banach Space of Operators
24.3 The Decimation Map
24.4 The Renormalization Map
24.5 Dynamics of RG and Spectra of Hamiltonians
24.6 Supplement: Group Property of Rρ
25
Mathematical Supplement: Elements of Operator Theory
25.1 Spaces
25.2 Operators on Hilbert Spaces
25.3 Integral Operators
25.4 Inverses and their Estimates
25.5 Self-adjointness
25.6 Exponential of an Operator
25.7 Projections
25.8 The Spectrum of an Operator
25.9 Functions of Operators and the Spectral Mapping Theorem
25.10 Weyl Sequences and Weyl Spectrum
25.11 The Trace, and Trace Class Operators
25.12 Operator Determinants
25.13 Tensor Products
25.14 The Fourier Transform
26
Mathematical Supplement: The Calculus of Variations
26.1 Functionals
26.2 The First Variation and Critical Points
26.3 The Second Variation
26.4 Conjugate Points and Jacobi Fields
26.5 Constrained Variational Problems
26.6 Legendre Transform and Poisson Bracket
26.7 Complex Hamiltonian Systems
26.8 Conservation Laws
27
Comments on Literature, and Further Reading
References
Index

Citation preview

Universitext

Stephen J. Gustafson Israel Michael Sigal

Mathematical Concepts of Quantum Mechanics Third Edition

Universitext Series Editors Sheldon Axler, San Francisco, USA Carles Casacuberta, Barcelona, Spain John Greenlees, Coventry, UK Angus MacIntyre, London, UK Kenneth Ribet, Berkeley, USA Claude Sabbah, Palaiseau, France Endre Süli, Oxford, UK Wojbor A. Woyczyński, Cleveland, USA

Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master’s level and beyond. The books, often well classtested by their author, may have an informal, personal even experimental approach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teaching curricula, into very polished texts. Thus as research topics trickle down into graduate-level teaching, first textbooks written for new, cutting-edge courses may make their way into Universitext. More information about this series at http://www.springer.com/series/223

Stephen J. Gustafson · Israel Michael Sigal

Mathematical Concepts of Quantum Mechanics 3rd edition

Stephen J. Gustafson Department of Mathematics University of British Columbia Vancouver, Canada

Israel Michael Sigal Department of Mathematics University of Toronto Toronto, Canada

ISSN 0172-5939 ISSN 2191-6675  (electronic) Universitext ISBN 978-3-030-59561-6 ISBN 978-3-030-59562-3  (eBook) https://doi.org/10.1007/978-3-030-59562-3 Mathematics Subject Classification: 81S22, 81Q10, 81Q15, 81S40, 81S30, 81Q20, 81U05, 81U10, 81T17, 35Q40, 35Q70, 35P25, 35P15, 47A40, 47A11, 47A40, 47A75 © Springer-Verlag GmbH Germany, part of Springer Nature 2003, 2011, 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Preface to the third edition In this edition we have expanded the elementary part of the book to make it self-contained, and have added new intermediate level material related to recent developments. This new material fits nicely into the general structure of Quantum Mechanics, as well as to our book. In particular, on the intermediate level, we added sections on the timedependent Born-Oppenheimer approximation, adiabatic theory, geometrical phases, Aharonov-Bohm effect and density functional theory. We also added a sections on quantum open systems, expanded some others, and organized the relevant sections into a separate chapter under this name. This chapter develops some fundamental concepts lying at the heart of quantum information theory, presently perhaps the fastest growing area of physics. It also comes closest to issues of foundations, which are not considered in this book. However, we refer to an excellent recent paper [116] for an in-depth discussion of the issues involved, and many references. As in the previous editions, we tried to stay at the most elementary mathematical level possible, and did not pursue generalizations and mathematical questions arising naturally in the subject. The latter is done in the excellent books of F. Strocchi ([285]), L. Takhtajan ([287]), G. Teschl ([288]), J. Dimock ([79]), B. C. Hall, ([153]) and G. Dell’Antonio ([76]). Consequently, prerequisites for this edition are the same as for the previous ones: introductory real and complex analysis and elementary differential equations. This book could be used for senior level undergraduate, as well as graduate, courses in both mathematics and physics departments. The new material makes the book more flexible as a source in designing courses guided by different interests and needs. This edition could be used for introductory, intermediate and advanced courses; some of the sections would serve for introductions to geometrical methods in Quantum Mechanics, to quantum information theory and to quantum electrodynamics/field theory (Sections 7.5-7.7, 7.9, 12.4; Chapters 18-19 and 20-24, respectively).

VI

Preface

Acknowledgment: The authors are grateful to R. Frank, M. Lemm, B. Nachtergaele and S. Teufel for reading parts of the new material and making many pertinent remarks. Vancouver/Toronto, May 2020

Stephen Gustafson Israel Michael Sigal

Preface to the second edition One of the main goals motivating this new edition was to enhance the elementary material. To this end, in addition to some rewriting and reorganization, several new sections have been added (covering, for example, spin, and conservation laws), resulting in a fairly complete coverage of elementary topics. A second main goal was to address the key physical issues of stability of atoms and molecules, and mean-field approximations of large particle systems. This is reflected in new chapters covering the existence of atoms and molecules, mean-field theory, and second quantization. Our final goal was to update the advanced material with a view toward reflecting current developments, and this led to a complete revision and reorganization of the material on the theory of radiation (non-relativistic quantum electrodynamics), as well as the addition of a new chapter. In this edition we have also added a number of proofs, which were omitted in the previous editions. As a result, this book could be used for senior level undergraduate, as well as graduate, courses in both mathematics and physics departments. Prerequisites for this book are introductory real analysis (notions of vector space, scalar product, norm, convergence, Fourier transform) and complex analysis, the theory of Lebesgue integration, and elementary differential equations. These topics are typically covered by the third year in mathematics departments. The first and third topics are also familiar to physics undergraduates. However, even in dealing with mathematics students we have found it useful, if not necessary, to review these notions, as needed for the course. Hence, to make the book relatively self-contained, we briefly cover these subjects, with the exception of Lebesgue integration. Those unfamiliar with the latter can think about Lebesgue integrals as if they were Riemann integrals. This said, the pace of the book is not a leisurely one and requires, at least for beginners, some amount of work. Though, as in the previous two issues of the book, we tried to increase the complexity of the material gradually, we were not always successful, and first in Chapter 13, and then in Chapter 20, and especially in Chapter 21, there is a leap in the level of sophistication required from the reader. One may say the book proceeds at three levels. The first one, covering Chapters 1 -11, is elementary; the second one, covering Chapters 13 - 18, is intermediate; and the last one, covering Chapters 20 - 24, advanced.

Preface

VII

During the last few years since the enlarged second printing of this book, there have appeared four books on Quantum Mechanics directed at mathematicians: F. Strocchi, An Introduction to the Mathematical Structure of Quantum Mechanics: a Short Course for Mathematicians. World Scientific, 2005. L. Takhtajan, Quantum Mechanics for Mathematicians. AMS, 2008. L.D. Faddeev, O.A. Yakubovskii, Lectures on Quantum Mechanics for Mathematics Students. With an appendix by Leon Takhtajan. AMS, 2009. J. Dimock, Quantum Mechanics and Quantum Field Theory. Cambridge Univ. Press, 2011. These elegant and valuable texts have considerably different aims and rather limited overlap with the present book. In fact, they complement it nicely. Acknowledgment: The authors are grateful to I. Anapolitanos, Th. Chen, J. Faupin, Z. Gang, G.-M. Graf, M. Griesemer, L. Jonsson, M. Merkli, M. M¨ uck, Yu. Ovchinnikov, A. Soffer, F. Ting, T. Tzaneteas, and especially J. Fr¨ohlich, W. Hunziker and V. Buslaev for useful discussions, and to J. Feldman, G.-M. Graf, I. Herbst, L. Jonsson, E. Lieb, B. Simon and F. Ting for reading parts of the manuscript and making useful remarks. Vancouver/Toronto, May 2011

Stephen Gustafson Israel Michael Sigal

Preface to the enlarged second printing For the second printing, we corrected a few misprints and inaccuracies; for some help with this, we are indebted to B. Nachtergaele. We have also added a small amount of new material. In particular, Chapter 11, on perturbation theory via the Feshbach method, is new, as are the short sub-sections 14.1 and 14.2 concerning the Hartree approximation and Bose-Einstein condensation. We also note a change in terminology, from “point” and “continuous” spectrum, to the mathematically more standard “discrete” and “essential” spectrum, starting in Chapter 6. Vancouver/Toronto, July 2005

Stephen Gustafson Israel Michael Sigal

VIII

Preface

From the preface to the first edition The first fifteen chapters of these lectures (omitting four to six chapters each year) cover a one term course taken by a mixed group of senior undergraduate and junior graduate students specializing either in mathematics or physics. Typically, the mathematics students have some background in advanced analysis, while the physics students have had introductory quantum mechanics. To satisfy such a disparate audience, we decided to select material which is interesting from the viewpoint of modern theoretical physics, and which illustrates an interplay of ideas from various fields of mathematics such as operator theory, probability, differential equations, and differential geometry. Given our time constraint, we have often pursued mathematical content at the expense of rigor. However, wherever we have sacrificed the latter, we have tried to explain whether the result is an established fact, or, mathematically speaking, a conjecture, and in the former case, how a given argument can be made rigorous. The present book retains these features.

Vancouver/Toronto, Sept. 2002

Stephen Gustafson Israel Michael Sigal

Contents

1

Physical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Double-Slit Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 State Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The Schr¨odinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Classical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 4 5 5 7

2

Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Conservation of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Self-adjointness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Existence of Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Free Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Semi-classical Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 12 16 19 20

3

Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Position and Momentum Operators . . . . . . . . . . . . . . . . . . . . 3.2 General Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Heisenberg Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conserved Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 23 24 26 27 30

4

Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Quantization and Correspondence Principle . . . . . . . . . . . . . . . . . 4.3 A Particle in an External Electro-magnetic Field . . . . . . . . . . . . 4.4 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Many-particle Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Identical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Supplement: Hamiltonian Formulation of Classical Mechanics .

33 33 35 38 40 42 44 47

X

Contents

5

Uncertainty Principle and Stability of Atoms and Molecules 5.1 The Heisenberg Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . 5.2 A Refined Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Application: Stability of Atoms and Molecules . . . . . . . . . . . . . .

53 53 54 56

6

Spectrum and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Spectrum of an Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Spectrum and Measurement Outcomes . . . . . . . . . . . . . . . . . . . . . 6.3 Classification of Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Bound and Decaying States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Spectra of Schr¨odinger Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Particle in a Periodic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 59 60 62 63 66 71 74

7

Special Cases and Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 7.1 The Square Well and Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 7.2 Motion in a Spherically Symmetric Potential . . . . . . . . . . . . . . . . 81 7.3 The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.4 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 7.5 A Particle in a Constant Magnetic Field . . . . . . . . . . . . . . . . . . . . 87 7.6 Aharonov-Bohm Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 7.7 Linearized Ginzburg-Landau Equations of Superconductivity . . 95 7.8 Ideal Quantum Gas and Ground States of Atoms . . . . . . . . . . . . 97 7.9 Supplement: L−equivariant functions . . . . . . . . . . . . . . . . . . . . . . 100

8

Bound States and Variational Principle . . . . . . . . . . . . . . . . . . . . 103 8.1 Variational Characterization of Eigenvalues . . . . . . . . . . . . . . . . . 103 8.2 Exponential Decay of Bound States . . . . . . . . . . . . . . . . . . . . . . . . 109 8.3 Number of Bound States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

9

Scattering States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 9.1 Short-range Interactions: µ > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 9.2 Long-range Interactions: µ ≤ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 9.3 Wave Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 9.4 Appendix: The Potential Step and Square Well . . . . . . . . . . . . . . 124

10 Existence of Atoms and Molecules . . . . . . . . . . . . . . . . . . . . . . . . . 127 10.1 Essential Spectra of Atoms and Molecules . . . . . . . . . . . . . . . . . . 127 10.2 Bound States of Atoms and BO Molecules . . . . . . . . . . . . . . . . . . 129 10.3 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 11 Perturbation Theory: Feshbach-Schur Method . . . . . . . . . . . . . 135 11.1 The Feshbach-Schur Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 11.2 The Zeeman Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 11.3 Time-Dependent Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 11.4 Appendix: Proof of Theorem 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Contents

XI

12 Born-Oppenheimer Approximation and Adiabatic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 12.1 Problem and Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 12.2 Stationary Born-Oppenheimer Approximation . . . . . . . . . . . . . . . 152 12.3 Complex ψy and Gauge Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 12.4 Time-dependent Born-Oppenheimer Approximation . . . . . . . . . . 160 12.5 Adiabatic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 12.6 Geometrical Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 12.7 Appendix: Projecting-out Procedure . . . . . . . . . . . . . . . . . . . . . . . 167 12.8 Appendix: Proof of Theorem 12.11 . . . . . . . . . . . . . . . . . . . . . . . . . 167 13 General Theory of Many-particle Systems . . . . . . . . . . . . . . . . . 171 13.1 Many-particle Schr¨odinger Operators . . . . . . . . . . . . . . . . . . . . . . 171 13.2 Separation of the Centre-of-mass Motion . . . . . . . . . . . . . . . . . . . 173 13.3 Break-ups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 13.4 The HVZ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 13.5 Intra- vs. Inter-cluster Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 13.6 Exponential Decay of Bound States . . . . . . . . . . . . . . . . . . . . . . . . 180 13.7 Remarks on Discrete Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 13.8 Scattering States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 14 Self-consistent Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 14.1 Hartree, Hartree-Fock and Gross-Pitaevski equations . . . . . . . . 185 14.2 Appendix: BEC at T=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 15 The Feynman Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 15.1 The Feynman Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 15.2 Generalizations of the Path Integral . . . . . . . . . . . . . . . . . . . . . . . 198 15.3 Mathematical Supplement: the Trotter Product Formula . . . . . 199 16 Semi-classical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 16.1 Semi-classical Asymptotics of the Propagator . . . . . . . . . . . . . . . 202 16.2 Semi-classical Asymptotics of Green’s Function . . . . . . . . . . . . . . 206 16.2.1 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 16.3 Bohr-Sommerfeld Semi-classical Quantization . . . . . . . . . . . . . . . 209 16.4 Semi-classical Asymptotics for the Ground State Energy . . . . . . 211 16.5 Mathematical Supplement: The Action of the Critical Path . . . 213 16.6 Appendix: Connection to Geodesics . . . . . . . . . . . . . . . . . . . . . . . . 216 17 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 17.1 Complex Deformation and Resonances . . . . . . . . . . . . . . . . . . . . . 219 17.2 Tunneling and Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 17.3 The Free Resonance Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 17.4 Instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 17.5 Positive Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 17.6 Pre-exponential Factor for the Bounce . . . . . . . . . . . . . . . . . . . . . 230

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17.7 Contribution of the Zero-mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 17.8 Bohr-Sommerfeld Quantization for Resonances . . . . . . . . . . . . . . 232 18 Quantum Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 18.1 Density Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 18.2 Quantum Statistics: General Framework . . . . . . . . . . . . . . . . . . . . 239 18.3 Stationary States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 18.4 Hilbert Space Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 18.5 Semi-classical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 18.6 Generalized Hartree-Fock and Kohn-Sham Equations . . . . . . . . 248 19 Open Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 19.1 Information Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 19.2 Reduced dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 19.3 Some Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 19.4 Communication Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 19.5 Quantum Dynamical Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . 267 19.6 Irreversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 19.7 Decoherence and Thermalization . . . . . . . . . . . . . . . . . . . . . . . . . . 274 20 The Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 20.1 Fock Space and Creation and Annihilation Operators . . . . . . . . 277 20.2 Many-body Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 20.3 Evolution of Quantum Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 20.4 Relation to Quantum Harmonic Oscillator . . . . . . . . . . . . . . . . . 283 20.5 Scalar Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 20.6 Mean Field Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 20.7 Appendix: the Ideal Bose Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 20.7.1 Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . . . 291 21 Quantum Electro-Magnetic Field - Photons . . . . . . . . . . . . . . . . 295 21.1 Klein-Gordon Classical Field Theory . . . . . . . . . . . . . . . . . . . . . . . 295 21.1.1 Principle of minimum action . . . . . . . . . . . . . . . . . . . . . . . . 295 21.1.2 Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 21.1.3 Hamiltonian System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 21.1.4 Complexification of the Klein-Gordon Equation . . . . . . . 299 21.2 Quantization of the Klein-Gordon Equation . . . . . . . . . . . . . . . . . 300 21.3 The Gaussian Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 21.4 Wick Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 21.5 Fock Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 21.6 Quantization of Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . 309

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XIII

22 Standard Model of Non-relativistic Matter and Radiation . 315 22.1 Classical Particle System Interacting with an Electro-magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 22.2 Quantum Hamiltonian of Non-relativistic QED . . . . . . . . . . . . . . 318 22.2.1 Translation invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 22.2.2 Fiber decomposition with respect to total momentum . . 321 22.3 Rescaling and decoupling scalar and vector potentials . . . . . . . 323 22.3.1 Self-adjointness of H(ε) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 22.4 Mass Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 22.5 Appendix: Relative bound on I(ε) and Pull-through Formulae . 328 23 Theory of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 23.1 Spectrum of the Uncoupled System . . . . . . . . . . . . . . . . . . . . . . . . 332 23.2 Complex Deformations and Resonances . . . . . . . . . . . . . . . . . . . . 333 23.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 23.4 Idea of the proof of Theorem 23.1 . . . . . . . . . . . . . . . . . . . . . . . . . 337 23.5 Generalized Pauli-Fierz Transformation . . . . . . . . . . . . . . . . . . . . 338 23.6 Elimination of Particle and High Photon Energy Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 341 23.7 The Hamiltonian H0 (ε, z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 23.8 Estimates on the operator H0 (ε, z) . . . . . . . . . . . . . . . . . . . . . . . . 345 23.9 Ground state of H(ε) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 23.10Appendix: Estimates on Iε and HP¯ρ (ε) . . . . . . . . . . . . . . . . . . . . . 348 23.11Appendix: Key Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 24 Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 24.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 24.2 A Banach Space of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 24.3 The Decimation Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 24.4 The Renormalization Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 24.5 Dynamics of RG and Spectra of Hamiltonians . . . . . . . . . . . . . . . 361 24.6 Supplement: Group Property of Rρ . . . . . . . . . . . . . . . . . . . . . . . . 365 25 Mathematical Supplement: Elements of Operator Theory . . 367 25.1 Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 25.2 Operators on Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 25.3 Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 25.4 Inverses and their Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 25.5 Self-adjointness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 25.6 Exponential of an Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 25.7 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 25.8 The Spectrum of an Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 25.9 Functions of Operators and the Spectral Mapping Theorem . . . 388 25.10Weyl Sequences and Weyl Spectrum . . . . . . . . . . . . . . . . . . . . . . . 391 25.11The Trace, and Trace Class Operators . . . . . . . . . . . . . . . . . . . . . 397 25.12Operator Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 25.13Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

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25.14The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 26 Mathematical Supplement: The Calculus of Variations . . . . . 409 26.1 Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 26.2 The First Variation and Critical Points . . . . . . . . . . . . . . . . . . . . . 411 26.3 The Second Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 26.4 Conjugate Points and Jacobi Fields . . . . . . . . . . . . . . . . . . . . . . . . 416 26.5 Constrained Variational Problems . . . . . . . . . . . . . . . . . . . . . . . . . 420 26.6 Legendre Transform and Poisson Bracket . . . . . . . . . . . . . . . . . . . 421 26.7 Complex Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 26.8 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 27 Comments on Literature, and Further Reading . . . . . . . . . . . . 429 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

1 Physical Background

The main ingredients of any physical theory are the state space, the dynamical law giving the evolution of states, and the description of the results of measurements. The mathematical description of these structures is extracted from experiments. We describe here the foundational experiments and ideas of quantum mechanics. The starting point of quantum mechanics was Planck’s idea that electromagnetic radiation is emitted and absorbed in discrete amounts – quanta. Einstein ventured further by suggesting that the electro-magnetic radiation itself consists of quanta, or particles, which were then named photons. These were the first quantum particles and the first glimpse of wave-particle duality. Then came Bohr’s model of an atom, with electrons moving on fixed orbits and jumping from orbit to orbit without going through intermediate states. This culminated first in Heisenberg, and then in Schr¨odinger quantum mechanics, with the next stage incorporating quantum electro-magnetic radiation accomplished by Jordan, Pauli, Heisenberg, Born, Dirac and Fermi. To complete this thumbnail sketch we mention two dramatic experiments. The first one was conducted by E. Rutherford in 1911, and it established the planetary model of an atom with practically all its weight concentrated in a tiny nucleus (10−13 − 10−12 cm) at the center and with electrons orbiting around it. The electrons are attracted to the nucleus and repelled by each other via the Coulomb forces. The size of an atom, i.e. the size of electron orbits, is about 10−8 cm. The problem is that in classical physics this model is unstable. The second experiment is the scattering of electrons off a crystal conducted by Davisson and Germer (1927), G.P. Thomson (1928) and Rupp (1928), after the advent of quantum mechanics. This experiment is similar to Young’s 1805 experiment confirming the wave nature of light. It can be abstracted as the double-slit experiment described below. It displays an interference pattern for electrons, similar to that of waves. In this introductory chapter, we present a very brief overview of the basic structure of quantum mechanics, and touch on the physical motivation for © Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_1

2

1 Physical Background

the theory. A detailed mathematical discussion of quantum mechanics is the focus of the subsequent chapters.

1.1 The Double-Slit Experiment Suppose a stream of electrons is fired at a shield in which two narrow slits have been cut (see Fig. 1.1.) On the other side of the shield is a detector screen.

shield

electron

slits

gun screen Fig. 1.1. Experimental set-up. Each electron that passes through the shield hits the detector screen at some point, and these points of contact are recorded. Pictured in Fig. 1.2 and Fig. 1.3 are the intensity distributions observed on the screen when either of the slits is blocked.

P1 (brightness)

Fig.1.2. First slit blocked.

1.1 The Double-Slit Experiment

3

P2

Fig.1.3. Second slit blocked. When both slits are open, the observed intensity distribution is shown in Fig. 1.4.

P = P1 + P 2

Fig.1.4. Both slits open. Remarkably, this is not the sum of the previous two distributions; i.e., P 6= P1 + P2 . We make some observations based on this experiment. 1. We cannot predict exactly where a given electron will hit the screen, we can only determine the distribution of locations. 2. The intensity pattern (called an interference pattern) we observe when both slits are open is similar to the pattern we see when a wave propagates through the slits: the intensity observed when waves E1 and E2 (the waves here are represented by complex numbers encoding the amplitude and

4

1 Physical Background

phase) originating at each slit are combined is proportional to |E1 +E2 |2 6= |E1 |2 + |E2 |2 (see Fig. 1.5).

Fig.1.5. Wave interference. We can draw some conclusions based on these observations. 1. Matter behaves in a random way. 2. Matter exhibits wave-like properties. In other words, the behaviour of individual electrons is intrinsically random, and this randomness propagates according to laws of wave mechanics. These observations form a central part of the paradigm shift introduced by the theory of quantum mechanics.

1.2 Wave Functions In quantum mechanics, the state of a particle is described by a complex-valued function of position and time, ψ(x, t), x ∈ R3 , t ∈ R. This is called a wave function (or state vector). Here Rd denotes d-dimensional Euclidean space, R = R1 , and a vector x ∈ Rd can be written in coordinates as x = (x1 , . . . , xd ) with xj ∈ R. In light of the above discussion, the wave function should have the following properties: 1. |ψ(·, t)|2 is the probability distribution for the particle’s position. That is, that a particle is in the region RΩ ⊂ R3 at time t is R the probability 2 |ψ(x, t)| dx. Thus we require the normalization R3 |ψ(x, t)|2 dx = 1. Ω 2. ψ satisfies some sort of wave equation. For example, in the double-slit experiment, if ψ1 gives the state beyond the shield with the first slit closed, and ψ2 gives the state beyond the shield with

1.4 The Schr¨ odinger Equation

5

the second slit closed, then ψ = ψ1 + ψ2 describes the state with both slits open. The interference pattern observed in the latter case reflects the fact that |ψ|2 6= |ψ1 |2 + |ψ2 |2 .

1.3 State Space The space of all possible states of the particle at a given time is called the state space. For us, the state space of a particle will usually be the square-integrable functions: Z |ψ(x)|2 dx < ∞} L2 (R3 ) := {ψ : R3 → C | R3

(we can impose the normalization condition as needed). This is a vector space, and has an inner-product given by Z ¯ ψ(x)φ(x)dx. hψ, φi := R3

In fact, it is a Hilbert space (see Section 25.2 for precise definitions and mathematical details).

1.4 The Schr¨ odinger Equation We now give a motivation for the equation which governs the evolution of a particle’s wave function. This is the celebrated Schr¨ odinger equation. An evolving state at time t is denoted by ψ(x, t), with the notation ψ(t)(x) ≡ ψ(x, t). Our equation should satisfy certain physically sensible properties: 1. Causality: The state ψ(t0 ) at time t = t0 should determine the state ψ(t) for all later times t > t0 . 2. Superposition principle: If ψ(t) and φ(t) are evolutions of states, then αψ(t) + βφ(t) (α, β constants) should also describe the evolution of a state. 3. Correspondence principle: In “everyday situations,” quantum mechanics should be close to the classical mechanics we are used to. The first requirement means that ψ should satisfy an equation which is firstorder in time, namely ∂ ψ = Aψ (1.1) ∂t for some operator A, acting on the state space. The second requirement implies that A must be a linear operator. We use the third requirement – the correspondence principle – in order to find the correct form of A. Here we are guided by an analogy with the transition from wave optics to geometrical optics.

6

1 Physical Background

Wave Optics

→ Geometrical Optics

l

l

Quantum Mechanics → Classical Mechanics In everyday experience we see light propagating along straight lines in accordance with the laws of geometrical optics, i.e., along the characteristics of the equation ∂φ = ±c|∇x φ| (c = speed of light), (1.2) ∂t known as the eikonal equation. On the other hand we know that light, like electro-magnetic radiation in general, obeys Maxwell’s equations which can be reduced to the wave equation (say, for the electric field in the complex representation) ∂2u = c2 ∆u, (1.3) ∂t2 P3 2 where ∆ = j=1 ∂j is the Laplace operator, or the Laplacian (in spatial dimension three). The eikonal equation appears as a high frequency limit of the wave equation when the wave length is much smaller than the typical size of objects. iφ Namely we set u = ae λ , where a and φ are real and O(1) and λ > 0 is the ratio of the typical wave length to the typical size of objects. The real function φ is called the eikonal. Substitute this into (1.3) to obtain a ¨ + 2iλ−1 a˙ φ˙ − λ−2 aφ˙ 2 + iλ−1 aφ¨ = c2 (∆a + 2iλ−1 ∇a · ∇φ − λ−2 a|∇φ|2 + λ−1 a∆φ) (where dots denote derivatives with respect to t). In the short wave approximation, λ ≪ 1 (with derivatives of a and φ O(1)), we obtain −aφ˙ 2 = −c2 a|∇φ|2 , which is equivalent to the eikonal equation (1.2). An equation in classical mechanics analogous to the eikonal equation is the Hamilton-Jacobi equation ∂S = −h(x, ∇S), ∂t

(1.4)

where h(x, k) is the classical Hamiltonian function, which for a particle of mass 1 m moving in a potential V is given by h(x, k) = 2m |k|2 + V (x), and S(x, t) is the classical action. We would like to find an evolution equation which would lead to the Hamilton-Jacobi equation in the way the wave equation led to the eikonal one. We look for a solution to equation (1.1) in the form ψ(x, t) = a(x, t)eiS(x,t)/~ , where S(x, t) satisfies the Hamilton-Jacobi equation

1.5 Classical Limit

7

(1.4) and ~ is a parameter with the dimensions of action, small compared to a typical classical action for the system in question. Assuming a is independent of ~, it is easy to show that, to leading order, ψ then satisfies the equation i~

∂ ~2 ψ(x, t) = − ∆x ψ(x, t) + V (x)ψ(x, t). ∂t 2m

(1.5)

This equation is of the desired form (1.1). In fact it is the correct equation, and is called the Schr¨ odinger equation. The small constant ~ is Planck’s constant; it is one of the fundamental constants in nature. For the record, its value is roughly ~ ≈ 6.6255 × 10−27 erg sec. The equation (1.5) can be written as i~

∂ ψ = Hψ ∂t

(1.6)

where the linear operator H, called a Schr¨ odinger operator, is given by Hψ := −

~2 ∆ψ + V ψ. 2m

Example 1.1 Here are just a few examples of potentials. 1. 2. 3. 4. 5.

Free motion : V ≡ 0. A wall: V ≡ 0 on one side, V ≡ ∞ on the other (meaning ψ ≡ 0 here). The double-slit experiment: V ≡ ∞ on the shield, and V ≡ 0 elsewhere. The Coulomb potential : V (x) = −α/|x| (describes a hydrogen atom). 2 2 The harmonic oscillator : V (x) = mω 2 |x| .

We will analyze some of these examples, and others, in subsequent chapters.

1.5 Classical Limit Now we go in the opposite direction and pass from a quantum description to a classical one. Thus we assume that the ratio α := ~/(classical action) → 0 (the classical limit). Changing to dimensionless units, we arrive at the dimensionless Schr¨odinger equation iα

∂ α2 ψ(x, t) = − ∆x ψ(x, t) + V (x)ψ(x, t), ∂t 2m

(1.7)

where (abusing the notation) m and x, t stand for the dimensionless parameter and variables. For the classical limit, we would like to set α = 0 in (1.7), but we see that the limiting equation becomes trivial: V (x)ψ(x, t) = 0. So we have to do something more subtle.

8

1 Physical Background iS

Write ψ as ψ = ae α where a and S are real-valued functions. Plug this into the Schr¨odinger equation and take the real and imaginary parts of the result to obtain (assuming a−1 is not too singular) 1 α2 −1 ∂S =− |∇S|2 − V + a ∆a, ∂t 2m 2m 1 1 ∂a = − ∇S · ∇a − a∆S. ∂t m 2m

(1.8) (1.9)

Problem 1.2 Derive these equations. Now, we can set α = 0 in (1.8) and (1.9) (classical limit). If we denote S∗ := S α=0 and a∗ := a α=0 , then (1.8) gives ∂S∗ 1 + |∇S∗ |2 + V = 0, ∂t 2m

(1.10)

which is the Hamilton-Jacobi equation of Classical Mechanics. Hence S∗ is the classical action. The equation (1.9) can be rewritten as 1 ∂a2 = −div( (∇S)a2 ). ∂t m

(1.11)

This is a classical transport equation. Thus at α = 0, we arrive at two classical equations, (1.10) and (1.11). Let a∗ be a solution to (1.9) with S = S∗ . We show in Section 2.5 that, for small α, the function iS∗ ψ sc = a∗ e α (‘sc’ stands for the semi-classical), constructed from purely classical objects iS gives, gives a good approximation to the exact wave function ψ = ae α . 2 2 Let ρ = a = |ψ| . By the interpretation of Quantum Mechanics, this is the probability density. By Classical Mechanics, v = ∇S m is the classical velocity. 2 ∇S Hence, we interpret j := ρv = a m as the probability current density. Now, equation (1.11) can be rewritten as dρ = −divj, dt

(1.12)

This is the law of conservation of probability in differential form (for more details, see Subsection 3.5). The equations (1.10) and (1.12) (at α = 0) describe classical noninteracting particles of density ρ(x, t) = |a∗ |2 (x, t) α=0 and velocity v(x, t) =

∇S∗ (x,t) . Taking m ∇)∇S∗ and ∇S∗

the gradient of (1.10) and using that ∇|∇S∗ |2 = 2(∇S∗ · = mv, we obtain an equation for the velocity v: m

dv = −∇V, dt

(1.13)

1.5 Classical Limit

9

where dv dt = (∂t + v · ∇)v is the material derivative. This is the classical Newton’s equation for particle flow in the potential V (x). To take into account quantum corrections, one has to retain the last term on the r.h.s. of (1.8) (or treat it by a perturbation theory). We do not pursue this here, but take it up in the semi-classical analysis later on. iEt Next we consider a stationary state ψ(x, t) = e− α φ(x), where Hφ = Eφ. Then S = −Et+χ and a = |φ| where χ is the argument of φ: φ = |φ|eiχ = aeiχ . Hence, ∂S ∂a = −E and = 0. ∂t ∂t These equations, together with Eqs (1.8) and (1.9), imply that |∇S|2 − 2m(E − V ) =

α2 ∆a a

(1.14)

and div(a2 ∇S) = 0.

(1.15)

In the regime α = ~/(classical action) → 0 (the classical limit) we obtain ∗ a stationary flow of particle fluid. Hence in the classical limit, v = ∇S m is interpreted as velocity, and k =p∇S∗ as momentum. Note that (1.14) implies that in the classical limit, |k| = 2m(E − V ) (in the classically allowed region V (x) ≤ E).

2 Dynamics

The purpose of this chapter is to investigate the existence and a key property – conservation of probability – of solutions of the Schr¨odinger equation for a particle of mass m in a potential V . The relevant background material on linear operators is reviewed in the Mathematical Supplement Chapter 25. We recall that the Schr¨odinger equation, i~

∂ψ = Hψ ∂t

(2.1)

2

~ ∆ + V is the corresponding Schr¨odinger where the linear operator H = − 2m operator, determines the evolution of the particle state (the wave function), ψ. We supplement equation (2.1) with the initial condition

ψ|t=0 = ψ0

(2.2)

where ψ0 ∈ L2 (R3 ). The problem of solving (2.1)- (2.2) is called an initial value problem or a Cauchy problem. Both the existence and the conservation of probability do not depend on the particular form of the operator H, but rather follow from a basic property – self-adjointness. This property is rather subtle, so for the moment we just mention that self-adjointness is a strengthening of a much simpler property – symmetry. A linear operator A acting on a Hilbert space H is symmetric if for any two vectors in the domain of A, ψ, φ ∈ D(A), hAψ, φi = hψ, Aφi.

2.1 Conservation of Probability Since we interpret |ψ(x, t)|2 at a given instant in time as a probability distribution, we should have

© Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_2

12

2 Dynamics

Z

R3

2

|ψ(x, t)| dx =

Z

R3

|ψ(x, 0)|2 dx = 1

(2.3)

at all times, t. If (2.3) holds, we say that probability is conserved. Theorem 2.1 Solutions ψ(t) of (2.1) with ψ(t) ∈ D(H) conserve probability if and only if H is symmetric. Proof. Suppose ψ(t) ∈ D(H) solves the Cauchy problem (2.1)-(2.2). We compute d ˙ ψi + hψ, ψi ˙ = h 1 Hψ, ψi + hψ, 1 Hψi hψ, ψi = hψ, dt i~ i~ 1 = [hψ, Hψi − hHψ, ψi] i~ (here, and often below, we use the notation ψ˙ to denote ∂ψ/∂t). If H is symmetric then this time derivative is zero, and hence probability is conserved. Conversely, if probability is conserved for all such solutions, then by a version of the polarization identity, hψ, φi =

1 (kφ + ψk2 − kφ − ψk2 − ikφ + iψk2 + ikφ − iψk2 ), 4

(2.4)

d whose proof is left as an exercise below, we have dt hψ(t), φ(t)i = 0, for any two solutions ψ(t) and φ(t). This implies hHψ, φi = hψ, Hφi for all ψ, φ ∈ D(H) (since we may choose ψ0 = ψ and φ0 = φ). This, in turn, implies H is a symmetric operator. The latter fact follows from 

Problem 2.2 Prove (2.4). Problem 2.3 Show that the following operators on L2 (R3 ) (with their natural domains) are symmetric: 1. 2. 3. 4.

xj (that is, multiplication by xj ); pj := −i~∂xj ; ~2 ∆; H0 := − 2m the multiplication operator by f (x) (ψ(x) → f (x)ψ(x)), provided f : R3 → R is bounded; 5. f (p) := F −1 f (k)F , where f : R3 → R is bounded and F denotes Fourier transform; R 6. integral operators Kf (x) = K(x, y)f (y) dy with K(x, y) = K(y, x) and, say, K ∈ L2 (R3 × R3 ).

2.2 Self-adjointness As was mentioned above the key property of the Schr¨odinger operator H which guarantees existence of dynamics is its self-adjointness. We define this notion here. More detail can be found in Section 25.5 of the mathematical supplement.

2.2 Self-adjointness

13

Definition 2.4 A linear operator A acting on a Hilbert space H is self-adjoint if A is symmetric and Ran(A ± i1) = H. Note that the condition Ran(A ± i1) = H is equivalent to the fact that the equations (A ± i)ψ = f (2.5) have solutions for all f ∈ H. The definition above differs from the one commonly used (see Section 25.5 of the Mathematical Supplement and e.g. [244]), but is equivalent to it. This definition isolates the property one really needs and avoids long proofs which are not relevant to us. Example 2.5 The operators in Problem 2.3 are all self-adjoint. Proof. We show this for p = −i~∂x on the space L2 (R). This operator is symmetric, so we compute Ran(−i~∂x + i). That is, we solve (−i~∂x + i)ψ = f, which, using the Fourier transform (see Section 25.14), is equivalent to (k + ˆ i)ψ(k) = fˆ(k), and therefore ψ(x) = (2π~)−1/2

Z

R

eikx/~

fˆ(k) dk. k+i

Now for any such f ∈ L2 (R), ˆ (1 + |k|2 )1/2 |ψ(k)| = |fˆ(k)| ∈ L2 (R), so ψ lies in the Sobolev space of order one, H 1 (R) = D(−i~∂x ), and therefore Ran(−i~∂x + i1) = L2 . Similarly Ran(−i~∂x − i1) = L2 .  Problem 2.6 Show that xj , f (x) and f (p), for f real and bounded, and ∆ are all self-adjoint on L2 (R3 ) (with their natural domains). In what follows we omit the identity operator 1 in expressions like A − z1. The next result establishes the self-adjointness of Schr¨odinger operators. 2

~ ∆+V (x), Theorem 2.7 Assume that V is real and bounded. Then H := − 2m 2 3 with D(H) = D(∆), is self-adjoint on L (R ).

Proof. It is easy to see (just as in Problem 2.3) that H is symmetric. To prove Ran(H ± i) = H, we will use the following facts proved in Sections 25.4 and 25.5 of the mathematical supplement: 1. If an operator K is bounded and satisfies kKk < 1, then the operator 1 + K has a bounded inverse. 2. If A is symmetric and Ran(A − z) = H for some z, with Im z > 0, then it is true for every z with Im z > 0. The same is true for Im z < 0.

14

2 Dynamics

3. If A is self-adjoint, then A − z is invertible for all z with Im z 6= 0, and k(A − z)−1 k ≤

1 . | Im z|

(2.6)

Since H is symmetric, it suffices to show that Ran(H + iλ) = H, for some λ ∈ R, ±λ > 0, i.e. to show that the equation (H + iλ)ψ = f

(2.7)

has a unique solution for every f ∈ H and some λ ∈ R, ±λ > 0. Write ~2 H0 = − 2m ∆. We know H0 is self-adjoint, and so H0 + iλ is one-to-one and onto, and hence invertible. Applying (H0 + iλ)−1 to (2.7), we find ψ + K(λ)ψ = g, where K(λ) = (H0 +iλ)−1 V and g = (H0 +iλ)−1 f . By (2.6), kK(λ)k ≤ λ1 kV k. Thus, for |λ| > kV k, kK(λ)k < 1 and therefore 1 + K(λ) is invertible, according to the first statement above. Similar statements hold also for K(λ)T := V (H0 + iλ)−1 . Therefore ψ = (1 + K(λ))−1 g. Moreover, it is easy to see that (H0 + iλ)(1 + K(λ)) = (1 + K(λ)T )(H0 + iλ) and therefore ψ = (H0 + iλ)−1 (1 + K(λ)T )−1 f (show this). So ψ ∈ D(H0 ) = D(H). Hence Ran(H + iλ) = H and H is self-adjoint, by the second property above.  α is not bounded. Unbounded potentials. The Coulomb potential V (x) = |x| We can extend the proof of Theorem 2.7 to show that Schr¨odinger operators with real potentials with Coulomb-type singularities are still self-adjoint. More precisely, we consider a general class of potentials V satisfying for all ψ ∈ D(H0 ) kV ψk ≤ akH0 ψk + bkψk (2.8)

(H0 -bounded potentials) for some a and b with a < 1. α Problem 2.8 Show that V (x) = |x| satisfies (2.8) with a > 0 arbitrary and b depending on a. Hint : Write V (x) = V1 (x) + V2 (x) where ( ( V (x) |x| ≤ 1 0 |x| ≤ 1 V1 (x) = , V2 (x) = 0 |x| > 1 V (x) |x| > 1.

Use sup |ψ| ≤ R that kV1 ψk ≤ sup |ψ|kV1 k, that by the Fourier transform R ( (|k|2 + c)−2 dk)−1/2 (k∆ψk + ckψk), and the fact that (|k|2 + c)−2 dk → 0 as c → ∞.

2.2 Self-adjointness

15

Theorem 2.9 Assume that H0 is a self-adjoint operator and V is a symmetric operator satisfying (2.8) with a < 1. Then the operator H := H0 + V with D(H) = D(H0 ) is self-adjoint. Proof. As in the proof of Theorem 2.7, it suffices to show that kV (H0 − iλ)−1 k < 1, provided λ is sufficiently large. Indeed, (2.8) implies that kV (H0 − iλ)−1 φk ≤ akH0 (H0 − iλ)−1 φk + bk(H0 − iλ)−1 φk.

(2.9)

Now, since kH0 φk2 ≤ kH0 φk2 + |λ|2 kφk2 = k(H0 − iλ)φk2 and k(H0 − iλ)−1 φk ≤ |λ|−1 kφk, we have that kV (H0 − iλ)−1 φk ≤ akφk + b|λ|−1 kφk.

(2.10)

Since a < 1 we take λ such that a + b|λ|−1 < 1, which gives kV (H0 − iλ)−1 k < 1. After this we continue as in the proof of Theorem 2.7.  2

~ α Problem 2.10 Prove that the operator H := − 2m ∆ − |x| (the Schr¨odinger operator of the hydrogen atom with infinitely heavy nucleus) is self-adjoint.

Theorem 2.9 has the following easy and useful variant Theorem 2.11 Assume that H0 is a self-adjoint, positive operator and V is symmetric and satisfies D(V ) ⊃ D(H0 ) and hψ, V ψi ≤ ahψ, H0 ψi + bkψk,

(2.11)

with a < 1. Then the operator H := H0 + V with D(H) = D(H0 ), is selfadjoint. For a proof of this theorem see e.g. [245], Theorem X.17. Now we present the following more difficult result, concerning Schr¨odinger operators whose potentials grow with x: Theorem 2.12 Let V (x) be a continuous function on R3 satisfying V (x) ≥ 0, ~2 ∆ + V is self-adjoint on L2 (R3 ) and V (x) → ∞ as |x| → ∞. Then H = − 2m The proof of this theorem is fairly technical, and can be found in [162], for example. Remark 2.13 Here and elsewhere, the precise meaning of the statement “the operator H is self-adjoint on L2 (Rd )” is as follows: there is a domain D(H), with C0∞ (Rd ) ⊂ D(H) ⊂ L2 (Rd ), for which H is self-adjoint, and H (with ~2 ∆ + V (x), which is domain D(H)) is the unique self-adjoint extension of − 2m ∞ d originally defined on C0 (R ). The exact form of D(H) depends on V . If V is bounded or relatively bounded as above, then D(H) = D(∆) = H 2 (Rd ).

16

2 Dynamics

By definition, every self-adjoint operator is symmetric. However, not every symmetric operator is self-adjoint. Nor can every symmetric operator be extended uniquely to a larger domain on which it is self-adjoint. For example, the Schr¨odinger operator A := −∆ − c/|x|2 with c > 1/4 is symmetric on the domain C0∞ (R3 \{0}) (the infinitely differentiable functions supported away from the origin), but does not have a unique self-adjoint extension (see [245]). It is usually much easier to show that a given operator is symmetric than to show that it is self-adjoint, since the latter question involves additional domain considerations.

2.3 Existence of Dynamics We consider the Cauchy problem (2.1)- (2.2) for an abstract linear operator H on a Hilbert space H. Here ψ = ψ(t) is a differentiable path in H. Definition 2.14 We say the dynamics exist if for all ψ0 ∈ H the Cauchy problem (2.1)- (2.2) has a unique solution which conserves probability. The main result of this chapter is the following Theorem 2.15 The dynamics exist if and only if H is self-adjoint. We sketch here a proof only of the implication which is important for us, namely that self-adjointness of H implies the existence of dynamics, with details relegated to the mathematical supplement Section 25.6 (for a proof of the converse statement see [244]). We derive this implication from the following result: Theorem 2.16 If H is a self-adjoint operator, then there is a unique family of bounded operators, U (t) := e−itH/~ , having the following properties for t, s ∈ R: ∂ (2.12) i~ U (t) = HU (t) = U (t)H, ∂t U (0) = 1, (2.13) U (t)U (s) = U (t + s),

(2.14)

kU (t)ψk = kψk.

(2.15)

Theorem 2.16 implies the part of Theorem 2.15 of interest to us here. Indeed, the family ψ(t) := U (t)ψ0 is the unique solution of the Cauchy problem (2.1)(2.2), and also conserves probability. (The uniqueness follows from (2.3).) The operator family U (t) := e−itH/~ is called the propagator or evolution operator for the equation (2.1). The properties recorded in the equations (2.14) and (2.15) are called the group and isometry properties. The operator U (t) = e−iHt/~ is furthermore invertible (since U (t)U (−t) = 1). Moreover, one can show that it preserves the inner product:

2.3 Existence of Dynamics

17

hU (t)ψ, U (t)φi = hψ, φi, for all ψ, φ ∈ L2 , i.e. it is unitary (see Definition 25.29). Such a family is called a one-parameter unitary group. Sketch of a proof of Theorem 2.16. We begin by discussing the exponential of a bounded operator. For a bounded operator, A, we can define the operator eA through the familiar power series eA :=

∞ X An n! n=0

which converges absolutely since ∞ ∞ X kAn k X kAkn ≤ = ekAk < ∞. n! n! n=0 n=0

With this definition, for a bounded operator A, it is not difficult to prove (2.12) - (2.14) for U (t) = e−itA/~ , and if A is also self-adjoint, (2.15). Problem 2.17 For A bounded, prove (2.12) - (2.14) for U (t) = e−itA/~ , and, if A is self-adjoint, also (2.15). Now for an unbounded but self-adjoint operator A, we may define the bounded operator eiA by approximating A by bounded operators. Since A is self-adjoint, the operators Aλ :=

1 2 λ [(A + iλ)−1 + (A − iλ)−1 ] 2

(2.16)

are well-defined and bounded for λ > 0. Using the bound (2.6), implied by the self-adjointness of A, we show that the operators Aλ approximate A in the sense that Aλ ψ → Aψ

as

λ→∞

for ψ ∈ D(A).

(2.17)

Since Aλ is bounded, we can define the exponential eiAλ by power series as above. One then shows that the family {eiAλ , λ > 0} is a Cauchy family, in the sense that
k eiAλ′ − eiAλ ψk → 0 (2.18) as λ, λ′ → ∞ for all ψ ∈ D(A). This Cauchy property implies that for any ψ ∈ D(A), the vectors eiAλ ψ converge to some element of the Hilbert space as λ → ∞. Thus we can define eiA ψ := lim eiAλ ψ λ→∞

(2.19)

for ψ ∈ D(A). It follows from (2.15) that keiA ψk ≤ kψk for all ψ in D(A), which is dense in H. Thus we can extend this definition of eiA to all ψ ∈ H. This defines the exponential eiA for any self-adjoint operator A.

18

2 Dynamics

If H is self-adjoint, then so is Ht/~ for every t ∈ R. Hence the conclusions above apply to Ht/~. This defines the propagator U (t) = e−iHt/~ . Using Problem 2.17 and (2.19), we can prove that U (t) has the properties (2.12) – (2.15). This implies Theorem 2.16.  The theorem above, together with the Fourier transform (see Section 25.14), also provides one method of defining functions of self-adjoint operators: Definition 2.18 Let A be a self-adjoint operator, and R f (λ) be a function on R whose inverse Fourier transform, fˇ, is integrable: R |fˇ(t)|dt < ∞. Then the operator Z f (A) := (2π~)−1/2 fˇ(t)e−iAt/~ dt (2.20) R

is well-defined, is bounded, and is self-adjoint if f is real. It is a function of the operator A. Example 2.19 Theorem 2.16 allows us to define exponentials of the selfadjoint operators on L2 (R3 ) with which we are familiar: xj , pj := −i~∂xj , ~2 H0 := − 2m ∆, f (x) and f (p) (for f a real function). Problem 2.20 (i) Determine how the operators eixj and eipj act on functions in L2 (R3 ). (ii) Show that the families eixj a/~ , a ∈ R, and eipj b/~ , b ∈ R, are one parameter groups of unitary operators on L2 (R3 ). Problem 2.21 For any ψ0 in our Hilbert space, define the operator e− ψ0 → ψ, the solution to the Schr¨odinger equation, i~

∂ψ = Hψ, ∂t

iHt ~

iHt

ψ, e−

(b) the operators e− ~ and e iHt iHt of e− ~ is given by e ~ , i.e.

iHt ~

he−

iHt ~

iHt ~

:

(2.21)

with the initial condition ψ0 . Show that (a) e− he−

iHt ~

iHt ~

preserves the inner product,

φi = hψ, φi,

are inverses of each other and (c) the adjoint ψ, φi = hψ, e

iHt ~

φi. iHt

Hint: For the first relation in (a), prove first that e− ~ is isometry, preserves the norm, and then use the parallelogram relation between norms and inner products show that it preserves also the inner product. For the second relation in (a), consider the differential equation for e

iHt ~

e−

iHt ~

ψ

and for the third relation, show that it follows from the first two. (For H bounded, one can also use the representation

2.4 The Free Propagator

e− where the series

P∞ 0

iHt ~

1 iHt n n! (− ~ )

=

19

∞ X iHt n 1 (− ) , n! ~ 0

is absolutely convergent.)

To summarize, if H is self-adjoint, then the operators U (t) := e−iHt/~ exist and are unitary for all t ∈ R (since Ht/~ is self-adjoint). Moreover, the family ψ(t) := U (t)ψ0 is the unique solution of the equation (2.1), with the initial condition ψ(0) = ψ0 , and it satisfies kψ(t)k = kψ0 k. Thus for the Schr¨odinger equation formulation of quantum mechanics to make sense, the Schr¨odinger operator H must be self-adjoint. As was shown in Theorem 2.9, ~2 Schr¨odinger operators H := − 2m ∆ + V (x) with potentials V (x) satisfying (2.8) are self-adjoint, and therefore generate unitary dynamics.

2.4 The Free Propagator We conclude this chapter by finding the free propagator U (t) = eiH0 t/~ , i.e. the propagator for Schr¨odinger’s equation in the absence of a potential. Here ~2 ∆ acts on L2 (R3 ). The tool for doing this is the Fourier transform, H0 := − 2m whose definition and properties are reviewed in Section 25.14. a|k|2

Let g(k) = e− 2 (a Gaussian), with Re(a) ≥ 0. Then setting p := −i~∇ and using Definition 25.78 and Problem 25.75 (part 1) from Section 25.14, we have Z |x−y|2 2 −3/2 g(p)ψ(x) = (2πa~ ) e− 2a~2 ψ(y)dy.

Since −~2 ∆ = |p|2 , we can write this as (e

a~2 ∆/2

2 −3/2

ψ)(x) = (2πa~ )

Z

e−

|x−y|2 2a~2

ψ(y)dy.

(2.22)

it here, we obtain an expression for the Schr¨odinger evolution Taking a = m~ ~2 −iH0 t/~ operator e for the Hamiltonian of a free particle, H0 = − 2m ∆:

(e

−iH0 t/~

ψ)(x) =



2πi~t m

−3/2 Z

e

im|x−y|2 2~t

ψ(y)dy.

(2.23)

R3

One immediate consequence of this formula is the pointwise decay (in time) of solutions of the free Schr¨odinger equation with integrable initial data:  2π~t −3/2 Z −iH0 t/~ |ψ(y)|dy. (2.24) ψ(x) ≤ e m R3

As another consequence, we make a connection between the free Schr¨odinger evolution, and the classical evolution of a free particle. Using the relation |x − y|2 = |x|2 − 2x · y + |y|2 , we obtain

20

2 Dynamics

e

−iH0 t/~

ψ(x) =

Denoting ψt (y) := ei



2πi~t m

m|y|2 2~t

−3/2

e

Z

e

−3/2

ei

im|x|2 2~t

−i mx t ·y/~

  im|y|2 2~t e ψ(y) dy.

ψ(y), we have

e−iH0 t/~ ψ(x) =



it m

m|x|2 2~t

ψbt (mx/t)

(2.25)

where (as usual) ψb denotes the Fourier transform of ψ. One can show that if ˆ ψ(k) is localized near k0 ∈ Rd , then so is ψˆt (k) for large t, and therefore the right hand side of (2.25) is localized near the point x0 = v0 t,

where

v0 = k0 /m,

i.e., near the classical trajectory of the free particle with momentum k0 .

2.5 Semi-classical Approximation In this section, we describe the semi-classical approximation of the quantum propagator. As in Section 1.5, we consider the dimensionless Schr¨odinger equation in dimensionless units, iα

α2 ∂ ψ(x, t) = − ∆x ψ(x, t) + V (x)ψ(x, t), ∂t 2m

(2.26)

where m and x, t are the dimensionless parameter and variables and α is the ratio α := ~/(classical action) → 0 (the classical limit). As in Section 1.5, assuming an initial condition of the form ψ0 := a0 eiS0 /α , we write ψ as ψ = aeiS/α , where a and S are real-valued functions. Plug this into equation (2.26) and take the real and imaginary parts of the result to obtain (assuming a−1 is not too singular) 1 α2 −1 ∂S =− |∇S|2 − V + a ∆a, ∂t 2m 2m ∂a 1 1 = − ∇S · ∇a − a∆S. ∂t m 2m

(2.27) (2.28)

For a discussion of the physical meaning of Eqs. (2.27)-(2.28), see Section 1.5. Setting α = 0 in (2.27) and denoting the solution of the resulting equation by S∗ gives 1 ∂S∗ + |∇S∗ |2 + V = 0, (2.29) ∂t 2m which is the Hamilton-Jacobi equation of Classical Mechanics. Hence S∗ is the classical action. Now, we consider (2.28) with S = S∗ and denote by a = a∗ the solution of the resulting equation, i.e.

2.5 Semi-classical Approximation

∂a∗ 1 1 = − ∇S∗ · ∇a∗ − a∗ ∆S∗ , ∂t m 2m

21

(2.30)

Eq. (2.30) is a transport equation along the classical trajectories of the vector field ∇S·∇ (i.e. the solution a is obtained by transporting the initial conditions a0 along the characteristics of ∇S · ∇). Eqs. (2.29) and (2.30) are solved by the method of characteristics, with a∗ given by an explicit expression (see [77, 92, 141, 250]). With the purely classical objects S∗ and a∗ solving the equations (2.29) and (2.30), we associate the wave function ψ sc := a∗ eiS∗ /α , gives the leading order approximation to ψ, called the semi-classical or WKB (for Gregor Wentzel, Hendrik Anthony Kramers, and L´eon Brillouin) approximation. (‘sc’ for ‘semi-classical’.) We show below that, as long as Eqs. (2.29) and (2.30) have solutions, the difference of the exact solution and semiclassical approximation satisfies the estimate kψt − ψtsc k ≤ Ctα,

(2.31)

where we display the time-dependence as the subindex t. This is a remarkable estimate which shows that a function constructed from purely classical objects gives a good approximation to the quantum wave function. Since ψt = U (t)ψ0 , (2.31) gives a semi-classical estimate of the propagator U (t). We now prove estimate (2.31) justifying the semi-classical approximation sketched above. Instead of writing solutions to (2.26) as ψt = aeiS/α , with a and S real-valued functions (satisfying (2.27) and (2.28)), we look for them in the form ψt = aeiS∗ /α , where S∗ solves equation (2.29), which does not involve a and α. Plugging this into (2.26), we find the equation for a: ∂a = La, ∂t

L ≡ L(t) := −

1 1 iα ∇S∗ · ∇ − ∆S∗ + ∆. m 2m 2m

(2.32)

(L(t) depends on t through S∗ .) Thus, as long as Eq (2.29) has a solution, Eq. (2.32) is equivalent to Eq. (2.26). We observe that, by (2.30), a∗ solves (2.32) to the order O(α), (∂t − L)a∗ = −

iα ∆a∗ , 2m

(2.33)

and the semiclassical approximation ψtsc := a∗ eiS∗ /α solves (2.26) to the order O(α2 ): (iα

2 iS∗ α ∂ − Hα )ψtsc = −e α ∆a∗ ∂t 2m

(2.34)

22

2 Dynamics 2

α where Hα := − 2m ∆x + V (x). To estimate the difference between the exact solution ψt = aeiS∗ /α and the semiclassical approximation ψtsc := a∗ eiS∗ /α , we let U (t) := e−iHα t/α be the propagator for (2.26). Writing U (−t)ψtsc as the integral of its derivative, using iα∂t U (−t)ψtsc = U (−t)(iα∂t − Hα )ψtsc

and (2.34) and applying U (t) to the result, we obtain ψtsc

1 − U (t)ψ0 = iα

Z

0

t

U (t − s)(eiS∗ /α

α2 ∆a∗ )ds. 2m

Rt

Rt

Taking the norm in (2.35) and using that k U (t) is unitary yields kU (t)ψ0 − ψtsc k ≤

1 α

Z

0

t 0

k

f (s)dsk ≤

α2 ∆a∗ kds. 2m

0

(2.35)

kf (s)kds and that (2.36)

The last inequality, together with the relation ψt = U (t)ψ0 and the fact that k∆a∗ k < ∞, gives (2.31).

3 Observables

Observables are the quantities that can be experimentally measured in a given physical framework. In this chapter, we discuss the observables of quantum mechanics.

3.1 The Position and Momentum Operators We recall that in quantum mechanics, the state of a particle at time t is described by a wave function ψ(x, t). The probability distribution for the position, x, of the particle, is |ψ(·, t)|2 . Thus the mean value of the position R at time t is given by x|ψ(x, t)|2 dx (note that this is a vector in R3 ). If we define the coordinate multiplication operator xj : ψ(x) 7→ xj ψ(x) then the mean value of the j th component of the coordinate x in the state ψ is hψ, xj ψi. Recall that the evolution state ψ(x, t) obeys the Schr¨odinger equation i~

∂ψ = Hψ. ∂t

(3.1)

For ψ(x, t) solving (3.1), we compute d ˙ = h 1 Hψ, xj ψi + hψ, xj 1 Hψi ˙ xj ψi + hψ, xj ψi hψ, xj ψi = hψ, dt i~ i~ i i i = hψ, Hxj ψi − hψ, xj Hψi = hψ, [H, xj ]ψi ~ ~ ~ where [A, B] := AB − BA is the commutator of A and B. (More about the ~2 commutators later.) Since H = − 2m ∆ + V , and ∆(xψ) = x∆ψ + 2∇ψ, we find © Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_3

24

3 Observables

i i~ [H, xj ] = − ∇j , ~ m leading to the equation 1 d hψ, xj ψi = hψ, −i~∇j ψi. dt m As before, we denote the operator −i~∇j by pj . As well, we denote the mean value hψ, Aψi of an operator A in the state ψ by hAiψ . Then the above becomes m

d hxj iψ = hpj iψ dt

(3.2)

which is reminiscent of the definition of the classical momentum. We call the operator p the momentum operator. In fact, pj is a self-adjoint operator on L2 (R3 ). (As usual, the precise statement is that there is a domain on which ∂ ψ ∈ pj is self-adjoint. Here the domain is just D(pj ) = {ψ ∈ L2 (Rd ) | ∂x j 2 d L (R ).) Using the Fourier transform, we compute the mean value of the momentum operator Z 2 ˆ ˆ pd ˆ ˆ kj |ψ(k)| dk. hψ, pj ψi = hψ, j ψi = hψ, kj ψi = R3

2 ˆ This, and similar computations, show that |ψ(k)| is the probability distribution for the particle momentum.

3.2 General Observables Definition 3.1 An observable is a self-adjoint operator on the state space L2 (R3 ). We have already met six observables: the position operators, x1 , x2 , x3 , and the momentum operators, p1 , p2 , p3 (combined into vector-observables x and p). The Schr¨odinger operator, H =−

~2 ∆ + V, 2m

is self-adjoint and therefore, by our definition, is an observable. But what is the physical quantity it stands for (or ‘observes’)? We find the answer below. In general, we interpret hAiψ as the average of the observable A in the state ψ. The reader is invited to derive the following equation for the evolution of the mean value of an observable: d i hAiψ = h [H, A]iψ . dt ~

(3.3)

3.2 General Observables

25

Problem 3.2 Check that for any observable, A, and for any solution ψ of the Schr¨odinger equation, Eq (3.3) holds. d Eq (3.3) is analogous to the classical Hamilton equation, dt a = {h, a}, for a classical observable, a, i.e. a function on the phase space. Here {f, g} = P3 ∂f ∂g ∂f ∂g j=1 ( ∂kj ∂xj − ∂xj ∂kj ) is the Poisson bracket and h = h(x, k) is the classical Hamiltonian with h(x(t), k(t)) giving the energy of a solution (x(t), k(t)) (see Section 4.1 and supplemental Section 4.7 for more details). For this reason, H is interpreted as the observable of energy and is called the Hamiltonian operator, or quantum Hamiltonian. Now, we would like to apply (3.3) to the momentum operator, p = −i~∇. Simple computations give [∆, p] = 0 and [V, p] = i~∇V , so that ~i [H, p] = −∇V and hence d (3.4) hpj iψ = h−∇j V iψ . dt

This is a quantum mechanical mean-value version of Newton’s equation of classical mechanics. Or, if we include Equation (3.2), we have a quantum analogue of the classical Hamilton equations. Note also that, since −~2 ∆ = P3 2 2 2 |p| (here |p| = j=1 pj ), we have H=−

~2 1 ∆+V = |p|2 + V, 2m 2m

which is in an agreement with our interpretation of x and p as the observables of the co-ordinate and momentum, respectively, and H, as a Hamiltonian operator. Next, we address the question: What is the probability, Probψ (A ∈ Ω), that measured values of the physical observable represented by A in a state ψ land in an interval Ω ⊂ R? As in the probability theory, this is given by the expectation Probψ (A ∈ Ω) = hχΩ (A)iψ (3.5) of the observable χΩ (A), where χΩ (λ) is the characteristic function of the set Ω (i.e. χΩ (λ) = 1, if λ ∈ Ω and χΩ (λ) = 0, if λ ∈ / Ω) and the operatorfunction χΩ (A) can be defined according to the formula (2.20) and a limiting procedure which we skip here. We call χΩ (A) the characteristic function of the operator A. This definition can be justified using spectral decompositions of the type (25.53) of Section 25.11, but we will not go into this here. Finally, by the analogy with above, we introduce the angular momentum observable, L := (L1 , L2 , L3 ), with Lj = (x × p)j . The operator-vector L can be written as L = x × p.

26

3 Observables

3.3 The Heisenberg Representation The framework outlined up to this point is called the Schr¨ odinger representation of quantum mechanics. Chronologically, quantum mechanics was first formulated in the Heisenberg representation, which we now describe. For an observable A, define A(t) := eitH/~ Ae−itH/~ . Let ψ(t) be the solution of Schr¨odinger’s equation with initial condition ψ0 : ψ(t) = e−itH/~ ψ0 . Since e−itH/~ is unitary, we have, by simple computations which are left as an exercise,

and

hAiψ(t) = hA(t)iψ0

(3.6)

d i A(t) = [H, A(t)]. dt ~

(3.7)

Problem 3.3 Prove equations (3.6) and (3.7). This last equation is called the Heisenberg equation for the time evolution of the observable A. In particular, taking x and p for A, we obtain the quantum analogue of the Hamilton equations of classical mechanics: mx(t) ˙ = p(t),

p(t) ˙ = −∇V (x(t)).

(3.8)

In the Heisenberg representation, then, the state is fixed (at ψ0 ), and the observables evolve according to the Heisenberg equation. Of course, the Schr¨odinger and Heisenberg representations are completely equivalent (by a unitary transformation). Remark 3.4 A note of caution: defining commutators is a subtle business. We have to assume the set D(A) ∩ D(H) is dense in L2 (R3 ) and that A : D(H) ∩ D(A) → D(H) and H : D(H) ∩ D(A) → D(A). Checking the last two properties is cumbersome. To avoid this, we can choose to understand equations involving commutators in the sense of expectations or quadratic forms. For instance, we assume the set D(A) ∩ D(H) is dense in L2 (R3 ) d hψ, Aφi = and interpret (3.7) as (3.3), for all ψ ∈ D(H) ∩ D(A), or as dt i hψ, ~ [H, A]φi, for all ψ, φ ∈ D(H) ∩ D(A), by defining hψ, i[H, A]φi := i (hHψ, Aφi − hAψ, Hφi) , for any two self-adjoint operators and for all ψ, φ ∈ D(A) ∩ D(H).

3.4 Conservation Laws

27

3.4 Conservation Laws We say that an observable A (or, more precisely, the physical quantity represented by this observable) is conserved if its average, hAiψ(t) , is independent of t, hAiψ(t) = hAiψ(0) ,

(3.9)

for every ψ(t) which solves the Schr¨odinger equation i~∂t ψ = Hψ.

(3.10)

Differentiating relation (3.9) w.r. to and recalling (3.6)-(3.7), we conclude that an observable A is conserved if and only if hψ, [A, H]ψi = 0, ∀ψ, which in turn implies1 that A commutes with the Schr¨odinger operator H, i. e. [A, H] = 0. We consider several examples. Since obviously [H, H] = 0, we have hHiψ(t) = constant for any solution to (3.10), which is the mean-value version of the conservation of energy, or H(t) = H. As the second example, we take the obvious relation [1, H] = 0, which gives the conservation of the total probability: h1iψ(t) = kψ(t)k2 = constant for any solution to (3.10). Furthermore, ~i [xj , H] = pj 6= 0, which says that the particle position is never conserved, i.e. a quantum particle cannot be localized at a point. The relation ~i [pj , H] = −∂xj V 6= 0 implies that pj is conserved if and only if V is independent of xj . Taken for all j’s, this gives the first Newton’s law: a particle will be in the state of the motion with a constant velocity as long as no force is acting on it, i.e. ∇V (x) = 0. Problem 3.5 Prove that (a) the momentum is conserved if and only if the potential V (x) is constant; and (b) the angular momentum is conserved if and only if the potential V (x) is spherically symmetric, i.e. V (Rx) = V (x) for any rotation R in R3 around the origin, i.e. V (x) = W (|x|) for some function W (r) on the interval [0, ∞). Most of the conservation laws come from symmetries of physical systems. First, an operator U on a Hilbert space H is called unitary if it preserves the inner product: hU ψ, U φi = hψ, φi, for all ψ, φ ∈ H (see Definition 25.29). We say that a unitary operator U is a symmetry of (3.10) if U maps D(H) into itself and 1

Showing this is actually not trivial. If B is a non-negative operator, then using square roots one can show that hψ, Bψi = 0 implies B = 0. If B is self-adjoint but not necessary positive, then one can reduce the problem to proving the desired property for the spectral projections of B, which are non-negative operators.

28

3 Observables

ψt is a solution to (3.10) → U ψt is a solution to (3.10). If U is a symmetry, then inverting U in the equation i~∂t U ψt = HU ψt gives i~∂t ψt = U −1 HU ψt , which, together with the original Schr¨odinger equation, (3.10), implies U −1 HU ψ = Hψ for any ψ ∈ D(H). Hence, if U is a symmetry, then U commutes with H: U −1 HU = H. Most often symmetries are organized in continuous families which are, or could be reduced to, one-parameter groups Us , s ∈ R, of unitary operators. This means that each Us is a unitary operator, Us ψ is continuous in s, for every ψ, and U0 = 1,

Us Ut = Us+t

(3.11)

(cf. (2.13)-(2.14)). For a one-parameter group Us , one defines the generator as the operator, A, defined by Aψ := i∂s Us ψ s=0 , (3.12)

for those ψ’s for which the derivative on the r.h.s exist. (Note that the definition above differs from the standard one by the factor i and that Us satisfies the equation i∂s Us = AUs .) Then (ignoring domain questions) Us is a symmetry of (3.10) → A commutes with H ([H, A] = 0).

Indeed, the fact that Us is a symmetry implies that, after dropping i, Us−1 HUs = H. Differentiating the last equation with respect to s at s = 0 and using (3.12), we arrive, after dropping i, at [H, A] = 0. (For the definition of the commutators, see Remark 3.4.) Examples of one-parameter groups, which could serve as symmetry groups, and their generators: • •

•

Spatial translation: Ustransl : ψ(x) → ψ(x + sej ), s ∈ R, where e1 := (1, 0, 0), e2 := (0, 1, 0), e3 := (0, 0, 1), with generator ~1 pj = −i∇xj . Spatial rotation: Usrot : ψ(x) → ψ((Rsj )−1 x), where Rsj is the counterclockwise rotation around the j-axis by the angle s ∈ [0, 2π), with generator 1 Lj = (x × (−i∇x ))j . ~ Gauge transformation: Usgauge : ψ(x) → eis ψ(x), s ∈ R, with generator i1.

Problem 3.6 Show that the families, Ustransl , Usrot , Usgauge , defined above, are one-parameter groups of unitary operators, find their generators and write the differential equations for the groups.

3.4 Conservation Laws

29

Hence the translational, rotational and gauge symmetries imply the conservation of momentum, angular momentum and probability, respectively. The fact that Lj ’s are the generators of rotations justifies the term the angular momentum operator for L = (L1 , L2 , L3 ) = x × p. We summarize a part of the discussion above as • • • •

Time translation invariance (V is independent of t) ⇒ conservation of energy Space translation invariance (V is independent of x) ⇒ conservation of momentum Space rotation invariance (V is rotation invariant, i.e. is a function of |x|) ⇒ conservation of angular momentum Gauge invariance (invariance under the transformation ψ → eiα ψ) ⇒ conservation of probability.

Discussion. More generally, one can consider symmetries associated with groups. In the last three examples these are the groups of translations and rotations of R3 , denoted T 3 or R3 , and O(3), respectively (forming together the group of rigid motions of R3 ), and the group U (1) of complex numbers of unit modulus, serving as a gauge group. (For particles with internal degrees of freedom, specifically with a spin to be considered later on, the gauge group is the group SU (m) of complex, unitary, m × m matrices, for an appropriate m.) We represent these groups by unitary operators on the state space L2 (R3 ): transl Uy : ψ(x) → ψ(x + y), y ∈ R3 , URrot : ψ(x) → ψ(R−1 x), R ∈ O(3),

(3.13)

and Uαgauge : ψ(x) → eiα ψ(x), eiα ∈ U (1), for the spatial translations and rotations and the gauge transformations. These operators satisfy transl Uytransl Uytransl = Uy+y ′ ′

and

rot URrot URrot ′ = URR′ ,

and similarly for Uαgauge . The operators Uytransl and URrot give unitary representations of the groups of translations and rotations of R3 , respectively (the first one is commutative, or abelian, the second one is not). They can be written as products of one parameter groups, so that the analysis relevant for us can be reduced to the latter case. Problem 3.7 Find the conditions under which the above groups are symmetry groups of the Schr¨odinger equation. We will consider the group of rotations, O(3), in more detail. Note that the rotations, R, are represented by orthogonal matrices (i.e. real matrices satisfying RT R = 1) of determinant 1. The counter-clockwise rotations Rθω around the axis along a unit vector ω by the angles θ ∈ [0, 2π) form a oneparameter group, and its generator is given by

30

3 Observables

∂θ θ=0 Rθω x = ω × x.

e Problem 3.8 (a) ProveP the above statement. Hint: Prove that ∂θ θ=0 Rθj x = ej × x and use that ω = ωj ej . (b) Show that operators (matrices) Aj given by Aj x := ej × x satisfy the commutation relations [A1 , A2 ] = A3 , [A2 , A3 ] = A1 , [A3 , A1 ] = A2 . Now, ∂θ θ=0 ψ(Rθω x) = ∇ψ(x) · (ω × x) = ω · (x × ∇ψ(x)). Hence i~∂θ θ=0 URrot ω = −i~ω · (x × ∇) = ω · L, θ

(3.14)

(3.15)

which proves one of the statements in Problem 3.6. ~2 Furthermore, it is easy to see for H := − 2m ∆ + V (x) that URrot

−1

HURrot = −

~2 ∆ + V (Rx). 2m

(3.16)

Problem 3.9 Prove the last relation. Relations (3.15) and (3.16) imply the proof of statement (b) in Problem 3.5. Unlike the components of the position and momentum operators, xj and pj , the components of the angular momentum one do not mutually commute: i [Lk , Ll ] = ǫklm Lm . ~

(3.17)

Here ǫklm is the Levi-Chivita symbol: ǫ123 = 1 and ǫklm changes sign under the permutation of any two indices. This is related to the fact that URrot is a representation of the group of rotations of R3 and, unlike the group of translations, the group of rotations is non-abelian. 1 Lj are the generators of representation (3.13) of the Eq. (3.17) implies i~ 1 group SO(3). The vector space spanned by i~ Lj , equipped with the commutator relation [·, ·], is the representation on L2 (R3 ) of the Lie algebra so(3) of the group SO(3).

3.5 Conserved Currents With each conserved observable A and a state ψ, one can associate a density and a current (of A in ψ): ¯ ρA (ψ) := ψAψ,

jA (ψ) :=


i~ ¯ ¯ (∇ψ)Aψ − ψ∇Aψ . 2m

These functions satisfy an equation, called (the differential form of) the conservation law, along any solution ψ(t) of the Schr¨odinger equation:

3.5 Conserved Currents

31

Proposition 3.10 (Differential conservation laws) Let A be a conserved observable, i.e. D(H) ∩ D(A) is dense in L2 (R3 ) and [H, A] = 0. If ψ(t) is a solution of the Schr¨ odinger equation (3.1), then ∂t ρA (ψ(t)) = −divjA (ψ(t)).

(3.18)

Proof. We omit the arguments t and ψ and use the Leibniz rule and Schr¨odinger equation (3.10) to compute formally ¯ ¯ t ψ = 1 HψAψ + ψ 1 AHψ. ∂t ρA = (∂t ψ)Aψ + ψA∂ i~ i~ Since A and H commute, and since V is real and therefore 1 ψ¯ i~ V Aψ = 0, this gives ∂t ρA = −

1 i~ V

ψAψ +

i~ ¯ i~ ¯ ∆ψAψ + ψ∆Aψ. 2m 2m

¯ ¯ ¯ · ∇Aψ and ψ∆Aψ ¯ By the Leibnitz rule, ∆ψAψ = div((∇ψ)Aψ) − (∇ψ) =
i~ ¯ ¯ ¯ ¯ div(ψ∇Aψ)−(∇ ψ)·∇Aψ, which implies ∂t ρA = 2m div −(∇ψ)Aψ + ψ∇Aψ , giving (3.18).  Integrating equation (3.18) in x and using the Gauss theorem, one obtains the global conservation law, Z ∂t ρA (ψ(t)) = 0. We illustrate the general result on the ‘trivial’ observable A = 1 which gives the differential form of the conservation of probability. For A = 1, the associated density ρ1 (ψ) ≡ ρ(ψ) and the current j1 (ψ) ≡ j(ψ) are ρ(ψ)(x) := ~ Im(ψ(x)∇ψ(x)). Then (3.18) gives the differential |ψ(x)|2 and j(ψ)(x) = m form of the conservation of probability law: ∂t |ψ|2 = −divj(ψ).

(3.19)

This result shows how the probability distribution changes under the Schr¨odinger equation, and provides the formula for the probability current: j(ψ)(x) =

~ ¯ Im(ψ(x)∇ψ(x)). m

4 Quantization

In this chapter, we discuss the procedure of passing from classical mechanics to quantum mechanics. This is called “quantization” of a classical theory.

4.1 Quantization To describe a quantization of classical mechanics, we start with the Hamiltonian formulation of classical mechanics (see supplemental Section 4.7 for more details), where the basic objects are as follows: 1. The phase space (or state space): R3x × R3k . 2. The Hamiltonian: a real function h(x, k) on R3x × R3k (which gives the energy of the classical system). 3. Classical observables: (real) functions on R3x × R3k . 4. Poisson bracket: a bilinear form mapping each pair of classical observables, f, g, to the observable (function)  3  X ∂f ∂g ∂f ∂g {f, g} = − . ∂kj ∂xj ∂xj ∂kj j=1 5. Canonically conjugate variables: co-ordinate functions, xi , ki , satisfying {xi , xj } = {ki , kj } = 0;

{ki , xj } = δij .

(4.1)

6. Classical dynamics: Hamilton’s equations, x˙ = {h, x},

k˙ = {h, k}.

(4.2)

The corresponding fundamental objects in quantum mechanics are the following: © Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_4

34

4 Quantization

1. The state space: L2 (R3x ). 2. The quantum Hamiltonian: a Schr¨ odinger operator, H = h(x, p) acting on the state space L2 (R3 ). 3. Quantum observables: (self-adjoint) operators on L2 (R3x ). 4. Commutator : a bilinear form mapping each pair of operators acting on L2 (R3x ) into the commutator, ~i [·, ·]. 5. Canonically conjugate operators: co-ordinate operators xi , pi satisfying [xi , xj ] = [pi , pj ] = 0;

i [pi , xj ] = δij . ~

(4.3)

6. The dynamics of the quantum system can be described by the Heisenberg equations i i x˙ = [H, x], p˙ = [H, p]. ~ ~ The relations (4.3) are called the canonical commutation relations. To quantize classical mechanics we pass from the canonically conjugate variables, xi , ki , satisfying (4.1) to the canonically conjugate operators, xi , pi , i = 1, 2, 3, satisfying (4.3): xi , ki

−→

xi , pi .

(4.4)

Hence with classical observables f (x, k), we associate quantum observables f (x, p). This is a fairly simple procedure if f (x, p) is a sum of a function of x and a function of p, but rather subtle otherwise. It is explained in the next section. If the classical Hamiltonian function is h(x, k) = |k|2 /2m + V (x), the corresponding quantum Hamiltonian is the Schr¨odinger operator H = h(x, p) =

~2 |p|2 + V (x) = − ∆ + V (x). 2m 2m

Similarly, we pass from the classical angular momentum, lj = (x × k)j , to the angular momentum operators, Lj = (x × p)j . The following table provides a summary of the classical mechanical objects and their quantized counterparts:

4.2 Quantization and Correspondence Principle

Object state space evolution of state observable

35

CM QM 3 2 × Rk and L (R3x ) and Poisson bracket commutator path in path in phase space L2 (R3 ) real function self-adjoint operator on state space on state space result of measuring deterministic probabilistic observable object determining Hamiltonian Hamiltonian (Schr¨odinger) dynamics function operator canonical functions operators coordinates x and k x (mult.) and p (differ.) R3x

Quantization of classical systems does not lead to a complete description of quantum systems. As was noted in the previous chapter, quantum mechanical particles might have also internal degrees of freedom, such as spin, which have no classical counterparts and therefore cannot be obtained as a result of quantization of a classical system. To take these degrees of freedom into account one should modify ad hoc the quantization procedure above, or add new quantization postulates as is done in the relativistic theory.

4.2 Quantization and Correspondence Principle The correspondence between classical observables and quantum observables, f (x, k) → f (x, p), is a subtle one. It is easy to see that a classical observable f (x) is mapped under quantization into the operator of multiplication by f (x), and an observable g(k), into the operator g(p), defined for example using the Fourier transform and the three-parameter translation group, e−ip·x/~ : Z g(p) := (2π~)−3/2 gˇ(x)e−ip·x/~ dx where gˇ is the inverse Fourier transform of g (see Definition 2.18). However, the following simple example shows the ambiguity of this correspondence for more general functions of x and k. The function x·k = k ·x could, for example, be mapped into any of the following distinct operators: x · p,

p · x,

1 (x · p + p · x). 2

This ambiguity can be resolved by requiring that the quantum observables obtained by a quantization of real classical observables are self-adjoint (or

36

4 Quantization

at least symmetric) operators. This selects the symmetric term above, for example. The corresponding quantization is called the Weyl quantization. In this case the operator A = a(x, p) associated with the classical observable a(x, k) is given by ZZ ′ A = (2π~)−3 a ˆ(ξ, x′ )ei(ξ·x−x ·p)/~ dξdx′ , (4.5) where a ˆ is the Fourier transform of a in x and the inverse Fourier transform in k and therefore ZZ a(x, k) = (2π~)−3 a ˆ(ξ, η)ei(ξ·x−η·k)/~ dξdη. (4.6) Reversing the quantization procedure, one would like to show that classical mechanics arises from quantum mechanics in the limit as ~/(classical action) → 0. Assume we have passed to physical units in which a typical classical action in our system is 1, so that ~ is now the ratio of the Planck constant to the classical action. First one would like to show that a product of quantum observables is given by a product of classical ones, e.g. a(x, p)b(x, p) = (ab)(x, p) + O(~),

(4.7)

and therefore the former can be identified with latter. Assuming that the R classical observables a and b satisfy (|ξ|+|η|)|ˆ a(ξ, η)|dξdη < ∞, and similarly for b (this condition is considerably stronger than needed), one can easily prove (4.7). Indeed, we use the Baker-Campbell-Hausdorff formula 1

eA eB = eA+B+ 2 [A,B] ,

(4.8)

provided [A, B] is a multiple of the identity, which can be verified by computing ∂s (esA esB ) = (A + esA Be−sA )esA esB Z s = (A + B + drerA [A, B]e−rA )esA esB 0

= (A + B + s[A, B])esA esB .

Using (4.8), (4.5) and the relation [ξ · x − η · p, ξ ′ · x − η ′ · p] = ~ω, where ω := ξ · η ′ − η · ξ ′ , we compute for A := a(x, p) and B := b(x, p), Z Z −6 AB = (2π~) (4.9) ··· a ˆ(ξ, η)ˆb(ξ ′ , η ′ )eiΦ/~ eiω/~ dξdηdξ ′ dη ′ . where Φ := (ξ+ξ ′ )·x−(η+η ′ )·p. Now, we expand eiω = 1+O(|ω|) and evaluate the contribution of the first term using property 5 of the Fourier transform given in Section 25.14. Together with the definition of the convolution (f ∗

4.2 Quantization and Correspondence Principle

37

R g)(x) := Rd f (y)g(x − y)dy (see (25.56) of Section 25.14), this gives, after changing the variables of integration as ξ → ξ − ξ ′ , η → η − η ′ , Z Z (2π~)−6 · · · a ˆ(ξ, η)ˆb(ξ ′ , η ′ )eiΦ/~ dξdηdξ ′ dη ′ ZZ −6 = (2π~) (ˆ a ∗ ˆb)(ξ, η)ei(ξ·x−η·p)/~ dξdη = ab. The remainder is simply estimated by taking the absolute value under the integral. R Under the stronger condition (|ξ|2 +|η|2 )|ˆ a(ξ, η)|dξdη < ∞, one can prove the stronger statement, i a(x, p)b(x, p) = (ab − ~{a, b})(x, p) + O(~2 ). 2

(4.10)

To prove this, we expand eiω = 1 + iω + 21 (iω)2 + O(|ω|2 ) and use Properties 3 - 5 of the Fourier transform in Section 25.14, to evaluate the contribution of iω. This is done similarly to the first term above, once we write ξˆ a(ξ, η) = d d −i~∇x a(ξ, η), ηˆ a(ξ, η) = i~∇k a(ξ, η) and similarly for b. The remainder term here is treated similarly to the remainder above. Equation (4.10) implies that in the next order, the commutators give Poisson brackets: i[a(x, p), b(x, p)] = ~{a, b}(x, p) + O(~2 ).

(4.11)

Equation (4.11) allows one to connect the quantum and classical evolutions. Indeed, let φt be the flow generated by the Hamilton equations (4.2), i.e. the map φt : (x0 , k0 ) → the solution of (4.2) with the initial conditions (x0 , k0 ) (see supplemental Section 4.7), and let αcl t a = a ◦ φt and αt (A) := eiHt/~ Ae−iHt/~ be the evolutions of classical and quantum observables (αcl t is called the Liouville dynamics, and αt is nothing but the Heisenberg dynamics.) Denote the Weyl quantization map given in (4.5) by Q, so that A = Q(a). One can show that for a certain class of classical observables a, we have αt (Q(a)) = Q(αcl (4.12) t a) + O(~), for t ≤ C sup~|V | , as ~ → 0. Given (4.11), a proof of (4.12) is fairly simple. We give it here modulo one classical estimate. Using the Duhamel principle (i.e. cl writing α−t (Q(αcl t (a)))−Q(a) as the
integral of derivative ∂s α−s (Q(αs (a))) = cl cl α−s [H, Q(αs (a))] − Q({h, αs (a)}) ), we obtain αt (A) − Q(a ◦ φt ) =

Z

t

0

ds αt−s (R(a ◦ φs )),

(4.13)

where R(a) := ~i [H, A] − Q({h, a}). Since kαt (A)k = kAk, this gives kαt (A) − Q(a ◦

αcl t )k

≤

Z

0

t

dskR(a ◦ αcl s )k.

(4.14)

38

4 Quantization

Note that the full evolution αt drops out of the estimate. Using estimate (4.11) for the remainder R(a) and an appropriate estimate for the classical observable αcl s a = a ◦ φs , we arrive at (4.12). In mathematics, operators obtained by a certain quantization rule from functions f (x, k) satisfying certain estimates are called pseudodifferential operators, while the functions themselves are called symbols. Differential operators with smooth coefficients, as well as certain integral and singular integral operators are examples of pseudodifferential operators. The relation (4.11) is one of the central statements coming from pseudodifferential calculus.

4.3 A Particle in an External Electro-magnetic Field Here we apply the rules discussed in the previous sections to describe a quantum particle moving in an external electro-magnetic field. Of course, if the external field is purely electric, E, then it is a potential field, E = −∇Φ, for some Φ : R3 7→ R, and it fits within the framework we have considered above with V (x) = Φ(x). Suppose, then, that both electric and magnetic fields, E and B, are present. These are vector fields on R3 , which could be time-dependent: B, E : R3+1 7→ R3 . We know from the theory of electro-magnetism (Maxwell’s equations) that these fields can be expressed in terms of vector and scalar potentials A : R3 7→ R3 , and Φ : R3 7→ R, via E = −∇Φ − ∂t A,

B = curlA

(we are using units in which the speed of light, c, is equal to one; for more details see Subsection 4.7). For simplicity, in what follows we assume that the electric and magnetic fields E and B are time-independent. It is shown in Subsection 4.7 that the classical Hamiltonian function for a particle of charge e subject to the fields E and B is, h(x, k) =

1 (k − eA(x))2 + eΦ(x). 2m

According to our general quantization procedure, we replace the classical canonical variables x and k by the quantum canonical operators x and p. The resulting Schr¨odinger operator is H(A, Φ) =

1 (p − eA)2 + eΦ, 2m

(4.15)

acting on L2 (R3 ). The self-adjointness of H(A, Φ) can be established by using Kato’s inequality (see [73]). We now consider the Schr¨odinger equation with this Hamiltonian: i~∂t ψ = H(A, Φ)ψ.

(4.16)

4.3 A Particle in an External Electro-magnetic Field

39

It, like the Hamiltonian H(A, Φ), depends of the magnetic and electric potentials A and Φ, rather than on the magnetic and electric fields B := curlA and E := ∇Φ (in the static case). However, as we recall that in the theory of electro-magnetism, the vector potential A is not uniquely determined by the magnetic field B. In fact, if we add the gradient of any function χ to A (a gauge transformation), we obtain the same magnetic field B: curl(A + ∇χ) = curlA = B. Similar ambiguity exists for the electric potential Φ. This is distinct from the classical mechanics where A and Φ enter into the Newton’s equations only through the magnetic and electric fields B := curlA and E := ∇Φ. What saves the day is the gauge invariance of (4.16), which is its key feature: if ψ satisfies equation (4.16), then ψχ := eieχ/~ ψ satisfies i~∂t ψχ = H(A + ∇χ, Φ + ∂t χ)ψχ .

(4.17)

This property follows from the relation H(A + ∇χ, Φ) = eieχ/~ H(A, Φ)e−ieχ/~ .

(4.18)

Problem 4.1 Check that equation (4.18) holds. Thus if A and A˜ differ by a gradient vector-field ∇χ, then the operators ˜ Φ) are unitarily equivalent via the unitary map H(A, Φ) and H(A, ψ 7→ eieχ/~ ψ on L2 (R3 ). Thus the two Hamiltonians are physically equivalent. Of course, this is to be expected as A and A˜ correspond to the same magnetic field. One can impose restrictions (called gauge conditions) on the vector potential A in order to remove some, or all, of the freedom involved in the choice of A. A common choice is divA = 0, known as the Coulomb gauge. By an appropriate gauge transformation, the Coulomb gauge can always be achieved. The gauge invariance of (4.16) seems to say that physically, the magnetic potential does not matter, only the magnetic field is important. Indeed this is so, but only in physical spaces which have, like Rd , the special mathematical property of being simply connected in the sense that any closed path can be shrunk to a point; i.e. they have no holes. We return to this phenomenon in Section 7.6 on the Aharonov-Bohm effect.

40

4 Quantization

4.4 Spin Quantum mechanical particles may also have internal degrees of freedom, which have no classical counterpart. Mathematically, this mean that the wave functions ψ(x) take values not in C, but in a higher dimensional (complex) space V (with dimC V = n > 1). The state space in this case is L2 (R3 ; V ). For V we take a finite-dimensional, complex inner-product space. (When we want to be specific, we take V = Cn .) Then we have L2 (R3 ; V ) =

L2 (R3 ; C) | {z }

space of external degrees of freedom

⊗

V |{z}

.

space of internal degrees of freedom

Here, on the r.h.s. we have the tensor product of two Hilbert spaces, which can be thought of as the space with a basis {φi vj } given by products of basis elements of each factor, equipped with the corresponding inner product hφv, ψwi = hφ, ψihv, wi. It can be identified with the space of square integrable functions with values in V (x → φ(x) ∈ V ) on the l.h.s.. Now the Schr¨odinger equation is invariant under a larger group of gauge transformations, Uggauge : ψ(x) → gψ(x), g ∈ U (n), where U (n) is a group of complex, unitary, n × n matrices. Consider the simplest non-trivial case n = 2, and restrict ourselves to the subgroup SU (2) of U (2) with determinant one (the special unitary group). The Lie algebra su(2) of the group SU (2) is the space of trace zero, antihermitian (A∗ = −A) matrices acting on C2 , equipped with the commutator. We can choose a basis, A1 , A2 , A3 , in su(2), satisfying the commutation relations [A1 , A2 ] = A3 , [A2 , A3 ] = A1 , [A3 , A1 ] = A2 , (4.19) which have already come up in (3.14) while considering the Lie algebra so(3) of the group of rotations SO(3). In fact, the Lie algebras su(2) and so(3) are isomorphic, while one can find a Lie group homomorphism SU (2) → SO(3), with kernel {±1}; i.e., SU (2)/{±1} and SO(3) are Lie group isomorphic (see Remark 4.4 below). If we define the hermitian matrices Sj := −i~Aj , we obtain [Sk , Sl ] = i~ǫklm Sm ,

(4.20)

where, recall, ǫklm is the Levi-Chivita symbol: ǫ123 = 1 and ǫklm changes sign under permutation of any two indices. These are exactly the same relations as for the angular momentum operators (3.17). Consequently, the observable S = (S1 , S2 , S3 ) is thought of as internal angular momentum, and is called the spin. However, there is an important difference between the spin and angular momentum observables. We will show later on in Theorem 6.28 that the opP erator S 2 := j Sj2 has eigenvalues λ = ~2 r(r + 1), r = 0, 12 , 1, 23 , 2, . . . , of multiplicity 2r + 1. We see that, unlike the angular momentum, the spin can

4.4 Spin

41

take half-integer values. This means that if V is an eigenspace of S 2 with a half-integer r, then the spin observables Sj , j = 1, 2, 3, acting on V cannot be represented by generators of rotations on L2 (R3 ). We say that a particle has spin r if and only if the internal spin space V is an eigenspace, Vr , of S 2 (in some representation) with eigenvalue λ = ~2 r(r+1). By Theorem 6.28, Vr has dimension 2r+1 and r ∈ {0, 12 , 1, 23 , 2, . . . }. As a basis in Vr , it is convenient to use a basis of the eigenvectors of the third component, S3 , of the spin operator. We write elements of Vr in this basis as ψ(x, s), s = −r, . . . , r, where ψ(x, s) satisfies S3 ψ(x, s) = ~sψ(x, s), or as vectors ψ(x) = (ψ−r (x), . . . , ψr (x)), where each ψj belongs to the familiar one-particle space L2 (R3 ) = L2 (R3 ; C), with the identification ψ(x, s) ↔ ψs (x). (Usually such functions are written as columns, but for typographical simplicity we write them as rows.) This identifies Vr with the space C2r+1 and L2 (R3 ; C) ⊗ Vr = L2 (R3 ; Vr ), with L2 (R3 ; C) ⊗ C2r+1 = L2 (R3 ; C2r+1 ). For r = 21 , it is convenient to write Sj as Sj = ~2 σj , where σj are the Pauli matrices       01 0 −i 1 0 σ1 = , σ2 = , σ3 = . (4.21) 10 i 0 0 −1 For r = 12 , the spin operators Sj act on V 21 as S1 ψ(x, s) = ~|s|ψ(x, −s), S2 ψ(x, s) = −i~sψ(x, −s), S3 ψ(x, s) = ~sψ(x, s). It is an experimental fact that all particles belong to one of the following two groups: particles with integer spins, or bosons, and particles with halfinteger spins, or fermions. (The particles we are dealing with – electrons, protons and neutrons – are fermions, with spin 21 , while photons, which we will deal with later, are bosons, with spin 1. Nuclei, though treated as point particles, are composite objects whose spin could be either integer or halfinteger.) Problem 4.2 1) Find the generators, r1 , r2 , r3 , of the rotations, R1 (ϕ), R2 (ϕ), R3 (ϕ), around the x1 −, x2 −, x3 −axes. 2) Show that these generators satisfy commutation relations (4.19). As an example, consider the rotations R3 (ϕ) around the x3 −axis, given by   cos ϕ sin ϕ 0 R3 (ϕ) =  − sin ϕ cos ϕ 0  , (4.22) 0 0 1 The generators of these rotations are given by ri := ∂ϕ Ri (ϕ) ϕ=0 . For R3 (ϕ), we have   0 10 r3 =  −1 0 0  . (4.23) 0 01

42

4 Quantization

As we mentioned above, the algebra su(2) is isomorphic to the algebra of the rotation group so(3). We can define this isomorphism, φ : so(3) → su(2), starting with φ(ri ) = Si and then extending to the entire so(3) by linearity. Problem 4.3 Show that the map φ : so(3) → su(2) defined above is an algebra isomorphism. Remark 4.4 Though the algebras so(3) and su(2) are isomorphic, the groups SU (2) and SO(3) are not: there is a Lie group homomorphism, φ : SU (2) → SO(3), s.t. φ−1 (1) = {±1}. Indeed, P define the map h : R3 → { traceless hermitian 2 × 2 matrices} by hx := j xj σj and define φ : u → R, where R = φ(u) solves the equation hRx = uhx u∗ , for all x ∈ R3 . Using the equations hRx = uhx u∗ and det hx = −|x|2 , we compute |Rx|2 = − det hRx = − det hx = |x|2 ,

giving |Rx| = |x|, i.e. R preserves the Euclidean norm |x|. Hence R is a rotation of the space R3 and therefore is an element of the orthogonal group O(3). One can show that φ(u) has determinant 1, i.e. φ(u) ∈ SO(3), and therefore φ : SU (2) → SO(3). To show that φ is a group homomorphism, we use the equation hRx = uhx u∗ to compute (uu′ )hx (uu′ )∗ = uu′ hx (u′ )∗ u∗ = uhR′ x u∗ = hRR′ x . Finally, since the transformation Tu h = uhu∗ satisfies Tu h = uhu∗ = T−u h, we see that φ maps u and −u into the same element of SO(3). The spin interacts with an external magnetic field. The energy of this interaction has a form similar to that of an orbiting classical charge. For a charge moving in a circular orbit, the classical energy of interaction with a e l, where e and m are the charge and magnetic field B is −µ · B(x), µ := g 2m mass of the particle and l is its (classical) angular momentum. In quantum mechanics, this interaction (in the case of spin r = 21 ) is −µ · B(x), µ := g

e S, 2m

where g is called the gyromagnetic ratio. Based on classical mechanics, one expects g = 1, but it turns out that g = 2 (plus small corrections, due to creation and annihilation of photons out of and into the vacuum, if the electromagnetic field is quantized).

4.5 Many-particle Systems Now we consider a physical system consisting of n particles which interact pairwise via the potentials Vij (xi − xj ), where xj is the position of the j-th

4.5 Many-particle Systems

43

particle. Examples of such systems include atoms or molecules – i.e., systems consisting of electrons and nuclei interacting via Coulomb forces. In classical mechanics such a system is described by the particle coordinates, xj and momenta kj , j = 1, . . . , n, so that the classical state of the system is given by the pair (x, k) where x = (x1 , . . . , xn ) and k = (k1 , . . . , kn ) 3n and the state space, also called the phase-space, of the system is R3n x × Rk , or a subset thereof. The dynamics of this system is given by the classical Hamiltonian function h(x, k) =

n X 1 |kj |2 + V (x) 2m j j=1

where mj is the mass of the j-th particle and V is the total potential of the system, and the standard Poisson brackets  n  X ∂f ∂g ∂f ∂g {f, g} = · − · ∂kj ∂xj ∂xj ∂kj j=1 ∂f ∂f where ∂x and ∂k are the gradients in xj and kj , respectively. Since in our j j case the particles interact only with each other and by two-body potentials Vij (xi − xj ), V is given by 1X V (x) = Vij (xi − xj ). (4.24) 2 i6=j

Quantizing this system in exactly the same way as the one-particle one, we associate with particle coordinates xj , and momenta kj , the quantum coordinates xj , and momenta pj := −i~∇xj , which are operators. And so the classical Hamiltonian h(x, k) leads to the Schr¨odinger operator Hn := h(x, p), p = (p1 , . . . , pn ), i.e. Hn =

n X 1 |pj |2 + V (x), 2m j j=1

(4.25)

acting on L2 (R3n ). This is the Schr¨odinger operator, or quantum Hamiltonian, of the n−particle system. Example 4.5 Consider a molecule with N electrons of mass m and charge −e, and M nuclei of masses mj and charges Zj e, j = 1, . . . , M . In this case, the Schr¨odinger operator, Hmol , is Hmol =

N M X X 1 1 |qj |2 + V (x, y) |pj |2 + 2m 2m j 1 1

(4.26)

acting on L2 (R3(N +M) ). Here x = (x1 , . . . , xN ) are the electron coordinates, y = (y1 , . . . yM ) are the nucleus coordinates, pj = −i~∇xj is the momentum of the j-th electron, qj = −i~∇yj is the momentum of the j-th nucleus, and

44

4 Quantization

V (x, y) =

X e2 Zj 1 X e2 Zi Zj 1X e2 − + 2 |xi − xj | |xi − yj | 2 |yi − yj | i,j i6=j

(4.27)

i6=j

is the sum of Coulomb interaction potentials between the electrons (the first term on the r.h.s.), between the electrons and the nuclei (the second term), and between the nuclei (the third term). For a neutral molecule, we have M X

Zj = N.

j=1

If M = 1, the resulting system is called an atom, or Z-atom (Z = Z1 ).

4.6 Identical Particles Quantum many-particle systems display a remarkable new feature. Unlike in classical physics, identical particles (i.e., particles with the same masses, charges and spins, or, more generally, which interact in the same way) in quantum physics are indistinguishable. Assume we have n identical particles (of spin 0). Classically, states of such a system are given Q by (x1 , k1 , . . . , xn , kn ), with the state space, also called the phase-space, being n1 (R3x × R3k ) = R3n x × R3n . Naively, we might assume that the state space of the quantum system is k L2 (R3n , C) ≡ ⊗n1 L2 (R3 , C),

(4.28)

where the second term is the tensor product of n Hilbert spaces L2 (R3 , C), defined as the Hilbert space spanned by the products of elements of orthonormal bases in each L2 (R3 , C) (see Section 25.13 in the mathematical supplement for a definition for abstract Hilbert spaces). However this is not so, as we explain below. The indistinguishability of the particles means that all probability distributions which can be extracted from an n-particle wave function Ψ (x1 , . . . , xn ) should be invariant with respect to permutations of the coordinates, xj . To illustrate this, consider bound states, which can be always taken to be realvalued wave functions. Since |Ψ (x1 , . . . , xn )|2 is invariant under permutations, this implies that Ψ (x1 , . . . , xn ) is invariant under particle permutations, modulo a change of sign. A permutation is a one-to-one and onto map π of {1, 2, . . . , n} into itself, and the collection of all permutations of n indices forms the symmetric group Sn . To a permutation π, we associate a transformation of R3n (also denoted by π) π(x1 , . . . , xn ) = (xπ(1) , ..., xπ(n) ). We consider the map of Sn into unitary (i.e. preserving the inner product) operators on L2 (R3n ) defined by

4.6 Identical Particles

45

(Tπ Ψ )(x) = Ψ (π −1 x), with the property Tπ1 Tπ2 = Tπ1 π2 (this is why we need π −1 on the r.h.s.), called a (unitary) representation of Sn . Let U (X) denote the space of unitary operators acting on a Hilbert space X. For a group G, a representation T : G → U (X) is called irreducible iff X has no non-trivial proper subspace invariant under all Tg , g ∈ G. The dimension of T is the dimension of X. There are exactly two one-dimensional irreducible representations of Sn , and they are of special interest in the spinless case: one with Ψ (x1 , . . . , xn ) totally invariant under permutations, Tπ Ψ = Ψ ∀π ∈ Sn , and one with Ψ (x1 , . . . , xn ) transforming as Tπ Ψ = (−1)#(π) Ψ ∀π ∈ Sn ,

(4.29)

where #(π) is the number of transpositions making up the permutation π (so (−1)#(π) is the parity of π ∈ Sn – see the end of this section for more details on irreducible representations of Sn .) The particles described by the wave functions of the first type are called bosons and by the second type, fermions. Correspondingly, we have the following subspaces of L2 (R3n ): total Hbose := {Ψ ∈ L2 (R3n , C) | Tπ Ψ = Ψ ∀π ∈ Sn }, total Hfermi

2

3n

#(π)

:= {Ψ ∈ L (R , C) | Tπ Ψ = (−1)

Ψ ∀π ∈ Sn }

(4.30) (4.31)

Including spin. To complete the picture, we have to take into account particle spins. The one-particle space for a particle of spin r is L2 (R3 × {−r, . . . , r}, C) = L2 (R3 , C2r+1 ),

(4.32)

with wave functions of the form Ψ (x, s) = (Ψ−r (x), . . . , Ψr (x)). Assume we have n identical particles of spin r. In this case, the state space is L2 (R3 , C2r+1 )⊗n ≡ L2 (R3n , C(2r+1)n )

(4.33)

(see the remark above, and Section 25.13 about tensor products), consisting of functions which can be written as Ψ (x1 , s1 , . . . , xn , sn ). Now, the operator Hn acts on such functions. The indistinguishability of the particles means that all probability distributions which can be extracted from an n-particle wave function (4.33) should be symmetric with respect to permutations of the coordinates and spins of the identical particles. Since for bound states we can restrict ourselves to real wave functions, this is equivalent to the property that Ψ (x1 , s1 , . . . , xn , sn ) is invariant under such permutations, modulo a change of sign. Recall that all elementary (and composite) particles are divided into two groups: particles with half integer spins, called fermions (e.g. electrons, protons, and neutrons have spin 1/2), and particles with integer spins, called

46

4 Quantization

bosons (particles related to interactions). For bosons, the wave functions, Ψ (x1 , s1 , . . . , xn , sn ), should be symmetric with respect to permutations of the coordinates and spins of identical particles, and for fermions, antisymmetric. In particular, the state space for fermions of spin 12 is Hf ermi := {Ψ ∈ L2 (R3n , C2n ) | Tπ Ψ = (−1)#(π) Ψ, ∀π ∈ Sn }

(4.34)

where, as above, π is a permutation of the n indices, π : (1, 2, . . . , n) → (π(1), π(2), ..., π(n)), #(π) is the number of transpositions making up the permutation π, and (Tπ Ψ )(x1 , s1 , . . . , xn , sn ) = Ψ (xπ(1) , sπ(1) , ..., xπ(n) , sπ(n) ). V Below, we will write the space (4.31) as ni=1 L2 (R3 , C2 ). Since the Hamiltonian Hn does not act on the spin variables, by separation of variables, we may consider it acting on functions ψ(x1 , . . . , xn ) of the coordinates only, which arise from (4.34) by, say, taking inner products in the spin variables with functions of s1 , . . . , sn . What are the symmetry properties of these functions with respect to permutations of the coordinates? To answer this question one has to dip into the theory of representations of the symmetric group Sn (the group of permutations of n indices). We do not do so here, but just formulate below the outcome of the theory. Here we summarize: •

for identical particles the state space is not L2 (R3n ), but rather a subspace of it, defined by certain symmetry properties with respect to permutations of the particles.

Now, we summarize relevant elements of the theory. Consider n fermions of spin r. Denote by α partitions of the integer n into ordered positive integers 2r+1 ≥ α1 ≥ α2 ≥ · · · ≥ αk ≥ 1, α1 +α2 +· · ·+αk = n. Denote the set of such α’s by Ar . These can be visualized as arrangements of n squares into k rows containing α1 , α2 , . . . , αk squares each, called Young diagrams. For example, with n = 3:

In particular, for spin one half, r = 12 , we have one- and two-column Young diagrams. To a Young diagram, one associates a Young tableau by filling in squares with particles. We associate with a given Young diagram α, the space Hα of wave functions which are symmetric with respect to permutations of particles in the same row, and antisymmetric with respect to those in the same column, of some tableau T (e.g. a canonical one) associated with α. One can show that the subspaces Hα are mutually orthogonal and satisfy Hfermi = ⊕α∈Ar Hα .

(4.35)

4.7 Supplement: Hamiltonian Formulation of Classical Mechanics

47

In technical terms, irreducible representations of the symmetric group Sn are in one-to-one correspondence with α ∈ Ar , and on the subspace Hα , the representation of Sn is a multiple of the irreducible one labeled by α. The spaces (4.30) and (4.31) correspond to the irreducible representations of Sn , with the one-row and one-column Young diagrams, respectively. The remaining subspaces Hα correspond to fermions with higher spins. Irreducible representations of Sn can be connected, via Weyl’s theory of dual pairs of groups, to irreducible representations of SU (2) carried by the spin space C(2r+1)n , and therefore determine the total spin of the corresponding wave functions. Remark 4.6 In dimension 2, the state spaces of n identical particles is classified not by irreducible representations of the permutation group Sn , but by irreducible representations of the braid group Bn . See [111, 112, 262] for reviews, and a book with a collection of articles and commentary.

4.7 Supplement: Hamiltonian Formulation of Classical Mechanics In this supplement we discuss briefly the hamiltonian formulation of classical mechanics. For more details and extensions see Mathematical Supplement 26. The starting point here is the principle of minimal action: solutions of physical equations minimize (more precisely, make stationary) certain functionals, called action functionals. It is one of the basic principles of modern physics. The action functional, S : φ 7→ S(φ), is the integral of the form S(φ) :=

Z

0

T


˙ L φ(t), φ(t) dt,

(4.36)

where L : X × V → R is a twice differentiable function, called a Lagrangian function, or Lagrangian, V is a finite-dimensional inner-product vector space, called the space of velocities, X is an open subset of V , called the position, or configuration, space, and φ(t) is a differentiable path in X. The functional S(φ) is defined on the space of paths Pa,b = {φ ∈ C 1 ([0, T ]; X) | φ(0) = a, φ(T ) = b}, for some a, b ∈ X. We can write Pa,b in the form Pa,b = {φ0 + φ | φ ∈ P0 } ≡ φ0 + P0 , where φ0 is a fixed element of Pa,b , and P0 := P0,0 . Now, P0 is a vector space and consequently Pa,b is an affine space. For classical mechanics, L(x, v) = 21 mv 2 − V (x) : R3 × R3 → R and φ(t) = x(t), and therefore the action functional is given by S(φ) =

Z

0

T


m ˙2 |φ| − V (φ) dt. 2

(4.37)

The dynamics is given by critical points of this functional, and the dynamical equation is the equation for critical points

48

4 Quantization

S ′ (φ) = 0,

(4.38)

called the Euler-Lagrange equation. Here S ′ (φ) is the differential or variational derivative of S at φ, defined as follows. Let V ∗ be the space dual to V , i.e. the space of bounded linear functions or functionals on V (see Section 25.1). The action of l ∈ V ∗ is denoted as hl, vi. We define S ′ (φ) : [0, T ] → V ∗ by the equation Z T d hS ′ (φ)(t), ξ(t)idt = S(φλ )|λ=0 , (4.39) dλ 0

where φλ := φ + λξ, for any ξ ∈ P0 . For simplicity, let V = Rm , and denote by ∂φ L and ∂φ˙ L the gradients ˙ in the variables φ and φ, ˙ respectively. Using (4.36), we compute of L(φ, φ) RT
d ˙ ∂φ˙ L(φ)ξ + ∂φ L(φ)ξ dt. Integrating the first term on the dλ S(φλ )|λ=0 = 0 d r.h.s. by parts and using that ξ(0) = ξ(T ) = 0, we arrive at dλ S(φλ )|λ=0 =
RT d S(φ )| = 0, for any ξ ∈ P0 , this − ∂ (∂ L(φ)) + ∂ L(φ) ξ(t) dt. If λ λ=0 t φ˙ φ dλ 0 implies the equation ˙ + ∂φ L(φ, φ) ˙ =0 − ∂t (∂φ˙ L(φ, φ))

(4.40)

(see Section 26.2 for more details and generalizations). Applying this to the classical mechanics action functional (4.37), we arrive at Newton’s equation of Classical Mechanics: mφ¨ = −∇V (φ).

Suppose now that the dynamics of a system are determined by the action principle, with a differentiable Lagrangian function/functional L : X × V → R defined on a space X × V . We define the energy of a path φ as energy (φ) :=

∂L ˙ · φ − L. ∂ φ˙

We have Lemma 4.7 (Conservation of energy) If φ¯ is a critical path of the action ¯ = const. (4.36), then the energy is conserved, energy (φ) Proof. We compute   ∂2L ¨ ˙ ∂2L ˙ ˙ d ∂L ˙ ·φ−L = φ·φ+ φ·φ ˙ ds ∂ φ˙ ∂ φ˙ 2 ∂ φ∂φ ∂L ¨ ∂L ¨ ∂L ˙ + ·φ ·φ− ·φ− ∂φ ∂ φ˙ ∂ φ˙   d ∂L ∂L ˙ · φ. − = ds ∂ φ˙ ∂φ ¯ the expression on the right hand side vanishes, the result follows. Since for φ, 

4.7 Supplement: Hamiltonian Formulation of Classical Mechanics

49

We pass now to the new variables (x, v) → (x, k), where k, as a function of x and v, is given by k = ∂v L(x, v). (4.41) Note that k belongs to the space, V ∗ , dual to V (see Section 25.1). We assume that the equation (4.41) has a unique solution for v (which holds if L is strictly convex in the second variable, which means L(x, sv + (1 − s)v ′ ) < sL(x, v) + (1 − s)L(x, v ′ ), ∀s ∈ (0, 1),

(4.42)

∀x ∈ X, v, v ′ ∈ V, and it is guaranteed by the inequality d2v L(x, v) > 0, ∀x ∈ X, v ∈ V ). With this in mind, we express the energy in the new variables, as
h(x, k) = hk, vi − L(x, v) |v:∂v L(x,v)=k (4.43)

where the notation h·, ·i stands for the coupling between V ∗ and V (Section 25.1). This defines the Hamiltonian function/functional, h : X × V ∗ → R.

Theorem 4.8 If L(x, v) and h(x, k) are related by (4.43), then the EulerLagrange equation (4.40) for the action (4.36) is equivalent to the equations (called Hamilton’s equations) x˙ = ∂k h(x, k), k˙ = −∂x h(x, k).

(4.44)

Proof. Assume (4.40) is satisfied. First, we note that the equations (4.43) and (4.41) imply ∂k h(x, k) = v + (k − ∂v L(x, v))∂k v = v = x˙ and ∂x h(x, k) = −∂x L(x, v) + (k − ∂v L(x, v))∂x v = −∂x L(x, v). Now, the last equation and the equations (4.40) and (4.41) imply k˙ = −∂x h(x, k), which gives (4.44). Now, assuming (4.44), we obtain (4.40) from (4.41), ∂x h(x, k) = −∂x L(x, x) ˙ and k˙ = −∂x h(x, k).  Applying (4.43) to the classical mechanics Lagrangian, L(x, v) = we arrive at the classical Hamiltonian function h(x, k) =

1 |k|2 + V (x), 2m

mv 2 2

− V (x), (4.45)

1 which leads to Hamilton’s equations x˙ = m k, k˙ = −∂x V (x), which are equivalent to Newton’s equations. Another example of a Lagrangian is that for a classical relativistic particle (in units with speed of light c = 1): p L(x, x) ˙ = −m 1 − x˙ 2 . √ (ds = 1 − x˙ 2 dt is the proper time of the particle.) The generalized momx˙ and the Hamiltonian is mentum in this case is k = ∂v L(x, v) = √1− x˙ 2 p 2 2 h(x, k) = |k| + m .

50

4 Quantization

Problem 4.9 Prove this. Next, we recognize that Hamilton’s equations (4.44) can be written as z˙t = {z, h}(zt),

zt = (x(t), k(t)) ∈ Z = X × V ∗ ,

(4.46)

where now xj and kj together are thought of as a path in the phase space Z, and for any pair f , g of differentiable functions on Z, {f.g} denotes the function {f, g} = ∇x f · ∇k g − ∇k f · ∇x g. (4.47) The map (f, g) → {f, g}, given by (4.47), is a bilinear map, which has the following properties: for any functions f , g, and h from Z to R, 1. {f, g} = −{g, f } (skew-symmetry) 2. {f, {g, h}} + {g, {h, f }} + {h, {f, g}} = 0 (the Jacobi identity) 3. {f, gh} = {f, g}h + g{f, h}. (Leibniz rule) Bilinear maps, (f, g) → {f, g}, having these properties, are called Poisson brackets. The map (4.47) also obeys {f, g} = 0 ∀ g =⇒ f = 0. Poisson brackets with the latter property are said to be non-degenerate. Note that a space of smooth functions (or functionals), together with a Poisson bracket, has the structure of a Lie Algebra. The space Z together with a Poisson bracket on C ∞ (Z, R) is called a Poisson space. A Hamiltonian system is a pair: a Poisson space, (Z, {·, ·}), and a Hamiltonian function, h : Z → R. In this case Hamilton’s equations are given by (4.46). Classical mechanics of one particle is a Hamiltonian system with the phase space Z = R3 × R3 , with bracket (4.47) and the Hamiltonian (4.45). Remark 4.10 Our definition of a Hamiltonian system differs from the standard one in using the Poisson bracket instead of a symplectic form. The reason for using the Poisson bracket is its direct relation to the commutator. Definition 4.11 Functions on a Poisson space, (Z, {·, ·}), are called classical observables. The classical evolution of observables is given by f (z, t) = f (zt ), where zt is the solution of (4.46) with the initial condition z. Note that f (z, t) solves the equation d f (z, t) = {f, h}(z, t). dt with the initial condition f (z, 0) = f (z). Conversely, a solution of this equation with an initial condition f (z) is given by f (z, t) = f (zt ). Problem 4.12 Prove this.

4.7 Supplement: Hamiltonian Formulation of Classical Mechanics

51

The equation above implies that an observable f (z) is conserved or is a constant of motion, i.e. f (zt ) is independent of t, if and only if its Poisson bracket with the Hamiltonian h vanishes: {f, h} = 0. The map t → φt , where φt (z) := zt and zt is the solution of (4.46) with the initial condition z, is called the flow associated with the differential equation (4.46). Particle coupled to an external electro-magnetic field. As an application of the above machinery, we consider a system of charges interacting with an electro-magnetic field. Of course, if the external field is purely electric, then it is a potential field, and fits within the framework we have considered already. Suppose, then, that a magnetic field B, and an electric field, E, are present, B, E : R3+1 7→ R3 . The law of motion of a classical particle of mass m and electric charge e is given by Newton’s equation with the Lorentz force, e m¨ x(t) = eE(x(t), t) + x˙ ∧ B(x(t), t). c

(4.48)

To find a hamiltonian formulation of this equation we first derive it from the minimum action principle. We know from the theory of electro-magnetism (Maxwell’s equations) that the electric and magnetic fields, E and B, can be expressed in terms of the vector potential A : R3+1 7→ R3 , and the scalar potential Φ : R3 7→ R via 1 E = −∇Φ − ∂t A, c

B = curlA.

The action functional which gives (4.48) is given by Z T  e m ˙2 |φ| − eΦ(φ) + φ˙ · A(φ) dt. S(φ) = 2 c 0

(4.49)

(4.50)

Indeed, we find the Euler-Lagrange equation (see (4.40)) for this functional. ˙ = m |φ| ˙ 2 − eΦ(φ) + e φ˙ · A(φ), we compute Using that L(φ, φ) 2 c ˙ = mφ˙ + e A(φ), ∂φ L(φ, φ) ˙ = −e∇Φ(φ) − e ∇φ (φ · A(φ)). ∂φ˙ L(φ, φ) c c

d ˙ A(φ) = ∂t A(φ)+(φ·∇)A(φ), ∇(v· Plug this into (4.40) and use the relations dt A) − (v · ∇)A(φ) = v ∧ curlA and (4.49) to obtain (4.48). Now, the generalized momentum is k = mx˙ + ec A(x) and the classical e , which gives Hamiltonian function is hA,Φ (x, k) = k · v − L(x, v)|mx=k− ˙ c A(x)

hA,Φ (x, k) =

1 e (k − A(x))2 + eΦ(x). 2m c

Defining the Poisson bracket as in (4.47), we arrive at the hamiltonian formulation for a particle of mass m and charge e moving in the external electric and magnetic fields E and B.

5 Uncertainty Principle and Stability of Atoms and Molecules

One of the fundamental implications of quantum theory is the uncertainty principle. It states that certain pairs of physical quantities cannot be measured simultaneously with arbitrary accuracy. Mathematically, it follows from the fact that the corresponding observables do not commute. The key example here is provided by the observables of position and momentum. They do not commute, as seen from the commutation relation: i [p, x] = 1 ~

(5.1)

(this is a matrix equation, meaning ~i [pj , xk ] = δjk ). In this chapter, we establish precise mathematical statements of the uncertainty principle for the position and momentum observables.

5.1 The Heisenberg Uncertainty Principle We consider a particle in a state ψ and think of the observables x and p ˆ 2 respectively. as random variables with probability distributions |ψ|2 and |ψ| Recall that the means of xj and pj in the state ψ (∈ D(xj ) ∩ D(pj )) are hxj iψ and hpj iψ , respectively. The dispersion of xj in the state ψ is (∆xj )2 := h(xj − hxj iψ )2 iψ and the dispersion of pj is (∆pj )2 := h(pj − hpj iψ )2 iψ . Theorem 5.1 (The Heisenberg uncertainty principle) For any state ψ ∈ D(xj ) ∩ D(pj ), ~ (5.2) ∆xj ∆pj ≥ . 2

© Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_5

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5 Uncertainty Principle and Stability of Atoms and Molecules

Proof. The basic ingredient is commutation relation (5.1). For notational simplicity, we assume hxiψ = hpiψ = 0. Note that for two self-adjoint operators A, B, and ψ ∈ D(A) ∩ D(B), hi[A, B]iψ = −2 ImhAψ, Bψi.

(5.3)

So assuming ψ is normalized (kψk = 1), and ψ ∈ D(xj ) ∩ D(pj ), we obtain i 2 1 = hψ, ψi = hψ, [pj , xj ]ψi = − Imhpj ψ, xj ψi ~ ~ 2 2 2 ≤ |hpj ψ, xj ψi| ≤ kpj ψkkxj ψk = (∆pj )(∆xj ). ~ ~ ~ This does it.  What are the states which minimize the uncertainty, i.e. the l.h.s. of (5.2)? Clearly, the states for which − Imhpj ψ, xj ψi = kpj ψkkxj ψk would do this. This equality is satisfied by states obeying the equation pj ψ = iµxj ψ for some Q µ
1/4 − P µj x2 /2~ j µ > 0. Solving the latter equation we obtain ψµ := j π~j e for any µj > 0. Of course, shifting these states in coordinate and momentum as Y  µj 1/4 P 2 e(iq·x+iϕ− µj (xj −yj ) /2)/~ (5.4) ψyqµϕ := π~ j would give again states minimizing the uncertainty principle. These states are called coherent states. They are obtained by scaling and translating the 2 Gaussian state φ := (π~)−3/4 e−|x| /2~ and can be written as
ψyqµϕ := ei q·y+ϕ /~ Tyq φ, (5.5)
where Tyq is the shift operator in coordinate and momentum: Tyq := ei q·x−p·y /~ . Note that hψyqµϕ , xj ψyqµϕ i = yj , hψyqµϕ , pj ψyqµϕ i = qj , ~ hψyqµϕ , (x − yj )2 ψyqµϕ i = , 2µ ~µ . hψyqµϕ , (p − qj )2 ψyqµϕ i = 2 Problem 5.2 Prove (5.5) - (5.8).

5.2 A Refined Uncertainty Principle The following result is related to the Heisenberg uncertainty principle.

(5.6) (5.7) (5.8)

5.2 A Refined Uncertainty Principle

55

Theorem 5.3 (Refined uncertainty principle) On L2 (R3 ), −∆ ≥

1 . 4|x|2

Recall that for operators A, B, we write A ≥ 0 if hψ, Aψi ≥ 0 for all ψ ∈ D(A), and we write A ≥ B if A − B ≥ 0. So by the above statement, we mean 1 hψ, (−∆ − 4|x| 2 )ψi ≥ 0 for ψ in an appropriate dense subspace of D(−∆). We will prove it for ψ ∈ C0∞ (Rd ). Proof. We will ignore domain questions, leaving these as an exercise for the reader. Compute d X

i[|x|

−1

j=1

pj |x|

−1

, xj ] = |x|

−1

d X

i[pj , xj ]|x|−1 = ~d|x|−2

j=1

(d = space dimension = 3). Hence, using (5.3) again, ~dk|x|

−1

2

ψk = −2

d X j=1

Imh|x|−1 pj |x|−1 ψ, xj ψi

and therefore, using pj |x|−1 = |x|−1 pj + [pj , |x|−1 ] = |x|−1 pj + i~ we obtain ~(d − 2)k|x|−1 ψk2 = −2 Im

d X j−1

hpj ψ,

xj , |x|3

xj ψi. |x|2

Now the Cauchy-Schwarz inequality implies |

d X j=1

hpj ψ,

xj ψi| ≤ hψ, |p|2 ψi1/2 k|x|−1 ψk |x|2

(prove this!), which together with the previous equality gives ~|d − 2|k|x|−1 ψk2 ≤ 2hψ, |p|2 ψi1/2 k|x|−1 ψk. Squaring this, and observing that k|x|−1 ψk2 = hψ, |x|−2 ψi and hψ, |p|2 ψi = ~2 hψ, −∆ψi yields (for d ≥ 3) hψ, −∆ψi ≥

1 |d − 2|2 hψ, 2 ψi 4 |x|

which, for d = 3, implies the desired result. 

56

5 Uncertainty Principle and Stability of Atoms and Molecules

5.3 Application: Stability of Atoms and Molecules In classical mechanics, atoms and molecules are unstable: as the electrons orbit the nuclei, they radiate away energy and fall onto the nuclei. The demonstration that this is not so in quantum mechanics was one of the first triumphs of the theory. The statement that a quantum system (with the particles interacting vial Coulomb potentials) is stable with respect to collapse is expressed mathematically by the property that the Hamiltonian H (and therefore the energy), is bounded from below. First we demonstrate the stability for the simplest quantum system – the hydrogen-type ion. The latter is described by the Schr¨odinger operator ~2 e2 Z Hhyd = − ∆− 2m |x|

on L2 (R3 ) (m and −e are the mass and charge of the electron respectively and eZ is the charge of the nucleus). The refined uncertainty principle gives e2 Z ~2 −1 = Hhyd ≥ 8m|x| 2 − |x| . The right hand side reaches its minimum at |x| 2 2 4me Z/~ and so 2me4 Z 2 . (5.9) Hhyd ≥ − ~2 Thus, the energy of the hydrogen atom is bounded from below, and the electron does not collapse onto the nucleus. Now we show how to extend the argument above to an arbitrary system of electrons and nuclei, by considering for simplicity an atom with N electrons and an infinitely heavy nucleus. According to Section 4.5, the Schr¨odinger operator of this system is given by Hat =

N X j=1

(−

~2 e2 Z 1X e2 ∆xj − )+ , 2m |xj | 2 |xi − xj |

(5.10)

i6=j

acting on L2 (R3N ). Here m and −e are the electron mass and charge, x = PN 2 (x1 , . . . , xN ) are the electron coordinates, and the term 1 (− e|xjZ| ) on the r.h.s. is the sum of Coulomb interaction potentials between the electrons and the nuclei (Ze is the charge of the nucleus) and the last term, between the electrons. For a neutral atom, Z = N . For the moment, we ignore the fact that the electrons are fermions. To prove a lower bound on Hat we observe that the electron-electron interaction potential is positive and therefore we have the following lower bound for Hat : indep Hat ≥ Hat , (5.11) PN 2 2 indep ~ ∆xj − e|xjZ| ). Using, for each term on the r.h.s., the where Hat := 1 (− 2m bound (5.9), we obtain

5.3 Application: Stability of Atoms and Molecules

57

2m(e2 Z)2 . (5.12) ~2 This bound works but it is rather rough. First of all it ignores the electronelectron repulsion, but most importantly, it ignores that the electrons are fermions (see Section 4.6). In section 14.1, we show how to take into account the second feature and how to improve on the first one. Hat ≥ −N

6 Spectrum and Dynamics

Given a quantum observable (a self-adjoint operator) A, what are the possible values A can take in various states of the system? The interpretation of hψ, Aψi as mean value of the observable A in a state ψ, which is validated by quantum experiments, leads to the answer. It is the spectrum of A. The most important observable is the energy – the Schr¨odinger operator, H, of a system. Hence the spectrum of H gives the possible values of the energy. The goal of this chapter is to develop techniques for finding the spectra of Schr¨odinger operators. A rough classification of the spectra is into discrete and continuous (also called essential) components. Such a classification is related to the space-time behaviour of solutions of the corresponding Schr¨odinger equations. Thus our main thrust is toward describing these components. We begin by presenting the general theory, and then proceed to applications. In particular, we explain how the spectrum of a Schr¨odinger operator gives us important information about the solutions of the Schr¨odinger equation. Details about the general machinery are presented in Mathematical Supplement 25.

6.1 The Spectrum of an Operator We begin by giving some key definitions and statements related to the spectrum. More details can be found in Section 25.8. (Note we will often omit the identity operator 1 in expressions like A − z1.) Definition 6.1 The spectrum of an operator A on a Hilbert space H is the subset of C given by σ(A) := {λ ∈ C | A − λ is not invertible (has no bounded inverse)}.

The complement of the spectrum of A in C is called the resolvent set of A: ρ(A) := C\σ(A). For λ ∈ ρ(A), the operator (A − λ)−1 , called the resolvent of A, is well-defined and bounded. The following exercise asks for the spectrum of our favorite operators. © Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_6

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6 Spectrum and Dynamics

Problem 6.2 Prove that as operators on L2 (Rd ) (with their natural domains), σ(1) = {1}. σ(pj ) = R. σ(xj ) = R. σ(V ) = Ran(V ), where V is the multiplication operator on L2 (Rd ) by a continuous function V (x) : Rd → C. 5. σ(−∆) = [0, ∞). 6. σ(f (p)) = Ran(f ), where f (p) := F −1 f F with f (k) the multiplication operator on L2 (Rd ) by a continuous function f (k) : Rd → C. 1. 2. 3. 4.

The following two results state important facts about the spectrum. They are proved in Section 25.8. Theorem 6.3 The spectrum σ(A) ⊂ C is a closed set. Theorem 6.4 The spectrum of a self-adjoint operator is real: A self-adjoint =⇒ σ(A) ⊂ R.

6.2 Spectrum and Measurement Outcomes We saw in Section 3.2 that the postulates of quantum mechanics imply that the probability that, in the state ψ, the physical quantity corresponding to an observable A is in a set Ω is given by Born’s rule Probψ (A ∈ Ω) = hψ, χΩ (A)ψi.

(6.1)

Here χΩ (λ) is the characteristic function of the set Ω (i.e. χΩ (λ) = 1, if λ ∈ Ω and χΩ (λ) = 0, if λ ∈ / Ω) and the operator χΩ (A) can be defined using an operator calculus (see Section 25.11, Eq (25.43)). We define it in an important special case below. Conclusion (6.1) is validated by quantum experiments. For A self-adjoint, the operator χΩ (A) is also self-adjoint. Motivated by (6.1), we call it the probability observable. Mathematically, it is a projection, i.e. it satisfies (see Exercise 6.5 below) χΩ (A)2 = χΩ (A).

(6.2)

One can show, using the formula (25.43) for functions of operators given in the Mathematical Supplement (we sketch the proof below), that χΩ (A) = χΩ∩σ(A) (A).

(6.3)

χΩ (A) 6= 0 ⇔ Ω ∩ σ(A) 6= ∅.

(6.4)

This relation implies

6.2 Spectrum and Measurement Outcomes

61

The latter property shows that the spectrum is equal to the set of all the values the physical quantity represented by the observable A could take, or in short σ(A) = {values of the physical quantity represented by A}.

(6.5)

Due to the relation (6.1), the above equation suggests that σ(A) can be interpreted as the set of all possible values of the observable A. The most important observable is the energy – the Schr¨odinger operator, H, of a system. Hence the spectrum of H gives the possible values of the energy. Now, we sketch a proof of (6.3). First, we recall from Section 25.11, H Eq 1 (25.43), that for Ω ⊂ C with ∂Ω ⊂ ρ(A), we use that χΩ (λ) := 2πi γ (z − λ)−1 dz to define χΩ (A) by the Cauchy integral I 1 (z − A)−1 dz, (6.6) χΩ (A) := 2πi γ where γ = ∂Ω. The integral here can be understood either as the Bochner integral of vector-valued functions with values in a Banach space, i.e. as the norm limit Riemann sum approximations, or in the weak sense: I 1 hψ, (A − z)−1 φidz, hψ, χΩ (A)φi := 2πi γ for any ψ, φ ∈ H. Here we use the fact that the knowledge of hψ, T φi for all ψ and φ determines the operator T uniquely. We call χΩ (A) the characteristic function of the operator A. This definition coincides with all other definitions for χΩ (A). Problem 6.5 Prove (6.2), or, more generally, that (f · g)(A) = f (A)g(A). We claim that with this definition we have (6.3) (assuming for simplicity that ∂Ω ⊂ ρ(A)). Indeed, if Ω∩σ(A) = ∅, then (A−z)−1 is analytic in H the resolvent −1 Ω, and therefore by Cauchy’s theorem γ (A − z) dz = 0, and so χΩ (A) = 0. One familiar reason for A − λ not to be invertible, is that (A − λ)ψ = 0 has a non-zero solution, ψ ∈ H. In this case we say that λ is an eigenvalue of A and ψ is called a corresponding eigenvector. If ψ is a normalized eigenvector of A with eigenvalue λ, then the equation (A−z)ψ = (λ−z)ψ implies (A−z)−1 ψ = (λ−z)−1 ψ for any z in the resolvent set of A. This and definition (6.6) imply χΩ (A)ψ = χΩ (λ)ψ, which, together with definition (6.1) gives Probψ (A ∈ Ω) = 1 if λ ∈ Ω and = 0 if λ ∈ / Ω.

(6.7)

Hence measuring the corresponding physical observable in the state ψ will always give the same answer, λ.

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6 Spectrum and Dynamics

P Now consider a superposition ψ = i ai ψi of several normalized eigenvectors ψi of A, with eigenvalues λi . By Problem 6.6(2) below, these ψi can be chosen to be mutually orthogonal. Then by χΩ (A)ψi = χΩ (λi )ψi and the orthonormality of the ψi , we have X |ai |2 Probψi (A ∈ Ω), Probψ (A ∈ Ω) = i

which is consistent with an interpretation of the the coefficients ai as probability ampltudes. Similar statements, though technically more complicated, could be made about the essential spectra discussed below. As a partial generalization of (6.7), we have that for any ψ ∈ RanχΩ ′ (A), we have Probψ (A ∈ Ω) = 1 if Ω ′ ⊂ Ω and = 0 if Ω ′ ∩ Ω = ∅.

(6.8)

6.3 Classification of Spectra We begin with the simplest type of spectrum. The discrete spectrum of an operator A is σd (A) = {λ ∈ C | λ is an isolated eigenvalue of A with finite multiplicity} (isolated meaning some neighbourhood of λ is disjoint from the rest of σ(A)). Here the multiplicity of an eigenvalue λ is the dimension of the eigenspace Null(A − λ) := {v ∈ H | (A − λ)v = 0}. Problem 6.6 1. Show Null(A − λ) is a vector space. 2. Show that if A is self-adjoint, eigenvectors of A corresponding to different eigenvalues are orthogonal. The rest of the spectrum is called the essential spectrum of the operator A: σess (A) := σ(A)\σd (A).

Remark 6.7 Some authors may use the terms “point spectrum” and “continuous spectrum” rather than (respectively) “discrete spectrum” and “essential spectrum’. Problem 6.8 For the following operators on L2 (Rd ) (with their natural domains), show that

6.4 Bound and Decaying States

63

1. σess (pj ) = σ(pj ) = R; 2. σess (xj ) = σ(xj ) = R; 3. σess (−∆) = σ(−∆) = [0, ∞). Hint: Show that these operators do not have discrete spectrum. Problem 6.9 Show that if U : H → H is unitary, then σ(U ∗ AU ) = σ(A), σd (U ∗ AU ) = σd (A), and σess (U ∗ AU ) = σess (A). Problem 6.10 Let A be a self-adjoint operator on H. If λ is an accumulation point of σ(A), then λ ∈ σess (A). Hint: use the definition of the essential spectrum, and the fact that the spectrum is a closed set. For a self-adjoint operator A the sets {span of eigenfunctions of A} and {span of eigenfunctions of A}⊥ , where W ⊥ := {ψ ∈ H | hψ, wi = 0 ∀ w ∈ W }, are invariant under A in the sense of the definition Definition 6.11 A subspace W ⊂ H of a Hilbert space H is invariant under an operator A if Aw ∈ W whenever w ∈ W ∩ D(A). Problem 6.12 Assume A is a self-adjoint operator. Show that 1. If W is invariant under A, then so is W ⊥ ; 2. The span, V , of the eigenfunctions of A, and its orthogonal complement, V ⊥ , are invariant under A; 3. Suppose further that A has only finitely many eigenvalues, all of them with finite multiplicity. Show that the restricted operator A|V has a purely discrete spectrum; 4. Show that the restricted operator A|V ⊥ has a purely essential spectrum. The spaces {span of eigenfunctions of A} and {span of eigenfunctions of A}⊥ is said to be the subspaces of the discrete and essential spectra of A.

6.4 Bound and Decaying States We show how the classification of the spectrum introduced in the previous section is related to the space-time behaviour of solutions of the Schr¨odinger equation ∂ψ i~ = Hψ ∂t with given initial condition ψ|t=0 = ψ0 , where H is a self-adjoint Schr¨odinger operator acting on L2 (R3 ). Naturally, we want to distinguish between states which are localized for all time, and

64

6 Spectrum and Dynamics

those whose essential support moves off to infinity. We assume all functions below are normalized. Suppose first that ψ0 ∈ { span of eigenfunctions of H}. Then for any ǫ > 0, there is an R such that Z inf |ψ|2 ≥ 1 − ǫ. (6.9) t

|x|≤R

iHt

To see this, note that if Hψ0 = λψ0 , then e− ~ ψ0 = e− Z Z 2 |ψ| = |ψ0 |2 → 0 |x|≥R

iλt ~

ψ0 , and so

|x|≥R

as R → ∞. Such a ψ is called a bound state, as it remains essentially localized in space for all time. A proof of (6.9) in the general case is given at the end of this section. On the other hand, if ψ0 ∈ {span of eigenfunctions of H}⊥ , where W ⊥ := {ψ ∈ H | hψ, wi = 0 ∀ w ∈ W }, then for all R, Z |ψ|2 → 0

(6.10)

|x|≤R

as t → ∞, in the sense of ergodic mean. Convergence f (t) → 0 in ergodic mean as t → ∞ means that Z 1 T f (t)dt → 0 T 0 as T → ∞. This result is called the Ruelle theorem. We sketch the proof below (see, eg, [73, 162] for a complete proof). Such a state, ψ, is called a decaying state, as it eventually leaves any fixed ball in space. We say that a system in a bound state is stable, while in a decaying state, unstable. This notion differs from the notion of stability in dynamical systems. Indeed, solutions of the Schr¨odinger equation are always dynamically orbitally stable (see Section 14.1 for the relevant definitions). The notion of stability here characterizes the space-time behaviour of the system, whether it stays essentially in a bounded region of the space, or falls apart with fragments departing from each other. Now, we saw above that solutions of the Sch¨ odinger equation with initial conditions in the discrete spectral subspace describe bound states, those in the essential spectral subspace describe decaying states. Hence the spectral classification for a Schr¨odinger operator H leads to a space-time characterization of the quantum mechanical evolution, ψ = e−iHt/~ ψ0 . Namely, the classification of the spectrum into discrete and essential parts corresponds to a classification

6.4 Bound and Decaying States

65

of the dynamics into localized (bound) states and locally-decaying (scattering) states. Finally, we give proof of (6.9) and (6.10). Proof of equation (6.9) in the general case: if ψ0 ∈ { span of eigenfunctions of H}, P P 2 then ψ0 can be written as ψ0 = j |aj | = 1, j aj ψj where aj ∈ C, (we assume in what follows that kψ0 k = 1) and {ψj } is an orthonormal set (hψj , ψk i = δjk ) of eigenfunctions of H: Hψj = λj ψj . We will assume PN that the above sum has only a finite number of terms, say ψ0 = j=1 aj ψj . Otherwise an additional continuity argument is required below. The solution ψ = e−iHt/~ ψ0 can be written as ψ=

N X

e−iλj t/~ aj ψj .

j=1

Multiplying this equation by the characteristic function of the exterior of the R−ball, |x| ≥ R, taking the L2 −norm and using the triangle inequality, we obtain Z Z 1/2 1/2 X N |aj | |ψ|2 ≤ |ψj |2 . (6.11) |x|≥R

|x|≥R

j=1

To estimate the second factor on the right hand side, for any ε > 0, we choose R such that for all j, Z 1/2 ε |ψj |2 ≤√ . N |x|≥R Using this estimate and applying the Cauchy-Schwarz inequality PN Pto the sum on the right hand side of (6.11) and using that j=1 1 = N and j |aj |2 = 1, we obtain Z  1/2

|x|≥R

|ψ|2

≤ε

and so (6.9) follows.  Sketch of proof of (6.10): in this proof we display the time dependence as a subindex t. We suppose the potential V (x) is bounded, and so as a multiplication operator kV k = kV k∞ , and therefore V ≥ −kV k1. Since also −∆ ≥ 0, the operator H is bounded below: H ≥ −kV k1. Let λ > kV k + 1 so that H + λ > 1, and let φt := (H + λ)ψt = e−iHt (H + λ)ψ0 . Defining B := (−∆ + λ)(H + λ)−1 ,

(6.12)

and using that B = 1 − V (H + λ)−1 , which shows that B is bounded, we see that ψt = (−∆ + λ)−1 Bφt . Let χΩ denote the characteristic function of a set Ω ⊂ R3 and let K(x, y) be the integral kernel of the operator χΩ (−∆ + λ)−1 . Using (6.12) and the notation Kx (y) := K(x, y), we obtain that

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6 Spectrum and Dynamics

χΩ ψt = χΩ (−∆ + λ)−1 Bφt = hKx , Bφt i = hB ∗ Kx , φt i.

(6.13)

We use the notation k · kx for the L2 −norm in the variable x and claim now that Z 1 T dtkhB ∗ Kx , φt ik2x → 0, as T → ∞. (6.14) T 0 R To prove this claim we use the fact that |K(x, y)|2 dxdy < ∞ to show that |hB ∗ Kx , φt i| ≤ kB ∗ Kx kkφt k ≤ kB ∗ kkKxkk(H + λ)ψ0 k ∈ L2 (dx)

(uniformly in t). Next, we want to prove that ∀x,

1 T

Z

0

T

dt|hB ∗ Kx , φt i|2 → 0, as T → ∞.

(6.15)

Then (6.14) follows from interchange of t− and x−integration on the l.h.s. and the Lebesgue dominated convergence theorem. The proof of (6.15) is a delicate one. First note that, since ψ0 ⊥ the eigenfunctions of H, so is φ0 := (H + λ)ψ0 (show this). Next, we write |hf, φt i|2 = hf ⊗ f¯, φt ⊗ φ¯t i = hf ⊗ f¯, e−itL/~ φ0 ⊗ φ¯0 i.

(6.16)

where L is an operator acting on H⊗H given by L := H ⊗1−1⊗H. The relation φ0 ⊥ (the eigenfunctions of H) implies that φ0 ⊗ φ¯0 ⊥ (the eigenfunctions of H ⊗ 1 − 1 ⊗ H). Hence we can compute the time integral of the r.h.s.: 1 T

Z

0

T

1 e−iT L/~ − 1 φ0 ⊗ φ¯0 i. dthf ⊗ f¯, e−itL/~ φ0 ⊗ φ¯0 i = hf ⊗ f¯, T −iL/~

(6.17)

The delicate point here is to show that, for nice f and φ0 , the r.h.s. is welldefined and is bounded by CT −δ , δ > 0. We omit showing this here (this can be done, for example. by using the spectral decomposition theorem, see [244]). Then (6.15) holds, which completes the proof (since this shows that for any bounded set Ω ⊂ R3 we have that kχΩ ψt k → 0, in the sense of ergodic mean, as t → ∞, which is equivalent to (6.10)). 

6.5 Spectra of Schr¨ odinger Operators We now want to address the question of how to characterize the essential spectrum of a self-adjoint operator A. Is there a characterization of σess (A) similar to that of σd (A) in terms of some kind of eigenvalue problem? In particular, we address this question for Schr¨odinger operators. We begin with Definition 6.13 Let A be an operator on L2 (Rd ). A sequence {ψn } ⊂ D(A) is called a spreading sequence for A and λ if

6.5 Spectra of Schr¨ odinger Operators

67

1. kψn k = 1 for all n 2. for any bounded set B ⊂ Rd , supp(ψn ) ∩ B = ∅ for n sufficiently large 3. k(A − λ)ψn k → 0 as n → ∞. Clearly, if not for the second condition, a sequence consisting of a repeated eigenfunction would fit this definition. The second condition implies that we can choose a subsequence {ψn′ } so that supp ψn′ ⊂ {|x| ≥ Rn }, with Rn → ∞. In what follows, we always assume that the sequence {ψn } has this property. ~2 As an example, a spreading sequence on L2 (R3 ) for − 2m ∆ and any λ ≥ 0 −3/2 ik·x/~ is given by ψn (x) := n f (|x|/n)e , where f is a smooth (suitably 1 normalized) function s.t. f (x) = 0 for |x| ≤ 1, and 2m |k|2 = λ. Problem 6.14 Show that the sequence constructed above is a spreading se1 ~2 ∆ and λ = 2m |k|2 . quence for − 2m 2

~ ∆+V is a Schr¨odinger Theorem 6.15 (Weyl-Zhislin theorem) If H = − 2m operator, with real potential V (x) which is continuous and bounded from below, then

σess (H) = {λ | there is a spreading sequence for H and λ}. We sketch a proof of this result later. Now, we describe the spectra of self-adjoint Schr¨odinger operators H =−

~2 ∆ + V. 2m

Our first result covers the case when the potential tends to zero at infinity. Recall that we have proved in Section 2.2 that H is self-adjoint. Theorem 6.16 Let V : Rd → R be continuous, with V (x) → 0 as |x| → ∞. Then σess (H) = [0, ∞) (so H can have only negative isolated eigenvalues, possibly accumulating at 0). Proof. We have, by the triangle inequality, k(H − λ)ψn k − kV ψn k ≤ k(−

~2 ∆ − λ)ψn k ≤ k(H − λ)ψn k + kV ψn k. 2m

Suppose {ψn } is a spreading sequence. Then the term kV ψn k goes to zero as n → ∞ because V goes to zero at infinity and {ψn } is spreading. So λ is ~2 ∆). We have (see in the essential spectrum of H if and only if λ ∈ σess (− 2m 2 ~ Problem 6.8) σess (− 2m ∆) = σess (−∆) = [0, ∞), and consequently σess (H) = [0, ∞). 

68

6 Spectrum and Dynamics

Problem 6.17 Extend Theorems 6.15 and 6.16 to real potentials V (x) satisfying (2.8). The bottom of the essential spectrum (0 in the present case), is called the ionization threshold, since above this energy the particle is no longer localized, but moves freely. Our next theorem covers confining potentials – that is, potentials which increase to infinity with x. As we have mentioned in Section 2.2, Schr¨odinger operators with such potentials are self-adjoint. Theorem 6.18 Let V (x) be a continuous function on Rd satisfying V (x) ≥ 0, and V (x) → ∞ as |x| → ∞. Then σ(H) consists of isolated eigenvalues {λn }∞ n=1 with λn → ∞ as n → ∞. Moreover, the corresponding eigenfunctions form an (orthonormal) basis in L2 (Rd ). Proof. Suppose λ is in the essential spectrum of H, and let {ψn } be a corresponding spreading sequence. Then as n → ∞, ~2 0 ← hψn , (H − λ)ψn i = hψn , − ∆ψn i + hψn , V ψn i − λ 2m Z 2 Z ~ |∇ψn |2 + V |ψn |2 − λ = 2m ≥ inf V (y) − λ → ∞ y∈supp(ψn )

(because {ψn } is spreading), which is a contradiction. Thus the essential spectrum is empty. Now we show that H has an infinite number of eigenvalues, with finite multiplicities, tending to +∞. Suppose, on the contrary, that H has a finite number of eigenvalues with finite multiplicities, and let f be a non-zero element of L2 which is orthogonal to all the eigenfunctions of H. Then, for any z ∈ C, the equation (H − z)ψ = f has a unique solution ψ = (H − z)−1 f in {span of eigenfunctions of H}⊥ . The function (H − z)−1 f is analytic in z ∈ C and satisfies k(H − z)−1 f k ≤ | Im z|−1 kf k. Hence, by a straightforward extension of the Liouville theorem, (H − z)−1 f = 0, a contradiction. Since the essential spectrum is empty, the eigenvalues have finite multiplicities and cannot accumulate, and since H ≥ 0, they must tend to +∞. The above argument shows also that the eigenfunctions of H form a basis.  Problem 6.19 What are the essential spectra, and what are the possible locations of the discrete spectra, of the following operators (justify your answer): ~2 ∆ − 10|x|3 + |x|4 , (a) H = − 2m ~2 (b) H = − 2m ∆ − (1 + |x|)−2 . Sketch of the proof of Theorem 6.15. We will prove the theorem for Schr¨odinger operators with H0 -bounded potentials V – that is, potentials satisfying the estimate (2.8).

6.5 Spectra of Schr¨ odinger Operators

Let {ψn } be a spreading sequence for H and λ, and let φn =

69

(H−λ)ψn k(H−λ)ψn k .

ψn , and k(H − λ)ψn k → 0, Evidently, kφn k = 1. Since (H − λ)−1 φn = k(H−λ)ψ nk we obtain that k(H − λ)−1 φn k → ∞,

as n → ∞. Therefore (H − λ)−1 is unbounded, which implies that λ ∈ σ(H). We will prove that λ ∈ / σd (H). Indeed, suppose on the contrary that λ ∈ σd (H). Let M denote the eigenspace of λ. Then (H − λ) is invertible on M ⊥ (show this). Let P and P ⊥ be the orthogonal projections on M and M ⊥ , respectively. (For the definition and properties of projection operators see Section 25.7.) We have P +P ⊥ = 1. Since P the sequence {ψn } is spreading and the operator P can be written as P = j |φj ihφj |, where {φn } is an orthonormal basis in M = RanP , we have kP ψn k → 0 and therefore kP ⊥ ψn k → 1. Hence P ⊥ ψn ⊥ for the normalized sequence ψn⊥ := kP ⊥ ψ k , we have that (H − λ)ψn → 0 and n therefore (H − λ) is not invertible on M ⊥ , a contradiction. Hence λ ∈ / σd (H) and therefore λ ∈ σess (H). Suppose now that λ ∈ σess (H). Then there is a sequence φn with kφn k = 1 (H−λ)−1 φn and k(H − λ)−1 φn k → ∞. Let ψn = k(H−λ) −1 φ k . We claim that for every n bounded set Ω we have that kχΩ ψn k → 0 as n → ∞. Indeed, we can assume without loss of generality that V ≥ 0, which implies that (H + 1) is invertible. So χΩ ψn = χΩ (−∆ + 1)−1 (−∆ + 1)(H + 1)−1 (H + 1)ψn . (6.18) Now, we have χΩ (−∆ + 1)−1 f = with K ∈ L2 (R3 × R3 ).

Z

K(x, y)f (y)dy,

(6.19)

Problem 6.20 Use properties of the Fourier transform to show that K(x, y) = −|y| χΩ (x)G(x − y), where G(y) = C e |y| and C is a constant. Define B := (−∆ + 1)(H + 1)−1 . The relation B = 1 − V (H + 1)−1 ,

(6.20)

and the assumption that V is H0 -bounded, imply that B is bounded. Using this definition, and using that (H + 1)ψn =

φn + (λ + 1)ψn , k(H − λ)−1 φn k

(6.21)

we obtain that χΩ ψn = χΩ (−∆+1)−1 B(λ+1)ψn +χΩ (−∆+1)−1 B

φn . (6.22) k(H − λ)−1 φn k

We consider the first term on the r.h.s.. We write this term as (λ + 1)hKx , Bψn i, where Kx (y) := K(x, y). Since B is bounded, we have that

70

6 Spectrum and Dynamics

∀x, hKx , Bψn i = hB ∗ Kx , ψn i. If the vector B ∗ Kx were in the domain of (H − λ)−1 , then we would have hKx , Bψn i = r1n h(H − λ)−1 B ∗ Kx , φn i, where rn := k(H − λ)−1 φn k → ∞, and therefore ∀x, hKx , Bψn i → 0.

(6.23)

In general, B ∗ Kx might not be in the domain of (H − λ)−1 , but we can show that the latter domain is dense and therefore B ∗ Kx can be approximated by vectors from this domain: ∀ǫ > 0, there is fǫ such that kB ∗ Kx − fǫ k ≤ ǫ. Now, one can modify the argument above to show (6.23). (A different argument showing (6.23) goes along the lines of the proof of Theorem 25.57 R in the mathematical supplement.) Furthermore, since |K(x, y)|2 dxdy < ∞, we have |hB ∗ Kx , ψn i| ≤ kB ∗ Kx kkψn k ≤ kB ∗ kkKx k ∈ L2 (dx). Hence, by the Lebesgue dominated convergence theorem, we have, using the notation k · kx for the L2 −norm in the variable x, that khKx , B(λ + 1)ψn ikx = |λ + 1|khB ∗ Kx , ψn ikx → 0.

(6.24)

For the second term on the r.h.s. of (6.22), we use that B is bounded, to obtain that kχΩ (−∆ + 1)−1 B

φn φn k ≤ kBkk k → 0. k(H − λ)−1 φn k k(H − λ)−1 φn k

(6.25)

Thus we conclude that for any bounded set Ω ⊂ R3 , kχΩ ψn k → 0.

(6.26)

Let B(R) be a ball of radius R centered at the origin and let Rm → ∞ as m → ∞. Since kχΩ ψn k → 0 as n → ∞ for any bounded set Ω we have that ∀m, kχB(Rm ) ψn k → 0 as n → ∞. Hence using a diagonal procedure and passing to a subsequence, if necessary, we obtain that kχBn ψn k → 0, as n → ∞, for Bn := B(Rm(n) ) and some subsequence m(n), satisfying m(n) → ∞, as n → ∞. Let (1 − χBn )ψn fn = . k(1 − χBn )ψn k

Evidently kfn k = 1 and supp(fn ) ∩ Ω = ∅ for all bounded Ω provided that n is large (depending on Ω, of course). To finish the proof it suffices to show that k(H − λ)fn k → 0. (6.27) To prove this relation, we compute (H − λ)(1 − χBn )ψn = (1 − χBn )(H − λ)ψn − [H, χBn ]ψn and [H, χBn ] = −((2∇χBn · ∇ + ∆χBn ). Therefore k(H − λ)(1 − χBn )ψn k ≤ k(1 − χBn )(H − λ)ψn k + k[H, χBn ]ψn k ≤ k(1 − χBn )kk(H −

6.6 Particle in a Periodic Potential

71

λ)ψn k + k(2∇χBn · ∇ + ∆χBn )ψn k → 0, as n → ∞. Since k(1 − χBn )ψn k → 1, (6.27) follows. Hence fn is a spreading sequence for H and λ.  To conclude this section, we present a result on the spectra of Schr¨odinger operators on bounded domains with Dirichlet boundary conditions. Theorem 6.21 Let Λ be a parallelepiped in Rd , and V a continuous function on Λ. Then the Schr¨odinger operator H = −∆ + V , acting on the space L2 (Λ) with Dirichlet or periodic boundary conditions, has purely discrete spectrum, accumulating at +∞. To be precise, the operator “H on L2 (Λ) with Dirichlet boundary conditions” should be understood as the unique self-adjoint extension of H from C0∞ (Λ). This theorem is proved in Section 25.10.

6.6 Particle in a Periodic Potential We consider a particle moving in a periodic potential. A primary example of this situation is an electron moving in the potential created by ions or atoms of a solid crystal lattice. Such a particle is described by a self-adjoint Schr¨odinger operator ~2 H=− ∆+V 2m on L2 (R3 ), with the potential V (x), having certain periodicity properties which we now explain. First, we identify the notion of physical crystal lattice with the mathematical (Bravais) lattice, L, which is defined as the subset of R3 given by L = {m1 s1 + m2 s2 + m3 s3 | m1 , m2 , m3 ∈ Z} for some three linearly independent vectors s1 , s2 , s3 ∈ R3 , called a basis of L. A basis in L is not unique. We say V (x) is periodic with respect to a lattice L if V (x + s) = V (x) for any s ∈ L. This implies that the operator H commutes with the lattice translations, Ts H = HTs , ∀s ∈ L, (6.28) where Ts is the translation operator, given by Ts f (x) = f (x + s). Problem 6.22 Show that Ts are unitary operators satisfying Tt Ts = Tt+s . Due to (6.28), the operator H can be decomposed into a direct fiber integral, as follows. First, let Ω be the lattice cell given by Ω := {x1 s1 + x2 s2 + x3 s3 | 0 ≤ x1 , x2 , x3 ≤ 1}. Let L∗ be the lattice dual to L, i.e. the lattice with the basis s∗1 , s∗2 , s∗3 , satisfying s∗i · sj = δij , and let Ω ∗ be a lattice cell in L∗ . Now, we define the Hilbert space H = L2 (Ω ∗ , dk; L2 (Ω)) of vector functions ψ : Ω ∗ ∋ k → ψk ∈ L2 (Ω), with the standard inner product

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6 Spectrum and Dynamics

R hψ, φi := Ω ∗ dkhψk , φk iL2 (Ω) R , where dk is the standard Lebesgue measure on Ω ∗ , normalized so that Ω ∗ dk = 1. We call H the direct fiber integral, denoted as Z H=

⊕

Ω∗

Hk dk,

where Hk := L2 (Ω), for each k ∈ Ω ∗ , while its elements are written as ψ = R⊕ ψ dk. Ω∗ k Given operators Hk , k ∈ RΩ ∗ , acting on Hk , we define R ⊕ the operator ⊕ ψk (x) → Hk ψk (x) on H = Ω ∗ Hk dk and denote it Ω ∗ Hk dk, so that R⊕ R⊕ R⊕ Ω ∗ Hk dk Ω ∗ ψk dk = Ω ∗ Hk ψk dk. Define the operator U : L2 (R3 ) → H on smooth functions with compact support by the formula X (U v)k (x) = χ−1 k (t)Tt v(x). t∈L

where χk (t) = eik·t , the character of L, i.e., a homomorphism from L → U (1) (see the remark below). We now have the following Bloch decomposition result: Proposition 6.23 The operator U extends uniquely to a unitary operator and Z ⊕ U HU −1 = Hk dk (6.29) Ω∗

∗

where Hk , k ∈ Ω , is the restriction of operator H to Hk , with domain consisting of vectors v ∈ Hk ∩ H 2 satisfying the boundary conditions Tt v(x) = χk (t)v(x),

(6.30)

for the basis elements t = s1 , s2 , s3 . Proof. We begin by showing that U is an isometry on smooth functions with compact support. Using Fubini’s theorem we calculate Z Z Z X

2

(U v)k 2 dk = kU vk2H = χ−1 k (t)Ty v(x)dx dk H k

Ω∗

=

Z  X Ω

=

Tt v(x)τs v(x)

t,s∈L

Z X Ω t∈L

Ω∗

|Tt v(x)|2 dx =

Z

Z

R2

Ω

Ω∗

t∈L

χ−1 k (t)χk (s)dk



dx

|v(x)|2 dx.

Hence kU vkH = kvkH and U extends to an isometry on all of H. To show that U is in fact a unitary operator we define U ∗ : H → H by the formula Z ∗ χk (t)gk (x)dk, U g(x + t) = Ω∗

6.6 Particle in a Periodic Potential

73

for t ∈ L and x ∈ Ω. Straightforward calculations show that U ∗ is the adjoint of U and that it too is an isometry, proving that U is a unitary operator. For (6.29), we need to first show that (U v)k is in the domain of Hk . For (U v)k we have X Tt (U v)k (x) = χ−1 k (s)Tt Ts v(x) s∈L

=

X

χ−1 k (s)Tt+s v(x)

s∈L

X

= χk (t)

χ−1 k (t + s)Tt+s v(x)

s∈L

= χk (t)(U v)k (x). Hence if v ∈ D(H), then U v ∈ D(Hk ). Next, we have that X (Hk (U v)k )(x) = χ−1 k (t)HTt v(x) t∈L

=

X

χ−1 k (t)Tt Hv(x)

t∈L

= (U Hv)k (x), which establishes (6.29).  Since the resolvents of the operators H and Hk are related as U (H − R⊕ z)−1 U −1 = Ω ∗ (Hk − z)−1 dk, we deduce that

Theorem 6.24

σ(H) =

[

σ(Hk ).

(6.31)

k∈Ω ∗

Since Ω is compact, by Theorem 6.21, the spectra of Hk are purely discrete, say, {λn (k)}. This shows that the spectrum of H is the union of the sets {λn (k) |k ∈ Ω ∗ }, called the bands. This is a key result in solid state physics. Mathematically, it reduces the investigation of the essential spectrum of a periodic quantum hamiltonian to the eigenvalue problem (the Bloch eigenvalue problem) Hk ψk = λ(k)ψk , ψk ∈ Hk .

(6.32)

Remark: eigenvalue problem (6.32) involves boundary conditions (6.30). We can define the Hilbert space Hk without reference to a specific fundamental cell (which is not unique) and incorporating boundary conditions (6.30) directly. Namely, we define Hk := {v ∈ L2loc (R3 ) | Tt v = χk (t)v ∀t ∈ L}, with an inner product that of L2 (Ω) for some fundamental cell Ω.

(6.33)

74

6 Spectrum and Dynamics

To connect the above theory with the Bloch theory of solid state physics, we consider eigenvalue problem (6.32) for the hamiltonian Hk on Hk . Since ψk satisfy Tt ψk = χk (t)ψk ∀t ∈ L, we have that e−ik·x ψk (x) is Lperiodic. Indeed, by the property of the exponentials, we have Ts e−ik·x ψk (x) = e−ik·x−ik·s Ts ψk (x) = e−ik·x e−ik·s χk (t)eik·s ψk (x) = e−ik·x ψk (x). Hence the eigenfunctions ψk (x) have the representation ψk (x) = eik·x φk (x), where φk (x) are L-periodic. This result is the celebrated Bloch theorem of condensed matter physics. Remark: equation (6.30) is an eigen-equation for the operators Ts . Remark: the dual group to L is the group consisting of all characters of L, i.e., all homomorphisms from L → U (1). Explicitly, for k ∈ Ω ∗ , we have the character χk given by χk (t) = eik·t . Since χk+k′ (t) = χk (t), ∀k ′ ∈ L∗ , the dual group of L can be identified with the fundamental cell, Ω ∗ , of L∗ .

6.7 Angular Momentum In this section we study the angular momentum operators (or angular momentum observables, or just angular momenta) Lj = (x × p)j ,

(6.34)

where p = −i~∇, as usual. The term is justified in Section 3.4 on conservation 1 Lj are the generators of the rotation group SO(3). laws, by observing that i~ They satisfy the commutation relations [Lk , Ll ] = i~ǫklm Lm .

(6.35)

Here ǫklm is the Levi-Chivita symbol: ǫ123 = 1 and ǫklm changes sign under the permutation of any two indices. Problem 6.25 Prove (6.35). (Hint: Use that [pj , xk ] = −i~δij .) L22

We define the squared magnitude of the angular momentum, L2 = L21 + + L23 . Note that L2 commutes with Lk : [L2 , Lk ] = 0, ∀k.

Problem 6.26 Prove (6.36). (Hint: Use that [pj , xk ] = −i~δij .)

(6.36)

6.7 Angular Momentum

75

Furthermore, L2 is a homogeneous degree zero operator, in the sense that it commutes with the scalings Tλscal : ψ(x) → λ3/2 ψ(λx), λ > 0, and therefore it does not act on the radial variable r = |x|. Theorem 6.27 (a) The spectrum of the operator L2 is purely discrete; (b) it consists of isolated eigenvalues λ = ~2 l(l + 1), where l = 0, 1, 2, . . . , of the multiplicities 2l + 1. A proof of (a) is somewhat involved and we skip it here (see however the remark after the proof).We will derive the proof of (b) from the following Theorem 6.28 Let Lj be operators satisfying the commutation relations (6.35). Assume the operator L2 = L21 + L22 + L23 has purely discrete spectrum. Then this spectrum consists of the eigenvalues λ = ~2 l(l + 1), where l = 0, 12 , 1, 32 , 2, . . . , of the multiplicities 2l + 1. Proof (of Theorem 6.28). Let λ be an eigenvalue of L2 with eigenspace Vλ . It follows from (6.36) that the eigenspace Vλ is invariant under L3 , i.e. L3 Vλ ⊂ Vλ . (Indeed, if L2 φλ = λφλ , then L2 L3 φλ = L3 L2 φλ = λLk φλ .) Hence one can choose a basis in Vλ consisting of common eigenfunctions φλ,µ for these operators: L2 φλ,µ = λφλ,µ , L3 φλ,µ = µφλ,µ . (By choosing an arbitrary orthonormal basis in the finite dimensional subspace Vλ , one can reduce this problem to one for matrices.) We still would like to use the remaining operators L1 , L2 and so we form the combinations L± := L1 ± iL2 . (6.37) The virtue of these new operators is the following commutation relations, which are easily checked: [L2 , L± ] = 0, [L+ , L− ] = 2~L3 , [L3 , L± ] = ±~L± .

(6.38)

Problem 6.29 Prove (6.38). (Hint: Use that [pj , xk ] = −i~δij .) The last relation can be rewritten as L3 L± = L± (L3 ±~) which shows that L± are raising/lowering operators in the following sense: apply the operators L± to the eigenfunctions φλ,µ and use L3 φλ,µ = µφλ,µ and L3 L± = L± (L3 ±~) to obtain L3 L± φλ,µ = L± (L3 ± ~)φλ,µ = (µ ± ~)φλ,µ . Hence we see that the operators L± raise/lower eigenvalues of L3 . More precisely, assuming the eigenfunctions φλ,µ are non-degenerate (correspond to different eigenvalues), we have L± φλ,µ = Cφλ,µ±~ , for some constants C. Next, we claim that µ2 ≤ λ, for given λ. Indeed, we use L2 = L21 +L22 +L23 ≥ 2 L3 to conclude that µ2 = hφλ,µ , L23 φλ,µ i ≤ hφλ,µ , L2 φλ,µ i = λ. Furthermore, ¯ 3 = −L3 , if µ is an eigenvalue of L3 on Vλ , then so is −µ. since L

76

6 Spectrum and Dynamics

Denote µ∗ := max{|µ| : given λ}. Then by the definition of µ∗ and the raising/lowering property of the operators L± , we find that L± φλ,±µ∗ = 0. Next, we use L± L∓ = L21 + L22 ± ~L3 to obtain the following relation between the operators L2 , L± and L3 : L2 = L± L∓ ∓ ~L3 + L23 .

(6.39)

Taking the expectation of this equation in the eigenfunction φλ,−µ∗ , and using that L− φλ,−µ∗ = 0, we find λ = hφλ,−µ∗ , L2 φλ,−µ∗ i = hφλ,−µ∗ , (−~L3 + L23 )φλ,−µ∗ i = ~µ∗ + µ2∗ . This gives λ = µ∗ (µ∗ + ~). If we denote µ∗ = ~l, then λ = ~2 l(l + 1). Finally, since, using the raising operator L+ , we can go from φλ,−µ∗ to φλ,µ∗ in an integer number of steps, we conclude that 2µ∗ /~ is a nonnegative integer, µ∗ = ~l, l ∈ 12 Z+ . For each l, the index m runs through the values m = −l, . . . , l. Hence the multiplicity of the eigenvalue λ = ~2 l(l + 1) of the operator L2 is 2l + 1. Proof (of Theorem 6.27(b).). To prove statement (b), it remains to show that only integer l’s are realized now. To this end, we note that we can solve the eigenvalue problem for L3 easily by using spherical coordinates (r, θ, φ), in which L3 = ~∂φ , to obtain L3 eimφ = ~meimφ , so that µ = ~m, where m must be an integer. So we have shown that every eigenvalue λ of L2 is of the form λ = ~2 l(l + 1), with l = 0, 1, . . . , and has multiplicity 2l + 1. This proof allows one to construct all eigenfunctions of L2 , by solving the equation L− φλ,−µ∗ = 0, for λ = ~2 l(l + 1) and µ∗ = ~l, and then using the raising operator L+ to obtain all other eigenfunctions with the same l. (In spherical co-ordinates, we have L± = ~e±iφ (±∂θ + i cot θ∂φ ).) Thus we determined completely the eigenvalues of the operators L2 and L3 : λ = ~2 l(l+1) and µ = ~m, where m, l ∈ Z and m = −l, . . . , l. Denote the joint eigenfunction of the operators L2 and L3 , corresponding to the eigenvalues λ = ~2 l(l + 1) and µ = ~m, by Ylm (θ, φ): L2 Ylm = ~2 l(l + 1)Ylm ,

L3 Ylm = ~mYlm .

(6.40)

The functions Ylm (θ, φ) are the celebrated spherical harmonics. We know it is of the form Ylm (θ, φ) = Y˜lm (θ)eimφ , m ∈ Z, where Y˜lm (θ) is an eigenfunction of L2 : L2 Y˜lm (θ) = ~2 l(l + 1)Y˜lm (θ). More precisely, |k|

Ylk (θ, φ) = clk Pl (cos(θ))eikφ

(6.41)

where l = 0, 1, . . .; k ∈ {−l, −l + 1, . . . , l − 1, l}; clk is a constant; and the Legendre function Plk can be written as

6.7 Angular Momentum

Plk (u) =

(1 − u2 )k/2 d l+k 2 ( ) (u − 1)l . 2l l! du

77

(6.42)

This completes our excursion into the spectral theory of the operator L2 . Problem 6.30 Show that in the spherical coordinates (r, θ, φ), L2 (f (r)g(θ, φ)) = −f (r)∆Ω g(θ, φ), where ∆Ω is the Laplace-Beltrami operator on S2 , given in spherical coordinates (θ, φ), by ∆Ω =

∂2 1 ∂ ∂ 1 . (sin(θ) ) + 2 sin(θ) ∂θ ∂θ sin (θ) ∂φ2

One could interpret ∆Ω as a quantum hamiltonian of a ‘free’ particle on the sphere. Using the Weyl criterion of essential spectrum and the fact that the sphere S2 is a compact space, one can show that ∆Ω has purely discrete spectrum. Above, we found this spectrum explicitly using commutation relations, i.e. purely algebraic methods.

7 Special Cases and Extensions

In this chapter we consider several specific quantum systems which either fall outside of the general theory considered in Chapter 6, or are accessible to explicit computations providing the desired results. The results presented below not only illustrate some of the general arguments presented above, but also form a basis for our intuition about quantum behaviour.

7.1 The Square Well and Torus The Square Well We consider a one-dimensional potential well of finite depth V0 , and width a (see Fig. 7.1). V(x) −a/2

a/2 x −V

Fig. 7.1. The finite well. The determination of the discrete and essential spectra is straightforward, and is left as an exercise. Problem 7.1 Show 1. σess (H) = [0, ∞) 2. σd (H) ⊂ (−V0 , 0) © Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_7

80

7 Special Cases and Extensions

3. the equations for the eigenvalues E, for −V0 < E < 0, are k tan(ak/2) = K where

k cot(ak/2) = −K

r 2m(E + V0 ) 2mE K= − 2 k= ~ ~2 If we shift the well upward as HV0 := H + V0 , then we can take the limit as V0 → ∞ to obtain the quantum hamiltonian H∞ of a particle in an infinite well. In three dimensions, the potential of the infinite well can be thought of as  0 x∈W V (x) = ∞ x 6∈ W r

where W := [0, L]3 ⊂ R3 . This means we take ψ ≡ 0 outside W , and that we impose Dirichlet boundary conditions ψ|∂W = 0

(7.1)

on the wave function inside W . It is a simple matter to solve the eigenvalue equation ~2 ∆ψ = Eψ (7.2) − 2m in W with the boundary condition (7.1), using the method of separation of variables. Doing so, we obtain eigenvalues (energy levels) En =

3 ~2 π 2 X 2 n 2mL2 j=1 j

with corresponding eigenfunctions (bound states) ψn (x) =

3 Y

j=1

sin(

πnj xj ) L

for each integer triple n = (n1 , n2 , n3 ), nj ≥ 1. WeP see thatP the eigenvalue En occurs with degeneracy equal to #{(m1 , m2 , m3 )| m2j = n2j }. We remark

that the ground-state (lowest) energy E(1,1,1) =

3~2 π 2 2mL2

is non-degenerate.

The Torus Now we consider a particle on a torus, say T3 = R3 /LZ3 , and with no external potential, V ≡ 0. This corresponds to considering the Schr¨odinger ~2 operator − 2m ∆ in the cube W = [0, L]3 (or any other fundamental cell of the lattice LZ3 ), with periodic boundary conditions. Solving the eigenvalue equation (7.2) with boundary conditions

7.2 Motion in a Spherically Symmetric Potential

(

ψ(x)|xj =0 = ψ(x)|xj =L ∂ψ/∂xk |xj =0 = ∂ψ/∂xk |xj =L ,

81

(7.3)

for all j, k, leads to (separation of variables again) eigenfunctions ψn (x) =

3  2πnj xj  Y ) sin(

j=1

cos(

L 2πnj xj ) L

with eigenvalues En =

3 2π 2 ~2 X 2 n , mL2 1 j

for nj ≥ 0. The ground state energy, E(0,0,0) = 0 is non-degenerate, with eigenfunction ψ(0,0,0) ≡ 1. The spacing between energy levels is greater than for the infinite well, and the degeneracy is higher. Problem 7.2 Determine the spectrum of x, and the eigenvalues of p, on L2 (W ) with periodic boundary conditions. Challenge: show that p has no essential spectrum (hint – use the Fourier transform for periodic functions).

7.2 Motion in a Spherically Symmetric Potential We consider a particle moving in a potential which is spherically symmetric, i.e. V = V (|x|) depends only on r = |x|. Then the corresponding Schr¨odinger operator is ~2 ∆ + V (|x|), (7.4) H =− 2m acting on the Hilbert space L2 (R3 ). If V (r) → 0 as r → ∞, then from Section 6.5, we know that H is selfadjoint and has essential spectrum filling in the half-line [0, ∞). Hence the discrete spectrum, if it exists, is negative with only one possible accumulation point at 0. Our goal here is to use the spherical symmetry of V to reduce the eigenvalue problem for (7.4) to a problem in the radial variable r = |x|, only; i.e. an ODE problem. Because the potential is spherically symmetric, the Schr¨odinger operator H commutes with rotations and, consequently, with their generators, the angular momenta Lj = (x × p)j , (7.5) where p = −i~∇ as usual. Hence the angular momentum is conserved. The squared magnitude of the angular momentum operator L2 = L21 + 2 L2 + L23 also commutes with H. Furthermore, L2 is a homogeneous degree zero operator in the sense that it commutes with the scalings Tλscal : ψ(x) → λ3/2 ψ(λx), λ > 0, and therefore it does not act on the radial variable r = |x|. Indeed, if we introduce spherical coordinates (r, θ, φ), where

82

7 Special Cases and Extensions

x1 = r sin(θ) cos(φ),

x2 = r sin(θ) sin(φ),

x3 = r cos(θ),

0 ≤ θ < π, 0 ≤ φ < 2π, then L2 acts only on the angles (θ, φ). A straightforward computation gives − ~2 ∆ = −~2 ∆r +

1 2 L , r2

(7.6)

where ∆r is the radial Laplacian, acting only on the radial variable r = |x|, given by 1 ∂ ∂2 2 ∂ ∂ ∆r = 2 r2 = 2+ . (7.7) r ∂r ∂r ∂r r ∂r Problem 7.3 Prove (7.6). (Hint: Compute L2 = L21 + L22 + L23 as follows. 2 2 L2 = (x2 p3 − x3 p2 )2 + cyclic permutations 2 p3 x3 p2 − P = (x2 p3 )P+ (x3 p2 ) − xP x3 p2 x2 p3 + cyclic permutations = r2 p2 − j x2j p2j − ij pi xi xj pj + j pj x2j pj . This gives L2 = r2 p2 + (p · x)(x · p), from which one can easily derive (7.6).) Problem 7.4 Show that in spherical coordinates (r, θ, φ) the Laplacian becomes 1 ∆ = ∆r + 2 ∆Ω r where ∆r is the radial Laplacian introduced in (7.7), and ∆Ω is the LaplaceBeltrami operator on S2 , given in spherical coordinates by ∆Ω =

∂ 1 ∂2 1 ∂ (sin(θ) ) + . 2 sin(θ) ∂θ ∂θ sin (θ) ∂φ2

We now return to our original problem - the eigenvalue problem for the spherically symmetric Schr¨odinger operator, (7.4). In view of (7.6) and the fact that L2 commutes with ∆r and H, we seek eigenfunctions of H in the separated-variables form ψ(r, θ, φ) = R(r)Ylk (θ, φ) where Ylk is an eigenfunction of L2 corresponding to the eigenvalue ~l(l + 1) (see (6.40)), a spherical harmonic. Plugging this into the eigenvalue equation Hψ = Eψ, we obtain (

~2 l(l + 1) ] + V (r))R = ER. [−∆r + 2m r2

(7.8)

Usually, one cannot solve this equation explicitly with exception of a very few cases. The Schr¨odinger operator of the hydrogen atom is one of these cases, to which we now proceed.

7.3 The Hydrogen Atom

83

7.3 The Hydrogen Atom A hydrogen atom consists of a proton and an electron, interacting via a Coulomb force law. Let us make the simplifying assumption that the nucleus (the proton) is infinitely heavy, and so does not move. We also consider more generally the hydrogen-type atom, or ion, with nucleus of charge eZ, where e is the charge of the proton, and −e that of the electron. Placing the nucleus at the origin, we have the electron moving under the influence of the external (Coulomb) potential V (x) = −Ze2 /|x|. The appropriate Schr¨odinger operator is therefore H=−

~2 e2 Z ∆− , 2m |x|

(7.9)

acting on the Hilbert space L2 (R3 ). In Section 13.1 we will see how to reduce the problem of the more realistic hydrogen atom - when the nucleus has a finite mass (a two-body problem) - to the problem studied here. As usual, we want to study the spectrum of H. The first step is to invoke Theorems 2.9 and 6.16 and Problem 6.17 to conclude that H is self-adjoint and has essential spectrum filling in the half-line [0, ∞). Our goal, then, is to find the bound-states (eigenfunctions) and bound-state energies (eigenvalues). It is a remarkable fact that we can find these explicitly. Indeed, aside from the infinite well, the only multi-dimensional potentials for which the Schr¨odinger eigenvalue problem can be solved explicitly are the harmonic oscillator and the Coulomb potential. Because the Coulomb potential is spherically symmetric (depends only on r = |x|), the results of the previous section can be applied to the Schr¨odinger operator (7.9). It is a remarkable fact that, in this case, we can solve the equation (7.8) explicitly and find the eigenvalues explicitly. Indeed, aside from the infinite well, the only multi-dimensional potentials for which the Schr¨odinger eigenvalue problem can be solved explicitly are the harmonic oscillator and the Coulomb potential. For the Coulomb potential, the equation (7.8) reads (

l(l + 1) ~2 [−∆r + ] − e2 Z/r)R = ER. 2m r2

(7.10)

Eq. (7.10) is an ordinary differential equation (ODE) and is well-studied (see, eg, [186]). Without going into details, we remark that one can show (by powerseries methods) that (7.10) has square-integrable solutions only for r e2 Z −m n := ∈ {l + 1, l + 2, . . .}. ~ 2E The corresponding eigenfunctions, Rnl are of the form Rnl (r) = ρl e−ρ/2 Fnl (ρ)

84

7 Special Cases and Extensions 2

Z where ρ = 2me n~2 r, and Fnl is a polynomial. In full, then, the solutions of the eigenvalue problem Hψ = Eψ are

ψ(r, θ, φ) = Rnl (r)Ylk (θ, φ) where k ∈ {−l, −l + 1, . . . , l};

l = 0, 1, 2, . . . ;

n ∈ {l + 1, l + 2, . . .};

and the eigenvalues are E (= En ) = −



me4 Z 2 2~2



1 . n2

(7.11)

So we see that the hydrogen atom has an infinite number of bound states below the essential spectrum (which starts at zero), which accumulate at zero (this result is obtained in Section 8.3 by a general technique, without solving the eigenvalue problem). The ground state energy, attained when l = k = 0, n = 1, is E1 = −me4 Z 2 /2~2 . An easy count finds the degeneracy of the energy level En to be n−1 X

(2l + 1) = n2 .

(7.12)

l=0

Finally, we note that the expression (7.11) is in agreement with the empirical formula (“Balmer series”) ∆E = R(

1 1 − 2 ). n2f ni

Here 1 ≤ nf < ni are integers labeling the final and initial states of the atom in a radiation process, R is a constant, and ∆E is the difference of the two energy levels. This formula predates quantum mechanics, and was based on measurements of absorption and emission spectra.

7.4 The Harmonic Oscillator The Hamiltonian of the quantum harmonic oscillator in r dimensions is r

Hho = −

~2 1 X 2 2 ω x , ∆+ m 2m 2 i=1 i i

acting on the space L2 (Rr ). By Theorem 6.18, σ(H) consists of isolated eigenvalues, increasing to infinity. We will solve the eigenvalue problem explicitly for this operator.

7.4 The Harmonic Oscillator

85

Theorem 7.5 The spectrum of Hho is σ(Hho ) = {

r X i=1

~ωi (ni + 1/2) | ni = 0, 1, 2, . . .}.

To prove this result, we derive a representation of the operator H which facilitates its spectral analysis. It also prepares us for a similar technique we will encounter in the more complex situation of second quantization and quantum electrodynamics (see Chapters 20 and 21). We introduce the creation and annihilation operators 1 1 aj := p (mωj xj + ipj ) and a∗j := p (mωj xj − ipj ). 2m~ωj 2m~ωj

(7.13)

These operators are adjoint of each other (for the definition of the operator adjoint to a given operator see Definition 25.23) and they satisfy the commutation relation [ai , a∗j ] = δij . (7.14)
Pr 2 1 ~ 2 2 2 Using that H = i=1 − 2m ∂xi + 2 mωi xi , the Hamiltonian H can be ∗ re-written in terms of ai and ai ’s as follows: Hho =

  1 ~ωi a∗i ai + . 2 i=1

r X

(7.15)

We say that this expression is in normal form because a∗ appears to the left of a. Now, we are ready to prove Theorem 7.5. Proof (of Theorem 7.5). First we find the ground state and ground state energy of Hho . We define the particle number operators Ni := a∗i ai , so that, by the expression (7.15), Hho =

r X

1 ~ωi (Ni + ), 2 i=1

Now, because a∗i is the adjoint of a1 , Ni are non-negative, Ni ≥ 0: hψ, Ni ψi = kai ψk2 ≥ 0

P for any ψ. Therefore the ψ with the smallest eigenvalue, ri=1 12 ~ωi (i.e. the one that minimizes the average energy, hψ, Hho ψi), satisfies Ni ψ = 0, ∀i, which holds if and only if ai ψ = 0 ∀i. These equations, 1 (mωi xi + ~∂/∂xi )ψ = 0, ai ψ = √ 2m~ωi

86

7 Special Cases and Extensions

can be easily solved, giving the unique (up to a multiplicative constant which we denote c) family of solutions 2

ψ0i (xi ) := ce−mωi xi /2~ We choose c to have ψ0i normalized as kψ0i k = 1 which gives c := (2πm~ωi )−1/4 . To find the (normalized) ground state of Hho , we have to multiply these functions to obtain r Y Pr 2 (2πm~ωi )−1/4 e−m i=1 ωi xi /2~ . (7.16) ψ0 := i=1

P This is the ground state of Hho , with the ground state energy ri=1 12 ~ωi . To find the exited states we observe that (7.14) imply that the operators Ni satisfy the relations Ni ai = ai (Ni − 1), (7.17) Ni a∗i = a∗i (Ni + 1).

The commutation relation (7.18) implies Ni a∗ ψ0 = a∗i ψ0 and in general

Ni (a∗i )n

(7.18)

=

(a∗i )n (Ni +n),

which gives

Ni (a∗i )n ψ0 = ni (a∗i )n ψ0 . Thus φni := (a∗iQ )ni ψ0 is an eigenfunction of Ni with eigenvalue ni and therefore φn := ri=1 (a∗i )ni ψ0 is an eigenfunction of Hho with eigenvalue P r 1 i=1 ~ωi (ni + 2 ). Qr Problem 7.6Q Show that kφn k2 = |c|2 i=1 ni !. Hint: write r kφn k2 = hψ0 , i=1 (ai )ni (a∗i )ni ψ0 i, then pull the ai ’s through the a∗i ’s (including the necessary commutators) until they hit ψ0 and annihilate it. Qr So ψn := i=1 √1n ! (a∗i )ni ψ0 , n := (n1 , . . . , nr ), is a normalized eigenfunction i P of Hho with eigenvalue ri=1 ~ωi (ni + 1/2). We now show that these are the only eigenfunctions. To simplify the notation we do this in dimension 1, i.e. for r = 1. It follows from the commutation relation (7.17) that N an = an (N − n) (this is adjoint of N (a∗ )n = (a∗ )n (N + n)). Hence if ψ is any eigenfunction of N with eigenvalue λ > 0, then N am ψ = (λ − m)am ψ. (7.19)

If we choose m so that λ − m < 0 we get a contradiction to N ≥ 0 unless am ψ = 0. Let j be the largest integer s.t. aj ψ 6= 0 and aj+1 ψ = aaj ψ = 0. This implies aj ψ = cψ0 (7.20)

where c 6= 0 is a constant. Hence by N ψ0 = 0, (7.20) and (7.19), we have 0 = N aj ψ = (λ − j)aj ψ and therefore λ = j. Thus ψ corresponds to the eigenvalue j. If ψ is not proportional to ψj , then we can choose it to satisfy hψ, ψj i = 0. However, by aj ψ = cψ0 and ψj = (a∗ )j ψ0 , we have that hψ, ψj i = haj ψ, ψ0 i = ckψ0 k2 6= 0, a contradiction. So we are done.

7.5 A Particle in a Constant Magnetic Field

87

In the proof we obtained the following result Theorem 7.7 Pr The eigenfunctions of the operator Hho corresponding to the eigenvalues i=1 ~ωi (ni + 1/2) | ni = 0, 1, 2, . . . , are ψn =

r Y

i=1

p (1/ ni !)(a∗i )ni ψ0 , n := (n1 , . . . , nr ),

(7.21)

with ψ0 given in (7.16) and a∗i the first order differential operators defined in (7.13).

Remark 7.8 One can extend the last part of the proof above to show that any function f ∈ L2 (Rr ) can be written as f (x) =

X

cn ψn =

n

X n

cn

r Y

1 √ (a∗i )ni ψ0 . ni ! i=1

Hence the set {ψn } form an orthonormal basis in L2 (Rr ) and the space L2 (Rr ) n n is isometric to the space F (r) := ⊕∞ n=0 Csym , where Csym is equal to {0} for n n = 0 and to C /Sn for n ≥ 1. Here Sn is the symmetric group of permutations of n indices. R Remark 7.9 Since ψ0 is positive and normalized, ψ02 = 1, the operator U : −1 f → ψ0 f maps unitarily the space L2 (Rr , dr x) into the space L2 (Rr , ψ02 dr x). Under this map the operator Hho is mapped into L := U HU −1 acting on the space L2 (Rr , ψ02 dr x). We compute L=

r X j=1

~(−

~ ∂x + ωj xj )∂xj . 2m j

(7.22)

The operator L, with ~ = 1, 2m = 1, is the generator of the OrnsteinUhlenbeck stochastic process (see [135]). Problem 7.10 Find the ground state energy of the ideal quantum gas in R3 , with the Schr¨odinger operator given by (7.56), with potential U (y) := |y|2 , for (a) n = 27 identical spinless fermions and (b) n = 12 bosons (in terms of the one-particle energies).

7.5 A Particle in a Constant Magnetic Field Consider a charged quantum particle moving in a constant magnetic field, B, with no electric field present. According to (4.15), the quantum Hamiltonian of such a system is

88

7 Special Cases and Extensions

H(A) =

1 (p − eA)2 , 2m

(7.23)

acting on L2 (R3 ), where A is the vector potential of the magnetic field B and therefore satisfies curlA = B.

(7.24)

Recall that the corresponding Schr¨odinger equation has the gauge symmetry H(A + ∇χ) = eieχ/~ H(A)e−ieχ/~ .

(7.25)

We fix the gauge by choosing a special solution of equation (7.24). A possible choice for A is 1 (7.26) A′ (x) = B × x. 2 If, we chose the x3 axis along B so that B = (0, 0, b), then A′ becomes A′ (x) = 1 2 (−x2 , x1 , 0). Another possibility, again supposing B = (0, 0, b), is A′′ (x) = b(−x2 , 0, 0).

(7.27)

Problem 7.11 Check that both (7.26) and (7.27) yield the magnetic field B, and that the two are gauge-equivalent. Note that if the magnetic field is directed along the x3 axis, i.e. B = (0, 0, b), then H(A) is of the form H(A) = HA +

1 2 p , 2m 3

(7.28)

where the operator HA acts only on the variables x1 and x2 and is of the form HA :=

2 X 1 ~2 (pi − eAi )2 = − ∆A , 2m 2m 1

(7.29)

where ∆A := ∇∗A ∇A , with ∇A := ∇−ieA/~, on R2 . We say that the magnetic field is perpendicular to the plane (i.e. along the x3 −direction). Problem 7.12 Prove (7.28) - (7.29). The spectrum of the operator H(A) in (7.28) can be found by separation of variables: σ(H(A)) = σ(HA ) + [0, ∞),

1 2 where [0, ∞) is the spectrum of 2m p3 . Thus it suffices to find the spectrum of HA . From now on we deal with the operator HA in 2D, acting only on the variables x1 and x2 . Let ωb := eb m denote the cyclotron frequency (the frequency of the classical circular motion of a particle of charge e in a constant magnetic field of magnitude b). We have

7.5 A Particle in a Constant Magnetic Field

89

Theorem 7.13 The spectrum of HA consists of the eigenvalues 1 ~ωb (n + ), 2

n = 0, 1, . . . .

(7.30)

with infinite multiplicity. (These eigenvalues are called Landau levels.) Proof. We may introduce the harmonic oscillator annihilation and creation operators, α and α∗ , with α := ~[(∇A )1 + i(∇A )2 ],

α∗ := ~[−(∇A )1 + i(∇A )2 ],

(7.31)

where ∇A := ∇ − i ~e A(x). These operators satisfy the standard commutation relations: [α, α∗ ] = 2e~ curl A = 2be~.

(7.32)

Indeed, ~12 [α, α∗ ] = [(∇A )1 , i(∇A )2 ] + [i(∇A )2 , −(∇A )1 ] = 2 ~e (∂1 A2 − ∂2 A1 ). The operator HA can be expressed in terms of these operators as HA =

1 (α∗ α + be~). 2m

(7.33)

Indeed, ~12 α∗ α = −(∇A )21 − (∇A )22 − i[(∇A )1 , (∇A )2 ] = −∆A − be/~. The the argument used above for the standard harmonic oscillator shows that the spectrum of HA consist of the eigenvalues (7.30). Problem 7.14 Prove (7.32), (7.33) and that (7.30) are the eigenvalues of HA . Now, we show that the eigenvalues (7.30) have infinite multiplicities. First, we note that the gauges (7.26) and (7.27) become A′ (x) =

1 ⊥ ⊥ bx , x := (−x2 , x1 ), A′′ (x) = b(−x2 , 0). 2

(7.34)

We find the eigenfunctions of the operator HA′ , with A′ given in (7.34). First we write the annihilation operator, α, given in (7.31), explicitly as α = 2~∂¯ +

be z, 2

1 ∂¯ := (∂x1 + i∂x2 ), z := x1 + ix2 . 2

(7.35)

Next, a straightforward computation shows that 2 2 be be be ¯ e 4~ |z| (∂¯ + z)e− 4~ |z| = ∂. 4~

Problem 7.15 Prove (7.35) and (7.36).

(7.36)

90

7 Special Cases and Extensions be

2

Hence φ solves the equation αφ = 0 iff f (z) := e 4~ |z| φ solves the equation ¯ = 0. The latter equation implies that f is a holomorphic function. Hence ∂f the eigenspace of the operator HA′ corresponding to the lowest eigenvalue ~ω2 b is be

2

V0 := {f (z)e− 4~ |z| : f is a holomorphic function s.t. be

2

f (z)e− 4~ |z| ∈ L2 (R2 )}.

(7.37)

V0 is the eigenspace corresponding to the lowest eigenvalue ~ω2 b . It is called the zeroth Landau level subspace. Higher Landau level subspaces are obtained by applying powers of the creation operator α∗ to V0 : the eigenspace corresponding to the eigenvalue ~ωb (n + 12 ), n ≥ 1, the nth Landau level, is given by Vn := (α∗ )n V0 . Clearly, all these subspaces are infinite dimensional. This completes the proof of Theorem 7.13 To find a convenient basis in the infinite dimensional space V0 , we use the creation and annihilation operators β := ~[(πA )1 − i(πA )2 ],

β ∗ := ~[−(πA )1 − i(πA )2 ],

(7.38)

¯ A . We can compute as before where πA := ∇ + i ~e A(x) = ∇ β = 2~∂ +

be z¯, 2

β ∗ = −2~∂¯ +

be z. 2

(7.39)

Problem 7.16 Prove (7.39), that [β, β ∗ ] = 2be~ and that α, α∗ and β, β ∗ commute mutually. be

2

Now, the equations αψ0 = 0 and βψ0 = 0 have the solution ψ0 := e− 4~ |z| , unique up to multiplicative constant. Applying (β ∗ )m to this solution, one generates an orthogonal basis in V0 . An orthogonal basis in Vn is given by (β ∗ )m (α∗ )n ψ0 , m = 0, 1, . . . . The elements, ψm := (β ∗ )m ψ0 , m = 0, 1, . . . , of the basis in V0 are eigenfunctions of the angular momentum operator L3 = x1 p2 − x2 p1 . Problem 7.17 Prove this. Hint: show that (a) α∗ α − β ∗ β = −2ebL3, (b) in ¯ (c) ψm = cm z m ψ0 , for some cm . The last the z variables L3 = ~(z∂ − z¯∂), two properties give L3 ψm = ~mψm .

7.5 A Particle in a Constant Magnetic Field

91

Problem 7.18 Show that Schr¨odinger operator HA in the first gauge in (7.34) is of the form HA′ :=

1 2 1 [p + p22 + e2 b2 (x21 + x22 ) − ebL3 ] 2m 1 4

(7.40)

where L3 = x1 p2 − x2 p1 is the third component of the angular momentum operator. For the second gauge, the corresponding Schr¨odinger operator is HA′′ :=

1 [(p1 + ebx2 )2 + p22 ]. 2m

(7.41)

To analyze HA′′ , we apply the Fourier transform to only the first variable (x1 → 7 k1 ). This results in the unitarily equivalent operator 2 ˜ A′′ = 1 p22 + mωb (x2 + 1 k1 )2 , H 2m 2 eb

(7.42)

where, recall, ωb := eb m is the cyclotron frequency, and k1 acts as a multi˜ A′′ acts as a harmonic oscillator (centred plication operator. We see that H 1 k1 ) in the variable x2 , and as a multiplication operator in k1 . In the at − eb following problem you are asked to determine the spectrum of this operator. Problem 7.19 1. Show (7.41) and (7.42), and use (7.42) to prove directly that the energy levels of HA′′ are given by (7.30). 2. Show that the corresponding eigenfunctions are of the form Z ψn,α (x1 , x2 ) = (2π~)−1/2 eix1 k1 /~ f (k1 )φn (x2 + k1 /eb)dk1 where φn is the nth eigenfunction of the harmonic oscillator and f is an arbitrary function from L2 (R). Problem 7.20 Let A be a vector potential with zero magnetic field, curlA = 0. Show that H(A) is unitarily equivalent to H(A = 0), i.e. there is a unitary operator U s.t. H(A) = U ∗ H(A = 0)U . (Hint: Show that there is a function χ s.t. A = ∇χ and that the gauge transformation, ψ(x) → eiχ(x) ψ(x) provides the desired unitary operator. To conclude this subsection, we consider the simple but instructive case of a particle on a flat torus, say T2 = R2 /L,

L := {m1 L1 e1 + m2 L2 e2 : m1 , m2 ∈ Z} for some L1 , L2 > 0,

with no external electrostatic potential, V ≡ 0, and with a constant vector potential, A = (A1 , A2 ). The corresponding Schr¨odinger operator is still (7.23), acting on

92

7 Special Cases and Extensions

L2 (T2 ) := {ψ ∈ L2loc (R2 ) : ψ(x + Li ei ) = ψ(x), i = 1, 2}

(or on L2 (Ω), where Ω is a fundamental cell of L, with the domain H 2 (Ω), with the boundary conditions (7.3)). Here e1 := (1, 0), e2 := (0, 1). Since B = curlA = 0, on L2 (R2 ) the Schr¨odinger operator (7.23) with such an A is gauge 1 equivalent to the one without the vector potential, i.e. H(A = 0) = 2m |p|2 . But not on the torus. The gauge transformation, ψ(x) → eiA·x ψ(x),

which removes A on L2 (R2 ) (see Problem 7.20), does not leave L2 (T2 ) invariant. Hence, though the magnetic field corresponding to A is zero, B = curlA = 0, the operator H(A) is not unitary equivalent to H(A = 0). In particular, the spectrum of H(A) depends on A: it is straightforward to check that the eigenvalues are 3

En (A) =


2 1 X 2π~
2 nj − Φj /Φ0 , 2m 1 Lj

for nj ≥ 0, where Φj := Aj Lj and Φ0 := 2π~ e . Thus even for zero magnetic field, the vector potential, which is classically irrelevent, affects the Schr¨odinger operator. This surprising result is due to the subtle topology of the torus. T2 is not a simply connected space: one could draw two (classes of) loops on T2 which cannot be contracted to points (or to each other), say, γ1 (t) := (L1 t, 0) and γ2 (t) := (0, L2 t), 0 ≤ t ≤ 1 (remember that for each i, the points and γi (0), and γi (1) are identified by taking the quotient by L). Topologically, T2 is equivalent to a hollow donut in R3 and the loops above, to the loops around the two ‘holes’ of the donut. Assuming that the magnetic field B is confined to these holes, we see that though it is zero on the donut’s surface, its vector potential cannot be gauged R away. But what really matters are the magnetic fluxes through the holes, ‘ Di B ’, where Di is a cross-section of the i-th hole, which, by the Stokes theorem, are Z Ai = Aj Lj := Φj . ∂Di

Only when these fluxes are integers times the ‘flux quantum’ Φ0 does the vector potential A have no effect on the spectrum of H(A). Remark 7.21 Mathematically, A is a flat connection on a line bundle over the torus T2 (see e.g. [83]) and eiΦj are holonomies for this connection (cf. Section 7.6).

7.6 Aharonov-Bohm Effect Consider a charged quantum particle in 2 dimensions moving in external electric and magnetic fields with the potentials V and A, respectively. Let H(A) denote the corresponding Schr¨odinger operator, i.e.

7.6 Aharonov-Bohm Effect

H(A) := −

93

~2 ∆A + V, 2m

where, recall −~2 ∆A := (−i~∇ − eA)2 . Furthermore, we assume that V is radially symmetric, V (x) = U (|x|), and A is of the form1 A(x) := f (|x|)x⊥ /|x|2 ,

x⊥ := (−x2 , x1 ).

(7.43)

Note that divA = 0, and B = curlA = 2g + rg ′ , where g(r) := f (r)/r2 . (Indeed, curlA = 2g + (x1 x1 /r + x2 x2 /r)g ′ = 2g + rg ′ .) The operator H(A) with A and V specified above will be denoted by Hf . The computation above implies that, for f (r) = b (a constant), the magnetic field is B = 2πbδ0 , with 2πb = the total flux of B through R2 : R R2 B = 2πb. Using that (p − eA)2 = |p|2 − 2eA · p + |eA|2 , where, recall, p := −i~∇, and passing to polar co-ordinates, (r, θ), we find that (−i~∇ − eA)2 = −~2 ∆r +


1 −~2 ∂θ2 + 2ef (r)(i~∂θ ) + e2 f 2 (r) 2 r

= −~2 ∆r +

1 (i~∂θ + ef (r))2 , r2

where ∆r is the radial Laplacian, ∆r := 1r ∂r r∂r . Using this, we obtain ~2 1 ∆r + (i~∂θ + ef (r))2 + U (r). 2m 2mr2 P Expanding in a Fourier series, ψ(r, θ) = m∈Z ψm (r)eimθ and using separation of variables, we see that Hf = −

Hf = ⊕m∈Z Hf,m , 2

~ ∆r + where Hf,m := − 2m

1 2mr 2 (−~m

+ ef (r))2 + U (r). This yields

σ(Hf ) = ∪m σ(Hf,m ). Thus, writing f (r) ≡ ~e α, we have Hα ≡ Hf =~α/e = ⊕m∈Z Hα,m , where Hα,m := −

~2 ~2 (m − α)2 + U (r). ∆r + 2m 2mr2

We see that Hα is periodic in α: at each period, every Hα,m in the decomposition Hα = ⊕m∈Z Hα,m moves one step down resulting in Hα+1 = Hα . 1

The vector field A(x) is said to be spherically equivariant in the sense that it satisfies the equation A(gx) = gA(x) for any g ∈ O(2).

94

7 Special Cases and Extensions

(The corresponding eigenvalues move accordingly, which is referred to in the literature as the spectral flow.) The general theory developed in Section 6.5 (with the caveat that special ~2 2 care be taken of the singular potential 2mr 2 (m − α) ), implies that the spectrum of Hα consists of: the continuum [0, ∞) and possibly, depending on U (r), negative eigenvalues, if U (r) → 0 as r → ∞; and of isolated eigenvalues with finite multiplicities accumulating at ∞, if UR(r) → ∞ as r → ∞. Clearly, these eigenvalues depend on the magnetic flux B = 2π~α/e, even though there is no magnetic field in the physical space R2 /{0} in which particle moves. Moreover, one can show that in the first case, the quantities characterizing the essential spectrum, such as the scattering matrix, also depend on α. Mathematically, this is not surprising as the space R2 /{0} is not simply connected (it has a ‘hole’ at x = 0) and there are vector fields A, with zero magnetic fields, which are not gauge equivalent to 0 (flat connections in mathematical language). Such vector potentials are called topological vector potentials, and the corresponding fluxes (in the ambient space), topological fluxes. Note that we have already encountered the effect of the topology of the physical space on the properties of Schr¨odinger operators at the end of Section 7.5, where we considered the movement of a particle on a torus with a constant vector potential having zero magnetic field. The set-up above can be generalized to several topological fluxes: Hα = H(A), α := (α1 , . . . , αr ) for A given by A(x) :=

r X ~αi (x − ai )⊥ 1

e |x − ai |2

,

(7.44)

⊥ where, P recall, x := (−x2 , x1 ), with the magnetic field given by B = 2π ~e i αi δai . (One can also take individual vector potentials in different gauges.) In this case, the operator Hα does not decompose into one-dimensional ones, but the conclusions reached above about the spectrum of Hα still hold for Hα . Furthermore, Hα is periodic in each αi :

Hα+ei = Hα , where {ei } is the standard basis in Rr . In particular, we can think of α as varying in the standard flat torus Tr := Rr /Zr . Remark 7.22 The structure of the Aharonov-Bohm magnetic potential (7.43) with f (r) ≡ ~α/e, or (7.44), is especially simple in the complex representation. For this, we go from the real vector variable x = (x1 , x2 ) to the complex one z = x1 + ix2 and from the real vector potential A = (A1 , A2 ) to the complex function Ac := A1 − iA2 . Then, e.g. (7.43), with f (r) ≡ ~α/e, becomes Ac (z) = −(~α/e)i/z.

(7.45)

7.7 Linearized Ginzburg-Landau Equations of Superconductivity

95

7.7 Linearized Ginzburg-Landau Equations of Superconductivity One of the greatest achievements in condensed matter physics is A.A. Abrikosov’s discovery of vortex lattice solutions of the Ginzburg-Landau theory of superconductivity (Abrikosov was awarded the Nobel prize for this discovery). We formulate the key mathematical problem of Abrikosov’s paper. Let L be a Bravais lattice in R2 ; i.e. for some basis {ω1 , ω2 } in R2 , L := {α =

2 X i=1

mi ωi : mi ∈ Z}.

Let ∆A stand for the covariant Laplacian defined as ∆A := (∇ − iA)2 . The Abrikosov problem can be formulated as the eigenvalue problem −∆A ψ = κ2 ψ,

(7.46)

where A is a vector potential of a constant external magnetic field b, curlA = b, κ > 0 is a material constant (different for different materials) and ψ is a twice differentiable function satisfying ψ(x + s) = eigs (x) ψ(x),

(7.47)

for some functions gs ∈ C 2 (R2 ; R) and each s ∈ L. We can fix the gauge as A(x) =

b ⊥ x , 2

gs (x) =

b ⊥ s · x + cs , 2

(7.48)

where x⊥ = (−x2 , x1 ) for x = (x1 , x2 ), and cs satisfies 1 cs+t − cs − ct − bs ∧ t ∈ 2πZ. 2 (For more details and for the relation of the Abrikosov problem to equations on line bundles see [268].) Let Ω be a fundamental cell 2 of the lattice L, say ( 2 ) X 1 1 ai ω i | − < ai ≤ Ω := . 2 2 i=1 (7.47)-(7.48) are consistent with the fact that ψ is a single valued function if and only if the magnetic flux, b|Ω|, through the fundamental cell Ω is quantized: b|Ω| = 2πn, 2

(7.49)

A fundamental cell of L ⊂ C is a subset Ω ⊂ C such that (i) Ω ∩ (Ω + s) = ∅ ∀s ∈ L/{0} and (ii) ∪s∈L Ω = C. (The second property says that Ω tessellates C.)

96

7 Special Cases and Extensions

for some integer n. (Performing translations by s and then by t should lead to the same value as by t and then by s. For more details see Supplement 7.9.) This is the celebrated quantization condition. To finish the formulation of the eigenvalue problem (7.46)-(7.47), we have to define the space on which ∆A acts, and its domain. We define L2per(R2 ) to be the space of locally L2 functions satisfying the gauge-periodicity condition (7.47)-(7.48). It becomes a Hilbert space under the inner product of L2 (Ω) for an arbitrary fundamental cell Ω of the lattice L. We consider the eigenvalue 2 problem (7.46)- (7.47) on the corresponding Sobolev space Hper (R2 ) of index 2. A key step in proving the existence of Abrikosov vortex lattice solutions is the following Theorem 7.23 The values of b for which the problem (7.46)- (7.47), with the quantization condition b|Ω| = 2πn, has a non-trivial solution, are b = κ2 /(2k + 1), k = 0, 1, 2, · · · . Furthermore, for the largest value b = κ2 , it has exactly n linearly independent solutions. 2 Proof. One can show that the operator ∆A with domain Hper (R2 ) is selfadjoint. We denote it by H. By Theorem 7.13, with ~ = 1, m = 1/2 and e = 1, the spectrum of H is given by

σ(H) = { (2k + 1)b : k = 0, 1, 2, . . . }. Moreover, the eigenspace of the lowest eigenvalue satisfies Null(H − b) = Null α.

(7.50)

Now, identifying x ∈ R2 with z = x1 + ix2 ∈ C, the arguments of the proof of Theorem 7.13 imply that b

2

Null α = {ψ(z) := f (z)e− 4 |z| | f is holomorphic, ψ satisfies (7.47)-(7.48)}. (7.51) Hence we look for ψ(z) in the form b

ψ(z) = e 4 (z

2

−|z|2 )

Θ(z),

(7.52)

where Θ(z) is an entire function, and ψ satisfies (7.47)-(7.48). Viewing L as a subset of C, and using the rotation symmetry, if necessary, we can assume that L has a basis of the form { r, rτ }, where r is a positive real number and τ ∈ C, Im τ > 0.3 Setting r = 1 (by rescaling) and taking cr = cτ = 0, the quasiperiodicity (7.47)-(7.48) of ψ transfers to Θ as follows: 3

We identify x ∈ R2 with z = x1 + ix2 ∈ C, and view L as a subset of C. Then set ′ τ = rr for a basis r, r ′ . Although the basis is not unique, the value of τ is, and it is used that as a characterization of the shape of the lattice. Using the rotation symmetry we can assume that if L has as a basis { r, rτ }, where r is a positive real number and τ ∈ C, Im τ > 0.

7.8 Ideal Quantum Gas and Ground States of Atoms

97

Θ(z + 1) = Θ(z),

(7.53a)

Θ(z + τ ) = e−2πinz e−iπnτ Θ(z).

(7.53b)

To complete the proof, we now need to show that the space of analytic functions which satisfy these relations form a vector space of dimension n. The first relation ensures that Θ has an absolutely convergent Fourier expansion of the form Θ(z) =

∞ X

ck e2πikz .

k=−∞

The second relation, on the other hand, leads to a relation for the coefficients of the expansion: ck+n = einπτ e2kiπτ ck .

(7.54)

That means such functions are determined solely by the values of c0 , . . . , cn−1 , and therefore form an n-dimensional vector space. This completes the proof of Theorem 7.23. Problem 7.24 Prove (7.53) and (7.54). Holomorphic functions satisfying (7.53) are called the theta functions. The proof implies also the form of the corresponding eigenfunctions: (7.52). Historical Remark. Abrikosov found the vortex lattice solutions solutions using the linearized Ginzburg-Landau equations, and taking into account the nonlinearity in the first order of perturbation theory. For this, he considered the eigenvalue problem (7.46)- (7.47) in the case n = 1. In this case, the space (7.50) is one-dimensional and spanned by the function (setting b = 1 for simplicity) i

ψ := e 2 x2 (x1 +ix2 )

∞ X

k=−∞

ck e2πik(x1 +ix2 ) , ck = ceikπτ

k−1 Y

ei2mπτ .

(7.55)

m=1

This is the leading approximation to the Abrikosov lattice solution ([6]). The normalization coefficient c cannot be found from the linear theory and is obtained by taking into account nonlinear terms by perturbation theory (see [294] and references therein for a rigorous treatment and [146, 267] for a review).

7.8 Ideal Quantum Gas and Ground States of Atoms Consider a system of N identical particles not interacting with each other in an external potential U (x). It is called the ideal quantum gas. The Schr¨odinger operator for such a system is

98

7 Special Cases and Extensions

Higas =

N X

(−

j=1

~2 ∆x + U (xj )). 2m j

(7.56)

2

~ Assume the Schr¨odinger operator − 2m ∆x + U (x) has an isolated eigenvalue at the bottom of its spectrum – the ground state energy. In this case, so does Higas and, for bosons, its ground state and ground state energy are given by Φ(x1 , . . . , xn ) = φ1 (x1 ) . . . φ1 (xN ), and (N )

Eigas = N e1 ,

(7.57)

respectively, where where φ1 (x) and e1 are a ground state and the ground ~2 state energy of the Schr¨odinger operator − 2m ∆x + U (x). Problem 7.25 Prove the above statement. On the other hand, for spinless fermions, for N ≥ Q 2, the ground state N of the operator Higas cannot be given by the product 1 φ1 (xj ). A little contemplation shows that it is given by the anti-symmetric product, N ^

φj (xj ),

1

of N bound states, φj (x), of the operator Higas , corresponding to the N lowest energies e1 , . . . , eN , counting multiplicities. In this case, the ground state (N ) energy, Eigas , of Higas is given by (N )

Eigas =

N X

ej

1

Problem 7.26 Prove the above statement. This is, in general a much higher number. For example, for non-degenerate eigenvalues converging quickly to 0, this number remains O(1) for large N . Now we consider the most important case of spin 12 . Recall that the state space of a system of N fermions of spin 12 is given by (4.31), Hf ermi :=

N ^ 1

(L2 (R3 ) ⊗ C2 ).

(7.58)

Hence, after separation of spin variables and for N > 2, the ground state of PN ~2 the operator Higas := 1 (− 2m ∆xj + U (xj )) is given by the anti-symmetric product, N/2 ^ (φ2j−1 (x2j−1 )φ2j (x2j )), 1

7.8 Ideal Quantum Gas and Ground States of Atoms

99

(if N is even) of N2 bound states, φj (x), corresponding to the N2 lowest energies, say E1 , . . . , EN/2 . (The case of N odd requires a slight modification.) In P (N ) this case, the ground state energy of Higas is given by Eigas := 2 N/2 Ej , so 1 that N/2 X Higas ≥ 2 Ej . (7.59) 1

Now we apply the above analysis to an estimation of the ground state energies of atoms. We have shown in Subsection 5.3 that the Schr¨odinger operator, Hat , of the atom (or ion) with N electrons and an infinitely heavy nucleus of the charge Ze is estimated from below as

el where Higas :=

PN 1

el Hat ≥ Higas , 2

~ ∆xj − (− 2m

e2 Z |xj | ).

(7.60)

The latter operator is the quantum 2

Hamiltonian of the ideal gas of N electrons in the external potential − e|x|Z created by the nucleus. If we ignore the fact that the electrons are fermions, then we have according to (7.60), (7.57) and (7.11), Hat ≥ −N

2m(e2 Z)2 . ~2

(7.61)

Now, we estimate Hat from below using the fact that the electrons are P el fermions. Then by (7.60) and (7.59), Hat ≥ Higas ≥ 2 N/2 j=1 Ej , where the Ej are given by (7.11). To estimate the r.h.s. we should take into account the multiplicities of the eigenvalues. We relabel {Ej , j = 1, 2, . . . } as {Ek,m , m = 1, 2, . . . , mk , k = 1, 2, . . . }, with Ek,m distinct for different k, and equal for the same k. Thus N independent particles, in the lowest energy state, occupy 2 ~2 ∆x − e|x|Z , where K is defined the K lowest energy levels of the operator − 2m PK by 1 mk = N/2. The corresponding energy of Higas is (N )

Eigas = 2

K X

mk Ek,m1 , with

k=1

K X

mk = N/2.

1

4

2

We know from (7.11) and (7.12) that Ek,m − me2~2Z k −2 with the multiplicities mk = k 2 . Then the previous two equations yield K ∼ N 1/3 and (N )

Eigas ∼ −Z 2 K ∼ −Z 2 N 1/3 . For large N , this gives a much higher number than the one on the r.h.s. of (7.61). PN Note that it is easy to give a realistic estimate of the sum 2 12 Ej , when 2 ~2 Ej are the eigenvalues of the operator − 2m ∆x − e|x|Z . However, if one wants to obtain a more realistic estimate on the ground state energy of (5.10),

100

7 Special Cases and Extensions

then one should not throw away the electron repulsion. To keep the problem manageable, one replaces it by some mean field term (see Section 14.1). In PN this case we want to bound 12 Ej , where Ej are eigenvalues of the effective one-particle operator containing the mean-field potential. This is substantially more difficult ,and to do this, one uses the Lieb-Thirring inequalities. Problem 7.27 Find the ground state energy of Higas (in terms of the oneparticle energies) for (a) 11 identical fermions of spin 3/2 on the subspace corresponding to a Young diagram with with columns of lengths 4, 4, 2, 1; (b) 17 identical fermions of spin 1/2 on the subspace corresponding to a Young diagram with with columns of lengths 10, 7.

7.9 Supplement: L−equivariant functions Consider the two dimensional Bravais lattice L = {mω1 + nω2 : m, n ∈ Z}, where (ω1 , ω2 ) is some basis in R2 . A function Ψ and a vector field A are said to be L−equivariant if and only if there are differentiable functions, gs : R2 → R, s ∈ L, s.t. 4 Ψ (x + s) = eigs (x) Ψ (x) and A(x + s) = A(x) + ∇gs (x), ∀s ∈ L.

(7.62)

Since Tstrans is a group, the family of functions gs should satisfy gs+t (x) − gs (x + t) − gt (x) ∈ 2πZ,

s, t ∈ L.

(7.63)

This can be seen by evaluating the effect of translation by s+t in two different ways (as Ψ ((x + t) + s) and Ψ (x + (t + s))). We list some important properties of gs : •

If (Ψ, A) satisfy (7.62) with gs (x), then Tχgauge (Ψ, A) satisfies (7.62) with gs (x) → gs′ (x), where gs′ (x) = gs (x) + χ(x + s) − χ(x).

•

(7.64)

The following functions satisfy (7.63): gsb (x) =

b s ∧ x + cs , 2

(7.65)

where b satisfies the flux quantization relation b|Ω| ∈ 2πZ and cs are numbers satisfying

4

1 cs+t − cs − ct − bs ∧ t ∈ 2πZ. 2 R x x0

(7.66)

Eq. (7.62) implies that gs (x) = As (y)dy, for some x0 , where As (x) := A(x + s) − A(x). Since curlA (x) = curlA(x + s) − curlA(x) = 0 ∀s ∈ L, the integral s Rx A (y)dy is independent of the path between x0 and x. For a constant magnetic s x0 field vector potential Ab (x) = 2b x⊥ , where x⊥ = (−x2 , x1 ), this gives gsb (x) = b s ∧ x + cs , where cs is a constant of integration (cf. (7.65) below). 2

7.9 Supplement: L−equivariant functions

• •

101

Every exponential gs satisfying the cocycle condition (7.63) is equivalent to the exponential (7.65) with (7.66). The exponentials gs satisfying the cocycle condition (7.63) are classified by the irreducible representation of the group of lattice translations.

Indeed, the first and second statements are straightforward. For the third property, which was shown by A. Weil and generalized by R. Gunning, we refer to [144]. We show that under (7.62) the flux of the magnetic field is quantized: Z curlA ∈ 2πZ. (7.67) Ω

Indeed, let, as above, Ω be a fundamental cell of the lattice L. Take Ω := P2 { 1 ai ωi : − 21 < ai ≤ 12 }. We compute, by Stokes’ theorem Z Z curlA = A. Ω

∂Ω

Breaking the integral on the r.h.s. into the integrals over the four sides of the parallelogram Ω and using that due to (7.62), the sum of integrals R over parallel sides give gω2 (ω1 )− gω2 (0) and gω1 (0)− gω1 (ω2 ), which yields ∂Ω A = gω2 (ω1 ) − gω2 (0) − (gω1 (ω2 ) − gω1 (0)). By (7.63), this implies (7.67). If we plug A(x) from (7.48) into this formula, we find the flux quantization relation b|Ω| = 2πn, n ∈ Z.

8 Bound States and Variational Principle

In this chapter we develop powerful techniques for proving existence of bound states (eigenfunctions) corresponding to isolated eigenvalues. We also give estimates of their number.

8.1 Variational Characterization of Eigenvalues We consider, for the moment, a self-adjoint operator H, acting on a Hilbert space H. The main result of this chapter is the following important characterization of eigenvalues of H in terms of the minimization problem for the “energy” functional hψ, Hψi. Theorem 8.1 (1) The Ritz variational principle: inf

ψ∈D(H), kψk=1

hψ, Hψi = inf σ(H).

(2) The left hand side has a minimizer ψ if and only if Hψ = λψ, with λ := inf σ(H). (3) If there is a ψ ∈ D(H) (called a test function) with kψk = 1 and hψ, Hψi < inf σess (H), then H has at least one eigenvalue below its essential spectrum. As an example application of this theorem, we consider the bound state problem for the Hydrogen atom. The Schr¨odinger operator for the Hydrogen atom is e2 ~2 ∆− H=− 2m |x|

acting on the Hilbert p space L2 (R3 ) (see Section 5.3). Take the (normalized) test function ψ(x) = µ3 /πe−µ|x| for some µ > 0 to be specified later and compute (passing to spherical coordinates) © Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_8

104

8 Bound States and Variational Principle

Z Z 1 ~2 |ψ|2 |∇ψ|2 − e2 2m |x| Z ∞ Z ∞ ~ 2 µ3 µ3 4π e−2µr r2 dr − e2 4π e−2µr rdr. = 2m π π 0 0

hψ, Hψi =

To compute the above integrals, note Z ∞ Z ∞ d2 −αr 2 e−αr dr e r dr = dα2 0 0 and

Z

0

∞

e−αr rdr = −

d dα

Z

∞

e−αr dr.

0

R∞ R∞ R∞ Since 0 e−αr dr = α−1 , we find 0 e−αr r2 dr = 2α−3 , and 0 e−αr rdr = α−2 . Substituting these expressions with α = 2µ into the formula for hψ, Hψi, we obtain ~2 2 µ − e2 µ. hψ, Hψi = 2m The right hand side has a minimum at µ = me2 /~2 , which is equal to hψ, Hψi|µ=me2 /~2 = −

me4 . 2~2

Since σess (H) = [0, ∞) (according to Theorem 6.16, which can be extended to cover singular potentials like the Coulomb potential – see Problem 6.17 in Section 6.5) we conclude that H has negative eigenvalues, and the lowest negative eigenvalue, λ1 , satisfies the estimate λ1 ≤ −

me4 . 2~2

This should be compared with the lower bound λ1 > −

2me4 ~2

found in Section 5.3. The rest of this section is devoted to the proof of this theorem and a generalization. We begin with some useful characterizations of operators in terms of their spectra. Theorem 8.2 Let H be a self-adjoint operator with σ(H) ⊂ [a, ∞). Then H ≥ a (i.e. hu, Hui ≥ akuk2 for all u ∈ D(H)). Proof. Without loss of generality, we can assume a = 0 (otherwise we can consider H − a1 instead of H). First we suppose H is bounded (we will pass to the unbounded case later). If b > 2kHk, then the operator (H + b)−1 is positive, as follows from the Neumann series

8.1 Variational Characterization of Eigenvalues

(H + b)−1 = b−1 (1 + b−1 H)−1 = b−1

∞ X

105

(−b−1 H)j

j=0

(see (25.42)). For all λ > 0, the operator (H + λ)−1 is bounded, self-adjoint, and differentiable in λ (in fact it is analytic – see Section 25.9). Compute ∂ (H + λ)−1 = −(H + λ)−2 < 0, ∂λ Hence for any 0 < c < b, (H + c)−1 = (H + b)−1 +

Z

b

(H + λ)−2 dλ

c

−1

−1

and therefore (H + c) > (H + b) > 0. Now any u ∈ D(H) can be written in the form u = (H + c)−1 v for some v ∈ H (show this). Since hu, (H + c)ui = hv, (H + c)−1 vi > 0 for any c > 0, we conclude that hu, Hui ≥ 0 for all u ∈ D(H), as claimed. In order to pass to unbounded operators, we proceed as follows. Let c > 0 and A := (H + c)−1 , a bounded operator since −c ∈ / σ(H). For any λ 6= 0, we have A + λ = (H + c)−1 + λ = λ(H + c)−1 (H + c + λ−1 ). (8.1) Hence for λ > 0, the operator A + λ is invertible, and so σ(A) ⊂ [0, ∞) (in fact, one can see from (8.1) that A + λ is also invertible if λ < −c−1 , and so σ(A) ⊂ [0, c−1 ]). By the proof above, A := (H + c)−1 ≥ 0. Repeating the argument at the end of this proof, we find that H + c ≥ 0. Since the latter is true for any c > 0, we conclude that H ≥ 0. . Theorem 8.3 Let S(ψ) := hψ, Hψi for ψ ∈ D(H) with kψk = 1. Then inf σ(H) = inf S. Moreover, λ := inf σ(H) is an eigenvalue of H if and only if there is a minimizer for S(ψ) among ψ ∈ D(H) with the constraint kψk = 1. Proof. As in the proof of Lemma 25.21, we compute, for ψ ∈ D(H) and z ∈ R satisfying z < inf S =: µ, k(H − z1)ψk2 = h(H − z1)ψ, (H − z1)ψi

= k(H − µ)ψk2 + 2(µ − z)hψ, (H − µ)ψi + |µ − z|2 kψk2 . (8.2)

Since hψ, (H − µ)ψi ≥ 0, this gives k(H − z1)ψk ≥ |µ − z|kψk.

(8.3)

As in the proof of Lemma 25.21, this shows that the operator H − z1 is invertible and therefore z ∈ / σ(H). Therefore inf S ≤ inf σ(H). Now let λ :=

106

8 Bound States and Variational Principle

inf σ(H). By Theorem 8.2, hψ, Hψi ≥ λkψk2 for any ψ ∈ D(H). Hence inf S ≥ λ = inf σ(H), and therefore inf S = inf σ(H) as required. Now if λ = inf σ(H) is an eigenvalue of H, with normalized eigenvector ψ0 , then S(ψ0 ) = hψ0 , Hψ0 i = λ = inf S,

and therefore ψ0 is a minimizer of S. On the other hand, if ψ0 is a minimizer of S, among ψ ∈ D(H), kψk = 1, then it satisfies the Euler-Lagrange equation (see Section 26.5) S ′ (ψ0 ) = 2λψ0 for some λ. Since S ′ (ψ) = 2Hψ (see again Section 26.5), this means that ψ0 is an eigenvector of H with eigenvalue λ. Moreover, S(ψ0 ) = hψ0 , Hψ0 i = λkψ0 k2 = λ. Since S(ψ0 ) = inf S = inf σ(H), we conclude that λ = inf σ(H) is an eigenvalue of H (with eigenvector ψ0 ). 

Proof. of Theorem 8.1: Theorem 8.3 gives the proof of the first two parts of the theorem – the Ritz variational principle: for any ψ ∈ D(H), hψ, Hψi ≥ λ = inf σ(H) and equality holds iff Hψ = λψ. To obtain the third part of the theorem, stating that if we can find ψ with kψk = 1 and hψ, Hψi < inf σess (H) then we know that H has at least one eigenvalue below its essential spectrum, we note that, by Theorem 8.3, inf σ(H) = inf S < hψ, Hψi < inf σess (H), so λ = inf σ(H) must be an (isolated) eigenvalue of H.  The variational principle above can be extended to higher eigenvalues. Theorem 8.4 (Min-max principle) The operator H has at least n eigenvalues (counting multiplicities) below inf σess (H) if and only if λn < inf σess (H), where the number λn is given by λn =

inf

max

hψ, Hψi.

{X⊂D(H) | dim X=n} {ψ∈X | kψk=1}

(8.4)

In this case, the n-th eigenvalue (labeled in non-decreasing order) is exactly λn . Sketch of proof. We prove only the “if” part of the theorem. The easier “only if” part is left as an exercise. We proceed by induction. For n = 1, the statement coincides with that of Theorem 8.3. Now assume that the “if” statement holds for n ≤ m − 1, and we will prove it for n = m. By the induction assumption, the operator H has at least m − 1 eigenvalues, λ1 , . . . , λm−1 (counting multiplicities), all < inf σess (H). We show that H has at least m eigenvalues. Let Vm−1 denote the subspace spanned by the (normalized) eigenvectors ψ1 , . . . , ψm−1 , corresponding to λ1 , . . . , λm−1 . Then

8.1 Variational Characterization of Eigenvalues

107

⊥ 1. the subspace Vm−1 , and its orthogonal complement, Vm−1 , are invariant under the operator H, and 2. the spectrum of the restriction, H|Vm−1 ⊥ , of H to the invariant subspace ⊥ Vm−1 is σ(H)\{λ1 , . . . , λm−1 }.

Problem 8.5 Prove statements 1 and 2. Now apply Theorem 8.3 to the operator H|Vm−1 ⊥ to obtain inf

⊥ {ψ∈Vm−1 ∩D(H) | kψk=1}

hψ, Hψi = inf{σ(H)\{λ1 , . . . , λm−1 }}.

(8.5)

On the other hand, let X be any m-dimensional subspace of D(H). There exists φ ∈ X such that φ ⊥ Vm−1 , and kφk = 1. We have hφ, Hφi ≥

inf

⊥ {ψ∈Vm−1 ∩D(H) | kψk=1}

hψ, Hψi.

Hence for any such X max

hψ, Hψi ≥

{ψ∈X | kψk=1}

inf

⊥ {ψ∈Vm−1 ∩D(H) | kψk=1}

hψ, Hψi

and therefore λm defined by (8.4) obeys λm ≥

inf

⊥ {ψ∈Vm−1 ∩D(H) || kψk=1}

hψ, Hψi.

(8.6)

Since λm < inf σess (H) by assumption, and due to (8.5), we have λ′m := inf{σ(H)\{λ1 , . . . , λm−1 }} < inf σess (H)

(8.7)

is the m-th eigenvalue of H. Moreover, Equations (8.5)-(8.7) imply that λm ≥ λ′m . Now we show that λm ≤ λ′m . Let ψm be a normalized eigenvector corresponding to λ′m , and let Vm = span{ψ1 , . . . , ψm }. Then λm ≤

max

hψ, Hψi = λ′m .

ψ∈Vm ,kψk=1

Hence λm = λ′m . Thus we have shown that H has at least m eigenvalues (counting multiplicities) < inf σess (H), and these eigenvalues are given by (8.4).  This theorem implies that if we find an n-dimensional subspace X, such that µn := sup hψ, Hψi < inf σess (H), ψ∈X,kψk=1

then H has at least n eigenvalues less than inf σess (H), and the largest of these eigenvalues satisfies the bound

108

8 Bound States and Variational Principle

λn ≤ µn . This result will be used in Section 8.3. There is another formulation of the min-max (more precisely sup-inf) principle, in which Equation 8.4 is replaced by the equation λn =

sup

{X⊂D(H) | dim X=n} {ψ∈X

inf

⊥

hψ, Hψi.

| kψk=1}

A proof of this theorem is similar to the proof above. The following useful statement is a simple consequence of the min-max principle. Suppose A and B are self-adjoint operators with A ≤ B. Denote the j-th eigenvalue of A below its essential spectrum (if it exists) by λj (A) (and similarly for B). Suppose also that the eigenvectors of B corresponding to the eigenvalues λ1 (B), . . . , λj (B) lie in D(A). Then λj (A) ≤ λj (B). To see this, let Vj denote the span of the first j eigenvectors of B, and observe that max{ψ∈Vj |kψk=1} hψ, Bψi = λj (B). Since Vj ⊂ D(A), (8.4) gives λj (A) ≤ ≤

max

hψ, Aψi

max

hψ, Bψi = λj (B).

{ψ∈Vj | kψk=1} {ψ∈Vj | kψk=1}

Another useful criterion for finding eigenvalues of self-adjoint operators goes as follows. If for some λ ∈ R and ǫ > 0, there is a function ψ ∈ D(A) such that k(A − λ)ψk ≤ ǫkψk, (8.8) then the operator A has spectrum in the interval [λ − ǫ, λ + ǫ]: σ(A) ∩ [λ − ǫ, λ + ǫ] 6= ∅. To prove this statement we use the inequality k(A − z)−1 k ≤ [dist(z, σ(A))]−1 for z ∈ ρ(A), which extends the inequality (25.11). For a proof of this inequality see [162, 244]. Now if A has no spectrum in [λ − ǫ, λ + ǫ], i.e. if [λ − ǫ, λ + ǫ] ⊂ ρ(A), then k(A − λ)−1 k
0, m

a contradiction to (8.12). Thus the operator H, with V (x) satisfying x · ∇V (x) ≤ 0 has no eigenvalues. Problem 8.7 Show (8.13). We turn our attention now from a situation where there are no bound states, to one where there are many. In particular, we will prove that the Schr¨odinger operator ~2 q H=− ∆− (8.14) 2m |x|

8.3 Number of Bound States

111

with q > 0, has an infinite number of bound states. The potential V (x) = −q/|x| is called the (attractive) Coulomb potential. For appropriate q, the operator H describes either the the Hydrogen atom, or a one-electron ion. In Section 7.3, we went further, solving the eigenvalue problem for H exactly, finding explicit expressions for the eigenfunctions and (infinitely many) eigenvalues. Nevertheless, it is useful to be able to prove the existence of infinitely many bound states using an argument that does not rely on the explicit solvability of the eigenvalue problem. We would first like to apply Theorem 6.16 to locate the essential spectrum. However, the fact that the Coulomb potential is singular at the origin is a possible obstacle. In fact, Theorem 6.16 can be extended to cover this case (see Problem 6.17 and, eg, [73]), and we may conclude that H is self-adjoint, with essential spectrum equal to the half-line [0, ∞). To prove that the operator (8.14) has an infinite number of negative eigenvalues, we will construct an infinite sequence of normalized, mutually orthogonal test functions, un (x), such that hun , Hun i < 0.

(8.15)

The “min-max principle” (described in Section 8.1) then implies that H has an infinite number of eigenvalues. We begin by choosing a single function, u(x), which is smooth, and which satisfies kuk = 1

and

supp(u) ⊂ { x ∈ R3 | 1 < |x| < 2 }.

Then we set un (x) := n−3/2 u(n−1 x) for n = 1, 2, 4, 8, . . .. Problem 8.8 Show that hum , un i = δmn , and that hum , Hun i = 0 if m 6= n. Given the results of the exercise, it remains to show (8.15) for n sufficiently large. Indeed, changing variables to y = n−1 x, we compute Z Z ~2 1 hun , Hun i = |un (x)|2 dx |∇un (x)|2 dx − q 2m |x| Z Z 1 ~2 |u(y)|2 dy |∇u(y)|2 dy − n−1 q = n−2 2m |y| 0 otherwise for n sufficiently large. Problem 8.9 Prove these last two statements. Let Xn be the n-dimensional subspace spanned by the functions um , . . . , um+n , for m sufficiently large. The results of Problems 8.8 and 8.9 imply that sup

hψ, Hψi < 0

ψ∈Xn ,kψk=1

for all n, provided α < 2 and c < 0. Invoking the min-max principle again, we see that if α < 2 and c < 0, then the operator H has an infinite number of bound states. It is shown below that H has only a finite number of bound states if α > 2. The borderline for the question of the number of eigenvalues is given by the inequality 1 −∆ ≥ 4|x|2 (the Uncertainty Principle – see Section 5.2). This inequality shows that the ~2 1 kinetic term − 2m ∆ dominates at ∞ if α > 2, or if α = 2 and 2m ~2 c > − 4 . Otherwise, the potential term favouring eigenvalues wins out. This simple intuition notwithstanding, there is presently no physically motivated proof of the finiteness of the discrete spectrum for α > 2. The proof presented below uses mathematical ingenuity rather than physical intuition. We now prove finiteness of the number of eigenvalues for α > 2. To simplify the argument slightly, we assume the potential V (x) is non-positive and denote U (x) := −V (x) ≥ 0. Let λ < 0 be an eigenvalue of H with eigenfunction φ. The eigenvalue equation (H − λ)φ = 0 can be re-written as (−

~2 ∆ − λ)φ = U φ. 2m 2

~ ∆, we can invert Since λ < 0 is in the resolvent set of the operator − 2m ~2 (− 2m ∆ − λ) to obtain

φ = (−

~2 ∆ − λ)−1 U φ, 2m

8.3 Number of Bound States

113

the homogeneous Lippmann-Schwinger equation. By introducing the new function v := U 1/2 φ, this equation can be further re-written to read v = K(λ)v where the operator K(λ) is defined as K(λ) := U 1/2 (−

~2 ∆ − λ)−1 U 1/2 . 2m

We can summarize the above derivation as follows: λ < 0 EV H ↔ 1 EV K(λ) and so #{λ < 0 | λ EV H} = #{λ < 0 | 1 EV K(λ)}.

(8.17)

The next step is to prove that #{λ < 0 | 1 EV K(λ)} = #{ν > 1 | ν EV K(0)}.

(8.18)

To prove (8.18), we begin by showing that ∂ K(λ) > 0 ∀ λ ≤ 0 ∂λ

(8.19)

K(λ) → 0 as λ → −∞.

(8.20)

and Writing

~2 ∆ − λ)−1 U 1/2 φi 2m and differentiating with respect to λ, we obtain hφ, K(λ)φi = hU 1/2 φ, (−

~2 ∂ hφ, K(λ)φi = hU 1/2 φ, (− ∆ − λ)−2 U 1/2 φi ∂λ 2m ~2 ∆ − λ)−1 U 1/2 φk2 > 0 = k(− 2m which proves (8.19). To establish (8.20), we need to derive the integral kernel ~2 of the operator K(λ). Using the fact that the operator (− 2m ∆ − λ)−1 has integral kernel (−

m ~2 − ∆ − λ)−1 (x, y) = e 2 2m 2π~ |x − y|

q

2m|λ| |x−y| ~2

(see Equation (25.57)), we find that the integral kernel for K(λ) is K(λ)(x, y) = U (x)1/2

m − e 2 2π~ |x − y|

(see Section 25.3). Using the estimate

q

2m|λ| |x−y| ~2

U (y)1/2

114

8 Bound States and Variational Principle

 K ≤

1/2 |K(x, y)| dxdy 2

R3 ×R3

(see again Section 25.3), we obtain  1/2  m U (x)U (y) −2 2m|λ| |x−y| 2 K(λ) ≤ e dxdy . 2 2π2 R3 ×R3 |x − y| % Since exp(− 2m|λ| 2 |x − y|) → 0 as λ → −∞, Equation (8.20) follows. Now we show that the relations (8.19) and (8.20) imply (8.18). By (8.20), for all λ sufficiently negative, all of the eigenvalues of K(λ) are less than 1. By (8.19), the eigenvalues, νm (λ), of K(λ) increase monotonically with λ. Hence if νm (λm ) = 1 for some λm < 0, then νm (0) > νm (λm ) = 1. Similarly, if νm (0) > 1, then there is a λm < 0 such that νm (λm ) = 1. In other words, there is a one-to-one correspondence between the eigenvalues νm (0) of K(0) which are greater than 1, and the points λm at which some eigenvalue νm (λ) crosses 1 (see Fig. 8.1).



00 00
0
0


0
0


0
0 00



00 00
0
0


0
0 00

00
0
0



00



00

00 00

00
0
0

00 1


0
0

00

00 00

0
00
0



λ

Fig. 8.1. Eigenvalues of K(λ). Thus (8.18) follows. The relations (8.17) and (8.18) imply #{λ < 0 | λ EV H} = #{ν > 1 | ν EV K(0)}.

(8.21)

The quantity on the left hand side of (8.21) is what we would like to estimate, while the quantity on the right hand side is what we can estimate. Indeed, we have ⎞1/2 ⎛   2 ⎠ 1≤⎝ νm #{ν > 1 | ν EV K(0)} = νm >1,νm EV K(0)

⎛

≤⎝



⎞1/2

2 ⎠ νm

νm >1,νm EV K(0)



1/2 = tr(K(0)2 )

νm EV K(0)

(8.22)

8.3 Number of Bound States

115

(see Section 25.11 for the definition of tr, the trace). On the other hand (see Section 25.3), Z  m 2 Z U (x)U (y) 2 2 dxdy (8.23) tr(K(0) ) = |K(0)(x, y)| dxdy = 2π~2 |x − y|2 Collecting equations (8.21)- (8.23) and recalling that V (x) = −U (x), we obtain Z 1/2 m |V (x)V (y)| #{λ < 0 | λ EV H} ≤ dxdy . 2π~2 |x − y|2 Under our assumption (8.16) on the potential V (x), with α > 2, Z |V (x)V (y)| dxdy < ∞, |x − y|2 so that the number of negative eigenvalues of the operator H is finite. This is the fact we set out to prove. The argument used above (Equation (8.21) in particular) is called the Birman-Schwinger principle, and the operator K(λ) is the Birman-Schwinger operator.

9 Scattering States

In this chapter we study scattering states in a little more detail. As we saw in Section 6.4, scattering states are solutions of the time-dependent Schr¨odinger equation ∂ψ i~ = Hψ (9.1) ∂t with initial condition orthogonal to all eigenfunctions of H: ψ|t=0 = ψ0 ∈ Hb⊥

(9.2)

where Hb := span{ eigenfunctions of H} is the subspace of bound states of H. We will have to make a more precise assumption on the potential V (x) en~2 ∆+V (x): we assume, for simplicity, tering the Schr¨odinger operator H = − 2m that |∂xα V (x)| ≤ C(1 + |x|)−µ−|α| (9.3) for |α| ≤ 2 and for some µ > 0 (and C a constant). The notation needs a little explanation: α is a multi-index α = (α1 , α2 , α3 ) with each αj a non-negative P3 integer, |α| = j=1 αj , and ∂xα =

3 Y

∂xαjj .

j=1

The question we want to address is what is the asymptotic behaviour of the solution ψ = e−iHt/~ ψ0 of (9.1)-(9.2) as t → ∞. First observe that (6.10) shows that ψ moves away from any bounded region of space as t → ∞. Hence, since V (x) → 0 as |x| → ∞, (9.4) we expect that the influence of the potential V (x) diminishes as t → ∞. Thus the following question arises: does the evolution ψ approach a free evolution,

© Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_9

118

9 Scattering States 2

~ say φ = e−iH0 t/~ φ0 , where H0 = − 2m ∆, for some φ0 ∈ L2 (R3 ), as t → ∞? ⊥ Put differently, given ψ0 ∈ Hb , is there φ0 ∈ L2 (R3 ) such that

ke−iHt/~ ψ0 − e−iH0 t/~ φ0 k → 0

(9.5)

as t → ∞? This (or a modification of this question discussed below) is the problem of asymptotic completeness. It is the central problem of mathematical scattering theory. It conjectures that all the possible free motions, together with the bound state motions, form a complete set of possibilities for the asymptotic behaviour of solutions of time-dependent Schr¨odinger equations. There are two principal cases depending on the decay rate, µ, of the potential V (x) (more precisely, the decay rate, µ + 1, of the force −∇V (x)), and it is only in the first of these cases that the asymptotic completeness property formulated above holds. We now discuss these cases in turn.

9.1 Short-range Interactions: µ > 1 In this case, asymptotic completeness can be proved under condition (9.3). We can reformulate the asymptotic completeness property by defining the wave operator, Ω + : Ω + φ := lim eiHt/~ e−iH0 t/~ φ. (9.6) t→∞

Below we will show that under the condition (9.3) with µ > 1, the limit exists for any φ ∈ L2 (R3 ). Further, the operator Ω + is an isometry: kΩ + φk = kφk.

(9.7)

Indeed, (9.6) implies kΩ + φk = lim keiHt/~ e−iH0 t/~ φk = kφk t→∞

since the operators eiHt/~ and e−iH0 t/~ are isometries. The existence of the wave operator Ω + means that given a free evolution −iH0 t/~ e φ0 , there is a full evolution e−iHt/~ ψ0 such that (9.5) holds. To see this, note that since eiHt/~ is an isometry, (9.5) can be re-written as kψ0 − eiHt/~ e−iH0 t/~ φ0 k → 0 as t → ∞, which is equivalent to the relation ψ0 = Ω + φ0 .

(9.8)

Thus the existence of the wave operator Ω + is equivalent to the existence of scattering states – i.e. states e−iHt/~ ψ0 for which (9.5) holds for some φ0 . We have Ran(Ω + ) ⊂ Hb⊥ .

9.1 Short-range Interactions: µ > 1

119

Indeed, for any φ0 ∈ L2 (R3 ) and g ∈ Hb such that Hg = λg, we have hg, Ω + φ0 i = lim hg, eiHt/~ e−iH0 t/~ φ0 i = lim he−iHt/~ g, e−iH0 t/~ φ0 i t→∞

t→∞

= lim eiλt/~ hg, e−iH0 t/~ φ0 i.

(9.9)

t→∞

There is a general theorem implying that the right hand side here is zero, but we will prove it directly. Recalling expression (2.23) for the action of the free evolution operator e−iH0 t/~ , we find hg, e−iH0 t/~ φ0 i =



2πi~t m

−3/2 Z

R3

Z

g¯(x)eim|x−y|

2

/(2~t)

φ0 (y)dydx

R3

and therefore, if φ0 and g are integrable functions, i.e., Z Z |φ0 (x)|dx < ∞ and |g(x)|dx < ∞, R3

R3

we have |hg, e

−iH0 t/~

φ0 i| ≤



2π~t m

−3/2 Z

R3

|g|

Z

R3

|φ0 |

and thus the right hand side of (9.9) vanishes. For general φ0 and g the result is obtained by continuity. This argument can also be extended to cover the case where g is a linear combination of eigenfunctions. We can similarly define the wave operator Ω − describing the asymptotic behaviour as t → −∞: Ω − φ := lim eiHt/~ e−iH0 t/~ φ. t→−∞

This operator maps free states e−iH0 t/~ φ0 into states e−iHt/~ ψ0 which approach these free states as t → −∞. We have (in an appropriate sense) HΩ ± = Ω ± H0 .

(9.10)

Indeed, by changing variables, we can obtain the following intertwining relations e−iHt/~ Ω ± = Ω ± e−iH0 t/~ . (9.11) Differentiating these relations at t = 0, we obtain 9.10. Let Hsc := RanΩ + , the set of scattering states. We have shown that Hsc ⊂ Hb⊥ . The property of asymptotic completeness states that Hsc = Hb⊥

or

Hb ⊕ RanHsc = L2 (R3 )

i.e., that the scattering states and bound states span the entire state space L2 (R3 ). We will prove this property under some restrictive conditions below. One can also define the scattering operator

120

9 Scattering States ∗

S := Ω + Ω − which maps asymptotic states at t = −∞ into asymptotic states at t = ∞: −iH t/ h e o φ+ −iH t/ h e o φ−

Fig.9.1. S : φ− → φ+ Equations (9.11) and (9.10) imply e−iH0 t/~ S = Se−iH0 t/~ and H0 S = SH0 . The property of asymptotic completeness implies that the Hamiltonian H restricted to the invariant subspace Hb⊥ is unitarily equivalent to the free Hamiltonian H0 : ∗ H = Ω ± H0 Ω ± on Hb⊥ .

Let Ω ± (x, y) be the integral kernels of the operators Ω ± , and let ψ ± (x, k) denote their Fourier transforms with respect to the second variable, y. Equation (9.10) implies that Hψ ± (x, k) =

|k|2 ± ψ (x, k). 2m

(9.12)

In other words, ψ ± (x, k) are generalized eigenfunctions of the operator H, with eigenvalue |k|2 /2m. These generalized eigenfunctions are of the form   1 ψ ± (x, k) = e±ik·x + O |x| as |x| → ∞ (see, eg., [302, 246]). They are called scattering eigenfunctions of H. Problem 9.1 Prove relation (9.12).

9.2 Long-range Interactions: µ ≤ 1 In this case, the relation (9.5) must be modified to read

9.3 Wave Operators

121

keiHt/~ ψ0 − e−iS(t) φ0 k → 0 as t → ∞, where for 12 < µ ≤ 1, the operator-family S(t) is defined to be S(t) = S(p, t), where the function S(k, t) satisfies the equation   ∂ 1 S(k, t) = H0 (k) + V kt . ∂t m 1 |k|2 is the classical free Hamiltonian function. The operHere H0 (k) = 2m ator S(p, t) is defined according to the rules described in Section 4.2. Note that in this equation, the coordinate x in the potential V (x) is replaced by 1 kt. With this modification, the asymptotic its free, classical expression, m completeness property can again be established under assumption (9.3) with 1 1 2 < µ ≤ 1. For 0 < µ ≤ 2 , the expression for the operator-family S(t) is more complicated (see [77]). As in the short-range case, one can introduce modified wave operators and investigate their properties (see the references given in Chapter 27).

9.3 Wave Operators As promised, we prove the existence and completeness of the wave operators Ω ± . To simplify the proof, we impose a somewhat stronger condition on the potential: V (x) ∈ L2 (R3 ). Below, we also make use of the space Z 1 3 3 L (R ) := {ψ : R → C | |ψ(x)|dx < ∞}. R3

Theorem 9.2 If V ∈ L2 (R3 ), then the wave operators Ω ± , introduced formally above, exist. Proof. Denote Ω t := eiHt e−iH0 t (we drop the ~ in what follows, just to simplify the notation). Since kΩ t k ≤ const, uniformly in t (in fact, kΩ t k = 1), it suffices to prove the existence of the limit (9.6) (and similarly for t → −∞) on functions φ ∈ L2 (R3 ) ∩ L1 (R3 ) (the existence of the limit for φ ∈ L2 will then follow by approximating φ by elements of the dense subspace L2 ∩ L1 ). For ′ t ≥ t′ , we write the vector-function Ω t φ−Ω t φ as the integral of its derivative: Z t ′ d s Ω φds. Ωtφ − Ωt φ = t′ ds Using the relation we find

d iHs ds e

= iHeiHs , and similarly for e−iH0 s (see Chapter 2),

d s Ω = iHeiHs e−iH0 s + eiHs (−iH0 )e−iH0 s = ieiHs V e−iH0 s , ds

122

9 Scattering States

asR H − H0 =R V . Since keiHs k = 1, and using the Minkowski inequality k φ(s)dsk ≤ kφ(s)kds (see eg. [106]), we have ′

kΩ t φ − Ω t φk ≤

Z

t′

t

kV e−iH0 s φkds.

(9.13)

We estimate the integrand as follows: Z
−iH0 s 2 |V (x) e−iH0 s φ (x)|2 dx kV e φk = R3 Z
≤ sup | e−iH0 s φ (x′ )|2 |V (x)|2 dx, x′ ∈R3

R3

yielding

kV e−iH0 s φk ≤ kV kL2 sup |(e−iH0 s φ)(x′ )|. x′

Finally, recalling the bound (see (2.24))   2π~s −3/2 Z  −iH0 s/~ φ (x) ≤ |φ(x)|dx e m R3

(9.14)

from Section 2.4, we obtain

kV e−iH0 s φk ≤ (const)s−3/2 kV kL2 kφkL1 and so

Z

∞ 1

kV e−iH0 s φkds ≤ (const)kV kL2 kφkL1 .

This shows that the right hand side in (9.13) vanishes as t′ , t → ∞. In other words, for any sequence tj → ∞, {Ω tj φ} is a Cauchy sequence, and so {Ω t φ} converges as t → ∞. Convergence for t → −∞ is proved in the same way.  Finally, as promised, we prove asymptotic completeness in a special case. Theorem 9.3 Assume V ∈ L∞ ∩ L1 with kV kL∞ and kV kL1 sufficiently small. Then RanΩ± = H. Consequently, Hb is empty, and asymptotic completeness holds. Proof. We begin by proving that for any ψ ∈ L1 ∩ L2 , e−itH ψ has some decay in time, as measured in the k · kL2 +L∞ norm, which is defined by kf kL2+L∞ := inf (kgkL2 + khkL∞ ). f =g+h

In fact, we claim ke−itH ψkL2 +L∞ ≤ (const)(1 + |t|)−3/2 .

(9.15)

9.3 Wave Operators

123

To show this, set M (t) := sup0≤s≤t (1 + |s|)3/2 ke−isH ψkL2 +L∞ . Writing, as above, Z s d iτ H0 −iτ H eisH0 e−isH ψ − ψ = e e ψdτ 0 dτ and applying e−isH0 , we arrive at the Duhamel formula Z s −isH −isH0 e ψ=e ψ−i e−i(s−τ )H0 V e−isH ψds. 0

We estimate as follows: Z s−1 ke−isH ψkL2 +L∞ ≤ ke−isH0 ψkL2 +L∞ + ke−i(s−τ )H0 V e−iτ H ψkL∞ dτ 0 Z s −i(s−τ )H0 + ke V e−iτ H ψkL2 dτ. s−1

Using (9.14) as above, together with H¨ older’s inequality, we find  ke−isH ψkL2 +L∞ ≤ (const) (1 + |s|)−3/2 kψkL1 ∩L2 Z s−1 |s − τ |−3/2 kV kL1 ∩L2 ke−iτ H ψkL2 +L∞ dτ + Z0 s  + kV kL2 ∩L∞ ke−iτ H ψkL2 +L∞ dτ s−1

≤(const)[(1 + |s|)−3/2 kψkL1 ∩L2 + kV kL1 ∩L∞ M (s)I(s)],

R s−1 Rs where I(s) := 0 |s − τ |−3/2 (1 + |τ |)−3/2 dτ + s−1 (1 + |τ |)−3/2 dτ , and so multiplying through by (1+|s|)3/2 , and taking supremum over s ≤ t and using sup0≤s≤t I(s) ≤ (const), we find M (t) ≤ (const) [kψkL1 ∩L2 + kV kL1 ∩L∞ M (t)] .

(9.16)

Thus if kV kL1 ∩L∞ is sufficiently small (so that (const)kV kL1 ∩L∞ ≤ 1/2, say), (9.16) implies M (t) ≤ (const) for all t, and so (9.15) holds. Now denote W (t) := eiH0 t e−iHt , and, as above, write ′

W (t) − W (t ) = i

Z

t

eiH0 s V e−iHs ds.

t′

Using H¨ older again, for ψ ∈ L1 ∩ L2 we find Z t k(W (t) − W (t′ ))ψkL2 ≤ kV kL2 ∩L∞ ke−iHs ψkL2 +L∞ ds t′ Z t ≤ (const) (1 + |s|)−3/2 ds → 0 t′

124

9 Scattering States

as t > t′ → ∞. Hence {W (t)ψ} is a Cauchy sequence in L2 , and therefore limt→∞ W (t)ψ exists for all ψ ∈ L1 ∩ L2 . Since the operator W (t) is bounded uniformly in t, this limit exists for all ψ ∈ L2 . Denote W± ψ := limt→±∞ W (t)ψ. We want to show that Ω± W± = 1, ∗ W± Ω± = 1 and W± = Ω± . Indeed, since Ω± = s − lim eiHt e−iH0 t and W± = iH0 t −iHt s − lim e e we have that Ω± W± = s − lim eiHt e−iH0 t eiH0 t e−iHt = 1 and similarly for the other relations. The relation Ω± W± = 1 implies that RanΩ± = H, i.e., Ω± is unitary. This implies the asymptotic completeness.

9.4 Appendix: The Potential Step and Square Well Potential step. In the one-dimensional case one can say much more about scattering process. In particular, one can introduce very useful reflection and transmission coefficients as illustrated in the example of the one-dimensional potential  V0 x > 0 V (x) = 0 x≤0 with V0 > 0. Consider the “eigenvalue” problem −

~2 ′′ ψ + V ψ = Eψ. 2m

(9.17)

We put the term eigenvalue in quotation marks because we will allow solutions, ψ, which are not L2 -functions. Solving this eigenvalue problem separately in the two different regions gives us a general solution of the form ( ~2 k 2 Aeik0 x + Be−ik0 x ( 2m0 = E) x < 0 . ψ= ~2 k 2 Ceik1 x + De−ik1 x ( 2m1 = E − V0 ) x > 0 There are no bound states (L2 solutions), but we can say something about the scattering states. p Suppose 0 < E < V0 , and take k1 = iK where K = (2m/~2 )(V0 − E) > 0. Then for a bounded solution, we require D = 0. Imposing the condition that ψ be continuously differentiable at 0, that is ψ|0− = ψ|0+ ,

∂ψ/∂x|0− = ∂ψ/∂x|0+ ,

leads to the equations A + B = C,

ik0 (A − B) = −KC.

After some manipulation, we find that B k0 − iK = , A k0 + iK

9.4 Appendix: The Potential Step and Square Well

125

or making the dependence on the energy E explicit, √ √ E − i V0 − E B . =√ √ A E + i V0 − E Similarly, if E > V0 , we obtain √ √ B E − E − V0 = √ , √ A E + E − V0 and

2 √ p B 1 R(E) := = 2 [2E − V0 − 2 E E − V0 ]2 A V0

(R(E) is called the reflection coefficient). In particular, if R(E) ≈ 1 − 4

r

E−V0 V0

≪ 1, then

E − V0 V0

and almost all of the wave is reflected. This is in spite of the fact that the energy of the particle lies above the barrier. In classical mechanics, the particle would pass over the barrier. The square well. We consider the “eigenvalue” problem, Hψ = Eψ, for the square well potential defined in Section 7.1, and for positive energies E > 0. The “eigenvalue” equation is given by the same expression (9.17), but we do not expect this equation to have L2 solutions, i.e. bound states. So we look for bounded solutions which converge to plane waves as x → ±∞. Such solutions were called scattering states above. Consider the situation where a plane wave ψinc (x) = Aeikx is incoming from the left. Then on the left of the well, ψ is a superposition of incoming and reflected plane waves: ψ = Aeikx + AB(E)e−ikx

x < −a/2

while on the right of the well, ψ is an outgoing (transmitted) plane wave: ψ = AC(E)eik(x−a)

x > a/2.

Problem 9.4 Show that ¯ −i C(E) = [cos(ka) where k¯ =

r

2m(E + V0 ) ~2

k¯2 + k 2 ¯ −1 sin(ka)] ¯ 2kk

and

k=

r

2mE . ~2

126

9 Scattering States

This implies  T (E) := |C(E)|2 = 1 +

−1 V02 ¯ sin2 (ak) 4E(E + V0 )

(T (E) is the transmission coefficient) which is sketched in Fig. 9.2.

1

0

Fig. 9.2. Transmission coefficient. ¯ = 0, i.e. We see that at the energies satisfying sin(ka) E = −V0 +

n2 π 2 ~2 > 0, 2ma2

n = 1, 2, . . .

T (E) has maxima (T (E) = 1) which are called resonances. The corresponding values of E are the resonance energies. We remark that for large n these are approximately equal to the energy levels of the infinite well of the same width.

10 Existence of Atoms and Molecules

In this chapter we prove existence of stationary, well localized and stable states of atoms and, in a certain approximation, molecules. These are the lowest energy states, and their existence means that our quantum systems exist as well-localized objects, and do not disintegrate into fragments under sufficiently small perturbations. General many-body systems are considered in Chapter 13.

10.1 Essential Spectra of Atoms and Molecules Recall from Section 4.5 that a molecule with N electrons of mass m and charge −e, and M nuclei of masses mj and charges Zj e, j = 1, . . . , M, is described by the Schr¨odinger operator Hmol

M N X X ~2 ~2 ∆y + V (x, y) ∆xj − =− 2m 2mj j 1 1

(10.1)

acting on L2sym (R3(N +M) ). Here L2sym (R3(N +M) ) is a symmetry subspace of L2 (R3(N +M) ) reflecting the fact that electrons and some of the nuclei are identical particles, x = (x1 , . . . , xN ) and y = (y1 , . . . yM ) are the electron and nucleus coordinates, respectively, and V (x, y) =

X e2 Zj 1X 1 X e2 Zi Zj e2 − + , 2 |xi − xj | |xi − yj | 2 |yi − yj | i,j i6=j

(10.2)

i6=j

the sum of Coulomb interaction potentials between the electrons (the first term on the r.h.s.), between the electrons and the nuclei (the second term), and between the nuclei (the third term). (See Section 4.5.) For a neutral molecule, we have M X Zj = N. j=1

© Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_10

128

10 Existence of Atoms and Molecules

If M = 1, the resulting system is called an atom, or Z-atom (Z = Z1 ). In the case of atoms, the last term in (10.2) is absent. Since nuclei are much heavier than electrons, in the leading approximation one can suppose that the nuclei are frozen at their positions. (For a more precise statement and discussion of this point see Section 12.1 below.) One then considers, instead of (10.1), the Schr¨odinger operator BO HN (y) = −

N X ~2 ∆x + V (x, y) 2m j 1

(10.3)

acting in L2sym (R3N ), the positions y ∈ R3M of the nuclei appearing as parameters. This is called the Born-Oppenheimer approximation. It plays a fundamental role in quantum chemistry, where most computations are done with BO the operator HN (y). We discuss the justification of this approximation in Section 12.2. BO Theorem 10.1 (Kato theorem) The operators Hmol and HN (y) are selfadjoint and bounded below. BO Proof. The fact that the operators Hmol and HN (y) are bounded below was shown in Section 5.3. Next, to fix ideas we prove the self-adjointness for Hmol BO only. The self-adjointness for HN (y) is proven similarly. As in Problem 2.8 we have 1 1 ψk ≤ ak ∆x ψk + bkψk, k |xi − xj | 2m i (10.4) 1 1 k ψk ≤ ak ∆x ψk + bkψk, |xi − yj | 2m i

with a > 0 arbitrary and b depending on a and for all ψ ∈ D(H0 ), where PM 1 PN 1 ∆xj − 1 2m ∆yj . (Prove this. Hint: Pick the variable xi H0 := − 1 2m j and follow the instructions in Problem 2.8, then integrate over the rest of the variables.) Let V be given in (10.2). Then the last two estimates imply kV ψk ≤ akH0 ψk + bkψk

(10.5)

with a > 0 arbitrary and b depending on a. By Theorem 2.9, this implies that the operator Hmol is self-adjoint.  To simplify the exposition, in what follows we consider only atoms and molecules with fixed (infinitely heavy) nuclei and denote the corresponding BO quantum Hamiltonians by HN . Thus HN is equal to HN (y), where in the case of atoms (M = 1) y ≡ y1 is set to 0. For atoms the assumption of fixed nuclei is a minor one and can be easily removed; for molecules, it is a crucial one. A rigorous existence theory for molecules, apart from hydrogen, still does not exist. We begin with the HVZ theorem describing the essential spectra of atoms and Born-Oppenheimer (BO) molecules.

10.2 Bound States of Atoms and BO Molecules

129

Theorem 10.2 (HVZ theorem for atoms and BO molecules) σess (HN ) = [ΣN , ∞), where ΣN = inf σ(HN −1 ). This theorem is a special case of the general HVZ theorem proven in Chapter 13, so we omit its proof. The energy ΣN = inf σ(HN −1 ) is called the ionization threshold. To obtain σess (HN ) we take one of the electrons to infinity and let it move freely there. The rest of the atom is placed in the ground state, so that the energy of the atom is Energy = ΣN +

1 |k|2 ∀k 2m

(10.6)

where k is the momentum of the electron which is placed at infinity. Varying |k| from 0 to ∞ we see that (10.6) ranges over [ΣN , ∞). For molecules with mobile nuclei, the bottom of the essential spectrum is likely below its ionization threshold. However, the theorem above would imply the existence of bound states of Hmol , with m/ min mj sufficiently small, likely smaller then it is for real molecules (see Section 12.1).

10.2 Bound States of Atoms and BO Molecules Are atoms or BO molecules stable? To answer this question we have to determine whether HN has at least one bound state. Theorem 10.3 For N < Z + 1, HN has infinite number of eigenvalues below (i) its ionization threshold ΣN = inf σ(HN −1 ). Bound states, ΨN (x1 , x2 , ..., xN ), (i) of HN , with energies EN < ΣN , satisfy the exponential bound Z q (i) (i) (10.7) |ΨN (x)|2 e2α|x| dx < ∞, ∀α < ΣN − EN . Proof. To simplify the exposition, we assume the ground states, if they exist, are unique. In the case without statistics, i.e. on the entire L2 (R3N ), this is not hard to prove (see [247]). In the case with statistics, this is not known and is, probably, not true. However, a generalization of the proof below to the case of multiple ground states is straightforward. We prove this theorem by induction in N . (This is strictly speaking not necessary, but is convenient.) We have shown already that it holds for the hydrogen atom, i.e. N = 1. Assume now it holds for k ≤ N − 1 and prove it for k = N . Let ΨN −1 (x1 , x2 , ..., xN −1 ) be the normalized ground state of HN −1 with the ground state energy EN −1 < EN −2 . First, for simplicity, we ignore the statistics and consider HN on the entire space L2 (R3N ). We use the variational principle with the test function φ = ΨN −1 (x1 , ..., xN −1 )f (xN ), where f ∈ L2 (R3 ), kf k = 1. Using that HN = HN −1 −

(10.8) ~2 2m ∆xN

+ IN , with

130

10 Existence of Atoms and Molecules

IN (x) :=

N −1 X i=1

e2 Z e2 − , |xi − xN | |xN |

(10.9)

and HN −1 ΨN −1 = EN −1 ΨN −1 , we obtain HN φ = (EN −1 + IN )φ + ΨN −1 (−

~2 ∆x )f. 2m N

(10.10)

This implies that ~2 ∆f i + hφ, IN φi. (10.11) 2m R By the exponential bound (10.7) for ΨN −1 , i.e. |ΨN −1 (x)|2 e2α|x| dx < ∞, for any α < EN −2 − EN −1 , it follows that hφ, HN φi = EN −1 + hf, −

|hΨN −1 , (IN (x) − IN (xN ))ΨN −1 iL2 (RN −1 ) | ≤ (const)|xN |−2 , where IN (xN ) := IN (x)|xi =0 ∀i . Observe that IN (xN ) = −q/|xN |, where q := (Z − N + 1)e2 , which implies hΨN −1 , IN (x)ΨN −1 iL2 (RN −1 ) ≤ −

q const , + |xN | |xN |2

(10.12)

which, in turn, together with (10.11) and kφk = 1, gives     q const ~2 f . ∆x − + hφ, (HN − ΣN )φi ≤ f, − 2m N |xN | |xN |2 L2 (R3 ) Let f ∈ C0∞ (R3 ) and satisfy kf k = 1 and supp(f ) ⊂ {xN ∈ R3 | 1 < |xN | < 2}. Then the functions fn (xN ) = n−3/2 f (n−1 xN ),

n = 1, 2, 4, 8, . . . ,

are orthonormal, and have disjoint supports. Thus the corresponding trial states φn (x) = ΨN −1 (x1 , ..., xN −1 )fn (xN ) satisfy hφn , HN φm i = 0 for n 6= m, and q 1 hφn , (HN − ΣN )φn i ≤ −c1 + c2 2 < 0, n n R 1 |f (x)|2 dx and some positive constant c2 , if n is sufficiently large. for c1 = |x| Using this one can show that HN possesses infinitely many discrete eigenvalues below the threshold ΣN . Taking into account statistics. Now we show how to modify the proof above to the case of spinless fermions, i.e. for the state space L2sym (R3N ) :=

10.2 Bound States of Atoms and BO Molecules

131

1 2 3 ∧N 1 L (R ). (The fact that electrons have spin 2 can be accommodated similarly.) As before, let SN be the group of permutations of N indices and define the anti-symmetrization projection

PNA Ψ =

1 X (−1)#(π) Ψ (xπ(1) , ..., xπ(N ) ) N! π∈SN

where, recall, #(π) is the number of transpositions making up the permutation π ((−1)#(π) is the parity of π ∈ SN ). We replace the test function (10.8) by P (j) the function φ = PNA (ΨN −1 ⊗ f )/kPNA (ΨN −1 ⊗ f )k. This gives φ := N j=1 φ where, with the normalization constant c,

Let Ij :=

P

denote by i 6= j,

φ(j) := ±cΨN −1 (x1 , ..., xj−1 , xj+1 , ..., xN )f (xj ).

e2 i:i6=j |xi −xj | (i) φα and φα

−

e2 Z |xj | .

(10.13)

3

We take fα (x) = α 2 f (αx) with kf k = 1 and

the corresponding test functions. Then we have, for (j) |hφ(i) α , φα i| .

α3 , N

(10.14)

and, similarly,

~2 α3 ∆xj + Ij )φ(j) . (10.15) α i| . 2m N For α → 0, (10.14) implies that the normalization constant c in (10.13) is c ≍ O( √1N ). Then the equation (10.15) implies |hφ(i) α , (−

hφα , HN φα i ≤ EN −1 + hfα , (−

~2 q ∆x − )fα i + O(α3 N ). 2m |x| 2

q ~ f i/hf, − 2m ∆f i. This Hence hφα , HN φα i < EN −1 , if α ≪ √1N and α < hf, |x| proves the existence of the ground state energy for HN ∀N . Finally, the bound (10.7) is a special case of the general exponential bound for many-body bound states given in Theorem 13.7, Section 13.6 (see also Theorem 8.6 of Section 8.2). 

Problem 10.4 Go over the proof above and, where necessary, fill in the details. A more refined estimate of hφ, IN φi (replacing (10.12)). Consider the one-electron density Z ρN (y) := |ΨN (y, x2 , ..., xN )|2 dx2 ...dxN . (10.16) Assume that for any N , ρN (y) is spherically symmetric. Then we have a bound considerably stronger than (10.12):

132

10 Existence of Atoms and Molecules

hφ, IN φi ≤ −

(Z − N + 1)e2 . |xN |

Indeed, we write hφ, IN φi = hf, W f i where Z W (xN ) = IN |ΨN −1 (x1 , ..., xN −1 )|2 dN −1 x.

(10.17)

(10.18)

We compute W (xN ) = (N − 1)e

2

Z

ρN −1 (y) e2 Z dy − , |x − xN | |xN |

(10.19)

R where ρN −1 (x1 ) = |ΨN −1 |2 dx2 ...dxN −1 . It is not hard to show that ρN −1 is spherically symmetric. Hence we have by Newton’s theorem Z Z Z 1 1 ρN −1 (y)dy = ρN −1 (y)dy. ρN −1 (y)dy + |xN | |y|≤|xN | |y|≥|xN | |y| R3 |y − xN | (10.20) R Using that ρN −1 (y)dy = 1, this can be estimated as Z Z 1 1 ρN −1 (y)dy ≤ ρN −1 (y)dy = . (10.21) |y − x | |x | |x 3 3 N N N| R R R (Moreover, ρN −1 (x) = O(e−δ|x| ) =⇒ |y|≤|xN | ρN −1 (y)dy = 1 + O(e−δ|xN | ).) The equations (10.18) - (10.21) imply (10.17). Remark 10.5 The accumulation of eigenvalues at ΣN can be studied by similar arguments. Consider trial wave functions, φnm , constructed as above, with fnm (xN ) a hydrogen atom eigenfunction of energy −n−2 (in suitable units). Then one can show k(HN − En )φnm k ≤ (const) n−α for some α > 3, where En = Σ − n−2 . This implies that HN has groups of eigenvalues close to En compared to the spacing En+1 − En as n → ∞ (Rydberg states). This analysis can be easily extended to take into account the particle statistics. Problem 10.6 Show that the Schr¨odinger operator describing the Helium atom with infinitely heavy nucleus has at least one discrete eigenvalue (isolated eigenvalue of finite multiplicity).

10.3 Open Problems Though atoms and molecules have been studied since the advent of quantum mechanics, there are many open problems in their rigorous theory. We mention here three of these problems:

10.3 Open Problems

133

1. Existence of molecules. 2. Non-existence of negative atomic ions with more than a few (two?) extra electrons. 3. Uniqueness or non-uniqueness of the ground states. It is easy to prove the uniqueness of the ground states on the entire space, say L2 (R3N ), however there are no techniques available to deal with the fermionic subspace (see Section 4.5).

11 Perturbation Theory: Feshbach-Schur Method

As we have seen, many basic questions of quantum dynamics can be reduced to finding and characterizing the spectrum of the appropriate Schr¨odinger operator. Though this task, known as spectral analysis, is much simpler than the task of analyzing the dynamics directly, it is far from trivial. The problem can be greatly simplified if the Schr¨odinger operator H under consideration is very close an operator H0 whose spectrum we already know. In other words, the operator H is of the form H = Hκ , where Hκ = H0 + κW,

(11.1)

H0 is an operator at least part of whose spectrum is well understood, κ is a small parameter called the coupling constant, and W is an operator, called the perturbation. (All the operators here are assumed to be self-adjoint.) If the operator W is bounded relative to H0 , say in the sense that D(H0 ) ⊂ D(W ) and kW uk ≤ ckH0 uk + c′ kuk for some c, c′ ≥ 0, for all u ∈ D(H0 ),

(11.2)

then the “standard” perturbation theory applies, and allows us to find or estimate eigenvalues of Hκ (see [247, 176, 162]). In this chapter, we describe a powerful technique which allows us to estimate eigenvalues of Hκ even in cases where W is not bounded in terms of H0 . This is important in applications. We consider several examples of applications of this method, one of which is the hydrogen atom in a weak constant magnetic field B. Combining the expressions derived in Sections 7.3 and 7.5, we see that the Schr¨odinger operator for such an atom is HB :=

e2 1 (p − eA)2 − , 2m |x|

(11.3)

where e < 0 denotes the electron charge, and we have kept units in which the speed of light is c = 1. Recall that the vector potential A is related to © Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_11

136

11 Perturbation Theory: Feshbach-Schur Method

the magnetic field B by B = ∇ × A. In the notation of Section 7.5, HB = H(A, −e2 /|x|). Expanding the square in (11.3), we find HB = H0 + WB

(11.4)

1 2 where H0 = 2m p − e2 /|x| is the Schr¨odinger operator of the hydrogen atom (see Section 7.3), and

WB =

|e| e2 A·p+ |A|2 . m 2m

(11.5)

Here we have assumed the gauge condition ∇ · A ≡ 0. The small parameter κ here is the strength, |B|, of the magnetic field. Now we know from Section 7.3 that the operator H0 has a series of eigenvalues  4 1 me En := − , n = 1, 2, . . . 2 2~ n2 as well as continuous spectrum in [0, ∞). One would expect that for weak magnetic fields B, the operator HB has eigenvalues EB,n close to En , at least for the few smallest En ’s. This is not so obvious as it might seem at first glance, since the perturbation WB is not bounded relative to H0 – i.e., (11.2) does not hold for any c and c′ , small or large. (The perturbation (11.5) grows in x: take for example A := B2 (−x2 , x1 , 0).) The method we present below does show rigorously that such eigenvalues exist, though we will make only formal computations of EB,n . Two other examples, one we present below and a third, in the next chapter, display different physical phenomena, which conceptually and technically are considerably more complicated.

11.1 The Feshbach-Schur Method Before returning to our perturbation problem, we state a general result used below, which allows us to reduce a perturbation problem on a large space to one on a small space. (For some motivating discussion see Appendix 12.7.) Let P and P¯ be orthogonal projections (i.e. P ,P¯ are self-adjoint, P 2 = P , and P¯ 2 = P¯ ) on a separable Hilbert space X, satisfying P + P¯ = 1. Let H be a self-adjoint operator on X. We assume that RanP ⊂ D(H), that HP¯ := P¯ H P¯ ↾RanP¯ is invertible, and kRP¯ k < ∞ ,

kP HRP¯ k < ∞

and

kRP¯ HP k < ∞ ,

(11.6)

¯ where RP¯ = P¯ HP−1 ¯ P . We define the operator FP (H) := P (H − HRP¯ H)P ↾Ran P .

(11.7)

We call FP the Feshbach-Schur map. The key result for us is the following:

11.1 The Feshbach-Schur Method

137

Theorem 11.1 Assume (11.6) hold. Then the operators H and FP (H) are isospectral at 0, in the sense that (a) 0 ∈ σ(H) ⇐⇒ 0 ∈ σ(FP (H)), (b) Hψ = 0 ⇐⇒ FP (H)φ = 0 where ψ and φ are related by φ = P ψ and ψ = Qφ, with the (bounded) operator Q given by Q = Q(H) := P − RP¯ HP. (11.8) A proof of this theorem is given in an appendix, Section 11.4. Moreover, under the conditions above, H is self-adjoint

=⇒

FP (H) is self-adjoint.

(11.9)

The latter property is an example of transmission properties of the FeshbachSchur map. Now we return to our general perturbation problem. Thus we consider a family, Hκ , κ ≥ 0, of self-adjoint operators of the form (11.1). Assume the operator H0 has an isolated eigenvalue λ0 of finite multiplicity. Let P be the orthogonal projection onto the eigenspace Null(H0 − λ0 ) spanned by all the eigenfunctions of H0 corresponding to the eigenvalue λ0 , and let P¯ := 1 − P . We apply Theorem 11.1 to the family of operators H = Hκ − λ for some λ close to λ0 , with the projections P and P¯ defined as above. Observe that λ ∈ σd (Hκ ) ⇐⇒ 0 ∈ σd (Hκ − λ),

(11.10)

and that FP (Hκ − λ), if it is well-defined, is a family of m × m matrices, where m is the multiplicity of the eigenvalue λ0 of the operator H0 (indeed, P is a rank-m projection – i.e., dim RanP = m). Thus the perturbation problem reduces the problem of finding an eigenvalue (and an eigenfunction) of an (infinite-dimensional) operator H, to the problem of finding the values λ for which 0 ∈ σ(FP (Hκ − λ)). Such values are called singular values of the family FP (Hκ − λ). We need to discuss the problem of finding singular values of the family FP (Hκ − λ) of matrices, but first we address the issue of defining FP (Hκ − λ). Write HP¯ = H0P¯ + κWP¯ , where we are using the notation AQ := QAQ ↾RanQ . Assume the operators W and H0 satisfy the conditions WP P¯ Hκ P¯ − λ0

and

P W are bounded,

is invertible on

RanP¯

for

|κ| < κ0 .

(11.11) (11.12)

(If H0 and W are self-adjoint and if W P is bounded, then so is P W .) Since P¯ Hκ P = κP¯ W P , the condition (11.6) is satisfied for Hκ −λ, with λ sufficiently

138

11 Perturbation Theory: Feshbach-Schur Method

close to λ0 , and therefore FP (Hκ − λ) is well-defined. (In this case, RP¯ = P¯ (HP¯ − λ)−1 P¯ , HP¯ := P¯ Hκ P¯ ↾RanP¯ .) The operator H0P¯ := P¯ H0 P¯ ↾RanP¯ has no spectrum near λ0 . The same is true, as can be readily verified, for the operator HP¯ = H0P¯ + κP¯ W P¯ for |κ| sufficiently small, if the operator W is bounded relative to H0 in the sense (11.2). Otherwise, justifying the assumption (11.12) is a delicate matter, and is done on a case-by-case basis. For the examples of the hydrogen atom in a weak constant magnetic field, and of a particle system in a weak time-periodic (electric) field, we address this question below. Under the assumptions above, conditions (11.6) hold. Consequently, the operator FP (Hκ − λ) is well-defined for λ close to λ0 . Now let’s compute the operator FP (Hκ − λ). We write it as the sum of three terms FP (Hκ − λ) = HP − κ2 U (λ) − λ (11.13) where HP := P Hκ P ↾RanP and ¯ U (λ) := P W R(λ)W P ↾RanP ,

with

¯ R(λ) = P¯ (P¯ Hκ P¯ − λ)−1 P¯ . (11.14)

The matrix family U (λ) is called the level shift operator. Since P H0 = H0 P = λ0 P and P P¯ = 0, we have P H0 P = λ0 P,

P H P¯ = κP W P¯ ,

P¯ HP = κP¯ W P.

These relations yield FP (Hκ − λ) = κWP − κ2 U (λ) + λ0 − λ.

(11.15)

Theorem 11.1 and equations (11.10) and (11.15) imply the relation λ ∈ σd (Hκ ) ⇐⇒ λ − λ0 ∈ σd (κWP − κ2 U (λ)). Note that the operator on the right itself depends on the spectral parameter λ. Expanding the resolvent ¯ R(λ) = P¯ (H0P¯ − λ0 + κWP¯ + λ0 − λ)−1 P¯ in a Neumann series in κWP + λ0 − λ we obtain ¯ ¯ 0 + O(|κ| + |λ − λ0 |) R(λ) =R ¯ 0 = P¯ (H0P¯ − λ0 )−1 P¯ . Consequently, where R U (λ) = U0 + O(|κ| + |λ − λ0 |) ¯ 0 W P ↾RanP . Therefore any eigenvalue λκ of Hκ sufficiently where U0 := P W R close to λ0 has the form λκ = λ0 + κµk + O(|κ|3 )

11.1 The Feshbach-Schur Method

139

where µκ ∈ σd (WP − κU0 ). Similarly, we can obtain expressions for λκ to any order in κ. Observe that if λ0 is a simple eigenvalue (i.e. of multiplicity one) of H0 , with normalized eigenfunction ψ0 , then WP and U0 are just the real ¯ 0 W ψ0 i, and we have numbers hψ0 , W ψ0 i and hW ψ0 , R ¯ 0 W ψ0 i + O(|κ|3 ). λκ = λ0 + κhψ0 , W ψ0 i + κ2 hW ψ0 , R

(11.16)

Our next step is to examine the structure of the quadratic term in more detail. Suppose that the operator H0 is self-adjoint and has isolated eigenvalues λj , j = 0, 1, . . ., counting multiplicities, with λj < inf σess (H0 ). Let {ψj } be corresponding normalized eigenfunctions, and let Pψj be the rank-one orthogonal projections onto these eigenfunctions: Pψj = |ψj ihψj |, Then we have

P¯ =

or X

Pψj f = hψj , f iψj .

Pψj + Pess

j6=0

P where Pess := 1− j Pψj is the projection onto the essential spectral subspace of H0 . Then we can write X ¯ 0 P¯ W ψ0 i = hP¯ W ψ0 , R |hψj , W ψ0 i|2 (λj − λ0 )−1 j6=0 (11.17) + hPess W ψ0 , (H0,ess − λ0 )−1 Pess W ψ0 i where H0,ess := Pess H0 Pess is the essential spectral part of the operator H0 . Now we can interpret the coefficient hψ0 , W ψ0 i in (11.16) as due to a direct interaction of the bound state ψ0 with itself, the term |hψj , W ψ0 i|2 (λj −λ0 )−1 as due to the interaction of ψ0 with itself via the bound state ψj , and the last term in (11.17) as due to the interaction of ψ0 with itself via the essential spectral states of H0 . In conclusion of this section we sketch a proof of an extension of the central theorem of perturbation theory Theorem 11.2 Assume (11.11) hold, the operator H0 has an isolated eigenvalue, λ0 , of a finite multiplicity, m, and there is κ0 > 0 s.t. (11.12) holds. (i) Then for |κ| sufficiently small the operator Hκ has eigenvalues λκ near λ0 of the total multiplicity equal to m. Moreover, if Hκ is self-adjoint, then the (i) eigenvalues λκ have the expansions of the form (11.16) - (11.17). Sketch of proof. For |λ − λ0 | < k(P¯ Hκ P¯ − λ0 )−1 k−1 , the operator P¯ Hκ P¯ − λ is invertible on RanP¯ and therefore, due to (11.11), FP (λ) exist. Moreover, (i) the eigenvalues, λκ , of Hκ near λ0 , if they exist, are also eigenvalues of

140

11 Perturbation Theory: Feshbach-Schur Method (i)

FP (λκ ). Let νj (λ, κ) be the eigenvalues of FP (λ) (remember that, though we do not display this, the latter operator depends on κ). To simplify the exposition, assume they are simple. The general case is treated similarly. Then (i) the eigenvalues λκ must solve the equation λ = νi (λ, κ). To show that this equation has a unique solution for κ sufficiently small and this solution is close to λ0 , we observe that, due to (11.15), νi (λ, κ) is a differentiable function, λ0 = νi (λ0 , 0), |νi (λ, κ) − λ0 | . κ and ∂λ νi (λ, κ) = O(κ). Therefore by the implicit function theorem, the equation λ = νi (λ, κ) (i) (i) has a unique solution, λκ , and this solution satisfies λκ = λ0 +O(κ). Proving the expansions of the form (11.16) - (11.17) follows the arguments leading to this expansion.  As was mentioned above, the condition (11.12) is satisfied if W is H0 − bounded (show this). It also holds if H0 is self-adjoint and W is non-negative and either λ0 is the ground state energy or we take for P the orthogonal projection on the eigenspace corresponding to the eigenvalues ≤ λ0 . In the next chapter we consider the celebrated Born-Oppenheimer approximation, for which no condition of the theorem above holds, but which still can be handled by the present technique. This little discussion indicates the power of the method.

11.2 The Zeeman Effect We can apply the theory developed above to compute the energy levels of the Schr¨odinger operator of a hydrogen atom in a weak homogeneous magnetic field (the Zeeman effect). Recall that for any magnetic field this operator is given by the expression (11.4)- (11.5). For a constant magnetic field B, we can choose the vector potential to be A(x) = 21 B × x (see Equation (7.26)). Then the perturbation WB can be re-written as ˆ · L + m κ2 (B ˆ × x)2 WB = κB 2 where L := x × p is the operator of angular momentum (see Section (7.2)), |e| ˆ := |B|−1 B. Thus κ is a small parameter in our problem. |B|, and B κ = 2m Let λ0 = En be the n-th energy level of the Hydrogen atom Schr¨odinger operator H0 . One can show that if λ is close to λ0 , then the operator HB P¯ has no spectrum near λ0 , and therefore the Feshbach-Schur operator FP (HB − λ) is well-defined. We will not do this here, but mention only that this follows from the inequality (H0P¯ − λ0 )−1 WB (H0P¯ − λ0 )−1 ≥ −cκ on RanP¯ , for some uniform c > 0 and for κ sufficiently small. The idea of a proof of the latter inequality is that (H0P¯ − λ0 )−1 controls p entering WB through L and x 2 ˆ 2 entering WB through L is controlled by the positive term m 2 κ (B × x) .

11.2 The Zeeman Effect

141

We compute formally the perturbation expansion for the eigenvalue λκ = En,B of the operator HB near the eigenvalue λ0 = En of the operator H0 . This computation is slightly more complicated than the corresponding computation in the abstract case considered above, since our perturbation is of the form WB = κW1 + κ2 W2 ˆ · L and W2 := m (B ˆ × x)2 , rather than of the form κW where W1 := B 2 considered above. First, we take the x3 -axis to be in the direction of the magnetic field vector 2 2 2 2 B, so that W1 = L3 (see Section 7.2) and W2 = m 2 r⊥ , where r⊥ = x1 + x2 . Second, since the operator H0 is invariant under rotations, we know that L commutes with P : [L, P ] = 0, and therefore P WB P¯ = κ2 P W2 P¯ ,

P¯ WB P = κ2 P¯ W2 P.

and

Third, by the results of Section 7.3, the projection P on the eigenspace corresponding to the eigenvalue En is given by P =

n−1 X

l X

Pnlk

l=0 k=−l

where Pnlk = |ψnlk ihψnlk |, and ψnlk (x) = Rnl (r)Ylk (θ, φ) is the normalized eigenfunction derived in Section 7.3. Since L3 ψnlk = ~kψnlk , this gives P W1 P =

n−1 X

l X

~kPnlk

(11.18)

l=0 k=−l

and P W2 P =

n−1 X

min(l,l′ )

X

l,l′ =0 k=− min(l,l′ )

|ψnlk ihψnl′ k |hψnlk ,

m 2 r ψnl′ k i. 2 ⊥

(11.19)

Proceeding as in the abstract case, we find that En,B − En is an eigenvalue of ¯ 0 W1 P ) + O(|κ|3 ). κP W1 P + κ2 (P W2 P + −P W1 R

(11.20)

Thus for n 6= 1, we have in the leading order En,B = En + µ|B|k + O(|B|2 ), where µ := |e|~/2m is the Bohr magneton, for k = −l, . . . , l and l = 0, . . . , n − 1. Thus the magnetic field lifts the degeneracy of the energy levels in the direction of the angular momentum. This is called the Zeeman effect.

142

11 Perturbation Theory: Feshbach-Schur Method

For the ground state energy En=1 , the term linear in κ vanishes and the expansion yields E1,B = E1 +

me2 |B|2 r¯2 + a(e|B|)4 + O(|B|5 ) 3

where r¯2 =

Z

|x|2 |ψ100 |2

and

a=

m2 2 2 hr ψ100 , P¯ (H0P¯ − E0 )−1 P¯ r⊥ ψ100 i. 4 ⊥

Here we have used spherical coordinates to simplify the first integral.

11.3 Time-Dependent Perturbations Our second example is an atom placed in a spatially localized but timeperiodic electric field. We write the total Schr¨odinger operator as in (11.1): Hκ = H0 + κW, where H0 is the Schr¨odinger operator of an atom, or, to fix ideas, a one-particle Schr¨odinger operator ~2 ∆ + V (x) H0 = − 2m with V (x) continuous and decaying to zero as |x| → ∞ as in Theorem 6.16, so that σess (H0 ) = [0, ∞). (Of course, for an atom, the potential V (x) has singularities, but the analysis below can be easily generalized to this case.) We also suppose H0 has discrete eigenvalues Ej < 0 with corresponding normalized eigenfunctions φj (x), j = 0, 1, . . .. The perturbation W = W (x, t) is assumed to be smooth, vanishing as |x| → ∞, and time-periodic with period T = 2π ω . Since the perturbation W depends on time, so does the operator Hκ = Hκ (t), and the spectrum of Hκ (at each moment of time) does not tell us much about the dynamics – that is, the solution of the time-dependent Schr¨odinger equation ∂ψ = Hκ (t)ψ. (11.21) i~ ∂t So we have to deal with this equation directly. The question we address is what happens to the bound states ψj (x, t) := φj (x)e−iEj t/~ of the unperturbed equation ∂ψ = H0 ψ i~ ∂t when the perturbation κW (x, t) is “switched on”? More precisely, are there time-(quasi) periodic solutions ψκ,j (x, t) of (11.21) which are L2 functions of x, and such that ψκ,j → ψj as κ → 0 in, say, the L2 (R3 )-norm? If so, then we

11.3 Time-Dependent Perturbations

143

would like to find an approximate expression for such solutions. If not, what are the descendants of ψj (x, t)? We fix j and consider the perturbation theory for the j-th state ψj (x, t). We look for a solution to (11.21) of the form ψκ,j (x, t) = φκ,j (x, t)e−iEκ,j t/~

(11.22)

where the function φκ,j is time-periodic with period T , and L2 in x, and with φκ,j → φj (in the L2 -sense) and Eκ,j → Ej as κ → 0. Plugging the expression (11.22) into (11.21), we find the equation for φκ,j : (Hκ (t) − i~

∂ − Eκ,j )φ = 0. ∂t

(11.23)

We look for solutions of this equation in the space L2 (Rn × ST ) where ST is the circle of circumference T – i.e., we assume that the function φ(x, t) is periodic in t with period T , and satisfies Z TZ |φ(x, t)|2 dxdt < ∞. 0

Rn

Thus we can treat Equation (11.23) as an eigenvalue equation for the operator Kκ := Hκ (t) − i~

∂ ∂t

on

L2 (Rn × ST ).

We call this operator the Bloch-Floquet Hamiltonian, since the general approach we describe here, of reducing the time-dependent problem (11.21) to an eigenvalue problem on a larger space, follows the outlines of the theory laid out in parallel by F. Bloch in solid state physics and by Floquet in mathematics. Thus our task is to find out whether the operator Kκ on L2 (Rn × ST ) has an eigenvalue Eκ,j close to the eigenvalue Ej of the operator H0 . Since κ is assumed to be small, we treat this as a perturbation problem: Kκ = K0 + κW. ∂ on We begin by examining the spectrum of the operator K0 := H0 − i~ ∂t 2 n L (R × ST ). Using the facts that

σess (H0 ) = [0, ∞)

and

σd (H0 ) = {Ej }N j=0

and, on L2 (ST ),     ∂ ∂ ∞ = σd i~ = {~ωn}n=−∞ , σ i~ ∂t ∂t where, recall, ω = 2π/T (see Section 7.1), and separation of variables, we find that

144

11 Perturbation Theory: Feshbach-Schur Method ∞ [

σess (K0 ) =

[~ωn, ∞)

n=−∞

and

σpp (K0 ) = {~ωn + Em | n ∈ Z, m = 0, 1, . . .}. Here σpp denotes the full set of eigenvalues (including non-isolated ones). Thus the essential spectrum of K0 fills the entire real axis R, and the eigenvalues of K0 (which are infinite in number) lie on top of the essential spectrum, or, as it is said, are embedded in the essential spectrum. The eigenfunctions corresponding to the eigenvalues Emn = Em + ~ωn are given by ψmn (x, t) := φm (x)e−iωnt . Note that the Ej ’s themselves are eigenvalues of K0 , with eigenfunctions ψj0 (x) = φj (x): K0 ψj0 = Ej ψj0 . Now we apply the Feshbach projection method to the operator K0 + κW in order to find its eigenvalues near Ej . Let us assume, for simplicity, that Ej is a simple eigenvalue of H0 (for example, j = 0 and E0 is the ground state energy), and that Em + ~ωn 6= Ej

∀ (m, n) 6= (j, 0)

(in other words, that there is no “accidental” degeneracy in the spectrum of K0 ). As the projection P we use the orthogonal rank-one projection onto the eigenspace of K0 , spanned by the eigenfunction ψj0 = φj , corresponding to the eigenvalue Ej , that is P = |ψj0 ihψj0 |. Then we will find from (11.16) the expansion for the desired eigenvalue Eκ,j : Eκ,j = Ej + κEj1 + κ2 Ej2 + O(κ3 )

(11.24)

where Ej1 := hhψj0 , W ψj0 ii and

¯ 0 P¯ W ψj0 ii. Ej2 := hhP¯ W ψj0 , R

¯ 0 = P¯ (P¯ K0 P¯ − Ej − i0)−1 P¯ , and hh·ii is Here, recall that P¯ = 1 − P and R used to denote the inner product in the space L2 (Rn × ST ): hhu, vii :=

Z

0

T

Z

u¯vdxdt. Rn

Notice that we inserted −i0 into the resolvent above. This was not there in our previous discussion of the Feshbach perturbation theory, and the reason for it will become apparent shortly.

11.3 Time-Dependent Perturbations

145

Let Pmn = Pψmn be the rank-one orthogonal projections onto the eigenfunctions ψmn of the operator K0 . Then X Pess := 1 − Pmn mn

defines the projection onto the essential spectral subspace of K0 . Thus X Pmn + Pess . P = Pj0 and P¯ = (m,n)6=(j,0)

By the definition of the inner product we have Ej1 = hφj , W0 φj i

RT

where W0 (x) := 0 W (x, t)dt, and the inner product on the right is the L2 (Rn ) inner product. Now we compute the third term on the right hand side of (11.24) – that is, the coefficient of κ2 . It can be written as (see (11.17)) Ej2 =

X

(m,n)6=(j,0)

|hhψmn , W ψj0 ii|2 ¯ 0 Pess W ψj0 ii. + hhPess W ψj0 , R Emn − Ej

By the definition of the ψmn ’s, we have Z T hhψmn , W ψj0 ii = hφm , W (t)φj ieiωnt dt = hφm , Wn φj i

(11.25)

(11.26)

0

where Wn is the n-th Fourier coefficient of W : Z T Wn := W (t)eiωnt dt. 0

This simplifies the expression of the first term on the right hand side of (11.25). Now we analyze the second term. By separation of variables, X H0 Pess = Pess ⊗ Pn n

H0 is the projection onto the essential spectral subspace of the operwhere Pess ator H0 , X H0 Pess := 1 − Pψm , m

and Pn is the projection onto the eigenfunction e−iωnt of the operator i∂/∂t. Inserting this into the last term in (11.25), and using Z T H0 H0 H0 Pess ⊗ Pn W ψj0 = Wn φj Pess W (t)φj eiωnt dt = Pess 0

146

11 Perturbation Theory: Feshbach-Schur Method

and H0 H0 ¯ 0 (Pess hhPess f ⊗ e−iωnt , R f ⊗ e−iωnt )ii

H0 H0 = hPess f, (H0ess − ~ωn − Ej − i0)−1 Pess f i,

H0 where H0ess := H0 Pess , we find

¯ 0 Pess W ψj0 ii hhPess W ψj0 , R X H0 H0 = hPess Wn φj , (H0ess − ~ωn − Ej − i0)−1 Pess Wn φj i. n

Substituting (11.26) and (11.27) into (11.25), we obtain Ej2 =

X

(m,n)6=(j,0)

+

X n

|hφm , Wn φj i|2 Em + ~ωn − Ej

(11.27)

H0 H0 hPess Wn φj , (H0ess − Ej − ~ωn − i0)−1 Pess Wn φj i.

Thus we have obtained detailed expressions for the first and second order coefficients in the expansion of Eκ,j in κ. Now let’s analyze the expression for Ej,2 a little further. Start with the well-known formula Z ∞ Z ∞ f (λ) f (λ) dλ := lim+ dλ ǫ→0 −∞ λ + iǫ −∞ λ + i0 Z ∞ Z ∞ f (λ) dλ − 2πi f (λ)δ(λ)dλ = PV −∞ −∞ λ R∞ where P V −∞ denotes the principal value of the singular integral, defined by PV

Z

∞

−∞

f (λ) dλ := lim λ ǫ→0+

Z

−ǫ

+

−∞

Z

ǫ

∞

f (λ) dλ, λ

and δ(λ) denotes the Dirac delta function (centred at zero). Thus we find that Ej2 is a complex number of the form Re Ej2 = Ej2 − iΓj , Re whose real part, Ej2 , is the sum of the first term on the right hand side of (11.27) and the real part (or principal value) of the second term. The imaginary part is −Γj , where X H0 H0 Γj = 2π Wn φj i. hPess Wn φj , δ(H0ess − ~ωn − Ej )Pess n

The last expression is known as the Fermi golden rule. Since δ(λ) ≥ 0, we have Γ ≥ 0. Then for “generic” perturbations (that is, barring an accident), we expect

11.4 Appendix: Proof of Theorem 11.1

147

Γj > 0. This means that for κ sufficiently small, Eκ,j is a complex number of the form Re Eκ,j = Eκ,j − iκ2 Γj + O(κ3 ) Re Re where Eκ,j := κEj1 +κ2 Ej,2 . This suggests that for generic perturbations, and for κ sufficiently small, the operator Kκ has no eigenvalues near Ej . So what happened to the eigenvalue Ej of K0 (or H0 )? It is apparently unstable under small generic perturbations. But does it just disappear without a trace, or is something left behind? It turns out that the eigenvalue Ej of H0 (or K0 ) gives rise to a resonance of H0 (or K0 ). A theory of resonances is briefly described in Chapter 17. The method of complex deformations indicated there, together with the Feshbach method described in this section, can be used to establish the existence of resonance ’eigenvalues’ Eκ,j born out of Ej (see [73] and references therein). The number − Im Eκ,j is called the width of the resonance at Eκ,j , and Tκ,j := ~(− Im Eκ,j )−1 gives the lifetime of the resonance. The Fermi golden rule gives the leading order of the resonance width

− Im Eκ,j = κ2 Γj + O(κ3 ) (in fact O(κ3 ) can be replaced by O(κ4 )). A physical interpretation of the phenomenon of instability of bound states under time-periodic perturbations is that an atom in a photon field becomes unstable, as photons of sufficiently high energy can break it up. This photoelectric effect was predicted by Einstein in 1905. To develop a consistent theory of the photoelectric effect, one has to use the quantized electro-magnetic field (or Maxwell equations) given in Chapter 21.

11.4 Appendix: Proof of Theorem 11.1 First, in addition to (11.8), we define the operator Q# = Q# (H) := P − P HRP¯ .

(11.28)

The operators P , Q and Q# have the following properties: NullQ ∩ NullH ′ = {0},

NullP ∩ NullH = {0},

HQ = P H ′ , #

′

Q H = H P,

(11.29) (11.30) (11.31)

where H ′ = FP (H). We prove relations (11.29) - (11.31). The second relation in (11.29) is shown in Proposition 11.3, while the first one follows from the inequality kQuk2 = kP uk2 + kRP¯ HP uk2 ≥ kP uk2.

148

11 Perturbation Theory: Feshbach-Schur Method

In the first equality, we used the fact that the projections P and P¯ are orthogonal. Now we prove relations (11.30) - (11.31). Using the definition of Q(H), we transform ¯ HQ = HP − H P¯ H −1 ¯ P HP P

¯ ¯ ¯ −1 ¯ = P HP + P¯ HP − P H P¯ HP−1 ¯ P HP − P H P HP¯ P HP ¯ = P HP − P H P¯ H −1 ¯ P HP

(11.32)

P

= P FP (H).

Next, we have ¯ Q# H = P H − P H P¯ HP−1 ¯ PH ¯ ¯ −1 ¯ ¯ = P HP + P H P¯ − P H P¯ HP−1 ¯ P HP − P H P HP¯ P H P ¯ = P HP − P H P¯ H −1 ¯ P HP P

= FP (H)P.

This completes the proof of (11.29) - (11.31). Proposition 11.3 Assume conditions (11.6) are satisfied. Then (11.29) (11.31) imply that 0 ∈ σ(H) ⇒ 0 ∈ σ(H ′ ) (the part of property (a) which is crucial for us). Moreover, we have NullP ∩ NullH = {0}.

Proof. Let 0 ∈ ρ(H ′ ). Then we can solve the equation H ′ P = Q# H for P to obtain −1 P = H ′ Q# H . (11.33) ¯ ¯ ¯ The equation P + P = 1 and the definition HP¯ = P H P imply ¯ ¯ ¯ −1 ¯ ¯ P¯ = P¯ HP−1 ¯ P H P = P HP¯ (P H − P HP ) .

(11.34)

Substituting expression (11.33) for P into the r.h.s., we find ′ −1 # ¯ ¯ P¯ = P¯ HP−1 Q )H . ¯ (P − P HP H

Adding this to Equation (11.33) multiplied from the left by P , and using P + P¯ = 1, yields i h −1 ¯ ′ −1 # ′ −1 # ¯ ¯ H. 1 = P¯ H −1 P − P H P HP H Q + P H Q ¯ ¯ P

P

¯ Since by our conditions P¯ HP−1 ¯ P HP is bounded, the expression in the square brackets represents a bounded operator. Hence H has a bounded inverse. So 0 ∈ ρ(H). This proves the first statement. The second statement follows from the relation 1 = QP + RP¯ H ,

(11.35)

which, in turn, is implied by Equation (11.34) and the relation P + P¯ = 1. Indeed, applying (11.35) to a vector φ ∈ NullP ∩ NullH, we obtain φ = QP φ + RP¯ Hφ = 0. 

11.4 Appendix: Proof of Theorem 11.1

149

Now we proceed directly to the proof of Theorem 11.1. Statement (b) follows from relations (11.29)-(11.30). Proposition 11.3 implies that 0 ∈ ρ(H) if 0 ∈ ρ(H ′ ), where H ′ := FP (H), which is half of statement (a). Conversely, suppose 0 ∈ ρ(H). The fact that 0 ∈ ρ(H ′ ) follows from the relation H′

−1

= P H −1 P

(11.36)

which we set out to prove now. We have by definition −1 ¯ H ′ P H −1 P = P HP H −1 P − P H P¯ HP−1 P ¯ P HP H ¯ ¯ −1 P = P H(1 − P¯ )H −1 P − P H P¯ H −1 ¯ P H(1 − P )H P

=P .

Similarly one shows that P H −1 P H ′ = P . Hence H ′ has the bounded inverse P H −1 P . So we have shown that 0 ∈ ρ(H) ⇔ 0 ∈ ρ(FP (H)), which is equivalent to 0 ∈ σ(H) ⇔ 0 ∈ σ(FP (H)). 

12 Born-Oppenheimer Approximation and Adiabatic Dynamics

In this chapter, we discuss the Born-Oppenheimer approximation of molecular physics. This approximation lies at the foundation of quantum chemistry. It gave rise to adiabatic theory (see Section 12.5) in quantum mechanics and to multiscale analysis in mathematics and physics. It also led to the theory of geometrical phases discussed in Section 12.6. We begin with the stationary theory and then present informally the time-dependent one.

12.1 Problem and Heuristics We consider a molecule with N electrons and M nuclei of masses m1 , . . . , mM . Its state space is a symmetry subspace L2sym (R3(N +M) ) of L2 (R3(N +M) ) and its Schr¨odinger operator, acting on L2sym (R3(N +M) ), is given, in units such that ~ = 1 and c = 1, by Hmol = −

M N X X 1 1 ∆yj + V (x, y). ∆xj − 2m 2m j 1 1

(12.1)

Here x = (x1 , . . . , xN ) and y = (y1 , . . . , yM ) are the electron and nuclear coordinates, and V (x, y) is the sum of Coulomb interaction potentials between the electrons, between the electrons and the nuclei, and between the nuclei (see (10.2)). As we know, the evolution of the molecule is described by the timedependent Schr¨odinger equation (SE) i∂t Ψ = Hmol Ψ,

(12.2)

with Ψ (x, y, t) ∈ H 2 (R3(N +M) ), ∀t ∈ R. A key fact here is that nuclei are much heavier than electrons and therefore much slower. Consequently, the electrons adjust almost instantaneously to the positions of the nuclei (‘slave’ to the nuclei) and prefer to be in the state of © Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_12

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12 Born-Oppenheimer Approximation and Adiabatic Dynamics

lowest possible energy, i.e. the ground state, of the electronic Schr¨odinger operator. Thus, in the first step, one supposes that the nuclei are frozen at their positions and considers, instead of (12.1), the electronic Schr¨odinger operator Hel (y) := −

N X 1 ∆xj + V (x, y). 2m 1

(12.3)

This is the adiabatic regime. The operator Hel (y), called the Born-Oppenheimer Hamiltonian, acts on a symmetry subspace L2sym (R3N ) of L2 (R3N ) and depends on the coordinates, y, of the nuclei, as parameters, and therefore its eigenvalues are functions of y, as well. In the second step, the ground state energy, E(y), of Hel (y) is considered as the potential (interaction) energy of the nuclear motion. Replacing Hel (y) in (12.1) by its ground state energy, E(y), leads to the nuclear Hamiltonian Hnucl := −

M X 1 ∆yj + E(y). 2m j 1

(12.4)

where E = E(y) is the operator of multiplication by the function E(y). One expects that the eigenvalues of Hnucl give a good approximation to eigenvalues of Hmol and that the dynamics generated by (12.4) approximates the true dynamics of nuclei under exact evolution (12.2). Minimizing the ground state energy, E(y), of the operator Hel (y) with respect to y gives the equilibrium positions of the nuclei, i.e. the shape of the molecule. This provides key information for quantum chemistry. To proceed to our analysis, we use a rescaling to pass to dimensionless variables so that mj become the ratios of nuclear masses to the electronic one. The small parameter in our problem is κ := 1/ min mj , j

which depending on the nuclei varies from ≈ 1/1836 to ≈ 1/367000.

12.2 Stationary Born-Oppenheimer Approximation In the stationary Born-Oppenheimer approximation, eigenvalues of molecular Hamiltonian (12.1) are approximated by eigenvalues of the nuclear Hamiltonian (12.4). To justify this, we use the Feshbach - Schur method (see Chapter 11). Assume for simplicity that the ground state energy, E(y), of the electronic Hamiltonian, Hel (y), is non-degenerate (which is not hard to prove on L2 (R3N ) but is not known and might be, in certain cases, false on

12.2 Stationary Born-Oppenheimer Approximation

153

L2sym (R3N )), and denote the corresponding normalized ground state by ψy (x): Hel (y)ψy = E(y)ψy . Also, for technical reasons, we modify the original Hamiltonian by replacing the point nuclear charges by smeared ones. (This is often done in physical and computational literature. In technical jargon, the smeared charges are called form-factors.) This replaces the singular Coulomb interaction potential V (x, y) in (12.3) by a potential which is differentiable as many times as the form-factors are. This will be used in the proof. We keep the same notation H(y) = Hel (y) for the modified Hamiltonian. The main result of this section is the following Theorem 12.1 [Born-Oppenheimer approximation] To second order, O(κ2 ), the ground state energy, E0 , of Hmol is the ground state energy of the operator Heff := Hnucl + v,

(12.5)

acting on L2 (R3M ), where Hnucl is given in (12.4) and v = O(κ) is an operator given by v :=

Z M X 1 |∇yj ψy |2 dx, 2m j 1

(12.6)

if ψy is real, as for (12.3), and by (12.29) below, if ψy is complex, as in the case when a magnetic field is present. In the latter case, the operator Heff in (12.5) can be written as Heff = −

M X 1 (∇yj − iAj )2 + E(y) + v˜, 2m j 1

(12.7)

where Aj := ihψy , ∇yj ψy i, with, recall, h·, ·i standing for the L2 -inner product P 1 2 in the x variable, and v˜ is given by v˜ = M 1 2mj k[P, ∇yj ]ψy k .

Proof. As was mentioned above, we use the Feshbach - Schur method to prove this result. Recall that in this method, given a quantum Hamiltonian, H, we pick a projection P so that the Feshbach-Schur map FP is defined on H − λ and maps the latter operator to a simpler one. Let P be an orthogonal projection and P ⊥ = 1−P . Introduce the notation ⊥ H = P ⊥ HP ⊥ . Assume (a) The operator H ⊥ − λ is invertible; (b) The operator U (λ) := P HP ⊥ (H ⊥ − λ)−1 P ⊥ HP is well defined.

(12.8)

154

12 Born-Oppenheimer Approximation and Adiabatic Dynamics

The Feshbach-Schur method, as applied to the quantum Hamiltonian H, states that if Conditions (a) and (b) are satisfied, then the Feshbach-Schur map FP (H − λ) = (P HP − U (λ))|RanP

(12.9)

is well defined and λ eigenvalue of H ⇐⇒ λ eigenvalue of FP (H − λ).

(12.10)

Moreover, the eigenfunctions of H and FP (H − λ) corresponding to the eigenvalue λ are connected as Hψ = λψ

⇔

FP (H − λ)φ = λφ,

(12.11)

where φ, ψ are related by the following equations φ = P ψ, ψ = Q(λ)φ. Here the family of operators Q(λ) is defined as Q(λ) = P − P ⊥ (H ⊥ − λ)−1 P ⊥ HP. Recall that ψy (x) denotes the Rnon-degenerate ground state, Hel (y)ψy = E(y)ψy , of Hel (y), normalized as |ψy (x)|2 dx = 1. We define P to be the orthogonal projection Z (P Ψ )(x, y) = ψy (x) ψy (x)Ψ (x, y)dx. (12.12) Problem 12.2 Check that P is an orthogonal projection. Note that it has infinite rank. Remark 12.3 In the previous applications of the Feshbach-Schur method, the projection P was related to an isolated eigenvalue of some ‘unperturbed’ operator H0 . In this section, P is the spectral projection for the band of the spectrum {E(y) : y ∈ R3M }, which might be only partially isolated. The analysis of Chapter 10 shows that the function E(y) increases as |y| → ∞. Hence it has a minimum λ∗ := miny E(y). (The corresponding minimum points - there could be several of those - determine the shape of the molecule.) We apply the Feshbach - Schur map with projection (12.12) to Hmol − λ, with the spectral parameter λ close to λ∗ . Before we proceed, we introduce a useful concept and notation. For a family, {A(y)}, of operators, A(y), acting on the space L2 (R3N ) (or a symmetry R⊕ subspace thereof), we define the fiber integral, A(y)dy, as the operator on L2 (R3(N +M) ) acting as 1 1

We identify vectors Ψ ∈ L2 (R3(N+M ) ) as vector-functions y R∈ R3M → ⊕ ψ(y) ∈ L2 (R3N ), where ψ(y)(x) := Ψ (x, y), and denote Ψ = ψ(y)dy : R⊕ R⊕ R⊕ A(y)dy ψ(y)dy = A(y)ψ(y)dy.

12.2 Stationary Born-Oppenheimer Approximation

Z

⊕

155


A(y)dy Ψ (x, y) = (A(y)Ψ (·, y))(x).

In particular, the operator P defined in (12.12) can be written as the fiber R⊕ integral P (y)dy, where P (y) are the orthogonal projections on the ground state eigenspaces for the operators Hel (y). Moreover, using the fiber integral we can define the electronic Hamiltonian on the entire space L2 (R3(N +M) ) as Z ⊕ Hel := Hel (y)dy. (12.13) With this notation and by (12.1), the molecular Hamiltonian can be written as Hmol = Hel + Tnucl, (12.14) PM 1 where, recall, Hel is given in (12.13) and Tnucl := − 1 2mj ∆yj . R⊕ R⊕ Furthermore, given two fiber integral operators A(y)dy and B(y)dy, their product is also a fiber integral given by Z ⊕ Z ⊕ Z ⊕ A(y)dy B(y)dy = A(y)B(y)dy. (12.15) Below, we use that for twice differentiable form-factors, i.e. for twice differentiable potentials V (x, y) in (12.3), we have the estimate k∂yα ψy k ≤ C,

|α| ≤ 2,

(12.16)

for some constant C. This can be shown by using perturbation theory, which we omit here. Let E ′ (y) be the first excited state energy of Hel (y) above E(y) (i.e. ′ E (y) > E(y) ∀y) and let P¯ (y) := 1 − P (y). Then miny E ′ (y) > miny E(y) =: λ∗ . Next, Eq (12.14) and the relations Tnucl ≥ 0 and (12.15) imply Z ⊕ Hel (y)P¯ (y)dy P¯ Hmol P¯ ≥ Hel P¯ ≥ Z ⊕ ≥ E ′ (y)P¯ (y)dy > λ∗ P¯ . (12.17) Therefore, for λ close to λ∗ , HP¯ := P¯ (Hmol − λ)P¯ ↾RanP¯ is invertible, giving condition (a). To show that condition (b) above also holds, we first observe that, by the definition of RP , we have P Hmol RP = E(y)P RP = 0. Together with the above, this gives P Hmol RP = P Tnucl RP .

(12.18)

To show that P Tnucl RP is a bounded operator, we use a refined version of inequality (12.17) in which one keeps the term Tnucl to obtain

156

12 Born-Oppenheimer Approximation and Adiabatic Dynamics

P¯ Hmol P¯ ≥ Hel P¯ > λ∗ P¯ + P¯ Tnucl P¯ .

(12.19)

Next, since P RP = 0, we have P Tnucl RP = [P, Tnucl ]RP . Furthermore, to simplify the notation, we set 2mj = 1 for all j, so that Tnucl = −∆y . With this, we compute [P, Tnucl ] = 2(∇y P )∇y + (∆y P ). The operators (∇y P ) and (∆y P ) are bounded, so it remains to show that the operator ∇y RP is bounded. To this end, we write k∇y RP Ψ k2 = hRP Ψ, −∆y RP Ψ i ≤ kRP Ψ kkP¯ ∆y P¯ RP Ψ k and use that, by inequality (12.19) and our convention −∆y = Tnucl , the operator ∆y P¯ RP is bounded, which implies that so is ∇y RP . Hence by the above, the operator (12.18) is bounded. The latter implies that so is RP Hmol P = (P Hmol RP )∗ . This proves condition (b) above. Problem 12.4 Fill in details in the arguments above. Thus for λ close to λ∗ , FP (Hmol − λ) is well defined and according to (12.10), λ ∈ σd (Hmol ) ⇐⇒ 0 ∈ σd (FP (Hmol − λ)),

(12.20)

with the corresponding eigenfunctions related accordingly. Now, remember by the decomposition (12.9) of the Feshbach-Schur operator we have to compute the terms HP := P HP ↾RanP and U (λ), defined in (23.8). We begin with P HP . Using decomposition (12.14) and the eigenequation Hel (y)ψy = E(y)ψy , we compute for Ψ (x, y) ∈ H 2 (R3(N +M) )   (Hmol P Ψ )(x, y) = ψy (x)Hnucl + W f (y) (12.21) R where, recall, Hnucl := Tnucl + E(y) (see (12.4)), f (y) := ψy (x)Ψ (x, y)dx, so that P Ψ = ψy f , and, recall, W is a family of differential operators in y, with coefficients depending on x and y, given by W := [Tnucl , ψy (x)].

(12.22)

Problem 12.5 Show (12.21). Hint: use the decomposition Hmol = Hel +Tnucl of the molecular Hamiltonian (12.1), where (Hel Ψ )(x, y) = (Hel (y)Ψ (·, y))(x), and the eigen-equation Hel (y)ψy = E(y)ψy . Eqn (12.21) implies for any Ψ ∈ H 2 (R3(N +M) ), P Hmol P Ψ = ψy (Hnucl + v)f,

(12.23)

where f is as above and v the differential operator in the variable y given by Z Z v := Re ψy (x)W dx = ψy (x)[Tnucl , ψy (x)]dx. (12.24)

12.2 Stationary Born-Oppenheimer Approximation

157

To compute the operator v, we first evaluate the commutator in (12.22) to find W =−

M X 1 ((∇yj ψy ) · ∇yj + ∇yj · (∇yj ψy )). 2m j 1

(12.25)

Using that ψy ∇yj = ∇yj ψy − (∇yj ψy ), find furthermore that v=−

Z M X 1 (aj ∇yj + ∇yj aj − |∇yj ψy |2 dx), 2mj 1

(12.26)

where aj := hψy , ∇yj ψy i, with h·, ·i standing for the L2 -inner product in the x variable. We see that v = O(κ). Note that RanP can be identified with L2 (R3 ) and therefore the operator P Hmol P on RanP , with Hnucl + v on L2 (R3 ). Now we consider the term U (λ), defined in (12.8). Using (12.21) and P¯ = 1 − P , we compute P¯ (Hmol − λ)P Ψ = P¯ W P Ψ = Kf, where K is the first order differential operator given by K := P¯ W = W − v. This expression and definition (23.8) show that the U −term is formally of the order O(κ2 ) and is of the form U (λ)Ψ = ψy w(λ)f , where the operator w(λ) acts on L2 (R3M ) and is given by ⊥ w(λ) := h(W − v)ψy , P ⊥ (Hmol − λ)−1 P ⊥ (W − v)ψy i,

(12.27)

where the inner product is taken in the variable x. (Note that due (12.25), w(λ) involves derivatives in y up to the second order, it is a second order integro-differential operator.) Hence we obtain that the ground state energy, E0 , of Hmol is the ground state energy of the operator Hnucl + v + w with w := w(E0 ), acting on L2 (R3M ). Since w(E0 ) = O(κ2 ), this gives Theorem 12.1, with v given in (12.26). If ψyR real, then aj = ∇yj hψy , ψy i and therefore aj = 0, due to the normalization |ψy |2 dx = 1, which gives v=

Z M X 1 |∇yj ψy |2 dx. 2m j 1

(12.28)

This gives the first part of Theorem 12.1. In general, for ψy complex, (12.7) is equivalent to (12.5) with v given by v=−

M X 1 (aj ∇yj + ∇yj aj + a2j + v˜), 2m j 1

where, recall, aj := hψy , ∇yj ψy i and v˜ is given by

(12.29)

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12 Born-Oppenheimer Approximation and Adiabatic Dynamics

v˜ =

M X 1 k[P, ∇yj ]ψy k2 . 2m j 1

(12.30)

To prove (12.29), we note that (12.26) can be rewritten as (12.29), with v˜ is given by v˜ =

Z Z M X 1 ( |∇yj ψy |2 dx − ( ψ¯y ∇yj ψy dx)2 ). 2mj 1

(12.31)

Now, we show that this v˜ is equal the one in (12.30). To thisR end, we insert the partition of unity P + P¯ = 1, where P = hψy , · iψy , into |∇yj ψy |2 dx = h∇yj ψy , ∇yj ψy i to obtain Z |∇yj ψy |2 dx = h∇yj ψy , (P + P¯ )∇yj ψy i. (12.32) Using the definition of P , we find h∇yj ψy , P ∇yj ψy i = h∇yj ψy , ψy ihψy , ∇yj ψy i.

(12.33)

Since h∇yj ψy , ψy i = ∇yj hψy , ψy i − hψy , ∇yj ψy i and ∇yj hψy , ψy i = 0, this gives h∇yj ψy , P ∇yj ψy i = −a2j .

(12.34)

Using that P¯ ψy = 0 and therefore P¯ ∇yj ψy = [P, ∇yj ]ψy , the P¯ -part of can rewritten as h∇yj ψy , P¯ ∇yj ψy i = k[P, ∇yj ]ψy k2 . This equation together with (12.32), shows that the expressions on the r.h.s. of (12.30) and (12.31) are equal, which proves (12.7). The excited states of Hmol come from excited states of Hnucl + v and from bound states of the operator obtained by projecting onto the exited states of Hel (y). Remark 12.6 In the previous applications of the Feshbach-Schur method, the projection P was related to an isolated eigenvalue of some ‘unperturbed’ operator H0 . In this section, P is the spectral projection for the band of the spectrum {E(y) : y ∈ R3M }, which might be only partially isolated. Beyond the Born-Oppenheimer Approximation. We recall (12.20) and define Hnucl (λ) := FP (Hmol − λ) + λP. Then λ ∈ σd (Hmol ) ↔ λ ∈ σd (Hnucl (λ)),

12.3 Complex ψy and Gauge Fields

159

Let δ be the gap between the ground state energy, E(y), of HN (y) and the rest of its spectrum. Then, for λ ≤ E(y) − δ/2, the operator Hnucl (λ) acts on L2 (R3M ) and is of the form Hnucl (λ) := −

M X 1 ∆y + Eκ (y, λ), 2mj j j=1

where, recall, κ := 1/ minj mj and Eκ (y, λ) can be computed to any order of κ, e.g. in the second order we have Eκ (y, λ) = E(y) +

Z M X 1 |∇yj ψy |2 dx + O(κ2 ), 2m j 1

with O(κ2 ) standing for a non-local operator of the indicated order in κ, differentiable in λ, with derivatives of the same order in κ (but maybe nonuniform in y). The leading term in the energy Eκ (y, λ) gives the operator Hnucl defined in (12.4). The energy Eκ (y, λ) can be used to define the interaction energy in all orders as Wκ (y) := Eκ (y) − E(∞),

(12.35)

where Eκ (y) solves the equation λ = Eκ (y, λ). It is worth remembering however that Eκ (y, λ), and therefore Wκ (y), are not potentials anymore.

12.3 Complex ψy and Gauge Fields If ψy is not real, as is the case when magnetic fields are present, then we have shown that the operator Heff in (12.5) can be written as (12.7), which gives the effective time-independent Schr¨odinger equation Heff f = Ef,

Heff

M X 1 (∇yj − iAj )2 + E(y) + v˜, =− 2m j 1

(12.36)

where, recall, Aj := ihψy , ∇yj ψy i, with h·, ·i standing for the L2 -inner product P 1 2 in the x variable, and v˜ = M 1 2mj k[P, ∇yj ]ψy k .

Gauge invariance. Under the transformation f (y, t) → eiχ(y) f (y, t), ψy (x) → e−iχ(y) ψy (x), the function ΨBO (x, y, t) := ψy (x)f (y, t) is not changed. However, the vector potential Ay := ihψy , ∇y ψy ix changes as Ay → Ay + ∇χ(y). Thus an effective equation for f should be invariant under the gauge transformation f (y) → eiχ(y) f (y), Ay → Ay + ∇χ(y). As can be easily checked (12.36) is indeed invariant under (12.37).

(12.37)

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12 Born-Oppenheimer Approximation and Adiabatic Dynamics

Problem 12.7 Prove that (12.36) is invariant under the gauge transformations (12.37). Remark 12.8 (Degenerate eigenvalues) If the eigenvalue E(y) is degenerate, then A(y) := ihψy , dy ψy ix takes values in the k × k-matrices, where k is the degeneracy of E(y).2 In this case, Eq (12.36) is invariant under the gauge transformation f (y) → g(y)f (y), A(y) → g(y)A(y)g(y)−1 + dg(y)g(y)−1

(12.38)

where g(y) ∈ U (k), ∀y ∈ Rk . Hence A(y) transforms as a Yang-Mills gauge field, or a U (k)−connection.

12.4 Time-dependent Born-Oppenheimer Approximation We present a heuristic discussion of the time-dependent Born-Oppenheimer approximation. We look for the solution Ψ to the time-dependent Schr¨odinger equation (12.2) in the form Ψ (x, y, t) ≈ ΨBO (x, y, t) := ψy (x)f (y, t). We call the functions ΨBO (x, y, t), f ∈ L2 (R3M ), the Born-Oppenheimer states. Now, as in the Dirac-Frenkel theory (see e.g. [216] and references therein), projecting the time-dependent Schr¨odinger equation (12.2) on the manifold of the Born-Oppenheimer states, we arrive the effective time-dependent Schr¨odinger equation i∂t ΨBO = P Hmol ΨBO ,

(12.39)

where P is the projection defined in (12.12). Using that ΨBO = P ΨBO , so that P Hmol ΨBO = P Hmol P ΨBO and relation (12.23), we find the equation for f : i∂t f = Heff f,

Heff := Tnucl + E + v,

(12.40)

where v is the operator given by (12.26), or (12.29). Moreover, if ψy is real, then v is the operator of multiplication by the function (12.6), and, if ψy is complex, then the effective Hamiltonian Heff in (12.40) is given by (12.7), i.e. Heff := −

M X 1 (∇yj − iAj )2 + E(y) + v˜. 2m j 1

(12.41)

Using a time-dependent version of the Feshbach-Schur method of Section 12.2, as in [225], would give the exact effective quantum hamiltonian 2

We can also consider P to be a projection onto several eigenspaces. In this case, k is the total dimension of the eigenspaces considered.

12.5 Adiabatic Motion ′ Heff := Hnucl + v + w,

161

(12.42)

with w a time-dependent differential operator, acting on L2 (R3M ), of the order O(κ2 ), so that the original time-dependent Schr¨odinger equation would be equivalent to the effective one. Remark 12.9 (The effective action) The (effective) Lagrangian for the effective quantum Hamiltonian (12.41), with j = 1 and m = 1, is Leff = 1 ˙ 2 + y˙ · A(y) − E. (Indeed, we compute p · y˙ − Leff |p=y+A = [p · y˙ − 12 |y| ˙ 2− ˙ 2 |y| 1 2 |p − A| + E.) Let L := y ˙ · A(y). Using this in the y˙ · A(y) + E]|p=y+A = geom ˙ 2 path integral representation of the propagator, we obtain the factor eiSgeom = ei

R

A

= eiγ .

12.5 Adiabatic Motion As was mentioned in the introduction to this chapter, heavy nuclei move relatively slowly and in the Born-Oppenheimer approximation we fix them in their positions. The next step would be to let them move but take advantage of their slow motion. In this section, we explore the slow, or adiabatic, motion in an abstract context and at the end discuss application of the results to molecular dynamics. In its original context of thermodynamics, the notion of adiabatic evolution describes a slow dynamics for which at every moment of time the evolving state is approximately a stationary one, but with a possibly different set of parameters. Below, we formulate this statement rigorously in the context of quantum mechanics. Let Hs , s ∈ R, be a one-parameter family of Schr¨odinger operators and consider the adiabatic (slow) evolution iǫ∂s ψsǫ = Hs ψsǫ ,

ǫ ψs=0 = φ0 ,

(12.43)

with a small parameter ǫ (incorporating the Planck constant ~) quantifying the adiabaticity. Assume Hs have isolated, simple eigenvalues Es . We expect that, if the initial condition φ0 is an eigenvector of H0 with the eigenvalue E0 , then, for ǫ sufficiently small, the solution ψsǫ is close to an instantaneous eigenvector φs with the eigenvalue Es . The next result justifies this. Since even simple eigenvectors are defined up to a phase, we pick normalized eigenvectors: φs ∈ Null(Hs − Es ).

(12.44)

A family Hs of operators is said to be differentiable in the resolvent sense iff its resolvent (Hs − z)−1 is differentiable, as a family of bounded operators, for some z in each connected component of the resolvent set.

162

12 Born-Oppenheimer Approximation and Adiabatic Dynamics

Theorem 12.10 [Adiabatic theorem] Assume Hs is (k + 1)-times differentiable in the resolvent sense. Then there are phases γs s.t. ψsǫ = ei(γs −

Rs

Er dr/ǫ)

φs + O(ǫk ) as ǫ → 0.

(12.45)

We discuss the phases eiγs in the next section. Another formulation of the adiabatic regime is to consider the normal evolution for a slowly varying family of quantum Hamiltonians, say, Ht/τ , with τ large: τ i~∂t Ψtτ = Ht/τ Ψtτ , Ψt=0 = φ0 .

(12.46)

Rescaling (12.46) by setting s := t/τ and ψsǫ := Ψττs , where ǫ := ~/τ , we obtain (12.43). Then Theorem 12.10 implies Ψττs → φs as τ → ∞. Next, we reformulate Theorem 12.10 in a convenient and important way. Let Uǫ (s, r) be the propagator generated by Hs /ǫ (see Section 25.6, Theorem ǫ 25.32) and Uǫ (s) ≡ Uǫ (s, 0), so that ψsǫ = Uǫ (s)ψs=0 . It satisfies the equation iǫ∂s Uǫ (s) = Hs Uǫ (s), Uǫ (s = 0) = 1.

(12.47)

Let Ps be the orthogonal projection onto Null(Hs − Es ). Note that, unlike eigenvectors, eigenprojections are defined uniquely. We define the adiabatic hamiltonian HsA := iǫ[P˙ s , Ps ] + Es

(12.48)

and let UǫA (s, r) be the propagator generated by the hamiltonian HsA /ǫ i.e. the two-parameter family of bounded operators satisfying the initial value problem (see (25.30) - (25.31)) iǫ

∂ A U (s, r) = HsA UǫA (s, r), ∂s ǫ

U (r, r) = 1.

(12.49)

Let UǫA (s) ≡ UǫA (s, 0). The following (adiabatic) theorem implies Theorem 12.10: Theorem 12.11 Assume Hs is (k + 1)-times differentiable in the resolvent sense. Then we have Uǫ (s)P0 = UǫA (s)P0 + O(ǫk ).

(12.50)

A proof of Theorem 12.11 is given in Section 12.8. It is based on the Feshbach - Schur approach described in Section 11.1 (with the orthogonal projection Ps ). However, unlike Chapter 11, we apply to it to a time-dependent problem, specifically, to Eq. (12.47). Using this approach we derive the equation FPs (Ks,ǫ )Ps Uǫ (s) = 0,

(12.51)

12.5 Adiabatic Motion

163

where FPs (Ks,ǫ ) is the Feshbach - Schur map, with the projection Ps (cf. (11.7)), applied to the operator Ks,ǫ := Hs − iǫ∂s acting on s-dependent vectors. Using the Duhamel principle and the notation P¯s := 1 − Ps , we rewrite equation (12.51) as (see (12.72)-(12.73) of Section 12.8) Ps Uǫ (s) = UǫA (s, 0)P0 Z s Z + drUǫA (s, r)Pr P˙r P¯r 0

0

(12.52) r

¯ A (r, t)P¯t P˙t Pt Uǫ (t), dtU ǫ

(12.53)

¯ A (s, r) is the propagator generated by the hamiltonian H ¯ A := where U ǫ s ˙ ¯ ¯ ¯ ¯ iǫ[Ps , Ps ] + Hs , with Hs := Ps Hs Ps , and estimate the second term on the r.h.s., (12.53), using integration by parts. (In Section 11.3, we came close to this but then used Floquet theory (and the Fourier transform) to convert the time-dependent problem to a time-independent one.) The next proposition reveals a key property of the adiabatic propagator UǫA (s, r): Proposition 12.12 The (adiabatic) evolution UǫA (s, r) has the intertwining property UǫA (s, r)Pr = Ps UǫA (s, r).

(12.54)

Proof. Using that the propagator UǫA (s, r) satisfies the differential equation (12.49) and using the similar equation for UǫA (s, r)−1 , we find ∂ (U A (s, r)−1 Ps UǫA (s, r)) = UǫA (s, r)−1 Bs UǫA (s, r), ∂s ǫ

(12.55)

where Bs := −[P˙s , Ps ]Ps + Ps [P˙ s , Ps ] + P˙ s . Next, relations Ps P˙s Ps = 0 and P˙s = P˙s Ps + Ps P˙s (by differentiating Ps = Ps2 ) imply that Bs = 0. Using this in (12.55) and integrating the latter equation in s from r to s, we find UǫA (s, r)−1 Ps UǫA (s, r) = Pr , which gives (12.54). .  Remark 12.13 Because of the Coulomb singularity, in the Born-Oppenheimer case, the family H(y) = Hel (y) is not differentiable, but only H¨ older continuous with an arbitrary exponent less than 1. One can extend Theorems 12.10 and 12.11 to fractional derivatives and show that in the Born-Oppenheimer case, Theorems 12.10 and 12.11 hold for any k < 1. (Presumably, for curves whose velocity vector fields do not move the Coulomb singularities (the Hunziker vector fields) Theorems 12.10 and 12.11 hold for any k.) For physical and computational reasons, one often replaces the point nuclear charges by smeared ones (in technical jargon, introduces form-factors). Then the corresponding potential, which is differentiable as many times as the form-factors are, replaces the interaction potential V (x, y) in (12.3). In this case, the family H(y) = Hel (y) is differentiable the same number of times.

164

12 Born-Oppenheimer Approximation and Adiabatic Dynamics

12.6 Geometrical Phases We begin with the adiabatic dynamics given by the time-dependent Schr¨odinger equation (12.43), which we reproduce here iǫ∂s ψsǫ = Hs ψsǫ ,

ǫ ψs=0 = φ0 .

(12.56)

Then the adiabatic theorem of Section 12.5 (see Theorem 12.10) says that ψsǫ → ψs ∈ Null(Hs ), as ǫ → 0.

(12.57)

Recall that we have chosen normalized basis vectors φs in the eigenspaces Null(Hs − Es ), see (12.44). Proposition 12.14 γs of Theorem 12.10 and ψs of (12.57) are given by ψs = ei(γs − Proof. Introduce Ψs := ei that hΨs , Ψ˙ s i = 0. Indeed,

Rs

Rs

Er dr/ǫ)

Er dr/ǫ

φs ,

∂s γs = ihφs , φ˙ s i.

ψs and Ψsǫ := ei

Rs

Er dr/ǫ

(12.58)

ψsǫ . We show first

hΨs , Ψ˙ s i = lim hΨs , Ψ˙ sǫ i = lim hψs , −iǫ−1 (Hs − Es )ψsǫ i ǫ→0

ǫ→0

= lim h(Hs − Es )ψs , −iǫ−1 ψsǫ i = 0. ǫ→0

Since ψs and φs belong to Null(Hs − Es ) and are normalized, Ψs = Us φs , where Us is a complex number of unit modulus (U (1)-factor) defined by this expression. We compute 0 = hΨs , Ψ˙ s i = U˙ s Us∗ + hφs , φ˙ s i. Since Us is unitary (Us∗ = Us−1 ), this equation can be rewritten as U˙ s = −hφs , φ˙ s iUs , which can be solved as Us = eiγs with γs given by (12.58).

(12.59) 

Let Y be a parameter space (R3m , or a surface therein for the nuclear coordinates in the Born-Oppenheimer approximation, or a torus if y describes magnetic fluxes as in the integer quantum Hall effect, see [25]). We consider a family H(y), y ∈ Y, of Schr¨odinger operators. Assume ∀y ∈ Y , the operator H(y) has an isolated, simple eigenvalue E(y) and fix a corresponding normalized eigenfunction, φ(y). We consider a curve C ⊂ Y parametrized by y(s), s ∈ [0, 1] and the timedependent Schr¨odinger equation (12.56), with Hs := H(y(s)),

(12.60)

12.6 Geometrical Phases

165

with the initial condition ψs=0 = φ(0) ∈ Null(H(0) − E(0)). The adiabatic theorem implies that ψsǫ → ψs ∈ Null(Hs − Es ), Es = E(y(s)), as ǫ → 0. By replacing H(y) with H(y) − E(y), we may assume E(y) ≡ 0. Then by Proposition 12.14, ψs = eiγy(s) φ(y(s)), for γy(s) ≡ γs given by (12.58). We call ψ1 the adiabatic transport of φ(0) ∈ Null(H(0) − E(0)) along C. If C is a loop, then we have y(0) = y(1). Since E(y) and therefore Es are simple eigenvalues, this gives ψ1 = eiγC ψ0 ,

(12.61)

where γC = γy(1) . By (12.58), we have ˙ ∂s γy(s) = ihφ(y(s)), φ(y(s))i = ihφ(y(s)), ∇y φ(y(s))i · y(s). ˙ We introduce the vector field Ay := ihφ(y), ∇y φ(y)i. Then ∂s γRy(s) = Ay(s) · y(s). ˙ For a loop C, we can write γy(1) = y(s)ds ˙ = C Ay , and therefore Z γC = Ay .

(12.62) R1 0

Ay(s) ·

(12.63)

C

Geometric interpretation. A better way to think about Ay is as a covector, or a one-form, i.e. Ay := ihφ(y), dy φ(y)i, where d is the differential (exterior derivative) in y. Note that if φ(y) → e−iχ(y) φ(y), then Ay → Ay +dχ(y). Hence Ay could be viewed as a connection one-form on the vector (line) bundle, E, over Y , with the fibers being the eigenspaces Fy := Null(H(y) − E(y)) (or the associated frame bundle over Y with the fibers Fy := {ψ ∈ Null(H(y)− E(y)) : kψk = 1}). In the physics literature it is called the Berry (hermitian) connection. The curvature (‘magnetic field’) of A is given by FA := dA. By Stokes’ theorem, we have Z Z FA , dAy = γC = S

S

where S is a surface spanning C, i.e. ∂S = C. If S is a 2 dimensional torus, as in the case of the integer quantum Hall effect (see [20]), then the flux of the curvature FA over S is quantized: Z FA = 4πc1 , S

166

12 Born-Oppenheimer Approximation and Adiabatic Dynamics

where c1 is the first Chern number of the bundle F over S.3 It turns out FA is proportional to the Hall conductivity in the integer quantum Hall effect (IQHE), see [20, 25].There the surface S is a 2-torus spanned by the magnetic fluxes through the ‘holes’ (see the last paragraph of Subsection 7.5 and Subsection 7.6). Hence the above relation states that the Hall conductivity averaged over all fluxes is quantized (and proportional the first Chern number). If the eigenvalues E(y) are degenerate, then Ay is a matrix-valued (connection) one-form on the vector bundle, E, over Y with the fibers Fy = Null(H(y) − E(y)) (or the associated frame bundle, F , over Y with the fibers Fy consisting of orthonormal k-frames (ordered bases) (φ1 (y), ..., φr (y)) in Null(H(y) − E(y)), where r := dim Null(H(y) − E(y))). Considering the adiabatic evolution along the path y(s) parametrizing a curve C and letting Hs := H(y(s)), we see from Eq (12.54) that UǫA (s, r) can be viewed as a parallel transport on the frame bundle F along C. Moreover, A is the connection related to this parallel transport. Furthermore, solving Schr¨odinger equation (12.56) with quantum Hamiltonian (12.60) along a loop gives the map ψ0 → ψ1 of the fiber Fy = Null(H(y) − E(y)) into itself. This is the holonomy, UC , of the R connection A. If the eigenvalues E(y) are non-degenerate, then UC = ei C Ay . In the degenerate case, we have, instead, UC = P ei

R

C

Ay

,

(12.64)

R

where P ei C Ay stands for the ‘time-ordered exponential. (This is the Wilson loop, see [300] and the paragraph around (12.38) for more discussion.) Remark 12.15 If P (y) is the orthogonal projection onto Null(H(y) − E(y)), then the covariant derivative, or connection, on the vector bundle E is given by ∇ := P (y)dy = dy − [dy P (y), P (y)], where dy is the exterior derivative on Y . If {φj (y)}P is an orthonormal basis in P RanP (y), so that P (y) = j |φj (y)ihφj (y)| and j fj (y)φj (y) ∈ RanP (y), where fj (y) are functions of just y, then, due to the relation [dy P (y), P (y)] = dy P (y)P (y), the last formula becomes (cf. (12.62)) X X (dy fj (y)δij + hφi (y), dy φj (y)ifj (y))φi (y). fj (y)φj (y) = ∇ j

3

ij

This is a special case of the Chern-Weil theory. A 2 dimensional torus can be identified with a fundamental domain of a lattice L. In this case A is defined on R2 as an L-equivariant one-form in the sense of (7.62), i.e. it satisfies A(x + s) = A(x) + ∇χs (x),

for any s ∈ L and for some functions χs (x). For more details see Section 7.9.

12.8 Appendix: Proof of Theorem 12.11

167

12.7 Appendix: Projecting-out Procedure Let H be a Hilbert space, and P a projection on H (i.e. an operator satisfying P 2 = P ). Let A be an operator acting on H. Then P AP is an operator acting on Ran(P ): i.e. P AP : H → Ran(P ) and consequently P AP : Ran(P ) → Ran(P ). The simplest example is when P is a rank-one projection: P = |φihφ| for some φ ∈ H, kφk = 1. Here we have used Dirac’s bra-ket notation |φihφ| : ψ 7→ φhφ, ψi. Then P AP = hφ, AφiP acts on Ran(P ) = {zφ | z ∈ C} as multiplication by the complex number hφ, Aφi. PThis example can be easily generalized to a finite-rank projection, say P = m i=1 |φi ihφi | where {φj } is an orthonormal set in H. In this case the number hφ, Aφi is replaced by the matrix (hφi , Aφj i)ij . For another typical example, take H = L2 (R) and take P to be multiplication of the Fourier transform by the characteristic function, χI , of an interval I ⊂ R: Z ∞ Z −1/2 ikx −1/2 ˆ (P f )(x) = (2π) e χI (k)f (k)dk = (2π) eikx fˆ(k)dk, −∞

I

R∞ where fˆ(k) = (2π)−1/2 −∞ e−ikx f (x)dx is the Fourier transform of f . P is a projection since χ2I = χI . For a given operator A on H, the operator P AP (assuming it is well-defined) maps functions whose Fourier transforms are supported on I into functions of the same type. Of interest to us is the following situation: the projection P acts on a space H = H1 ⊗ H2 as P = P1 ⊗ 1, where P1 is a rank-one projection on H1 , say P1 = |φ1 ihφ1 |, with φ1 ∈ H1 . If A is an operator on H, then the operator P AP is of the form P AP = 1 ⊗ A2 where A2 is an operator on H2 which can formally be written as A2 = hφ1 , Aφ1 iH1 .

12.8 Appendix: Proof of Theorem 12.11 Applying the projections Ps and P¯s := 1 − Ps to equation (12.47) and using Ps iǫ∂s P¯s = −iǫPs P˙s P¯s and P¯s iǫ∂s Ps = iǫP¯s P˙ s Ps , we find (the Feshbach Schur splitting) Ps Ks,ǫ Ps Uǫ (s) = −iǫPs P˙s P¯s Uǫ (s), P¯s Ks,ǫ P¯s Uǫ (s) = iǫP¯s P˙ s Ps Uǫ (s).

(12.65) (12.66)

We compute the operators Ps Ks,ǫ Ps and P¯s Ks,ǫ P¯s . First, we differentiate the relation Ps = Ps2 and using the notation P˙s := ∂s Ps to obtain

168

12 Born-Oppenheimer Approximation and Adiabatic Dynamics

P˙s Ps + Ps P˙ s = P˙s and similarly for P¯s . These relations imply Ps P˙s Ps = 0 and P¯s P˙s P¯s = 0, which give Ps ∂s Ps = (∂s − [P˙s , Ps ])Ps ,

P¯s ∂s P¯s = (∂s − [P˙s , Ps ])P¯s .

Observe that the operator i[P˙s , Ps ] is self-adjoint. Using these relations, we compute Ps Ks,ǫ Ps = (iǫ∂s − HsA )Ps , ¯ sA )P¯s , P¯s Ks,ǫ P¯s = (iǫ∂s − H

(12.67) (12.68)

¯ sA := iǫ[P˙ s , Ps ] + H ¯ s , with H ¯ s := where, recall, HsA := iǫ[P˙ s , Ps ] + Hs and H ¯ ¯ Ps Hs Ps . Combining Eqs (12.65) and (12.67), we find (iǫ∂s − HsA )Ps Uǫ (s) = −iǫPs P˙s P¯s Uǫ (s). Using now the Duhamel formula, we find4 Z s Ps Uǫ (s) = UǫA (s, 0)P0 − UǫA (s, r)Pr P˙ r P¯r Uǫ (r)dr,

(12.69)

(12.70)

0

where, recall, UǫA (s, r) is the propagator generated by the hamiltonian HsA . Treating (12.66) similarly, we arrive at Z s ¯ A (s, 0)P¯0 − ¯ A (s, r)P¯r P˙ r Pr Uǫ (r)dr, P¯s Uǫ (s) = U (12.71) U ǫ ǫ 0

¯ A (s, r) U ǫ

¯ A . Plugging where is the propagator generated by the hamiltonian H s (12.71) into (12.70), we find Ps Uǫ (s) = UǫA (s, 0)P0 + Rs , where Rs is given by Z s Z A ˙ ¯ Rs = drUǫ (s, r)Pr Pr Pr 0

4

0

r

¯ǫA (r, t)P¯t P˙ t Pt Uǫ (t). dtU

(12.72)

(12.73)

The Duhamel formula could be rephrased as Ps Uǫ (s) = UǫA (s, 0)P0 + (iǫ∂s − HsA )−1 W P¯s Uǫ (s) (the sum of homogeneous and inhomogeneous solutions), with W := −iǫPs P˙s and Z s (iǫ∂s − HsA )−1 f (s) = (iǫ)−1 UǫA (s, r)f (r)dr. 0

To prove the last relation, we use that UǫA (s, r) problem (ie∂s − HsA )UǫA (s, r) = 0 and UǫA (r, r) = (iǫ∂s − HsA )(ie)−1

Z

s 0

¯ǫA (s, r)f (r)dr = (∂s U

Z

is defined by the initial value 1. Hence s 0

¯ǫA (s, r)f (r)dr = f (s). )U

12.8 Appendix: Proof of Theorem 12.11

169

Lemma 12.16 Rs = O(ǫ).

(12.74)

Proof. Using definition (12.73), the fact that UǫA (s, r) commutes with P˙r P¯r = −[P˙ s , Ps ]P¯r and the notation Gǫ (t) := P¯t P˙t Pt Uǫ (t), we rewrite the expression for Rs as Z s Z r ˙ ¯ Rs = drPr Pr dtVǫA (s, r, t)Gǫ (t), (12.75) 0

0

where VǫA (s, r, t) = UǫA (s, r)U¯ǫA (r, t). To extract e from this expression, we integrate by parts. First, we notice that because of the gap condition, Er is in ¯ A and the operator H ¯ A − Er is invertible. Next, using the resolvent set of H r r A A ¯ the equation (iǫ∂r − Hr + Er )Vǫ (s, r, t) = 0, we derive the formula ¯ rA − Er )−1 iǫ∂r VǫA (s, r, t), VǫA (s, r, t) = (H

(12.76)

¯ rA is differentiable in the resolvent sense, the proSince the operator family H ¯ A − Er )−1 are twice differentiable. Using the jections Pr and the resolvent (H r formula above and integrating by parts, we find Z s Z r ˙ ¯ ¯ A − Er )−1 iǫ∂r V A (s, r, t)Gǫ (t) Rs = drPr Pr dt(H r ǫ 0 0 Z r r=s ¯ A − Er )−1 = iǫ[P˙ r P¯r (H dtVǫA (s, r, t)Gǫ (t)] r=0 (12.77) r 0 Z s Z r ¯ rA − Er )−1 ) dtVǫA (s, r, t)Gǫ (t) (12.78) − iǫ dr∂r (P˙r P¯r (H 0 0 Z s ¯ A − Er )−1 V A (s, r, r)Gǫ (r). − iǫ drP˙r P¯r (H (12.79) r ǫ 0

¯ rA − Er )−1 is uniformly bounded. Hence estimating the Now, we use that (H expressions on the r.h.s. of the above relation gives (12.74). Relations (12.72) and (12.74) imply Ps Uǫ (s) = UǫA (s, 0)P0 + O(ǫ).

(12.80)

¯ A (s, r)P¯0 + O(ǫ). P¯s Uǫ (s) = U ǫ

(12.81)

Next, we claim that

Indeed, combining (12.71) with (12.80) gives

Rs

¯ǫA (s, 0)P¯0 − Rs′ + O(ǫ), P¯s Uǫ (s) = U

(12.82)

¯ A (s, r)P¯r P˙r Pr U A (r, 0)P0 dr. Using that U A (s, r) commutes where Rs′ := 0 U ǫ ǫ ǫ with P¯r P˙r Pr = [P˙s , Ps ]Pr , we write Rs′ as

170

12 Born-Oppenheimer Approximation and Adiabatic Dynamics

Rs′

=

Z

0

s

V¯ǫA (s, r)P¯r P˙r P0 dr,

(12.83)

¯ǫA (s, r)UǫA (r, 0). Next, using the equation iǫ∂r V¯ǫA (s, r) = where V¯ǫA (s, r) = U A A ¯ ¯ Vǫ (s, r)(Hr − Er ), we derive the formula ¯ rA − Er )−1 . V¯ǫA (s, r) = iǫ(∂r V¯ǫA (s, r))(H

(12.84)

Using the formula above and integrating the r.h.s. of (12.83) by parts, we find Z s ′ ¯ A − Er )−1 P¯r P˙r P0 dr Rs = iǫ (∂r V¯ǫA (s, r))(H r 0 ¯ A − Er )−1 P¯r P˙ r P0 ] r=s = iǫ[V¯ǫA (s, r)(H r r=0 Z s ¯ A − Er )−1 P¯r P˙ r )P0 dr. − iǫ (12.85) V¯ǫA (s, r)∂r ((H r 0

¯ A − Er )−1 , m = 0, 1, is uniformly bounded and Again, using that ∂rm (H r n ∂r Pr , n = 0, 1, 2, are bounded, we estimate the expressions on the r.h.s. of the above relation to obtain Rs′ = O(ǫ), which yields (12.81). Combining (12.80) and (12.81) gives (12.50) for k = 1. For a general k ≥ 1, one integrates (12.83) by parts k times.  Discussion. Consider the operator Ks,ǫ := iǫ∂s − Hs acting on s-dependent functions. Eq (12.69), with (12.71), can be rewritten FPs (Ks,ǫ )Ps Uǫ (s) = 0,

(12.86)

where FPs (Ks,ǫ ) is the Feshbach - Schur map, with the projection Ps (cf. (11.7)), applied to the operator Ks,ǫ . Indeed, according to (11.13) - (11.14), we have FPs (Ks,ǫ ) = [Ps Ks,ǫ Ps − W Ps ] ↾RanPs , ¯ ¯ = P¯s (P¯s Ks,ǫ P¯s )−1 P¯s . W := Ps (iǫ∂s )R(iǫ∂ R s )Ps ,

(12.87) (12.88)

This exactly (12.69), with (12.71) as can be seen from (12.67), (12.68) and the relation Z s ¯ sA )−1 f (s) = (iǫ)−1 ¯ǫA (s, r)f (r)dr. (iǫ∂s − H U (12.89) 0

13 General Theory of Many-particle Systems

In this chapter, we extend the concepts developed in the previous chapters to many-particle systems. Specifically, we consider a physical system consisting of N particles of masses m1 , . . . , mN which interact pairwise via the potentials Vij (xi − xj ), where xj is the position of the j-th particle. Examples of such systems include atoms or molecules – i.e., systems consisting of electrons and nuclei interacting via Coulomb forces. They were considered in Chapter 10.

13.1 Many-particle Schr¨ odinger Operators Recall from Section 4.1 that quantizing a system of n particles of masses m1 , . . . , mN interacting via pair potentials Vij (xi − xj ), we arrive at the Schr¨odinger operator (quantum Hamiltonian) Hn =

n X 1 2 p + V (x) 2mj j j=1

(13.1)

acting on L2 (R3n ). Here pj := −i~∇xj and V is the total potential of the system, given in this case by V (x) =

1X Vij (xi − xj ). 2

(13.2)

i6=j

A key example of a many-body Schr¨odinger operator, that of a molecule, was given in (4.26), Section 4.1. As in Theorem 10.1, one can show that if the pair potentials Vij satisfy the condition Vij ∈ L2 (R3 ) + L∞ (R3 )

(i.e. each Vij can be represented as the sum of an L2 function and an L∞ function) then the operator Hn is self-adjoint and bounded below. Indeed, the © Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_13

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13 General Theory of Many-particle Systems

above conditions that the potential V (x) satisfies the inequality (2.8), Pn imply 1 2 with H0 := p j=1 2mj j and a < 1. Observe that the Coulomb potentials −1 Vij (y) = ei ej |y| satisfy this condition: |y|−1 = |y|−1 e−|y| + |y|−1 (1 − e−|y| ), for example, with |y|−1 e−|y| ∈ L2 (R3 ) and |y|−1 (1 − e−|y| ) ∈ L∞ (R3 ). After this one proceeds exactly as in the proof of Theorem 10.1. The spectral analysis of the operator Hn for n ≥ 3 is much more delicate than that of the one-body Hamiltonians we have mostly considered so far. (We will show shortly that for n = 2, the operator Hn is reduced to a one-body Hamiltonian.) We are faced with the following issues: • • •

identical particle symmetries separation of the center-of-mass motion complicated behaviour of the potential V (x) at infinity in the configuration space R3n .

In Section 4.5 we commented on the first issue. In the subsequent sections, we will deal with the last two issues in some detail. In the remainder of this section, we comment briefly on them. In what follows, we shall assume Vij (x) → 0 as |x| → ∞, though for many considerations this condition is not required. Separation of the centre-of-mass motion. The Schr¨odinger operator (13.1) commutes with the operator of total translation of the system Th : ψ(x1 , . . . , xn ) 7→ ψ(x1 + h, . . . , xn + h) and one can show that, as a result, its spectrum is purely essential. So in order to obtain interesting spectral information about our system, we have to remove this translational invariance (“break” it). One way of doing this is by fixing the centre of mass of the system at, say, the origin: n X

mj xj = 0.

j=1

We will describe a general mathematical procedure for fixing the centre of mass below, but first will show how to do it in the case of two particles (n = 2). In this case, we change the particle variables as follows: x1 , x2 7→ y = x1 − x2 , z =

m1 x1 + m2 x2 . m1 + m2

(13.3)

Here y is the coordinate of the relative position of the two particles, and z is the coordinate of their centre of mass. Using this change of variables in the two-particle Schr¨odinger operator

13.2 Separation of the Centre-of-mass Motion

H2 =

173

1 2 1 2 p + p + V (x1 − x2 ) 2m1 1 2m2 2

acting on L2 (R6 ), we arrive easily at the operator ˜ 2 = 1 p2 + 1 P 2 + V (y) H 2µ 2M m2 (the reduced mass), and M = where p = −i~∇y , P = −i~∇z , µ = mm11+m 2 ˜ 2 are unitarily m1 +m2 (the total mass). In fact, it can be shown that H2 and H equivalent, with the equivalence given by a unitary realization of the change of coordinates (13.3). ˜ 2 . In The point now is that one can separate variables in the operator H ˜ 2 can be written in the form formal language, this means that H

˜ 2 = H ⊗ 1 + 1 ⊗ HCM H on L2 (R6 ) = L2 (R3y ) ⊗ L2 (R3z ) where H=

1 2 p + V (y) 2µ

acts on L2 (R3y ), and

1 2 P 2M acts on L2 (R3z ) (see the mathematical supplement, Section 25.13, for a description of the tensor product, ⊗). Clearly H and HCM are the Schr¨odinger operators of the relative motion of the particles, and of their centre of mass motion, respectively. It is equally clear that of interest for us is H, and not HCM . Note that H has the form of a one-particle Schr¨odinger operator with external potential V (y). All the analysis we developed for such operators is applicable now to H. Behaviour of V (x) at infinity. The second issue mentioned above arises from the geometry of the potential (13.2). The point is that V (x) does not vanish as x → ∞ in certain directions, namely in those directions where xi = xj for at least one pair i 6= j (we assume here that Vij (0) 6= 0). This property is responsible for most of the peculiarities of many-body behaviour. In particular, the spectral analysis of Chapter 6 does not work in the manybody case, and must be modified in significant ways by taking into account the geometry of many-body systems. HCM =

13.2 Separation of the Centre-of-mass Motion This section is devoted to a description of a general method for separating the centre-of-mass motion of a many-body system. After applying this method,

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13 General Theory of Many-particle Systems

one is left with a many-body system whose centre-of-mass is fixed at the origin. We begin by equipping the n-body configuration space R3n with the innerproduct n X mi xi · yi (13.4) hx, yi := i=1

where m1 , . . . , mn are the masses of the n particles, and xi · yi is the usual dot-product in R3 . Next, we introduce the orthogonal subspaces X := {x ∈ R3n | and 3n

n X

mi xi = 0}

(13.5)

i=1

X ⊥ := {x ∈ R3n | xi = xj ⊥

∀ i, j}

(13.6)

of R . Recall that orthogonality (X ⊥ X ) means hx, yi = 0 for any x ∈ X, y ∈ X ⊥ . To see that X ⊥ X ⊥ , suppose x ∈ X, and y ∈ X ⊥ . Since yj = y1 for all j, we have n n X X hx, yi = mi xi · yi = y1 · mi xi = 0 y=1

y=1

by the definition of X. We recall here the definition of the direct sum of subspaces.

Definition 13.1 If V1 , V2 are orthogonal subspaces of a vector space, V , with an inner product, then V1 ⊕ V2 denotes the subspace V1 ⊕ V2 := {v1 + v2 | v1 ∈ V1 , v2 ∈ V2 }. Problem 13.2 Show that R3n = X ⊕ X ⊥ .

(13.7)

X is the configuration space of internal motion of the n-particle system, and X ⊥ is the configuration space of the centre-of-mass motion of this system. The relation (13.7) implies that L2 (R3n ) = L2 (X) ⊗ L2 (X ⊥ )

(13.8)

(see Section 25.13 for an explanation of the tensor product, ⊗). Let ∆ denote the Laplacian on R3N in the metric determined by (13.4), i.e. n X 1 ∆= ∆xj . m j j=1 Under the decomposition (13.8), the Laplacian decomposes as ∆ = ∆X ⊗ 1X ⊥ + 1X ⊗ ∆X ⊥

(13.9)

where 1X and 1X⊥ are the identity operators on L2 (X) and L2 (X ⊥ ) respectively (see again Section 25.13).

13.3 Break-ups

Problem 13.3 Show that ∆X ⊥ = 1 Pn j=1 mj xj . M

1 M ∆xCM

where M =

Pn

j=1

175

mj and xCM =

Let πX be the orthogonal projection operator from R3n to X (see Section 25.7 for background on projections). Explicitly, (πX x)i = xi − PN

1

j=1

mj

n X

mj xj .

(13.10)

j=1

Problem 13.4 Show that (13.10) is the orthogonal projection operator from R3n to X. Find the orthogonal projection operator, πX ⊥ , from R3n to X ⊥ . Equation (13.10) implies that (πX x)i − (πX x)j = xi − xj . Hence the many-body potential (4.24) satisfies V (x) = V (πX x).

(13.11)

Equations (13.9) and (13.11) imply that the operator Hn given in (13.1) can be decomposed as Hn = H ⊗ 1X ⊥ + 1X ⊗ TCM (13.12) where

~2 ∆X + V (x) 2 is the Hamiltonian of the internal motion of the system, and H =−

TCM = −

~2 ∆ ⊥ 2 X

is the Hamiltonian of the motion of its centre-of-mass. Equation (13.12) is the centre-of-mass separation formula, and H is called the Hamiltonian in the centre-of-mass frame. It is a self-adjoint operator under the assumptions on the potentials mentioned above. It is the main object of study in many-body theory.

13.3 Break-ups Here we describe the kinematics of the break-up of an n-body system into noninteracting systems. First we introduce the notion of a cluster decomposition a = {C1 . . . Cs } for some s ≤ n. The Cj are non-empty, disjoint subsets of the set {1, . . . , n}, whose union yields the whole set:

176

13 General Theory of Many-particle Systems s [

j=1

Cj = {1, . . . , n}.

The subsets Cj are called clusters. An example of a cluster decomposition for n = 3 is a = {(12), (3)}. The number of clusters, s, in the decomposition a will be denoted by #(a). There is only one cluster decomposition with #(a) = 1, and one with #(a) = n. In the first case, the decomposition consists of a single cluster a = {(1 . . . n)}, and in the second case the clusters are single particles: a ¯ = {(1) . . . (n)}. In the first case the system is not broken up at all, while in the second case it is broken into the smallest possible fragments. To each cluster decomposition, a, we associate the intercluster potential X Vij (xi − xj ), Ia (x) := (ij)6⊂a

where the notation (ij) 6⊂ a signifies that the indices i and j belong to different clusters in the decomposition a. Similarly, we associate to a the intracluster potential X Vij (xi − xj ) Va (x) := (ij)⊂a

where (ij) ⊂ a signifies that i and j belong to the same cluster in a. Thus Ia (x) (resp. Va (x)) is the sum of the potentials between particles from different (resp. the same) clusters of a. The Hamiltonian of a decoupled system (in the total centre-of-mass frame) corresponding to a cluster decomposition, a, is Ha := −

~2 ∆X + Va (x) 2

acting on L2 (X). For a = {C1 , . . . , Cs }, the Hamiltonian Ha describes s noninteracting sub-systems C1 , . . . , Cs . For s = 1, Ha = H, and for s = N , 2 Ha = − ~2 ∆X . The operators Ha are also self-adjoint. Note that H = Ha + Ia .

(13.13)

If s > 1, then the system commutes with relative translations of the clusters C1 . . . Cs : T h H a = Ha T h , for h ∈ X satisfying

13.4 The HVZ Theorem

hi = hj

if

177

(ij) ⊂ a.

Here Th : ψ(x1 , . . . , xn ) → ψ(x1 + h1 , . . . , xN + hn ) for h = (h1 , . . . , hn ). As a result, one can show that the spectrum of Ha is purely essential. This is due to the fact that the clusters in a move freely. One can separate the centre-of-mass motions of the clusters C1 . . . Cs in a and establish a decomposition for Ha similar to that for Hn (equation (13.12)); we will do this later.

13.4 The HVZ Theorem In this section we formulate and prove the key theorem in the mathematical theory of n-body systems – the HVZ theorem. The letters here are the initials of W. Hunziker, C. van Winter, and G.M. Zhislin. This theorem identifies the location of the essential spectrum of the many-body Hamiltonian, H. Theorem 13.5 (HVZ Theorem) We have σess (H) = [Σ, ∞), where Σ := min Σa , #(a)>1

with

Σa := min(σ(Ha )).

Note that Σ is the minimal energy needed to break the system into independent parts. Proof. We begin by showing that σ(Ha ) ⊂ σ(H) for #(a) > 1. Suppose λ ∈ σ(Ha ). Then for any ε > 0, there is ψ ∈ L2 (X) with kψk = 1, such that k(Ha − λ)ψk < ε. Let h ∈ X satisfy hi = hj if (ij) ⊂ a, and hi 6= hj otherwise. For s > 0, let Tsh be the operator of coordinate translation by sh. Note that Tsh is an isometry. As remarked in the previous section, Tsh commutes with Ha , and so k(Ha − λ)Tsh ψk = kTsh (Ha − λ)ψk < ε. On the other hand, HTsh ψ → Ha Tsh ψ as s → ∞, because the translation Tsh separates the clusters in a as s → ∞. So for s sufficiently large, k(H − λ)Tsh ψk < ε. Since ε > 0 is arbitrary, and kTsh ψk = kψk = 1, we see λ ∈ σ(H). So we have shown σ(Ha ) ⊂ σ(H). As we remarked earlier, Ha has purely essential spectrum. In other words, σ(Ha ) = [Σa , ∞). Thus we have shown [Σ, ∞) ⊂ σ(H). It remains to prove that σess (H) ⊂ [Σ, ∞). To do this, we introduce a “partition of unity”, i.e., a family {ja } of smooth functions on X, indexed by all cluster decompositions, a, with #(a) > 1, such that

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13 General Theory of Many-particle Systems

X

#(a)>1

ja2 (x) ≡ 1.

(13.14)

We can use {ja } to decompose H into pieces which are localized in the supports of the ja , plus an error term: H=

X

#(a)>1

ja Hja −

X ~2 |∇ja |2 . 2

(13.15)

#(a)>1

Indeed, summing the identity [ja , [ja , H]] = ja2 H + Hja2 − 2ja Hja over a, and using (13.14), we obtain the relation known as the IMS formula (see [73]) X 1 (ja Hja + [ja [ja , H]]). H= 2 #(a)>1

Computing [ja , [ja , H]] = −~2 |∇ja |2 finishes the proof of (13.15). Now we construct an appropriate partition of unity, namely one satisfying min |xj − xk | ≥ ε|x|

(jk)6⊂a

|x| ≥ 1, x ∈ supp(ja )

for

(13.16)

for some ε > 0. Indeed, the sets Sa := {x ∈ X | |x| = 1; |xj − xk | > 0 ∀ (jk) 6⊂ a} form S an open cover of the unit sphere, S, of X (i.e., Sa are open sets and S ⊂ a Sa ). For each a, let χa be a smooth function supported in Sa , and equal to 1 in a slightly smallerPset (such that these smaller sets still cover S). Then the functions ja := χa /( χ2a )1/2 form a partition of unity on S, with supp(ja ) ⊂ Sa . In fact, (13.16) holds (with |x| = 1), because supp(ja ) is compact. We extend ja (x) to all of X by setting ja (x) := ja (x/|x|) for |x| > 1, and for |x| < 1, choosing any smooth extension of ja (x) which preserves (13.15). Thus the partition {ja } satisfies ja (λx) = ja (x)

for

|x| ≥ 1, λ ≥ 1

as well as (13.16). By (13.17), |∇ja (x)|2 → 0

as

|x| → ∞.

By (13.16), ja (H − Ha )ja = ja Ia ja → 0

as

Returning to (13.15), we conclude that X ja Ha ja + K H= #(a)>1

|x| → ∞.

(13.17)

13.5 Intra- vs. Inter-cluster Motion

179

where K is multiplication operator vanishing at infinity. An argument similar to the proof of the second part Theorem 6.16 shows that   X ja Ha ja  . σess (H) = σess  #(a)>1

Since H ≥ (inf σ(H))1 for any self-adjoint operator, H, we see hψ, ja Ha ja ψi = hja ψ, Hja ψi

≥ Σa kja ψk2 ≥ Σa kψk2

P for any ψ ∈ X (note P Σa ≤ 0). Thus ja Ha ja ≥ Σa , and therefore a ja Ha ja ≥ Σ, yielding σess ( a ja Ha ja ) ⊂ [Σ, ∞), and consequently σess (H) ⊂ [Σ, ∞). This completes the proof of the HVZ theorem. 

13.5 Intra- vs. Inter-cluster Motion As was mentioned in Section 13.3, the Hamiltonians Ha describing the system broken up into non-interacting clusters have purely essential spectra. This is due to the fact that the clusters in the decomposition move freely. To understand the finer structure of many-body spectra, we have to separate the centre-of-mass motion of the clusters, as we did with the centre-of-mass motion of the entire system. Proceeding as in Section 13.2, we define the subspaces X mj xj = 0 ∀ i} X a := {x ∈ X | j∈Ci

and Xa := {x ∈ X | xi = xj if (ij) ⊂ a}. Problem 13.6 Show that these subspaces are mutually orthogonal (X a ⊥ Xa ) and span X: X = X a ⊕ Xa . (13.18)

X a is the subspace of internal motion of the particles within the clusters of the decomposition a, and Xa is the subspace of the centre-of-mass motion of the clusters of a. As before, (13.18) leads to the decomposition L2 (X) = L2 (X a ) ⊗ L2 (Xa ) of L2 (X) and the related decomposition of the Laplacian on X: ∆X = ∆X a ⊗ 1Xa + 1X a ⊗ ∆Xa

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13 General Theory of Many-particle Systems

where ∆X a and ∆Xa are the Laplacians on the spaces X a and Xa (or L2 (X a ) and L2 (Xa )) in the metric (13.4). Again, if πX a is the orthogonal projection from X to X a , then Va (x) = Va (πX a x) and consequently we have the decomposition Ha = H a ⊗ 1Xa + 1X a ⊗ Ta ,

(13.19)

where

~2 ∆X a + Va (x) 2 is the Hamiltonian of the internal motion of the particles in the clusters of 2 a, and Ta = − ~2 ∆X a is the Hamiltonian of the centre-of-mass motion of the clusters in a. Applying the HVZ theorem inductively, we arrive at the following representation of the essential spectrum of H: [ [λ, ∞), σess (H) = Ha = −

λ∈τ (H)

where the discrete set τ (H), called the threshold set of H, is defined as [ [ σd (H a ) {0}, τ (H) := #(a)>1

the union of the discrete spectra of the break-up Hamiltonians and zero. The points of τ (H) are called the thresholds. Thus one can think of the essential spectrum of a many-body Hamiltonian H (in the centre-of-mass frame) as a union of branches starting at its thresholds and extending to infinity.

13.6 Exponential Decay of Bound States Theorem 13.7 If H has a bound state, Ψ (x), with an energy E < Σ (i.e. below the ionization threshold Σ, see Theorem 13.5), then Ψ (x) satisfies the exponential bound Z √ (13.20) |Ψ (x)|2 e2α|x| dx < ∞, ∀α < Σ − E. Proof. Let J be a real, bounded, smooth function supported in {|x| ≥ R}. Proceeding as in the prove of the HVZ theorem above, we show that there is ǫ = ǫ(R) → 0, as R → ∞, s.t. JHJ ≥ (Σ − ǫ)J 2 .

(13.21)

13.7 Remarks on Discrete Spectrum

181

Denote by ∇X the gradient on the space (13.5) with the metric (13.4). We assume now that ∇X J is supported in {R ≤ |x| ≤ 2R}. Let f be a bounded twice differentiable, positive function and define Hf := ef He−f . We compute Hf = H −

~2 [|∇X f |2 − ∇X f · ∇X − ∇X · ∇X f ]. 2

(13.22)

Then (Hf − E)Φ = 0, where Φ := ef Ψ , and therefore (Hf − E)JΦ = [Hf , J]Φ. On the other hand, by (13.22) and (13.21) and the fact that the operator ∇X f · ∇X + ∇X · ∇X f is anti-self-adjoint, we have RehJΦ, (Hf − E)JΦi = hJΦ, (H −

~2 |∇X f |2 − E)JΦi 2

≥ δkJΦk2 . 2

~ where δ := Σ − ǫ − E − supx∈supp J 2m |∇X f |2 . Then the last two equations imply δkJΦk2 ≤ RehJΦ, (Hf − E)JΦi ≤ kJΦkk[Hf , J]Φk.

Now we take for f √ a sequence of bounded functions approximating α(1 + |x|2 )1/2 , with α < Σ − E. Taking the limit in the last inequality gives (13.20). 

13.7 Remarks on Discrete Spectrum For n = 2, the results in Section 8.3, show that the discrete spectrum of H is finite if the potential V (x) is “short-range”, whereas a “long-range” attractive potential produces an infinite number of bound states. The borderline between short- and long-range potentials is marked by the asymptotic behaviour V (x) ∼ |x|−2 as |x| → ∞ (which is different than the borderline asymptotic behaviour of |x|−1 which we encountered in scattering theory in Chapter 9). For n > 2, however, the question of whether σd (H) is finite or infinite cannot be answered solely in terms of the asymptotic fall-off of the intercluster potentials, Ia (x); the nature of the threshold Σ at the bottom of the essential spectrum plays a decisive role. Now we comment on bound states of molecules with mobile nuclei. We restrict our attention to the case where Σ is a two-cluster threshold. This means that for energy Σ and slightly above, the system can only disintegrate into two bound clusters, C1 and C2 . This fits the case of molecules with dynamic nuclei which break up into atoms or stable ions. However, our next assumption, that the lowest energy break-ups originate from charged clusters (having, of course, opposite total charges), is not realistic. Ignoring particle statistics, the disintegration of the system into the clusters corresponding to a partition a can be represented by a product wave function

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13 General Theory of Many-particle Systems

φ(x) = Ψ (xa )f (xa ),

(13.23)

a

a

where Ψ is the eigenfunction of H with eigenvalue Σ, H Ψ = ΣΨ , and xa and xa denote the components of x along the subspaces Xa and X a respectively. Here f is chosen so that hf, Ta f i is arbitrarily small. The condition that Σ is a two-cluster threshold means that Σ is a discrete eigenvalue of H a , and as a consequence it can be shown that Ψ (xa ) decays exponentially as |xa | → ∞: |Ψ (xa )| ≤ (const)e−α|x

a

|

(13.24)

for some α > 0. Using states of the form (13.23) as trial states to make hφ, Hφi < Σ, we can show the existence of an infinite number of bound states, provided that the lowest energy break-ups originate from charged clusters (of opposite total charges). This means that the intercluster potential Ia (x) =

X

i∈C1 ,k∈C2

ei ek , |xi − xk |

P P satisfies ( i∈C1 ei )( k∈C2 ek ) < 0. Using the exponential bound (13.24), it follows that |hφ, (Ia (x) − Ia (xa ))φiL2 (X a ) | ≤ (const)|xa |−2 . Observe that Ia (xa ) = −q/|xa | with q > 0. Since H = H a ⊗ 1a + 1a ⊗ Ta + Ia (see (13.13) and (13.19)) and H a φ = Σφ, the last inequality implies     q hφ, (H − Σ)φi ≤ u, Ta − + (const)|xa |−2 f . |xa | L2 (Xa ) As in Section 8.3, we let f ∈ C0∞ (R3 ) satisfy kf k = 1 and supp(f ) ⊂ {xa | 1 < |xa | < 2}. Then the functions fk (xa ) = k −3/2 f (k −1 xa ),

k = 1, 2, 4, 8, . . .

are orthonormal, and have disjoint supports. Thus the corresponding trial states φk (x) = Ψ (xa )fk (xa ) satisfy hφk , Hφm i = 0 for k 6= m, and hφk , (H − Σ)φk i ≤ −(const)

q 1 + (const) 2 < 0, k k

for some positive constants, if n is sufficiently large. We can now apply the min-max principle (see Section 8.1) to conclude that H possesses infinitely many discrete eigenvalues below the threshold Σ.

13.8 Scattering States

183

13.8 Scattering States Unlike in the one-body case, the many-body evolution ψ = e−iHt/~ ψ0 behaves asymptotically as a superposition of several (possibly infinitely many) free evolutions, corresponding to different scenarios of the scattering problem. These scenarios, called scattering channels, can be described as follows. For a given scattering channel, the system is broken into non-interacting clusters C1 , . . . , Cs corresponding to some cluster decomposition a, and its motion in each cluster is restricted to a cluster bound state. (If a given decomposition contains a cluster for which the cluster Hamiltonian has no bound state, then this cluster decomposition does not participate in the formation of scattering channels.) In other words, a scattering channel is specified by a pair (a, m), where a is a cluster decomposition such that the operator H a has some discrete spectrum, and m labels the the eigenfunction ψ a,m of H a (we suppose that for fixed a, the ψ a,m are chosen orthonormal). The evolution in the channel (a, m) is determined by the pair (Ha,m , Ha,m ) where Ha,m := ψ a,m ⊗ L2 (Xa ) is the channel Hilbert space, and Ha,m := Ha ⌈Ha,m = E a,m + Ta where E a,m is the eigenvalue of H a corresponding to the eigenfunction ψ a,m . Thus the channel evolution is given by e−iHa,m t/~ (ψ a,m ⊗ f ) for f ∈ L2 (Xa ). As in the one-body case (see Chapter 9), the existence and asymptotic form of scattering states for t → ∞ depends crucially of the rate at which the potentials tend to zero for large separations. We suppose that the intercluster potentials satisfy ∂ (α) Ia (x) ≤ (const)|x|−µ−|α| (13.25) a as |x|a → ∞, for |α| ≤ 2. Here |x|a is the intercluster distance, i.e. |x|a := min |xj − xk |. (jk)6⊂a

As in the one-body case, µ = 1 marks the borderline between short-range and long-range systems, for which the scattering theory is quite different. Short-range systems: µ > 1. In this case, for a given scattering channel, (a, m), the wave operator + Ω(a,m) = s-limt→∞ eiHt/~ e−iHa,m t/~

can be shown to exist on Ha,m . The proof is similar to that for the one-body case (see Chapter 9). This wave operator maps free channel evolutions to full evolutions: setting

184

13 General Theory of Many-particle Systems + ψ := Ω(a,m) (ψ a,m ⊗ f ),

we have e−iHt/~ ψ → e−iHa,m t/~ (ψ a,m ⊗ f )

+ as t → ∞. The wave operator Ωa,m is an isometry from Ha,m to H. Moreover, + + the ranges Ha,m := Ran(Ωa,m ) satisfy + + Ha,m ⊥ Hb,n

if

a 6= b

or

m 6= n

which follows from the fact that lim he−iHa,m t/~ (ψ a,m ⊗ f ), e−iHb,n t/~ (ψ b,n ⊗ g)i = 0

t→∞

if a 6= b or m 6= n. Therefore, the outgoing scattering states form a closed subspace M + H+ := Ha,m ⊂ L2 (X). a,m

Under the condition (13.25) with µ > 1, it has been proved that the property of asymptotic completeness holds – i.e., that H+ = L2 (X) (see Chapter 27 for references). In other words, as t → ∞, every state approaches a superposition of channel evolutions and bound states (the bound states are the channel with #(a) = 1). Long-range systems: µ ≤ 1. As in the one-body case, it is necessary to modify the form of the channel evolutions in the long-range case. The evolution e−iHa,m t/~ with Ha,m = λa,m + Ta is replaced by e−iHa,m t/~−iαa,t (pa ) where pa = −i~∇X a . Here αa,t (pa ) is an adiabatic phase, arising from the fact that classically, the clusters are located at xa = pa t(1 + O(t−µ )) as t → ∞. The modification αa,t (pa ), whose precise form we will not give here, is similar to that for the one-body case (see Chapter 9). We refer the interested reader to the references listed in Chapter 27 for further details. We remark only that with this modification in place, the existence of the (modified) wave operators, and asymptotic completeness, have been proved for µ not too small.

14 Self-consistent Approximations

Even for a few particles the Schr¨odinger equation is prohibitively difficult to solve. Hence it is important to have approximations which work in various regimes. One such approximation, which has a nice unifying theme and connects to large areas of physics and mathematics, is the one approximating solutions of n-particle Schr¨odinger equations by products of n one-particle functions (i.e. functions of 3 variables). This results in a single nonlinear equation in 3 variables, or several coupled such equations. The trade-off here is the number of dimensions for the nonlinearity. This method, which goes under different names, e.g. the mean-field or self-consistent approximation, is especially effective when the number of particles, n, is sufficiently large.

14.1 Hartree, Hartree-Fock and Gross-Pitaevski equations For simplicity we consider a system of n identical, spinless bosons. It is straightforward to include spin. To extend our treatment to fermions requires a simple additional step (see discussion below). The Hamiltonian of the system of n bosons of mass m, interacting with each other and moving in an external potential V is Hn :=

n X

hx i +

j=1

1X v(xi − xj ), 2

(14.1)

i6=j

2

~ where hx = − 2m ∆x + V (x), acting on the state space sn1 L2 (Rd ), d = 1, 2, 3. Here v is the interaction potential, and s is the symmetric tensor product. As we know, the quantum evolution is given by the Schr¨odinger equation

i~

∂Ψ = Hn Ψ. ∂t

© Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_14

186

14 Self-consistent Approximations

This is an equation in 3n + 1 variables, x1 , ..., xn and t, and it is not a simple matter to understand properties of its solutions. We give a heuristic derivation of the mean-field approximation for this equation. A rigorous derivation is sketched in Section 20.6. First, we observe that the potential experienced by the i-th particle is X W (xi ) := V (xi ) + v(xi − xj ). j6=i

Assuming v(0) is finite, it can be re-written, modulo the constant term v(0), which we neglect, as W (xi ) = V (xi ) + (v ∗ ρmicro )(xi ). Here, recall, f ∗ g denotes the convolution of the functions f and g, and ρmicro stands for the (operator of) microscopic density of the n particles, defined by X δ(x − xj ). ρmicro (x, t) := j

Note that the average quantum-mechanical (QM) density in the state Ψ is hΨ, ρmicro (x, t)Ψ i = ρQM (x, t)

R where ρQM (x, t) := n | Ψ (x, x2 , ..., xn , t) |2 dx2 ...dxn , the one-particle density in the quantum state Ψ . In the mean-field theory, we replace ρmicro (x, t) with a continuous function, MF ρ (x, t), which is supposed to be close to the average quantum-mechanical density, ρQM (x, t), and which is to be determined later. Consequently, it is assumed that the potential experienced by the i-th particle is W MF (xi ) := V (xi ) + (v ∗ ρMF )(xi ). Thus, in this approximation, the state ψ(x, t) of the i-th particle is a solution MF of the following one-particle Schr¨odinger equation i~ ∂ψ )ψ ∂t = (h + v ∗ ρ 2 ~ micro (x, t) is where, recall, h = − 2m ∆x + V (x). OfR course, the integral of ρ equal to the total number of particles, R3R ρmicro (x, t)dx = n. We require that the same should be true for ρMF (x, t): R3 ρMF (x, t)dx = n. We normalize the one-particle state, ψ(x, t), in the same way Z |ψ(x, t)|2 dx = n. (14.2) R3

Consider a situation in which we expect all the particles to be in the same state ψ. Then it is natural to take ρMF (x, t) = |ψ(x, t)|2 . In this case ψ solves the nonlinear equation i~

∂ψ = (h + v ∗ |ψ|2 )ψ. ∂t

This nonlinear evolution equation is called the Hartree equation (HE).

(14.3)

14.1 Hartree, Hartree-Fock and Gross-Pitaevski equations

187

If the inter-particle interaction, v, is significant only at very short distances (one says that v is very short range, which technically can be quantified by assuming that the “particle scattering length” a is small), we can replace v(x) → 4πaδ(x), and Equation (14.3) becomes ∂ψ = hψ + 4πa|ψ|2 ψ (14.4) ∂t (with the normalization (14.2)). This equation is called the Gross-Pitaevski equation (GPE) or nonlinear Schr¨ odinger equation. It is a mean-field approximation to the original quantum problem for a system of n bosons. The Gross-Pitaevski equation is widely used in the theory of superfluidity, and in the theory of Bose-Einstein condensation (see Appendix 14.2). Reconstruction of solutions to the n-particle Schr¨ odinger equation, i~

∂Ψ = Hn Ψ, Ψ |t=0 = ⊗n1 ψ0 . (14.5) ∂t How do solutions of (HE) or (GPE) relate to solutions of the original manybody Schr¨odinger equation (14.5)? It is shown rigorously (see [90, 91, 238]) that the solution of equation (14.5) satisfies, in some weak sense and and in an appropriate regime of n → ∞ and a → 0 with n4πa =: λ fixed, i~

Ψ − ⊗n1 ψ → 0

where ψ satisfies, depending on the limiting regime, either (HE) or (GPE) with initial condition ψ0 . For the mean-field regime (replacing for the moment v by gv) of n → ∞ and g → 0, with ng fixed, we have (HE) (see [159, 134, 117, 123, 33, 34]). It is not obvious how to extend the mean-field argument above to fermions. To do this, we explore another way to derive formally the Hartree equation from the n-particle Schr¨odinger one. It goes as follows. For complex ψ, we define dψ¯ E(ψ) = (dψ1 + idψ2 )E(ψ), where ψ = ψ1 + iψ2 and call the critical point equation dψ¯ E(u∗ ) = 0 the Euler-Lagrange equation. Now, observe

Proposition 14.1 The Schr¨ odinger equation is the Euler-Lagrange equation for stationary points of the action functional Z  − ~ ImhΨ, ∂t Ψ i − hΨ, Hn Ψ i dt, (14.6) S(Ψ ) :=

The proof is an exercise in the standard variational calculus – see Chapter 26 for material on variational calculus. We sketch it here: RR Proof (Sketch of proof ). We write S(Ψ ) := − Ψ¯ AΨ dxdt, where A := i~∂t + ¯ and Hn . We consider S(Ψ ) as a functional of Ψ and Ψ¯ , S(Ψ ) ≡ S(Ψ, Ψ) ∞ d ¯ differentiate it in Ψ along ξ ∈ C0 (R × R). We obtain Z Z Z Z ∂ ¯ (Ψ + λξ)AΨ = ξAΨ ∂λ λ=0 Since ξ ∈ C0∞ (Rd × R) is arbitrary, this gives the Schr¨odinger equation (14.5).

188

14 Self-consistent Approximations

Problem 14.2 Review the basic facts of variational calculus. Now, for bosons, we consider the the action functional (14.6) the space (not linear!) {Ψ := ⊗n1 ψ | ψ ∈ H 1 (R3 )},

(14.7)

where (⊗n1 ψ) is the function of 3n variables defined by (⊗n1 ψ)(x1 , ..., xn ) := ψ(x1 )...ψ(xn ). A simple computation gives Proposition 14.3 Let SH (ψ) := n−1 S(⊗n1 ψ) (‘H’ stands for Hartree), with kψk = 1, and w := (n − 1)v. Then we have Z Z  SH (ψ) = − ~ Imhψ, ∂t ψi − |∇ψ|2 − V |ψ|2 R3

1 − |ψ|2 w ∗ |ψ|2 dx. 2

(14.8)

Recall that the regime in which w := (n − 1)v = O(1) is called the mean-field regime. Proof. First, wePcompute hΨ, ∂t Ψ i for Ψ := ⊗n1 ψ. Denote ψj := ψ(xj ). The Q relation ∂t Ψ = i ( j6=i ψj )∂t ψi gives Z Y X XZ Y 2(n−1) ¯ ψ¯i ∂t ψi kψk ψj ( ψj )∂t ψi = hΨ, ∂t Ψ i = i

j

i

j6=i

= nkψk

2(n−1)

Z

¯ tψ ψ∂

Similarly, for Ψ := ⊗n1 ψ, we compute n Z n X X (−∆xi + V (xi ))Ψ i = (|∇xi Ψ |2 + V (xi )|Ψ |2 ) hΨ, i=1

i=1

=

n X i=1

kψk

2(n−1)

= nkψk2(n−1)

Z

Z

(|∇xi ψi |2 + V (xi )|ψi |2 )

(|∇ψ|2 + V (x)|ψ|2 )

Finally, we compute the particle pair interaction terms Z Y 1X 1X hΨ, ( v(xi − xj ))Ψ i = ( |ψk |2 )|ψi |2 v(xi − xj )|ψj |2 2 2 i6=j i6=j k6=i,j Z Z 1X 2(n−2) kψk |ψi |2 v(xi − xj )|ψj |2 = 2 i6=j Z Z 1 2(n−2) = n(n − 1)kψk |ψ(x)|2 v(x − x′ )|ψ(x′ )|2 2 Collecting the terms above and using that v = (n − 1)w, we arrive at (14.8).

14.1 Hartree, Hartree-Fock and Gross-Pitaevski equations

189

The Euler-Lagrange equation for stationary points of the action functional (14.8) considered on the first set of functions is i~

∂ψ = (h + w ∗ |ψ|2 )ψ. ∂t

(14.9)

Here we assume the normalization kψk = 1. This nonlinear evolution equation is called the Hartree equation (HE). It is convenient to pass to the normalization kψk = n − 1 ≈ n, which leads to the equation i~

∂ψ = (h + v ∗ |ψ|2 )ψ, ∂t

(14.10)

where hx = −∆x + V (x). It is relatively easy to extend this approach to fermions. For (spinless) fermions, we consider the action S(Ψ ) on the following function space {Ψ := ∧n1 ψj | ψi ∈ H 1 (R3 ) ∀i = 1, ..., n},

(14.11)

where ∧n1 ψj := det[ψi (xj )] is the determinant of the n × n matrix [ψi (xj )]. It is called the Slater determinant. Then the Euler-Lagrange equation for S(Ψ ) on the latter set gives a system of nonlinear, coupled evolution equations i~

X X ∂ψj (v ∗ ψi ψ¯j )ψi , |ψi |2 )ψj − = (h + v ∗ ∂t i i

(14.12)

for the unknowns ψ1 , ..., ψn . This systems plays the same role for fermions as the Hartree equation does for bosons. It is called the Hartree-Fock equations (HFE). Finally, we mention another closely related nonlinear equation: the GinzburgLandau equations of superconductivity. Properties of (HE), (HFE) and (GPE). The Hartree and Gross-Pitaevski equations have the following general features 1. For space dimensions d = 2, 3 and assuming v is positive definite, (HE) and (HFE) have solutions globally in time; for (GPE) solutions exist globally (in time) if a > 0, but blow-up for certain initial conditions in finite time if a < 0. 2. (HE), (HFE) and (GPE) are Hamiltonian systems (see Section 20.6). 3. (HE), (HFE) and (GPE) are invariant under the gauge transformations, ψ(x) → eiα ψ(x), α ∈ R, and, for v = 0, the translations, ψ(x) → ψ(x+ y), y ∈ R, and the Galilean transformations, v ∈ R3 , ψ(x) → ei(mv·x+

mv2 t )/~2 2

ψ(x − vt).

190

14 Self-consistent Approximations

4. (HFE) is invariant under time-independent unitary transformations of {ψ1 , ..., ψn }. 5. The energy, E(ψ), and the number of particles, N (ψ), (see below) are conserved quantities. Moreover, (HFE) conserves the inner products, hψi , ψj i, ∀i, j. The fourth item shows that the natural object for (HFE) Pis the subspace spanned by {ψi }, or the corresponding projection γ := i |ψi ihψi |. Then (HFE) can be rewritten as an equation for γ: i

∂γ = [h(γ), γ] ∂t

(14.13)

P where h(γ) := h + v ∗ ργ − ex(γ), with ργ (x, t) := γ(x, x, t) = i |ψi (x, t)|2 and ex(γ) is P the operator with the integral kernel ex(γ)(x, y, t) := v(x − y)γ(x, y, t) = i (ψ¯i v ∗ ψi ). (We write ex(γ) as ex(γ) =: v#γ.) Here A(x, y) stands for the integral kernel of an operator A. This can be extended to arbitrary non-negative, trace-class operators (i.e. density operators γ, see Section 18.1). We address this in Section 18.6. To fix ideas, we will hereafter discuss mainly (GPE). For (HE) and (HFE) the results should be appropriately modified. For (GPE) the energy functional is  Z  2 ~ E(ψ) := |∇ψ|2 + V |ψ|2 + 2πa|ψ|4 dx. 2m R3 The number of particles for (GPE) and (HE) is given by Z N (ψ) := |ψ|2 dx R3

P R

while for (HFE), by N (ψ) := i R3 |ψi |2 dx. Note that the energy and number of particle conservation laws are related to the time-translational and gauge symmetries of the equations, respectively. The above notions of the energy and number of particles are related to corresponding notions in the original microscopic system. Indeed, let Ψ := √1 ⊗n ψ. Then 1 n 1
hΨ, Hn Ψ i = E(ψ) + O n where E(ψ) is the energy for (HE) and Z Z n | Ψ (x1 , ..., xn ) |2 dx1 ...dxn = | ψ(x) |2 dx. The notion of bound state can be extended to the nonlinear setting as follows. The bound states are stationary solutions of (HE) or (GPE) of the form ψ(x, t) = φµ (x)eiµt

14.1 Hartree, Hartree-Fock and Gross-Pitaevski equations

191

where the profile φµ (x) is in H 2 (R3 ). Note that the profile φµ (x) satisfies the stationary Gross-Pitaevski equation: hφ + 4πa|φ|2 φ = −~µφ

(14.14)

(we consider here (GPE) only). Thus we can think of the parameter −µ as a nonlinear eigenvalue. A ground state is a bound state such that the profile φµ (x) minimizes the energy for a fixed number of particles: φµ minimizes E(ψ)

under N (ψ) = n

(see Chapter 26 which deals with variational, and in particular minimization problems). Thus the nonlinear eigenvalue µ arises as a Lagrange multiplier from this constrained minimization problem. In Statistical Mechanics µ is called the chemical potential (the energy needed to add one more particle/atom, see Section 20.7). Remark 14.4 1. Mathematically, the ground state can be also defined as a stationary solution with a positive (up to a constant phase factor) profile, ψ(x, t) = φµ (x)eiµt with φµ (x) > 0. Let δ(µ) := kφµ k2 . Then we have (see [142]) δ ′ (µ) > 0 =⇒ φµ minimizes E(ψ) under N (ψ) = n. 2. The Lagrange multiplier theorem in Section 26.5 implies that the ground state profile φµ is a critical point of the functional Eµ (ψ) := E(ψ) + ~µN (ψ). In fact, φµ is a minimizer of this functional under the condition N (ψ) = n. If φµ is the ground state of (GPE), then ⊗n1 φµ is close to the ground state of the n−body Hamiltonian describing the Bose-Einstein condensate (see [74] for a review, and [209, 210] and the Appendix below for rigorous results). It is known that for natural classes of nonlinearities and potentials V (x) there is a ground state. Three cases of special interest are 2

~ ∆ + V (x) has a ground state, and 2m 1. h := − 2m ~2 n|a| ≪ 1 n|a| ≫ 1, and a < 0 2. V has a minimum, 2m 2 ~ 3. V (x) → ∞ as |x| → ∞ (i.e. V (x) is confining) and a > 0.

(The first and third cases are straightforward and the second case requires some work [105, 234, 11].) Stability. We discuss now the important issue of stability of stationary solutions under small perturbations. Namely, we want to know how solutions of our equation with initial conditions close to a stationary state (i.e. small perturbations of φµ (x)) behave. Do these solutions stay close to the stationary

192

14 Self-consistent Approximations

state in question, do they converge to it, or do they depart from it? This is obviously a central question. This issue appeared implicitly in Section 6.4 (and in a stronger formulation in Chapter 9) but has not been explicitly articulated yet. This is because the situation in the linear case that we have dealt with so far is rather straightforward. On the other hand, in the nonlinear case, stability questions are subtle and difficult, and play a central role. We say that a stationary solution, φµ (x)eiµt , is orbitally (respectively, asymptotically) stable if for all initial conditions sufficiently close to φµ (x)eiα (for some constant α ∈ R), the solutions of the evolution equation under consideration stay close (respectively, converge in an appropriate norm) to a ′ nearby stationary solution (times a phase factor), φµ′ (x)ei(µ t+β(t)) . Here µ′ is usually close to µ, and the phase β depends on time, t. The phase factors come from the fact that our equations have gauge symmetry: if ψ(x, t) is a solution, then so is eiα ψ(x, t) for any constant α ∈ R. One should modify the statement above if other symmetries are present. The notion of orbital stability generalizes the classical notion of Lyapunov stability, well-known in the theory of dynamical systems, to systems with symmetries. For the linear Schr¨odinger equation, all bound states, as well as stationary states corresponding to embedded eigenvalues, are orbitally stable. But they are not asymptotically stable in general. For most nonlinear evolution equations in unbounded domains, the majority of states are not even Lyapunov/orbitally stable. For (GPE), if V → ∞ as |x| → ∞ (i.e. V is confining), the ground states are orbitally stable, but not asymptotically stable. If V → 0 as |x| → ∞, the ground states can be proved to be asymptotically stable in some cases (see [277, 293, 145, 127, 128, 72] and references therein).

14.2 Appendix: BEC at T=0 In this appendix we consider briefly the phenomenon of Bose-Einstein condensation, predicted by Einstein in 1925 on the basis of analysis of ideal bose gases and experimentally discovered 70 years later in 1995 in real gases by two groups, one led by Wieman and Cornell at Boulder, and another by Ketterle at MIT. The Gross-Pitaevskii equation arises in the description of this phenomenon. We concentrate on zero temperature. First we consider a system of n non-interacting bosons in an external potential V . The state space of such a system is the Hilbert space sn1 L2 (Rd ), d = 1, 2, 3, and the Hamiltonian operator is  n  X ~2 ∆x + V (xi ) . − Hn := 2m i j=1 (acting on sn1 L2 (Rd )). By separation of variables, the lowest energy for this Hamiltonian is ne0 , where e0 is the lowest energy for the one-particle

14.2 Appendix: BEC at T=0

193

2

~ Schr¨odinger operator − 2m ∆x + V (x). The corresponding eigenfunction – the ground state of Hn – is given by sn1 φ0 , where φ0 is the ground state of the ~2 ∆x + V (x). And that’s it – at zero temperature, and in the operator − 2m ground state, all particles are in the same state φ0 ! Now we consider a system of n interacting bosons subject to an external potential V , which is described by the Hamiltonian  n  X ~2 1X − Hn := ∆xi + V (xi ) + v(xi − xj ), 2m 2 j=1 i6=j

acting on the Hilbert space sn1 L2 (Rd ). Here v is the potential of interaction between the particles. Let Φn,0 (x1 , . . . , xn ) be a ground state of Hn . How do we tell if in this state all (or the majority of) the particles are individually in some one- particle state, say φ0 ? To begin with we would like to describe, say, the coordinate or momentum distributions for a single particle. To extract one-particle information from Φn,0 , we use the information reduction principle elucidated in Section 19.1: we pass to density operators (Φn,0 → PΦn,0 = the rank 1 orthogonal projection on the state Φn,0 ) and contract (n−1)−particle degrees of freedom. This leads to the one-particle density operator: γ1n := T rn−1 PΦn,0 where T rn−1 is the trace over n − 1 of the 3 dimensional coordinates (see Section 19.1). The one-particle density matrix satisfies • • •

0 ≤ γ1n ≤ 1 T rγ1n = 1 Φn,0 = ⊗n1 φ0 ⇔ γ1n = Pφ0

Let λ1 ≥ λ2 ≥ · · · λj ≥ · · · be the eigenvalues of γn , counting their multiplicities, and let χ1 , χ2 , · · · χj , · · · be corresponding eigenfunctions. We have the spectral decomposition γ1n =

∞ X

∞ X

λj Pχj ,

λj = 1

j=1

j=1

(see Mathematical Supplement, Section 25.11).The eigenvalues λj are interpreted as the probabilities for a single particle to be in the states χj . The Penrose-Onsager criterion of BEC says that the property of the ground state Φn,0 of the Hamiltonian Hn max eigenvalue of γ1n → 1 as n → ∞

(P OC)

corresponds to 100% condensation. The criterion (POC) implies γ1n − Pχ1 → 0

as

n→∞

194

14 Self-consistent Approximations

so that for large n, almost all the particles are in the single state χ1 . A rigorous proof of BEC in the Gross-Pitaevski limit, in which the number of particles n → ∞ and the scattering length a → 0, so that na =: λ/(4π) is fixed, is given in [207] (see also [209]). Moreover, they show that in the trace norm, γ1n − PφGP →0 n where φGP is the Gross-Pitaevski ground state. They also prove convergence n for the ground state energies.

15 The Feynman Path Integral

In this chapter, we derive a convenient representation for the integral kernel of the Schr¨odinger evolution operator, e−itH/~ . This representation, the “Feynman path integral”, will provide us with a heuristic but effective tool for investigating the connection between quantum and classical mechanics. This investigation will be undertaken in the next section.

15.1 The Feynman Path Integral Consider a particle in Rd described by a self-adjoint Schr¨odinger operator H =−

~2 ∆ + V (x). 2m

Recall that the dynamics of such a particle is given by the Schr¨odinger equation ∂ψ i~ = Hψ. ∂t Recall also that the solution to this equation, with the initial condition ψ|t=0 = ψ0 , is given in terms of the evolution operator U (t) := e−iHt/~ as ψ = U (t)ψ0 . Our goal in this section is to understand the evolution operator U (t) = e−iHt/~ by finding a convenient representation of its integral kernel. We denote the integral kernel of U (t) by Ut (y, x) (also called the propagator from x to y). A representation of the exponential of a sum of operators is provided by the Trotter product formula (Theorem 15.2) which is explained in Section 15.3 at the end of this chapter. The Trotter product formula says that ~2 t

e−iHt/~ = ei( 2m ∆−V t)/~ = s-limn→∞ Knn © Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_15

196

15 The Feynman Path Integral i~t

iV t

where Kn := e 2mn ∆ e− ~n . Let Kn (x, y) be the integral kernel of the operator Kn . Then by Proposition 25.12, Z Z Ut (y, x) = lim · · · Kn (y, xn−1 ) · · · Kn (x2 , x1 )Kn (x1 , x)dxn−1 · · · dx1 . n→∞

(15.1)

Now (see Section 25.3) i~t∆

Kn (y, x) = e 2mn (y, x)e−

iV (y)t ~n

since V , and hence e−iV t/n~ , is a multiplication operator (check this). Using the expression (2.23), and plugging into (15.1) gives us Ut (y, x) = lim

n→∞

where Sn :=

Z

···

Z

e

iSn /~



2πi~t mn

−nd/2

dx1 · · · dxn−1

n−1 X k=0

(mn|xk+1 − xk |2 /2t − V (xk+1 )t/n)

with x0 = x, xn = y. Define the piecewise linear function φn such that φn (0) = x, φn (t/n) = x1 , · · · , φn (t) = y (see Fig. 15.1).

y

φ

x

t/n 2t/n

(n−1)t/n t

s

Fig. 15.1. Piecewise linear function. Then Sn =

n−1 X k=0

m

 |φn ((k + 1)t/n) − φn (kt/n)|2 − V (φ ((k + 1)t/n)) t/n. n 2(t/n)2

Note that Sn is a Riemann sum for the classical action Z tn o m ˙ S(φ, t) = |φ(s)|2 − V (φ(s)) ds 2 0

15.1 The Feynman Path Integral

197

of the path φn . So we have shown Ut (y, x) = lim

n→∞

Z

n Px,y,t

eiSn /~ Dφn

(15.2)

n where Px,y,t is the (n − 1)-dimensional space of paths φn with φn (0) = x, φn (t) = y, and which are linear on the intervals (kt/n, (k + 1)t/n) for k = −nd/2 0, 1, . . . , n − 1, and Dφn = ( 2πi~t dφn (t/n) · · · dφn ((n − 1)t/n). nm ) Heuristically, as n → ∞ φn approaches a general path, φ, from x to y (in time t), and Sn → S(φ). Thus we write

Z

Ut (y, x) =

eiS(φ,t)/~ Dφ.

(15.3)

Px,y,t

Here Px,y,t is a space of paths from x to y, defined as Px,y,t := {φ : [0, t] → Rd |

Z

0

t

˙ 2 < ∞, φ(0) = x, |φ|

φ(t) = y}.

This is the Feynman path integral. It is not really an integral, but a formal expression whose meaning is given by (15.2). Useful results are obtained nonrigorously by treating it formally as an integral. Answers we get this way are intelligent guesses which must be justified by rigorous tools. n Note that Px,y,t is an (n − 1)-dimensional sub-family of the infiniten 2n n dimensional space Px,y,t . It satisfies Px,y,t ⊂ Px,y,t and limn→∞ Px,y,t = Px,y,t in some sense. We call such subspaces finite dimensional approximations of Px,y,t . In (non-rigorous) computations, it is often useful to use finitedimensional approximations to the path space other than the polygonal one above. We can construct more general finite-dimensional approximations as follows. Fix a function φxy ∈ Px,y,t . Then Px,y,t = φxy + P0,0,t . Note P0,0,t is a Hilbert space. Choose an orthonormal basis {ξj } in P0,0,t and define n P0,0,t := span {ξj }n1 and n n Px,y,t := φxy + P0,0,t . n Then Px,y,t is a finite dimensional approximation of Px,y,t . Typical choices of φxy and {ξj } are

1. φxy is piecewise linear and {ξj } are “splines”. This gives the polygonal approximation introduced above.

198

15 The Feynman Path Integral

2. φxy is a classical path (a critical point of the action functional S(φ)) and {ξj } are eigenfunctions of the Hessian of S at φxy (these notions are described in the Supplement on the calculus of variations, Chapter 26). n In this case, if η ∈ P0,0,t , then we can represent it as η=

n X

aj ξj ,

j=1

and we have Dη =



2πit~ m

−d/2 

It is reasonable to expect that if Z lim n→∞

2πn t

r

m ~

n Y n

daj .

j=1

eiS(φ,t)/~ Dφ

n Px,y,t

n exists, then it is independent of the finite-dimensional approximation, Px,y,t , that we choose.

Problem 15.1 1. Compute (using (15.3) and a finite-dimensional approximation of the path space) Ut for a) V (x) = 0 (free particle) 2 2 b) V (x) = mω 2 x (harmonic oscillator in dimension d = 1). 2. Derive a path integral representation for the integral kernel of e−βH/~ . 3. Use the previous result to find a path integral representation for Z(β) := tr e−βH/~ (hint: you should arrive at the expression (16.11)).

15.2 Generalizations of the Path Integral Here we mention briefly two extensions of the Feynman path integral we have just introduced. 1. Phase-space path integral: Z Ut (y, x) =

Px,y,t ×

anything

ei

Rt 0

˙ (φπ−H(φ,π))/~

where D /π is the path measure, normalized as Z Rt 2 i e− 2 0 kπk D /π = 1

DφD /π

15.3 Mathematical Supplement: the Trotter Product Formula

199

(in QM, /d 3 p = d3 p/(2π)3/2 ). To derive this representation, we use the Trotter product formula, the expression e−iǫH ≈ 1 − iǫH for ǫ small, and the symbolicR (pseudodifferential) composition formula. Unlike the representation eiS/~ Dφ, this formula holds also for more complicated H, which are not quadratic in p ! 2. A particle in a vector potential A(x). In this case, the Hamiltonian is H(x, p) =

1 (p − eA(x))2 + V (x) 2m

and the Lagrangian is L(x, x) ˙ =

m 2 x˙ − V (x) + ex˙ · A(x). 2

The propagator still has the representation Z eiS(φ)/~ Dφ, Ut (y, x) = Px,y,t

but with S(φ) =

Z

t

˙ L(φ, φ)ds =

0

Z

0

t

m ( φ˙ 2 − V (φ))ds + e 2

Z

0

t

˙ A(φ) · φds.

Since A(x) does not commute with ∇ in general, care should be exercised in computing a finite-dimensional approximation: one should take X

or

X1

2

and not X

1 A( (xi + xi+1 )) · (xi+1 − xi ) 2 (A(xi ) + A(xi+1 )) · (xi+1 − xi )

A(xi ) · (xi+1 − xi ) or

X

A(xi+1 ) · (xi+1 − xi ).

15.3 Mathematical Supplement: the Trotter Product Formula Let A, B, and A+B be self-adjoint operators on a Hilbert space H. If [A, B] 6= 0, then ei(A+B) 6= eiA eiB in general. But we do have the following. Theorem 15.2 (Trotter product formula) Let either A and B be bounded, or A, B, and A + B be self-adjoint and bounded from below. Then for Re(λ) ≤ 0, A B eλ(A+B) = s − lim (eλ n eλ n )n n→∞

200

15 The Feynman Path Integral

Remark 15.3 The convergence here is in the sense of the strong operator topology. For operators An and A on a Hilbert space H, such that D(An ) = D(A), An → A in the strong operator topology (written s-limn→∞ An = A) iff kAn ψ − Aψk → 0 for all ψ ∈ D(A). For bounded operators, we can take norm convergence. In the formula above, we used a uniform decomposition of the interval [0, 1]. The formula still holds for a non-uniform decomposition. Proof for A,B bounded: We can assume λ = 1. Let Sn = e(A+B)/n and Tn = eA/n eB/n . Now by “telescoping”, Snn − Tnn = Snn − Tn Snn−1 + Tn Snn−1 + · · · − Tnn =

n−1 X k=0

Tnk (Sn − Tn )Snn−k−1

so kSnn − Tnn k ≤ ≤

n−1 X

k=0 n−1 X

kTn kk kSn − Tn kkSn kn−k−1

(max(kTn k, kSn k))n−1 kSn − Tn k

k=0 kAk+kBk

≤ ne

kSn − Tn k.

Using a power series expansion, we see kSn − Tn k = O(1/n2 ) and so kSnn − Tnn k → 0 as n → ∞.  A proof for unbounded operators can be found in [244].

16 Semi-classical Analysis

In this chapter we investigate some key quantum quantities – such as quantum energy levels – as ~/(typical classical action)→ 0. We hope that asymptotics of these quantities can be expressed in terms of relevant classical quantities. This is called semi-classical (or semi-classical) analysis. To do this, we use the Feynman path integral representation of the evolution operator (propagator) e−iHt/~ . This representation provides a non-rigorous but highly effective tool, as the path integral is expressed directly in terms of the key classical quantity – the classical action. The heuristic power of path integrals is that when treated as usual convergent integrals, they lead to meaningful and, as it turns out, correct answers. Thus to obtain a “semi-classical approximation”, we apply the method of stationary phase. Recall that the (ordinary) method of stationary phase expands the integral in question in terms of the values of the integrand at the critical points of the phase, divided by the square root of the determinant of the Hessian of the phase at those critical points. The difference here is that the phase – the classical action – is not a function of several variables, but rather a “functional”, which (roughly speaking) is a function of an infinite number of variables, or a function on paths. Critical points of the classical action are the classical paths (solutions of Newton’s equation) and the Hessians are differential operators. Thus we need some new pieces of mathematics: determinants of operators and elements of the calculus of variations. These are presented in supplementary Section 25.12 and Chapter 26 respectively. Below we consider a particle in Rd described by a Schr¨odinger operator H =−

~2 ∆x + V (x). 2m

(16.1)

We want to pass to physical units in which a typical classical action in our system is 1, so that ~ is now the ratio of the Planck constant to the classical action, so that the regime we are interested in is the one for which ~ → 0. Let L be a length scale for the potential, and g its size. So roughly, g = supx |V (x)| and L = g(supx |∇V (x)|)−1 . Re-scaling the variable as x → x′ = x/L, we find © Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_16

202

16 Semi-classical Analysis

H = gH ′ where

~′2 ∆x′ + V ′ (x′ ) 2m′ √ √ where V ′ (x′ ) = g −1 V (Lx′ ) and ~′ / m′ = ~/(L mg). Now the potential V ′ (x′ ) is essentially of unit size √ and varies on a unit length scale. The parameter ~′ is dimensionless. If h′ / m′ ≪ 1, we can consider it a small parameter. As an example suppose V (x) is of the order 100me4/~2 , where me4 /~2 is twice the ionization energy of the ground state of the hydrogen atom, and varies on √ the scale of the Bohr radius (of the hydrogen atom) L = ~2 /(me2 ). ′ Then ~ / m′ = 1/10. In the expansions we carry out below, we always have in mind the operator H ′ and the dimensionless parameter ~′ with the primes omitted; that is, we think of (16.1) in dimensionless variables. H′ = −

16.1 Semi-classical Asymptotics of the Propagator The path integral (15.3) has the form of oscillatory integrals extensively studied in physics and mathematics. One uses the method of stationary phase in order to derive asymptotic expressions for such integrals. It is natural, then, to apply (formally) this method – with small parameter ~ – to the path integral, in order to derive a semi-classical expression for the Schr¨odinger propagator e−itH/~ (y, x). We do this below. But first, we quickly review the basics of the method of stationary phase (in the finite-dimensional setting, of course). The stationary phase method. We would like to determine the asymptotics of oscillatory integrals of the form Z eiS(φ)/~ dφ Rd

as ~ → 0 (here φ is a finite dimensional variable). The basic idea is that as ~ → 0, the integrand is highly oscillating and yields a small contribution, except where ∇S(φ) = 0 (i.e., critical points). We now make this idea more precise. Set Z f (φ)eiS(φ)/~ dφ

I(~) :=

Rd

where f ∈ C0∞ (Rd ) and S is smooth, and consider two cases:

Theorem 16.1 (stationary phase method) 1. If supp(f ) contains no critical points of S, then I(~) = O(~N ) ∀ N

as

~ → 0.

2. If supp(f ) contains precisely one non-degenerate critical point of S, i.e. ¯ = 0 and the matrix of second derivatives S ′′ (φ) ¯ is invertible, then ∇S(φ) as ~ → 0,

16.1 Semi-classical Asymptotics of the Propagator

203

¯ ¯ (φ)e ¯ iS(φ)/~ ¯ −1/2 ei π4 sgn(S ′′ (φ))f [1 + O(~)] I(~) = (2π~)d/2 | det S ′′ (φ)| (16.2) where for a matrix A, sgn(A) denotes the number of positive eigenvalues minus the number of negative ones.

Proof. To prove the first statement, define the operator L :=

~ ∇S(φ) · ∇. i |∇S(φ)|2

Note that LeiS(φ)/~ = eiS(φ)/~ , and that for smooth functions f and g, Z Z f Lgdx = (LT f )g Rd

where

Rd

  ~ ∇S LT f := − ∇ · f . i |∇S|2

So for any positive integer N , Z Z N iS(φ)/~ |I(~)| = | f (φ)L e dφ| = | [(LT )N f (φ)]eiS(φ)/~ dφ| Rd

Rd

N

≤ (const)~ ,

establishing the first statement. Turning to the second statement, suppose supp(f ) contains only one critical point, φ¯ of S, which is non-degenerate. We begin with a formal calculation, √ and then explain how to make it rigorous. Writing φ − φ¯ = ~α, we obtain √ 1 ¯ + O( ~|α|3 ). ¯ S(φ)/~ = S(φ)/~ + αT S ′′ (φ)α 2

So I(~) = ~

¯ d/2 iS(φ)/~

e

Z

f (φ¯ +

(16.3)

√ √ 3 T ′′ ¯ ~α)eiα S (φ)α/2 eiO( ~|α| ) dα.

Rd

Now we use the formula Z T ′′ ¯ ¯ −1/2 . lim eiα S (φ)α/2 dα = (2πi)d/2 [det S ′′ (φ)] R→∞

(16.4)

|α|≤R

R T ′′ We can derive this expression by analytically continuing e−aα S α from Re(a) > 0, though the integral is not absolutely convergent. Some care is needed in choosing the right branch of the square root function. An unambiguous expression for the right hand side is ¯ ¯ −1/2 eiπ·sgn(S ′′ (φ))/4 (2π)d/2 | det S ′′ (φ)|

204

16 Semi-classical Analysis

√ √ ¯ + O( ~), yields (16.2). Though this computation Using f (φ¯ + ~α) = f (φ) shows what is going on, it is not rigorous, since the integral in the remainder diverges. A more careful computation is based on the Fourier transform, and proceeds as follows. First, as a replacement for (16.3), we use the fact, known as the Morse Lemma, that there is a change of variables which makes S(φ) ¯ More precisely, there exists a smooth function Φ : Rd → Rd quadratic near φ. ¯ ¯ ¯ = 1, and with Φ(φ) = φ, DΦ(φ) ¯ T S ′′ (φ)(φ ¯ ¯ ¯ + 1 (φ − φ) − φ) S(Φ(φ)) = S(φ) 2 ¯ around φ. ¯ A proof of this can be found for φ in a sufficiently small ball, Bǫ (φ), in [92], for example. By the first statement of Theorem 16.1, we can assume ¯ with δ small enough so that Φ−1 (Bδ (φ)) ¯ ⊂ Bǫ (φ). ¯ Then that (supp f ) ⊂ Bδ (φ) we have Z Z f (φ)eiS(φ)/~ dφ = f (Φ(y))eiS(Φ(y)/~) | det DΦ(y)|dy I(~) = ¯ ¯ Φ−1 (Bδ (φ)) Bδ (φ) Z ¯ ¯ T ′′ ¯ ¯ = eiS(φ)/~ ei(y−φ) S (φ)(y−φ)/(2~) f (Φ(y))| det DΦ(y)|dy −1 ¯ Φ (Bδ (φ)) Z T ′′ ¯ ¯ iS(φ)/~ =e eix S (φ)x/(2~) f (Φ(φ¯ + x))| det DΦ(φ¯ + x)|dx. Φ−1 (Bδ (0))

Now we use the Plancherel formula Z Z a(x)b(x)dx = Rd

a ˆ(ξ)ˆb(−ξ)dξ

Rd

together with the fact (see Section 25.14) that for an invertible symmetric matrix A, (eix

T

Ax/(2~) ˆ

) (ξ) = ~d/2 | det A|−1/2 eiπsgn(A)/4 e−i~ξ

to obtain ¯ −1/2 eiπsgn(S I(~) = ~d/2 | det S ′′ (φ)|

′′

¯ ¯ (φ)/4 iS(φ)/~

e

¯ ¯ ¯ −1/2 eiπsgn(S ′′ (φ)/4 = ~d/2 | det S ′′ (φ)| eiS(φ)/~

Z

e−i~ξ

T

T

A−1 ξ/2

,

¯ −1 ξ/2 ˆ (S ′′ (φ))

b(−ξ)dξ

Rd

Z

(1 + O(~|ξ|2 ))ˆb(−ξ)dξ,

Rd

where b(x) := f (Φ(φ¯ + x))| det DΦ(φ¯ + x)|χ|x|≤δ . Finally, observing that R ˆb(−ξ)dξ = (2π)d/2 b(0) = (2π)d/2 f (φ), ¯ we arrive at the second statement Rd of Theorem 16.1. Now we would like to formally apply the method outlined above to the infinite-dimensional integral (15.3). To this end, we simply plug the path integral expression (15.3) into the stationary phase expansion formula (16.2). The result is

e−iHt/~ (y, x) =

Z

16.1 Semi-classical Asymptotics of the Propagator

eiS(φ)/~ Dφ

Px,y,t

= ¯ φ

205

X

cp of

√ ¯ ¯ −1/2 eiS(φ)/~ Mφ¯(det S ′′ (φ)) (1 + O( ~))

(16.5)

S

where Mφ¯ is a normalization constant, and the sum is taken over all critical ¯ of the action S(φ), going from x to y in time t. Critical points and points, φ, Hessians of functionals are discussed in the mathematical supplement, Chap¯ is a differential operator. The problems ter 26. Note that the Hessian S ′′ (φ) of how to define and compute determinants of Hessians are discussed in the mathematical supplement Section 25.12. We will determine M := Mφ¯, assuming it is independent of φ¯ and V . For V = 0, we know the kernel of the propagator explicitly (see (2.23)): e−iH0 t/~ (y, x) = (2πi~t/m)−d/2 eim|x−y|

2

/2~t

.

So in particular, e−iH0 t/~ (x, x) = (2πi~t/m)−d/2 . Now the right-hand side of the expression (16.5) for e−itH0 /~ (x, y) is (to leading order in ~) M (det S0′′ (φ0 ))−1/2 eiS0 (φ0 )/~ where the unique critical point is φ0 (s) = x + (y − x)s/t. Thus S0 (φ0 ) = m|y − x|2 /2t, and S0′′ (φ0 ) = −m∂s2 , an operator acting on functions satisfying Dirichlet boundary conditions. Comparison thus gives us M = (det(−m∂s2 ))1/2 (2πi~t/m)−d/2 and therefore X  2πit~ −d/2  det(−m∂ 2 ) 1/2 √ ¯ s e (y, x) = eiS(φ)/~ (1 + O( ~)) ′′ ¯ m det S (φ) ¯ cp S φ (16.6) as ~ → 0. This is precisely the semi-classical expression we were looking for. We now give a “semi-rigorous” derivation of this expression. We assume ¯ going from x to y in for simplicity that S has only one critical point, φ, ∞ ¯ acting time t. Let {ξj }j=1 be an orthonormal basis of eigenfunctions of S ′′ (φ) 2 on L ([0, t]) with zero boundary conditions (the eigenfunctions of such an ¯ j = µj ξj operator are complete – see the remark in Section 25.11). So S ′′ (φ)ξ for eigenvalues µj . For the n-th order finite dimensional approximation to the space of paths in the path integral, we take the n-dimensional space of functions of the form n X aj ξj φ(n) = φ¯ + −iHt/~

j=1

with aj ∈ R. Expanding S(φ(n) ) around φ¯ gives

206

16 Semi-classical Analysis

¯ + 1 hξ, S ′′ (φ)ξi ¯ + O(kξk3 ) S(φ(n) ) = S(φ) 2 where ξ := φ(n) − φ¯ = We also have Dφ(n) = Cn

n X

aj ξj .

j=1

n Y

daj

j=1

(Cn some constant). Now using the fact that ¯ = hξ, S ′′ (φ)ξi we have Z

eiS(φ

(n)

)/~

n X

i,j=1

¯ ji = ai aj hξi , S ′′ (φ)ξ

¯

Dφ(n) = eiS(φ)/~

φ(n) :x→y

Z

ei

P

n X

µj a2j ,

j=1

µj a2j /2~

(1 + O(a3 /~))Cn dn a

(as in Section 16.1, the integrals here are not absolutely convergent). Setting √ bj := aj / ~ this becomes Z P √ 2 ¯ ~n/2 Cn eiS(φ)/~ ei µj bj /2 (1 + O(b3 ~))dn b which is (see (16.4)) √ ¯ ¯ F ))−1/2 eiS(φ)/~ Cn (2πi~)n/2 (det(S ′′ (φ)| (1 + O( ~)) n Pn ¯ F ) = Qn µj . To avoid determining where Fn := { 1 aj ξj } so that det(S ′′ (φ)| n 1 the constants Cn arising in the “measure” Dφ, we compare again with the free (V = 0) propagator. Taking a ratio gives us ¯ ¯ F ))−1/2 eiS(φ)/~ √ e−iHt/~ (y, x) Cn (2πi~)n/2 (det(S ′′ (φ)| n ~)) = lim (1 + O( −d/2 n→∞ Cn (2πi~)n/2 (det(−m∂s2 |Fn ))−1/2 ( 2πi~t m )

which reproduces (16.6).

16.2 Semi-classical Asymptotics of Green’s Function Definition 16.2 Green’s function GA (x, y, z) of an operator A is (A − z)−1 (y, x), the integral kernel of the resolvent (A − z)−1 .

16.2 Semi-classical Asymptotics of Green’s Function

207

For A self-adjoint, (A − z)−1 =

i ~

Z

∞

e−iAt/~+izt/~ dt

0

converges if Im(z) > 0. Taking z = E + iǫ (E real, ǫ > 0 small), and letting ǫ → 0, we define Z i ∞ −iAt/~ e (y, x)eiEt/~ dt. (A − E − i0)−1 (y, x) = ~ 0 Note that the −i0 prescription is essential only for E ∈ σess (A), while for E ∈ R\σess (A), it gives the same result as the +i0 prescription. Here we are interested in the second case, and so we drop the −i0 from the notation. The above formula, together with our semi-classical expression (16.6) for the propagator e−iHt/~ , yields in the leading order as ~ → 0 Z i ∞ X ¯ −1 (H − E) (y, x) = Kφ¯ei(S(φ)+Et)/~ dt ~ 0 ¯ φ cp S ¯ from x to y in time t, and where the sum is taken over the critical paths, φ,  m d/2  det(−m∂ 2 ) 1/2 s Kφ¯ := . ¯ 2πit~ det S ′′ (φ) We would like to use the stationary phase approximation again, but this time in the variable t. Denote by t¯ = t¯(x, y, E) the critical points of the phase ¯ + Et. They satisfy the equation S(φ) ¯ ∂S(φ)/∂t = −E. ¯ t=t¯ is a classical path at energy E (see Lemma 16.13 of The path ωE := φ| ¯ for a classical path Section 16.5). Introduce the notation S0 (x, y, t) := S(φ) going from x to y in time t. Then the stationary phase formula gives (in the leading order as ~ → 0) (H − E)−1 (y, x) =

i X 1/2 iWω /~ D ωE e E , ~ ω

(16.7)

E

where the sum is taken over classical paths going from x to y at energy E. 1/2 Here we have used the notation DωE := KωE (2πi~)1/2 (∂ 2 S0 /∂t2 )−1/2 |t=t¯ and we have defined WωE (x, y, E) := (S0 (x, y, t)+Et)|t=t¯ (so WωE is the Legendre transform of S0 in the variable t). Lemma 16.3   d−1  2 1 ∂ 2 S0 ∂ 2 WωE ∂ 2 WωE ∂ WωE ∂ 2 WωE D ωE = . − det 2πi~ ∂t2 ∂x∂y ∂E 2 ∂x∂E ∂y∂E

(16.8)

208

16 Semi-classical Analysis

Proof. We just sketch the proof. The first step is to establish  2  det(−m∂s2 )  m −d ∂ S0 det = − det(S ′′ (ω)) t ∂x∂y

(16.9)

(we drop the subscript E for ease of notation). To see this, we use the fact that if for an operator A we denote by JA the d × d matrix solving AJ = 0 ˙ (the Jacobi equation) with J(0) = 0 and J(0) = 1, then (see (25.54)) det J−m∂s2 (t) det(−m∂s2 ) = . det(S ′′ (ω)) det JS ′′ (ω) (t) Next, we use −

1 JS ′′ (ω) (t) = m



∂ 2 S0 ∂x∂y

−1

(Equation (16.13) of Section 16.5), and for the free classical path φ0 = x + (y − x)(s/t), −1  2  m −1 1 ∂ S(φ0 ) 1 − J−m∂s2 (t) = =− m ∂x∂y t to arrive at (16.9). We can then show that det



∂ 2 S0 ∂x∂y



−∂ 2 S0 ∂t2

−d

t=t¯

equals the determinant on the right hand side in equation (16.8) (see the appendix to this section for details).  We will show in Section 16.5 (see Lemma 16.14) that the function WωE (the action at energy E) satisfies the Hamilton-Jacobi equation h(x, −

∂WωE ) = E. ∂x

Differentiating this equation with respect to y gives ∂h ∂ 2 WωE = 0, ∂k ∂x∂y and we see that the matrix (∂ 2 WωE /∂x∂y) has a zero-eigenvalue. Thus its determinant is zero. So if d = 1, (16.8) yields D ωE = −

∂ 2 WωE ∂ 2 WωE ∂x∂E ∂y∂E

(d = 1).

(16.10)

Formula (16.7), together with (16.8) or (16.10), is our desired semi-classical expression for Green’s function (H − E)−1 (y, x).

16.3 Bohr-Sommerfeld Semi-classical Quantization

209

16.2.1 Appendix Proposition 16.4 At t = t¯, det



∂ 2 S0 ∂x∂y



=



−∂ 2 S0 ∂t2

d

 ∂ 2 WωE ∂ 2 WωE ∂ 2 WωE ∂ 2 WωE . − det ∂x∂y ∂E 2 ∂x∂E ∂y∂E 

Proof. We drop the subscripts from S0 and WωE to simplify the notation. Differentiating W = S + Et|t=t¯ with respect to x, we obtain ∂W ∂S ∂S ∂t ∂t = + +E , ∂x ∂x ∂t ∂x ∂x which due to the relation ∂S/∂t = −E gives ∂W ∂S = . ∂x ∂x Similarly,

∂W ∂S = ∂y ∂y

∂W = t. ∂E

and

This last equation, together with ∂S/∂t = −E yields ∂t ∂2W = =− ∂E 2 ∂E



∂2S ∂t2

−1

.

Furthermore, ∂2W ∂ 2 W ∂E ∂t ∂2S = + ∂x∂y ∂x∂y ∂x∂E ∂t ∂y   ∂2W ∂2W ∂2W ∂2W ∂2S − + = ∂t2 ∂x∂y ∂E 2 ∂x∂E ∂E∂y and the result follows. 

16.3 Bohr-Sommerfeld Semi-classical Quantization In this section we derive a semi-classical expression for the eigenvalues (energy ~2 levels) of the Schr¨odinger operator H = − 2m ∆ + V . We use the Green’s function expansion (16.7) from the last chapter. For simplicity, we will assume d = 1. Application of the expression (16.7) requires a study of the classical paths at fixed energy. Consider the trajectories from x to y at energy E. We can write them (using informal notation) as φn = φxy ± nα

210

16 Semi-classical Analysis

where α is a periodic trajectory (from y to y) of minimal period, at the energy E, while φxy is one of the four “primitive” paths from x to y at energy E, sketched in Fig. 16.1.

V(x)

V(x) E

E x

y

x

y

Fig. 16.1. Primitive paths at energy E. All these paths are treated in the same way, so we consider only one, say the shortest one. The space time picture of φn in this case is sketched in Fig. 16.2.

t

x

φn

y

Fig. 16.2. Turning points of φn . For this path we compute Wφn = Wφ + nI where φ = φxy and I=

Z

t

L(α, α)ds ˙ + Et.

0

But α is a critical path so its energy is conserved (see Lemma 4.7) m 2 α˙ + V (α) = E 2 and so I=

Z

0

t

2

{mα˙ − E} + Et =

where k(s) = mα(s) ˙ and dx = α(s)ds. ˙

Z

α

k · dx,

16.4 Semi-classical Asymptotics for the Ground State Energy

211

Let ω be a classical path at energy E, and let us determine Dω . We will show later (see (16.18) and Lemma 16.9) that

and

p ∂Wω ∂S0 = |t=t¯ = −k(0)|t=t¯ = ∓ 2m(E − V (x)) ∂x ∂x

p ∂Wω ∂S0 = |t=t¯ = k(t)|t=t¯ = ± 2m(E − V (y)). ∂y ∂y Differentiating these relations with respect to E and using (16.10), we obtain Dω = −

m2 . k(x)k(y)

At a turning point x0 , k(x0 ) = 0 and k changes sign (we think about √ k(x) as a multi-valuedp function, or a function on the Riemann surface of z, so at a turning point k(x) crosses to a different sheet of the Riemann surface). Because k changes sign at each of the two turning points of the periodic trajectory, we conclude that 1/2

1/2

Dφn = Dφ (−1)n . So our semi-classical expression (16.7) for Green’s function GE (y, x) is   Z ∞ X 1 k · dx − π )] GE (y, x) = N exp[i(Wφ /~ + n ~ α n=0 = N eiWφ /~

1

1−e

i(

R

α

k·dx/~−π)

(N is a constant). We conclude that as ~ → 0, GE (y, x) has poles (and hence H has eigenvalues) when Z k · dx = 2π~(j + 1/2) α

for an integer j. This is the Bohr-Sommerfeld semi-classical quantization condition (for d = 1). It is an expression for the quantum energy levels (the energy E appears in the left hand side through the periodic path α at energy E), which uses purely classical data! Problem 16.5 Show that for the harmonic oscillator potential, the BohrSommerfeld condition gives all of the energy levels exactly.

16.4 Semi-classical Asymptotics for the Ground State Energy Here we derive a semi-classical expression for the ground state energy (the ~2 lowest eigenvalue) of the Schr¨odinger operator H = − 2m ∆ + V when V (x) →

212

16 Semi-classical Analysis

∞ as |x| → ∞ (as fast as some power of |x|, say); i.e., V (x) is a confining potential. We first define a couple of quantities which are familiar from statistical mechanics (see Chapter 18 for details and discussions). Definition 16.6 The partition function, Z(β), at inverse temperature β > 0 is Z(β) := tr e−βH (the trace is well-defined since σ(H) = {En }∞ 0 with En → ∞ sufficiently fast). Definition 16.7 The free energy, F , is F (β) := −

1 ln Z(β). β

The free energy is a useful quantity for us here because of the following connection with the ground state energy of the Schr¨odinger operator H: lim F (β) = E0 .

β→∞

This is the Feynman-Kac theorem of Section 18.3. Our goal is to find the semi-classical asymptotics for E0 by deriving an asymptotic expression for Z(β) from a path integral. As we have seen (Problem 15.1), the path integral expression for Z(β) is Z e−Se (φ)/~ Dφ (16.11) Z(τ /~) = φ a path of period τ Rβ ˙2 where Se (φ) = 0 { m 2 |φ| + V (φ)} (note that this is not the usual action - the potential enters with the opposite sign). Remark 16.8 The path integral appearing in (16.11) can be rigorously defined. We refer the reader to [272, 135, 245] for details. Mimicking the procedure we used for the Schr¨odinger propagator (i.e., the stationary phase method, which in the present context is called the Laplace method), we see that the semi-classical expression for Z(τ /~) is X Z(τ /~) ≈ N Bω1/2 e−Se (ω)/~ (16.12) minimal paths ω (N a constant) where by a minimal path, we mean a path minimizing Se , and where det S0′′ (ω0 ) Bω := det Se′′ (ω) Rτ ˙ 2 . A critical path for Se is a classical path for the with S0 (φ) = 0 (m/2)|φ| inverted potential −V . We specialize to d = 1 for simplicity, and we assume V

16.5 Mathematical Supplement: The Action of the Critical Path

213

has only one minimum, at x0 . Then the minimal path is ω(s) ≡ x0 (a constant path), and Se′′ (ω) = −m∂s2 + V ′′ (x0 ).

Because x0 minimizes V , V ′′ (x0 ) > 0 and we write V ′′ (x0 ) = mω 2 . Then using the method (25.54) of computing ratios of determinants (see Section 25.12 below), we easily obtain 2ωτ Bω = ωτ . e − e−ωτ Also, Se (ω) = τ V (x0 ). In this way, we arrive at the leading-order expression F (τ /~) ≈ V (x0 ) + ~ω/2 + O(1/τ ) as ~ → 0. Letting β = τ /~ → ∞ and using the Feynman-Kac formula, we obtain 1 E0 ≈ V (x0 ) + ~ω 2

which is the desired asymptotic (as ~ → 0) expression for the ground state energy. It is equal to the classical ground state energy, p V (x0 ), plus the ground state of the harmonic oscillator with frequency V ′′ (x0 )/m. This suggests that the low energy excitation spectrum of a particle in the potential V (x) is the low energy spectrum of this harmonic oscillator.

16.5 Mathematical Supplement: The Action of the Critical Path In this section we consider the situation of Example 26.1 no. 6, and its special case, Example 26.1 no. 5. Thus we set X = {φ ∈ C 1 ([0, t]; Rm ) | φ(0) = x, φ(t) = y}, and S(φ) =

Z

t

˙ L(φ(s), φ(s))ds.

0

¯ ¯ = y. We will Suppose φ¯ is a critical path for S with φ(0) = x and φ(t) ¯ (the action from x to y in time denote the action of φ¯ by S0 (x, y, t) := S(φ) t). ˙ φ(s), ¯ ¯˙ φ(s)). Recall that the momentum at time s, is k(s) := (∂L/∂ φ)( Lemma 16.9 We have ∂S0 = −k(0) ∂x

and

∂S0 = k(t). ∂y

214

16 Semi-classical Analysis

˙ 2 /2 − V (φ). Using the chain rule and Proof. Again, we specialize to L = m|φ| integration by parts, we find Z t ¯˙ ¯ · ∂ φ/∂x}ds ¯ ¯ ∂S(φ)/∂x = {mφ¯˙ · ∂ φ/∂x − ∇V (φ) 0 Z t t ¯ − ∇V (φ)) ¯ · ∂ φ/∂x} ¯ ¯ = {(−mφ¨ + mφ¯˙ · ∂ φ/∂x| 0 0

¯ ¯ which, since φ¯ is a critical point, (∂ φ/∂x)(t) = 0, and (∂ φ/∂x)(0) = 1, is ˙ ¯ just −mφ(0) = −k(0), as claimed. The corresponding statement for ∂S0 /∂y ¯ ¯ is proved in the same way, using (∂ φ/∂y)(0) = 0, and (∂ φ/∂y)(t) = 1.  This lemma implies ∂k(0)/∂y = −∂ 2 S0 (x, y, t)/∂x∂y. On the other hand, −1 ˙2 for L = m = mJ −1 (t) (as m(∂y/∂k(0)) 2 |φ| − V (φ), ∂k(0)/∂y = (∂y/∂k(0)) ¯ is the derivative of the classical path φ at t with respect to the initial velocity ¯˙ k(0)/m = φ(0)). This gives ∂ 2 S0 (x, y, t) = −mJ −1 (t) ∂x∂y

(16.13)

which establishes the following result: Proposition 16.10 If y is a conjugate point to x then det( ∂

2

S0 (x,y,t) ) ∂x∂y

= ∞.

The following exercise illustrates this result for the example of the classical harmonic oscillator. Problem 16.11 Consider the one-dimensional harmonic oscillator, whose ˙ 2 mω2 2 Lagrangian is L = m 2 φ − 2 φ . Compute S0 (x, y, t) =

ω [(x2 + y 2 ) cos(ωt) − 2xy] 2 sin(ωt)

and so compute

ω . sin ωt Note that this is infinite for t = nπ/ω for all integers n. Thus the points φ(nπ/ω) are conjugate to φ(0). ∂ 2 S0 (x, y, t)/∂x∂y = −

Lemma 16.12 (Hamilton-Jacobi equation) The action S0 (x, y, t) satisfies the Hamilton-Jacobi equation ∂S0 /∂t = −h(y, ∂S0 /∂y)

(16.14)

where h is the classical Hamiltonian function associated with L. Proof. The integrands below depend on s (as well as the parameters x,y, and R ¯˙ ¯ ¯ = t L(φ, ¯ φ)ds, we have Since S0 = S(φ) t), and φ¯˙ denotes ∂ φ/∂s. 0

16.5 Mathematical Supplement: The Action of the Critical Path

¯ ¯ φ)| ¯˙ s=t + ∂S(φ)/∂t = L(φ,

Z

t

0

215

¯˙ ¯ (∂L/∂ φ¯ · ∂ φ/∂t + ∂L/∂ φ¯˙ · ∂ φ/∂t)ds

s=t ¯ φ)| ¯˙ s=t + ∂L/∂ φ¯˙ · ∂ φ/∂t| ¯ = L(φ, s=0 Z t ¯˙ · ∂ φ/∂t. ¯ + (∂L/∂ φ¯ − d/ds(∂L/∂ φ)) 0

¯ = y + φ(t)(s ¯˙ ¯ = y), we have Since φ(s) − t) + O((s − t)2 ) (here we used φ(t) ∂ φ¯ ¯˙ |s=t = −φ(t). ∂t ¯ Using this, the fact that φ¯ is a critical point of S, and ∂ φ/∂t| s=0 = 0, we find ¯ ¯˙ · φ¯˙ − L(φ, ¯ φ))| ¯˙ s=t ∂S(φ)/∂t = −((∂L/∂ φ) ¯ ∂L/∂ φ)| ¯˙ = −h(φ, . s=t

¯ s=t = y and, by Lemma 16.9, (∂L/∂ φ)| ¯˙ s=t = ∂S0 /∂y, the result Since φ| follows.  We want to pass from a time-dependent to a time-independent picture of classical motion. We perform a Legendre transform on the function S0 (x, y, t) to obtain the function Wφ¯(x, y, E) via Wφ¯(x, y, E) = (S0 (x, y, t) + Et)|t:∂S0 /∂t=−E .

(16.15)

We denote by t¯ = t¯(x, y, E), solutions of ∂S0 /∂t|t=t¯ = −E.

(16.16)

There may be many such solutions, so in the notation Wφ¯, we record the ¯ ¯ t¯) = y). classical path φ¯ we are concerned with (for which φ(0) = x, φ( · φ˙ − L = Note that by the definition of the Hamilton function, ∂L ∂ φ˙

h(φ, ∂L ), and from the energy conservation law (see Lemma 4.7 of Section ∂ φ˙  ∂L 4.7) ∂ φ˙ · φ˙ − L |φ=φ¯ is constant, and therefore 

By (16.15), Wφ¯(x, y, E) =

 ∂L ˙ · φ − L |φ=φ¯ = E. ∂ φ˙

R t¯ ¯˙ ¯ φ(s)) + E)ds and therefore we have (L(φ(s), 0

Wφ¯(x, y, E) = where

R

¯k φ

· dx :=

Rt 0

Z

0

t

∂L ¯˙ · φds = ∂ φ¯˙

Z

¯ φ

k · dx,

¯˙ ˙ and, recall, k(s) = ∂L/∂ φ| k(s) · φ(s)ds ¯. φ=φ

216

16 Semi-classical Analysis

¯ t=t¯ is a classical path at energy E. Lemma 16.13 φ| Proof. By the Hamilton-Jacobi equation (16.14) and the conservation of en¯ t=t¯ is a classical path with energy −∂S0 (x, y, t)/∂t|t=t¯, ergy (Lemma 4.7), φ| which, by (16.16), is just E.  Lemma 16.14 Wφ¯ satisfies the Hamilton-Jacobi equation h(x, −∂Wφ¯/∂x) = E.

(16.17)

Proof. Using (16.16) and Lemma 16.9, we compute ∂Wφ¯ ∂S0 ∂S0 ∂ t¯ ∂S0 = |t=t¯ + ( + E)|t=t¯ = |t=t¯ = −k(0). ∂x ∂x ∂t ∂x ∂x

(16.18)

So by conservation of energy, E = h(x, k(0)) = h(x, −∂Wφ¯/∂x). 

16.6 Appendix: Connection to Geodesics The next theorem gives a geometric reinterpretation of classical motion. We consider a classical particle in Rd with a potential V (x). Recall the notation f (x)+ := max(f (x), 0). Theorem 16.15 (Jacobi theorem) The classical trajectory of a particle at an energy E is a geodesic in the Riemannian metric hu, vix = 2(E − V (x))+ u · v (where u · v is the inner product in Rn ) on the set {x ∈ Rn |V (x) ≤ E} (the classically allowed region). Proof. By the conservation of energy (Lemma 4.7), a classical path φ¯ has a ¯˙ 2 /2 + V (φ). ¯ Hence φ¯ is a critical point of the action fixed energy E = m|φ| R 2 ˙ ˙ 2 /2+V (φ) ≡ E}. S(φ) = (m|φ| /2−V (φ))ds among paths in M := {φ | m|φ| 2 ˙ /2 + V (φ) = E, we can write m|φ| ˙ 2 /2 − V (φ) as Using the relation m|φ| p ˙ ˙ 2 /2 − V (φ) = m|φ| ˙ 2 = m 2(E − V (φ))/m|φ|. m|φ| Hence φ¯ is a critical point of the functional Z p ˙ 2(E − V (φ))/mds L(φ) := m|φ|

on M . This functional gives the length of the path in the metric above. On the other hand, we can re-parameterize any path with V (φ) < E so ˙ 2 /2 + V (φ) = E. Indeed, replacing φ(s) with φ(λ(s)), we that it satisfies m|φ| 2 2 ˙ ˙ ˙ note |φ(λ)| = (λ(s))2 |φ(λ(s))| . We must solve

16.6 Appendix: Connection to Geodesics 2 ˙ (λ(s)) =

which we can re-write as Z s 0

217

2 E − V (φ(λ(s))) 2 ˙ m |φ(λ(s))|

′ ˙ ˙ ′) |φ(λ(s ))|λ(s p ds′ = E − V (φ(λ(s′ )))

r

2 s m

which can be solved. Since the functional L(φ) is invariant under reparame∂λ ∂s ˙ ˙ ˙ terizations (if λ = α(s), α′ > 0, then |φ(s)|ds = |φ(λ)| ∂s ∂λ dλ = |φ(λ)|dλ), ˜ ˜ if φ is a critical point of L(φ), then so are different reparameterizations of φ, ¯ ¯ and in particular the one, φ, with the energy E. This φ is also a critical point of L(φ) on M , and, by the above, a critical point of S(φ) at energy E. Thus classical paths are geodesics up to re-parameterization (so they coincide as curves).  Problem 16.16 Check that the Euler-Lagrange equation for critical points of L(φ) on M yields Newton’s equation.

17 Resonances

The notion of a resonance is a key notion in quantum physics. It refers to a metastable state – i.e., to a state which behaves like a stationary (bound) state for a long time interval, but which eventually breaks up. In other words, the resonances are states of the essential spectrum (i.e. scattering states), which for a long time behave as if they were bound states. In fact, the notion of a bound state is an idealization: most of the states which are (taken to be) bound states in certain models, turn out to be resonance states in a more realistic description of the system. In this chapter, we sketch briefly the mathematical theory of resonance states and apply it to the analysis of the important physical phenomenon of tunneling, on which we illustrate some of the mathematics and physics involved. In Chapter 23, we apply the resonance theory to the problem of radiation.

17.1 Complex Deformation and Resonances In this section we introduce the powerful tool of complex deformations, which allows for an efficient way to define resonances. We begin with a definition. Definition 17.1 A family of operators, H(θ), for θ in a complex disk {|θ| < ǫ}, will be called a complex deformation of H if H(0) = H, H(θ) is analytic in {|θ| < ǫ}, and H(θ) is an analytic continuation of the family H(θ) = U (θ)−1 HU (θ)

(17.1)

for θ ∈ R, where U (θ), θ ∈ R, is a one-parameter unitary group, leaving the domain of H invariant. (We say that the family H(θ) of unbounded operators is analytic iff all H(θ) have the same domain, D, for every ψ ∈ D, the family of vectors H(θ)ψ is analytic and the family (H(θ) − z)−1 of bounded operators is analytic, as long as the spectrum of H(θ) stays away from z.) © Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_17

220

17 Resonances 2

~ For Schr¨odinger operators, H := − 2m ∆ + V (x), acting on L2 (Rd ), the choice of the family H(θ) depends on analytic properties of the potentials V (x). The simplest and most important choice is provided by the notion of dilatation analyticity. Let U (θ) be the one-parameter group of dilatations:

U (θ) : ψ(x) 7→ edθ/2 ψ(eθ x) for ψ ∈ L2 (Rd ). This is a unitary implementation of the rescaling x 7→ eθ x for θ ∈ R and is the key example of a one-parameter unitary group U (θ) used for complex deformation. We compute U (θ)−1 HU (θ) = e−2θ (−

~2 ∆) + V (eθ x). 2m

(17.2)

Assume the family θ → V (eθ x) has an analytic continuation into a complex disk {|θ| < ǫ} as operators from the Sobolev space H 2 (Rd ) to L2 (Rd ) (the corresponding potentials are called dilatation analytic). Then the family on the r.h.s. of (17.2) is analytic in {|θ| < ǫ} and therefore is a complex deformation, H(θ), of H. As an example of the above procedure we consider the complex deformaα 1 ∆ − |x| : tion of the hydrogen atom Hamiltonian Hhydr := − 2m Hhydrθ = e−2θ (−

~2 α ∆) − e−θ . 2m |x|

Let ehydr be the eigenvalues of the hydrogen atom. Then the spectrum of this j deformation is } ∪ e−2 Im θ [0, ∞). σ(Hhydrθ ) = {ehydr j In general, one can show that: 1) The real eigenvalues of Hθ , Imθ > 0, coincide with eigenvalues of H and complex eigenvalues of Hθ , Imθ > 0, lie in the complex half-plane C− ; 2) The complex eigenvalues of Hθ , Imθ > 0, are locally independent of θ. The typical spectrum of Hθ , Imθ > 0, is shown in Fig. 17.1

Fig. 17.1. Typical spectrum of Hθ . We call complex eigenvalues of Hθ , with Imθ > 0, the resonances of H.

17.1 Complex Deformation and Resonances

221

Often resonances arise as a result of perturbation of eigenvalues embedded into the essential (continuous) spectrum; that is, when the operator H is of the form H = H0 +κW , where H0 is a self-adjoint operator with an eigenvalue λ0 embedded in its essential spectrum, W is a symmetric operator and κ is a small real parameter (coupling constant). This happens, for example, for an atom in a constant electric field (the Stark effect ). Another example is the problem of time-periodic perturbations, we considered in Section 11.3. There the isolated eigenvalues Em of the unperturbed operator H0 lead to the eigenvalues, Emn , of the unperturbed Bloch-Floquet Hamiltonian K0 embedded into the essential spectrum of K0 . The computations of Section 11.3 suggest that they turn into resonance eigenvalues, Eκmn , of the perturbed operator, Kκ . (In Section 11.3 we considered only the case of n = 0.) This can be proved rigorously by using the complex deformation theory above, which allows also to derive the expansions (11.24) and (11.27). We explain this briefly. Apply the complex deformation with the dilatation group to the Bloch-Floquet Hamiltonian Kκ to obtain, as in (17.2), U (θ)−1 Kκ U (θ) := Hκθ (t) − i~

∂ ∂t

on

L2 (Rn × ST ). 2

~ Here, for θ ∈ R, Hκθ := H0θ + κWθ (t), H0θ = −e−2θ 2m ∆ + V (eθ x), Wθ (t) := θ W (e x, t). If the potentials V (x) and W (x, t) are dilatation analytic in the sense of the definition above, then the family on the r.h.s. which we denote Kκθ can be continued analytically into a neighbourhood {|θ| < ǫ}. Again, the spectrum of K0θ can be easily computed:

σess (K0θ ) =

∞ [

n=−∞

~ωn + e−2 Im θ [0, ∞)

plus the collection of the eigenvalues Emn = Em + ~ωn, n ∈ Z, m = 0, 1, . . . , where, recall, ω = 2π/T, T is the time-period of W , and Em are the eigenvalues of H0 . Now, barring an accidental degeneracy (i.e. Emn = ~ωn′ for some m, n, n′ ), the eigenvalues Emn = Em + ~ωn are isolated and have finite multiplicity. The application of the Feshbach-Schur map becomes a standard affair and gives, for κ sufficiently small, the eigenvalues, Eκmn of Kκθ emerging from Emn . (Here we assumed for simplicity that the eigenvalues Em are of multiplicity one.) These eigenvalues are, in general, complex, and, as we saw above, are independent of θ and in general have negative imaginary parts, Im Eκ,j . They can be computed by the perturbation expansion (11.16) - (11.17) to give (11.24) and (11.27). Resonances as poles. We know from Section 25.9 of the mathematical supplement that the resolvent (H − z)−1 of the Hamiltonian H is analytic away from its spectrum. One can show that H has an isolated eigenvalue at a point z0 iff matrix elements of the resolvent (H − z)−1 have a pole at z0 . Similar to eigenvalues, we would like to characterize the resonances in terms

222

17 Resonances

of poles of matrix elements of the resolvent (H − z)−1 . To this end we have to go beyond the spectral analysis of H. Let Ψθ = Uθ Ψ , etc., for θ ∈ R and z ∈ C+ . Use the unitarity of Uθ for real θ, to obtain hΨ, (H − z)−1 Φi = hΨθ¯, (Hθ − z)−1 Φθ i.

(17.3)

Assume now that for a dense set of Ψ ’s and Φ’s (say, D, defined below), Ψθ and Φθ have analytic continuations into a complex neighbourhood of θ = 0, and continue the r.h.s of (17.3) analytically, first in θ into the upper half-plane, and then in z across the continuous spectrum (the Combes argument). This meromorphic continuation has the following properties: • •

The real eigenvalues of Hθ give real poles of the r.h.s. of (17.3) and therefore they are the eigenvalues of H. The complex eigenvalues of Hθ are poles of the meromorphic continuation of the l.h.s. of (17.3) across the spectrum of H onto the second Riemann sheet.

The complex poles manifest themselves physically as bumps in the scattering cross-section or poles in the scattering matrix. An example of the dense set D mentioned above is given by [
D := Ran χ|T |≤a . (17.4) a>0

Here T is the self-adjoint generator of the one-parameter group Uθ , θ ∈ R. (It is not hard to show that it is dense: ∀ψ ∈ H, χ|T |≤a ψ → ψ, as a → ∞.)

Resonance states as metastable states. While bound states are stationary solutions of the Schr¨odinger equation, one expects that resonance eigenvalues lead to almost stationary, long-living solutions. This is proven, so far, only for resonances arising from a perturbation of bound states with eigenvalues embedded into the essential (continuous) spectrum. In this case, for initial condition ψ0 localized in a small energy interval around the unperturbed eigenvalue, λ∗ , or a small perturbation of the corresponding eigenfunction φ∗ , one shows that the solutions, ψ = e−iHt/~ ψ0 , of the time-dependent Schr¨odinger equation, i~∂t ψ = Hψ, are of the form ψ = e−iz∗ t/~ φ∗ + Oloc (t−α ) + Ores (κβ ),

(17.5)

for some z∗ ∈ C− , z∗ = λ∗ + O(κ), and α > 0 (depending on ψ0 ). These are metastable states with resonance eigenvalue z∗ ∈ C− . Here κ is the perturbation parameter (the coupling constant) and the error term Oloc (t−α ) satisfies, for some ν > 0, |(1 + |x|)−ν Oloc (t−α )k ≤ Ct−α .

Eqn (17.5) implies that the negative of the imaginary part of the resonance eigenvalue, −Imz∗ , called the resonance width, gives the decay probability per unit time, and (−Imz∗ )−1 , can be interpreted as the life-time of the resonance.

17.2 Tunneling and Resonances

223

17.2 Tunneling and Resonances Consider a particle in a potential V (x), of the form shown in Fig. 17.2; i.e., V (x) has a local minimum at some point x0 , and V (x0 ) > lim supx→∞ V (x), for x in some cone, say. If V (x) → −∞ as x → ∞ (in some cone of directions), then the corresponding Schr¨odinger operator, H, is not bounded from below. "would be bound state" V(x)

tunnels under barrier escape to

x

Fig. 17.2. Unstable potential. If the barrier is very thick, then a particle initially located in the well spends lots of time there, and behaves as if it were a bound state. However, it eventually tunnels through the barrier (quantum tunneling) and escapes to infinity. Thus the state of the particle is a scattering state. It is intuitively reasonable that 1. the energy of the resonance ≈ the energy of a bound state in the well 2. the resonance lifetime is determined by the barrier thickness and height, and ~. Since the resonances are very close to bound states if the barrier is large or ~ is small (there is no tunneling in classical mechanics), we try to mimic our semi-classical treatment of the ground state (Section 16.4). But right away we run into a problem: if V (x) → / ∞ as x → ∞ in some directions, then Z(β) = Tre−βH = ∞. The paradigm for this problem is the divergence of the integral Z ∞ 2 e−λa /2 da Z(λ) = 0

for λ ≤ 0. However, we can define this integral by an analytic continuation. Z(λ) is well-defined for Re(λ) > 0, and it can be continued analytically into λ ∈ R− as follows. Move λ from Re(λ) > 0 into Re(λ) ≤ 0, at the same time deforming the contour of integration in such a way that Re(λa2 ) > 0 (see Fig. 17.3).

224

17 Resonances

λ

Fig. 17.3. Contour deformation. Of course, in this particular case we know the result: Z(λ) =



2λ π

−1/2

= −i



2|λ| π

−1/2

for λ < 0 (which is purely imaginary!). There is a powerful method of rotating the contour which is applicable much beyond the simple integral we consider. It goes as follows. For θ ∈ R, we change variables via a = e−θ b. This gives Z ∞ −2θ 2 −θ Z(λ) = e e−λe b /2 db. (17.6) 0

The integral here is convergent and analytic in θ as long as Re(λe−2θ ) > 0.

(17.7)

We continue it analytically in θ and λ, preserving this condition. In particular, for λ ∈ R− , we should have π/4 < Im(θ) < 3π/4. Now observe that the right hand side of (17.6) is independent of θ. Indeed, it is analytic in θ as long as (17.7) holds, and is independent of Re(θ) since the latter can be changed without changing the integral, by changing the variable ′ of integration (b 7→ e−θ b, θ′ ∈ R). Thus we have constructed an analytic continuation of Z(λ) with Re(λ) > 0 into a region with Re(λ) < 0. In fact, we have continued this function onto the second Riemann sheet! Finally, we define Z(λ), for λ < 0, by (17.6) with θ obeying (17.7).

17.3 The Free Resonance Energy With a bit of wisdom gained, we return to the problem of defining the partition function Z(β) and free energy F (β) (see Definitions 16.6) in the case when

17.3 The Free Resonance Energy

225

V (x) → / ∞, as x → ∞, in some directions (or more precisely, supx∈Γ V (x) < ∞ for some cone Γ ). Assume that we can construct a complex deformation, H(θ), of H, such that Z(β) = Tre−βH(θ) < ∞ (17.8) for Im(θ) > 0, or more generally for |θ| ≤ ǫ, Im(θ) > 0. ( Let, for example, V (x) = −Cx3 for x ≥ 0 and = 0 for x ≤ 0. Then V (eθ x) = −Ce3θ x3 for x ≥ 0 and = 0 for x ≤ 0. Take θ = −iπ/3. Then V (eθ x) = Cx3 is positive for x ≥ 0. In fact, it is not a simple matter to define the exponential e−βH(θ) rigorously – see [269]. Below we will deal formally with e−βH(θ) , assuming it has all the properties which can be derived from the power series expression for the exponential.) Proposition 17.2 If Tre−βH(θ) < ∞ for θ ∈ Ω ⊂ {|θ| < ǫ}, then Tre−βH(θ) is independent of θ. Proof. e−βH(θ) is analytic in {|θ| ≤ ǫ}, and satisfies e−βH(θ+s) = U (s)−1 e−βH(θ) U (s) for s ∈ R. This last relation can be derived using the expression H(θ + s) = U (s)−1 H(θ)U (s) (which follows from (17.1)) and a power series expansion of the exponential (or Equation (25.43)). By cyclicity of the trace (Tr(AB) = Tr(BA)), Tre−βH(θ+s) = Tre−βH(θ) . Hence Tre−βH(θ) is independent of Re(θ), and so is independent of θ.  If there is a complex deformation, H(θ), of H, such that (17.8) holds, we call Z(β) = tre−βH(θ) an adiabatic partition function for H, and F (β) = −(1/β) ln Z(β) the resonance free energy for H. We interpret E(β) := Re F (β) as the resonance energy at the temperature 1/β, Γ (β) := − Im F (β) as the resonance decay probability per unit time (or resonance width) at the temperature 1/β, and 1 T (β) := Γ (β) as the resonance lifetime at the temperature 1/β. The resonance eigenvalue for zero temperature is given by zr = Er − iΓr := lim F (β). β→∞

226

17 Resonances

Usually, | Im Z(β)| ≪ | Re Z(β)|. Hence, E(β) = Re F (β) ≈ − and Γ (β) = − Im F (β) =

1 ln(Re Z(β)) β

1 Im Z(β) 1 Im Z(β) Im ln(1 + i )≈ . β Re Z(β) β Re Z(β)

1 In fact, one can show that for ∆E ≪ t ≪ Γ −1 , where ∆E is the average gap 1 between eigenvalues of H(θ) ( ∆E gives a time scale for H),

e−iHt/~ ψ0 = e−izr t/~ ψ0 + small if ψ0 lies near Er in the spectral decomposition of H (see [269, 125]). Note that e−izr t/~ = e−Γr t/~ e−iEr t/~ exhibits exponential decay at the (slow) rate Γr . This is consistent with our picture of a resonance as a metastable state. Remark 17.3 The example given after Definition 17.1 does not lead to a ~2 unique self-adjoint Schr¨odinger operator H = − 2m ∆ + V (x). Presumably, F (β) is independent of the self-adjoint extension chosen. For a large class of self-adjoint Schr¨odinger operators, Condition (17.8) does not hold, and the trace has to be regularized (see [269]). In such a case, the potential can be modified at infinity in such a way that for the modified potential, Condition (17.8) holds. We expect that such a modification can be chosen so that it leads to a sufficiently small error in the tunneling probabilities. In any case, the results we discuss below (which are obtained by applying another non-rigorous technique – path integrals) coincide with those given by more involved rigorous analysis, wherever the latter is possible.

17.4 Instantons To compute Z(β) for the potential sketched in Fig. 17.2, we proceed as in the ground state problem; we represent Z(β) formally as a path integral, and then derive the formal semi-classical expansion (see (16.12)): X Z(τ /~) = N Bω1/2 e−Se (ω)/~ ω

√ (as usual, we ignore the factor (1 + O( ~))). The sum is taken over critical points ω of the “action” Se (φ), of period τ . N is a normalization factor independent of ω and H, and Bω =

det S0′′ (ω0 ) det Se′′ (ω)

17.4 Instantons

227

where the operators Se′′ (ω) and S0′′ (ω0 ) are defined on L2 ([0, τ ]) with zero boundary conditions. Now ω is a periodic classical path in imaginary time (or in the inverted potential −V (x)), with period τ , which we take large (see Fig. 17.4).

V(x)

"bounce"

static

a

x

Fig. 17.4. Paths in inverted potential. Two periodic solutions of arbitrarily large period are ωs (s) ≡ 0 (the subscript “s” for “static”) and ωb (s) : 0 7→ a 7→ 0 (“b” for “bounce”). The solution ωb is called an instanton or “bounce”. Since ωs is a minimum of V , V ′′ (ωs ) > 0, and so Se′′ (ωs ) = −∂s2 + Ω 2 where Ω 2 = V ′′ (0). We computed earlier Bω s =

Ωτ 2Ωτ ≈ Ωτ sinh(Ωτ ) e

for τ large. Moreover, Se (ωs ) = 0. We will show later (Section 17.6) that Bω1/2 b

=

−1/2 −iτ Sb

| det⊥ Se′′ (ωb )| det S0′′ (ω0 )

where Sb := Se (ωb ) =

Z

ωb

is the action of the “bounce”, and

!−1/2

(17.9)

k · dx

det ⊥ A := det(A|(Null

A)⊥ )

(17.10)

is the determinant of A restricted to the orthogonal complement of the null space, NullA, of A. Collecting these results, we have (for large τ )

228

17 Resonances

~Ω ~ E ≈ − ln(Re Z) ≈ τ 2 and | det⊥ Se′′ (ωb )| det Se′′ (ωs )

~ Im Z −1/2 Γ ≈− ≈ ~Sb τ Re Z

!−1/2

e−Sb /~ .

So the probability of decay of the state inside the well, per unit time, is Γ = (const)e−Sb /~ where Sb = Se (ωb ) is the action of the instanton (which equals the length of the minimal geodesic in the Agmon metric ds2 = (V (x) − E)+ dx2 ). This explains the sensitivity of the lifetimes of unstable nuclei to small variations of the parameters (for example, isotopes with different masses can have very different lifetimes). Finally, we note that eΩτ det Se′′ (ωs ) ≈ . 2Ω

17.5 Positive Temperatures Here we consider quantum tunneling at positive temperatures (T = β −1 > 0). We use the same approach as above, but let the parameter β be any positive number. Now we have to consider all three critical paths of period τ = ~β (see Fig. 17.5): ω1 = ωs ≡ xmin , ω2 , and ω3 ≡ xmax , where ω2 is a classical periodic trajectory in the potential −V (x) of period τ = ~β.

V(x) ω1

ω2

ω

3

a x Fig. 17.5. Paths of period τ . Since V ′′ (xmin ) > 0, ω1 is a minimal trajectory. As we will see, ω2 is a saddle point of Morse index 1 (see Section 26.4). Finally, V ′′ (xmax ) < 0, and so ω3 is also a saddle point. For k = 1, 3, Se (ωk ) = V (ωk )τ

17.5 Positive Temperatures

229

and for k = 2, 3, Se (ω1 ) < Se (ωk ). The semi-classical expression for the decay probability works out to be Γ =−

1 (Im Bω2 e−S(ω2 )/~ + Im Bω3 e−S(ω3 )/~ ). τ Bω 1

Through which trajectory, ω2 or ω3 , does the tunneling take place? ω3 corresponds to a thermally driven escape (due to thermal fluctuations), and ω2 corresponds to a quantum tunneling escape. If τ is very small (large temperature), the transition occurs through ω3 , as only ω3 can have arbitrarily small period. On the other hand, if τ is very large (small temperature), ω2 sits close to the bottom of the well, and one can show that Se (ω2 ) < Se (ω3 ). In this case, the transition occurs through ω2 . 2 There is a critical value of τ , τc ≈ 2π/Ωmax where Ωmax = −V ′′ (xmax ), at which a transition occurs; the transition is between the situations in which decay is due to tunneling, and in which it is due to thermal fluctuations. (Note τ that for τ < τc , the decay rate differs from Γ by the factor Ωmax (see [8])). 2π This transition can take place either continuously or discontinuously, depending on whether the energy of the periodic classical trajectory in the inverted potential −V (x) depends on its period continuously or discontinuously. In the first case, as temperature decreases below 1/τc (i.e. τ increases above τc ) the tunneling trajectory bifurcates from ω3 and slips down the barrier (see Fig. 17.6). For τ < τc tunneling takes place through ω3 .

jump for τ < τc

τc τ < τ =

Fig. 17.6. Continuous transition. In the second case (see Fig. 17.7), there are no closed trajectories with period > τc , so the transition is discontinuous: decay jumps from ω3 to a trajectory at the bottom of the barrier.

no trajectories for τ>τc τ =

τc

jump for τ
0 and µ ∈ R, solutions of the equation 1 γ = f ( (h(γ) − µ)), T

(18.34)

solve (18.33). Under some conditions, the converse is also true. (The parameters T > 0 and µ, the temperature and chemical potential, are introduced here for future reference.) The chemical potential µ is determined by the condition that Tr γ = some constant, say ν. The function f is selected on physical grounds by either a thermodynamic limit (Gibbs states) or by a contact with a reservoir (or imposing the maximum entropy principle). It is given by the Fermi-Dirac distribution f (h) =

1 , 1 + eh

(18.35)

as we are dealing with fermions. For bosons, fF D would be replaced by the Bose-Einstein distribution fBE (h) =

1 . eh − 1

(18.36)

Inverting the function f and letting f −1 =: g ′ , we rewrite the stationary generalized Hartree-Fock equation as hγ,µ − T g ′ (γ) = 0,

(18.37)

Here hγ,µ := h(γ) − µ = −∆ + V + v ∗ nγ − ex(γ) − µ, T ≥ 0 (temperature), and µ ≥ 0 (chemical potential). It follows from the equations g ′ = f −1 and (18.35), the function g is given by g(λ) = −(λ ln λ + (1 − λ) ln(1 − λ)), so that g ′ (λ) = − ln

λ . 1−λ

(18.38)

(18.39)

Kohn-Sham equation. According to the Hohenberg-Kohn, Levy and Lieb results ([163, 200, 196, 197]), the ground state energy functional for a quantum many-body system of identical particles (fermions) can be written as a functional of the one-particle density of the many-body ground state. However, this result says nothing about the form of this functional. The latter is suggested by replacing in the Hartree-Fock equation the HF exchange term, −ex(γ), which is an operator, by a one-particle density dependent function (or a multiplication operator) −xc(nγ (x)), called the exchange-correlation self-interaction, which is much handier for computations. One expects −xc(nγ ) is small in an appropriate sense. As a result we arrive at the equation

18.6 Generalized Hartree-Fock and Kohn-Sham Equations

i

∂γ = [hnγ , γ] ∂t

251

(18.40)

where, recall, nγ (x) := γ(x, x) and, with h := −∆ + V , hn := h + v ∗ n − xc(n).

(18.41)

The function xc(λ) is given empirically, with the expression xc(λ) = −cλ4/3 , going back to Dirac, used most often.2 (18.40) is the time-dependent KohnSham equation. The most studied is the static case. In this case, Eq (18.40) becomes 1 γ = f ( (hnγ − µ)), T

(18.42)

R where hno is given in (18.41). To this we add the condition nγ = const, which determines the chemical potential µ. Introducing the map den from operators, A, to functions, denA(x) := A(x, x), we can rewrite (18.42) as an equation for just the one-particle density n: 1 n = den f ( (hn − µ)), T

(18.43)

where f is given by (18.35) and hn by (18.41). Eqs (18.42), or (18.43), is the Kohn-Sham equation, the main equation of density functional theory. Note that because of the minimal coupling, there is no (pure) density functional theory when the system in question is coupled to a magnetic field.

2

xc(λ) = −cλ4/3 is the leading term in the semi-classical expansion for the HartreeFock term ex(γ) := −v ♯ γ.

19 Open Quantum Systems

In this chapter, we introduce and study the notion of open quantum system, generalizing that of the quantum system we have dealt with so far. This notion lies at the foundation of quantum information theory.

19.1 Information Reduction Consider a large system (total system T ) containing a subsystem we are interested in. We call the latter a system of interest, or just a system, and denote it symbolically as S. The rest of the total system will be called the environment and denoted E. We write T = S + E. Let x and y be the coordinates of the system and environment, respectively. The state spaces of S, E and T are the Hilbert spaces HS = L2 (dx), HE = L2 (dy) and HT = L2 (dxdy). We will be interested in system observables, i.e. operators acting on the space HS = L2 (dx). We will associate with them operators (denoted by the same letters) on HT = L2 (dxdy) acting on the variable x. Assume we are interested in measuring only properties of the system S, e.g. averages for its various observables. Assume the total system, T = S + E, is described by a wave function, say ψ(x, y). The question we would like to address is: is there a wave function ϕ(x) for S, such that measuring the average of any observable A associated with S, in the state ϕ(x), gives the same result as measuring it in the state ψ(x, y); i.e. is, for any A = Ax , hψ, AψiT = hϕ, AϕiS ? Here the inner-products on the l.h.s. and r.h.s. are in the spaces HT and HS , respectively. The answer is that this holds if and only if the subsystem and environment are not correlated, i.e. ψ(x, y) = ϕ(x)η(y) for some η. If we take, for example, ψ(x, y) = α1 ϕ1 (x)η1 (y) + α2 ϕ2 (x)η2 (y), then hψ, Aψi 6= hϕ, Aϕi for any ϕ(x). So what does it take to describe a state of S in this case without referring to the environment? The answer is: use density operators. Namely, with any © Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_19

254

19 Open Quantum Systems

total wave function, ψ(x, y), we can associate a density operator, ρ = ρψ , for our subsystem, so that hψ, Aψi = TrS (Aρψ ) (19.1) for any system observable A. Here TrS is trace of the system’s degrees of freedom. Indeed, the operator ρψ is defined by its integral kernel Z (19.2) ρψ (x, x′ ) = ψ(x, y)ψ(x′ , y)dy . Problem 19.1 Check that (19.1) holds for any operator A acting on the variable x, provided ρψ is given by (19.2). To develop an abstract theory, we define the important notion of the partial trace. For any trace-class operator, R, on HT , we define the operator TrE R on HS by X hφχi , Rψχi i, (19.3) hφ, (TrE R)ψi = i

for any φ, ψ ∈ HS and for any orthonormal basis {χj } in HE . Here (ψχ)(x, y) := ψ(x)χ(y). Problem 19.2 Show that the r.h.s. of (19.3) is independent of the choice of the orthonormal basis {χj }. Problem 19.3 Show that the integral kernel of TrE Pψ is given by (19.2).

We may consider TrE as a map from (linear) operators on HT to operators on HS . We record its properties. In what follows, TrS and TrT denote the standard traces on HS and HT and we omit the subindex T in TrT so that Tr ≡ TrT . We have 1) TrE is a linear map; 2) TrE is positive (or positivity preserving); 3) TrE is trace preserving: TrS ◦ TrE = Tr. In addition, TrE commutes with multiplication by system observables, i.e., for any system observable A, TrE (AR) = A TrE (R).

(19.4)

Here and below, we consider operators acting on HS = L2 (dx) also as operators on HT = L2 (dxdy) acting just on the variable x (see Remark 19.6 for a discussion). The properties of TrE listed above imply the following properties of the operator TrE R: a) TrE R is a linear operator; b) TrE R is positive if R is positive; c) TrS TrE R = Tr R.

19.1 Information Reduction

255

Problem 19.4 Show all the properties listed above. Next, we have the following key result: Theorem 19.5 For any system observable A we have Tr(AR) = TrS (Aρ), where ρ = TrE R.

(19.5)
Proof. By property (c) above, we have Tr(AR) = TrS TrE (AR) and by (19.4), we obtain furthermore TrE (AR) = A TrE (R), which, together with the previous relation, implies (19.5), since ρ = TrE R.  Thus if R is a density operator of the total system T=S+E, then ρ := TrE R is a density operator of the system S. It is called the reduced density operator. Now, as was mentioned above, the wave function, ψ, of the total system can be associated with a density operator acting on the total system coordinates, namely the rank-one projection R := Pψ , so that hψ, Aψi = Tr(APψ )

(19.6)

for any operator (observable) A. With this, if A is a system observable, i.e., an operator which acts only on the variables x, then (19.1), with ρψ := TrE Pψ , follows from Theorem 19.5. To summarize: if the total system is described by a density operator R, then the results of observations on the system S are given in terms the system’s density operator ρ = TrE R, without further reference to R. A system under consideration, whose interaction with the environment cannot be neglected but which is described in terms of its own degrees of freedom, is called an open system. In fact, every system, unless it is the entire universe, can be considered as a subsystem of a larger system. Hence, in reality every quantum system is an open system. Remark 19.6 In tensor product notation we write ψ ⊗ χ, and A ⊗ 1E , where 1E is the identity operator on HE , for ψ(x)χ(y) and an observable A acting only on x, respectively. Moreover, using the notion of tensor product (see Appendix 25.13), we can pass to a general framework of abstract Hilbert spaces, HT = HS ⊗ HE , without referring to the variables describing their degrees of freedom. One can also define the partial trace, ρ = TrE R, in terms of the integral kernels R(x, y, x′ , y ′ ) and ρ(x, x′ ) of R and ρ, by generalizing the formula (19.2) as Z ρ(x, x′ ) :=

R(x, y, x′ , y)dy.

(19.7)

Problem 19.7 Check that the definitions of partial trace given by (19.7) and (19.3) are equivalent.

256

19 Open Quantum Systems

The results of the next problem are useful in applications. Problem 19.8 Given a density matrix ρe of E, introduce the map Tr′E from operators on HT to operators on HT by Tr′E R := (TrE R) ⊗ ρe . Show that (i) Tr′E is a projection, (ii) its dual (on the space of bounded operators) is given by Tr∗E A := (TrE Aρe ) ⊗ 1. Consider also the symmetric version √ √ Tr′′E R := (TrE R ρe ) ⊗ ρe and show that the dual to Tr′′E is of the same form. Conditional Expectation. In Section 18.2, we explained that quantum statistics is a non-commutative extension of classical probability theory. We further elaborate this thesis by introducing and discussing the quantum (noncommutative) extension of the key probabilistic notion of conditional expectation. Given a density matrix R of T , we define, for any observable, A, the system observable (SO) ExSR (A) := TrE (AR)/ TrE (R).

(19.8)

This is a map from total observables to system ones. It extends the notion of conditional expectation from classical probability to the non-commutative setting, as confirmed by the following properties: (i) ExSR is a linear map; (ii) ExSR (1) = 1; (iii) ExSR (BAB ′ ) = BExSR (A)B ′ for any SOs B and B ′ ; (iv) ExSR (A∗ ) = ExSR (A)∗ ; (v) ExSR (A) ≥ 0, if A ≥ 0 . Problem 19.9 Prove properties (i) -(iii) above. Properties (ii)-(iii) imply that ExSR (B) = B for any SO B (take A = 1 and B ′ = 1 in (iii)), that ExSR (ExSR (A)) = ExSR (A) (follows from the previous equation by taking B = ExSR (A)), and that
Tr ExSR (A)R = Tr(AR). (19.9)

Problem 19.10 Prove (19.9).

19.1 Information Reduction

257

Properties (iv) and (v) above are more subtle and we prove them here. For brevity, we write ExSR (A) ≡ Ex(A) and ω(A) := Tr(AR). Using properties (iii) and (19.9) and letting B and C be any SOs, we obtain ω(BEx(A∗ )C) = ω(Ex(BA∗ C)) = ω(BA∗ C). Since BA∗ C = (C ∗ AB ∗ )∗ and ω(A∗ ) = ω(A), this yields ω(BEx(A∗ )C) = ω(C ∗ AB ∗ ). Now, moving in the reverse direction, we find ω(C ∗ AB ∗ ) = ω(F ), where F := Ex(C ∗ AB ∗ ), which, together with ω(F ) = ω(F ∗ ), F ∗ = B(Ex(A))∗ C and (iii), gives ω(C ∗ AB ∗ ) = ω(F ∗ ) = ω(B(Ex(A))∗ C). This and the previous conclusion imply ω(BEx(A∗ )C) = ω(B(Ex(A))∗ C). Since this is true for every SOs B and C, we have Ex(A∗ ) = (Ex(A))∗ . Indeed, since Ex(A) is a SO, this follows from the following Lemma 19.11 The relation ω(BEC) = 0, for a given SO E and for arbitrary SOs B and C, implies that E = 0. Proof. We use (19.5) to write ω(BEC) = TrS (BECρ), where ρ = TrE R, and then take B = |ψihχ| and C = |ϕihψ|, with ψ such that hψ, ρψi 6= 0, to obtain ω(BEC) = hχ, Eϕihψ, ρψi. This implies that hχ, Eϕi = 0, for every χ and ϕ, which yields that E = 0.  This concludes the proof of (iv). For property (v), proceeding as in the first part of the proof of (iv), we obtain ω(BEx(A)B ∗ ) = ω(BAB ∗ ) ≥ 0

if A ≥ 0. Since Ex(A) is a SO and since the above inequality is true for every SO B, we have that Ex(A) ≥ 0. Indeed, this follows from the following

Lemma 19.12 If ω(BEB ∗ ) ≥ 0, for a given SO E and for every SO B, then E ≥ 0.

Proof. Proceeding as above, we take B = |ψihϕ|, with ψ such that hψ, ρψi 6= 0, where ρ = TrE R. Then 0 ≤ ω(BEB ∗ ) = hϕ, Eϕihψ, ρψi, which implies that hϕ, Eϕi ≥ 0 for every ϕ, which yields that E ≥ 0.  Thus, we conclude that property (v) holds.

258

19 Open Quantum Systems

19.2 Reduced dynamics Consider a system of interest (S) interacting with an environment (E). We found in Section 19.1, that if the total system is described by a density operator R, acting on the total system state space HT = L2 (dxdy) (or HT = HS × HE ), then the density operator ρ = T rE R of S gives the same results for averages of the system’s observables as R. Here, recall, TrE is the partial trace over the environment degrees of freedom defined in (19.3). Then, for any observable A = Ax , associated with the system Tr(AR) = TrS (Aρ), ρ = TrE R.

(19.10)

(Recall that Tr ≡ TrT .) Recall that the operator ρ = TrE R, called the reduced density operator, acts on the system state space and has the following properties: 1) ρ is a linear operator on the state space of the system; 2) ρ is positive if R is positive; 3) TrS ρ = Tr R. Assume now our total system is described by the Schr¨odinger operator, HT , acting on the state space HT = L2 (dxdy). A model to keep in mind is the one given by HT = HS ⊗ 1E + 1S ⊗ HE + λv, (19.11) where HS and HE are Schr¨odinger operators of this system and environment acting on HS and HE , respectively, λ is a real parameter, called the coupling constant, and v is an operator acting on HT = HS ⊗ HE . Here λv describes the interaction of the system and the environment. The evolution of the total system is given by the von Neumann-Landau equation 1 ∂Rt = [HT , Rt ], Rt=0 = R0 . (19.12) i ∂t ~ We write the solution of this initial value problem as Rt = αt (R0 ). Let Ut := iHt e− ~ . It is shown in Homework 18.3 that αt (R) = Ut RUt∗ .

(19.13)

A map α on a space, B, of operators is called positive iff α(R) ≥ 0, whenever R ≥ 0. With this definition, we have Theorem 19.13 For each t, the map αt has the following properties 1) αt is linear; 2) αt is positive; 3) αt is trace preserving: Trαt (R) = TrR; 4) αt (R∗ ) = αt (R)∗ ; 5) αt is invertible.

19.2 Reduced dynamics

259

Problem 19.14 Prove this theorem. (Hint: To prove the trace preservation property, one can use either the definition of the trace or its cyclic property, iHT t Tr(AB) = Tr(BA), and properties of Ut := e− ~ , see Homework 2.21.) Here we prove the second statement. We omit the subindex tot at HT . Using the relations (19.13) and hψ, Ut∗ φi = hUt ψ, φi (the latter shown in Homework 2.21), we find hψ, αt (R)ψi = hψ, Ut RUt∗ ψi = hUt∗ ψ, RUt∗ ψi ≥ 0. Now, the reduced density operator of the system S at time t is given by ρt := TrE Rt . We consider the particular class of R0 of the form R0 = ρ0 ⊗ ρe for some fixed density operator ρe for E and define βt (ρ0 ) = TrE αt (ρ0 ⊗ ρe ).

(19.14)

The family βt is called the reduced evolution. We observe 1) The evolution βt depends on ρe . 2) Starting with a pure state, the evolution βt produces mixed states, unless S and E do not interact. 3) If S and E do not interact then βt = αSt ,where αSt (ρ) := Ut ρUt∗ , the evolution of S. (Indeed, in this case, αt := αSt ⊗ αE t and the result follows from the fact that αS and TrE commute.) 4) In general, βt (ρ0 ) = TrE αt (R0 ), where R0 is s.t. TrE R0 = ρ0 , is not even a linear map for fixed t. What can we say about the reduced evolution βt ? As we deal with a fixed t, we can concentrate on a single map α. For a fixed map α acting on density operators on the total state space HT = L2 (dxdy) and an environment’s density operator ρe , we define the reduced map β acting on system density operators on the system space HS = L2 (dx) as β(ρ) = TrE α(ρ ⊗ ρe ).

(19.15)

Moreover, we assume that α is of the form α(R) = U RU ∗ where U is a unitary (or more generally, invertible) operator on HT . Then we have Theorem 19.15 The map (19.15) has the following properties (a) β is linear; (b) β is positive; (c) β is trace preserving: Trβ(ρ) = Trρ; (d) β(ρ∗ ) = β(ρ)∗ . Proof. We show only property (b), leaving the other properties as the exercise. This property follows immediately as β is a composition of two positivity preserving maps, TrE and α.

260

19 Open Quantum Systems

Problem 19.16 Show (a) - (d). (Hint: To prove (c) show that TrS TrE = TrT and use the cyclic property of the trace, TrE (AB) = TrE (BA).) Theorem 19.17 The map β is of the form X β(ρ) = Vn ρVn∗

(19.16)

where Vn are bounded operators satisfying (strong convexity) X Vn∗ Vn = 1.

(19.17)

n

n

We prove this theorem at the end of this subsection. Problem 19.18 Show that (19.16) implies Theorem 19.13. Is the converse true, i.e. do the properties stated in Theorem 19.13 imply (19.16)? The answer is no. However, the following strengthening of the positivity property does the job. Definition 19.19 A map β on a space, B, of operators is called completely positive if and only if β ⊗ 1 is positive on B ⊗ Mk for any k ≥ 1. Here Mk is the space of k × k matrices. Problem 19.20 Show directly that maps of the form (19.16) are completely positive. The next theorem shows that the reverse, which is more subtle, is also true. Theorem 19.21 (Kraus) (i) Linear, completely positive maps β are of the form (19.16). If, in addition, β are trace preserving, then (19.17) holds also. (ii) (19.16) is unique up to a unitary transformation, i.e. if P P Representation ∗ ∗ n Wn ρWn , for every density matrix ρ, then there is a unitary n Vn ρVn = operator U s.t. Wn = U Vn , for every n. The operators Vn in (19.16) are called the Kraus operators. We prove this result below. This theorem illustrates the importance of the notion of complete positivity. This notion also differentiates between quantum and classical situations, see Remark 19.27. Definition 19.22 Linear, completely positive, trace preserving (TPCP) maps are called quantum (dynamical) maps or quantum (communication) channels. In the opposite direction, a quantum map can be lifted to a unitary evolution of a larger (total) system. To describe this, it is convenient to pass from quantum maps of density operators to the dual maps of observables. The map (automorphism) β ′ , acting on (an algebra of) bounded of operators on the system Hilbert space, is said to be dual to a map (automorphism) β on density operators if it satisfies

19.2 Reduced dynamics

(ρ, β ′ (A)) = (β(ρ), A),

261

(19.18)

where, recall, (ρ, A) is the coupling between density operators and observables: (ρ, A) = TrS (Aρ).

(19.19)

A map β ′ on (an algebra of) bounded of operators on the system Hilbert space, is said to be positive if and only if β ′ (A) ≥ 0, whenever A ≥ 0. Similarly, one defines the complete positivity. β ′ is called unital if and only if β ′ (1) = 1. A quantum map β satisfies the conclusions of Theorem 19.17 if and only if its dual, β ′ , satisfies 1) β ′ is linear; 2) β ′ is completely positive; 3) β ′ is unital: β ′ (1) = 1; 4) β ′ (A∗ ) =P β ′ (A)∗ ′ 5) β (A) = n Vn∗ AVn , P where Vn are bounded operators satisfying n Vn∗ Vn = 1 (strong convexity).

By Kraus’ theorem 19.21, properties 2) and 3) are equivalent to property 5). Furthermore, β ′ is unital if and only if β is the trace preserving. Indeed, Trβ(ρ) = (β(ρ), 1) = (ρ, β ′ (1)) and Trρ = (ρ, 1) imply that Trβ(ρ) = Trρ if and only if β ′ (1) = 1. Finally, property 3) shows that β ′ has the eigenvalue 1 (with the eigenvector 1). This suggests that β also has the eigenvalue 1. The corresponding eigenvector would be an equilibrium state. Problem 19.23 Show that (1) the composition of two quantum maps is again a quantum map and (2) the composition of two quantum maps with Kraus operators {Vn } and {Wn }, has the Kraus operators {Vm Wn }. In the direction opposite to the one of Theorem 19.17, we have Theorem 19.24 (Stinespring) β ′ : B(H) → B(H) is an adjoint quantum map if and only if it is of the form β ′ (A) = W ∗ (A ⊗ 1K )W,

(19.20)

where K is a Hilbert space and W : H → H ⊗ K, a bounded operator.

The artificial environment space K is called the ancilla space1 . Derivation of Theorem 19.24 from Theorem 19.21 (finite-dimensional H). d Let d := dimH. Define W : H ⊗ CP → H by W : f → ⊕i Vi f . Then W ∗ : H → d ∗ H ⊗ C is given by W : ⊕i fi → i Vi∗ fi and (19.20), with K := Cd , follows.  1

In quantum information theory, the ancilla is an auxiliary quantum system. One of its common uses is for indirect measurements, in which a measurement is done on the ancilla coupled to S. After the measurement is finished, the ancilla is discarded.

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19 Open Quantum Systems

For other proofs of this result see [93, 177]. ([93], Theorem 4.6, shows a more general result implying Theorem 19.24.) Problem 19.25 Show that the operators W and W ∗ P is the proof above (a) are mutually adjoint and (b) satisfy W ∗ (A ⊗ 1K )W = i Vi∗ AVi .

In the opposite direction, Stinespring’s theorem implies Kraus’ Theorem 19.21, as we show in Appendix 19.3 below. The first proof of Theorem 19.21 (finite-dimensional H). Let K be an abstract Hilbert space of the same dimension as H. We construct the function X θi ⊗ χi ∈ H ⊗ K, Ψ := i

where {θi } and {χi } are some orthonormal bases in H and K, respectively. The function Ψ provides the map from K to H as K : φ ∈ K → ψ := hΨ¯ , φiK ∈ H, whose inverse is given by K −1 : ψ ∈ H → φ := hΨ, ψiH ∈ K. (This demonstrates that any two Hilbert spaces of the same dimension are isometrically isomorphic.) Problem 19.26 Show that the two expressions above define operators inverse of each other. (Hint: use the expansion in the bases {θi } and {χi }.)

It is convenient to introduce the maps Tφ : H ⊗ K → H and Tφ∗ : H → H ⊗ K, defined by Tφ ξ := hφ, ξiK and Tφ∗ η := η ⊗ φ (the Dirac bra and ket operators). Then Kφ = Tφ Ψ . Let ψ = Kφ = Tφ Ψ . Then Pψ = Tφ PΨ Tφ∗ , where φ = K −1 ψ, and therefore β(Pψ ) = Tφ (β ⊗ 1)(PΨ )Tφ∗ .

(19.21)

Since the map β ⊗ 1 is completely positive, the operator (β ⊗ 1)(PΨ ) is positive and therefore can be written as X (β ⊗ 1)(PΨ ) = |Φk ihΦk |, (19.22) k

where Φk are non-normalized, but mutually orthogonal vectors from H ⊗ K (Φk are eigenvectors of the operator (β ⊗ 1)(PΨ ) with the norms equal to the square roots of the corresponding eigenvalues). Using relations (19.21) and (19.22), we find X β(Pψ ) = |Tφ Φk ihTφ Φk |, (19.23) k

19.3 Some Proofs

263

where, recall, φ = K −1 ψ. ¯ ξiK = hξ, ¯ φiK . For φ = K −1 ψ, this gives Tφ ξ = Now we have Tφ ξ := hφ, −1 ∗ −1 ¯ ¯ hξ, K ψiK = h(K ) ξ, ψiH and therefore Tφ ξ = T˜(K ∗ )−1 ξ¯ψ,

(19.24)

where T˜ψ : f → hψ, f iH mapping H ⊗ K to K. The last two relations imply X β(Pψ ) = |Vk ψihVk ψ|, k

with the operators Vk on H given by Vk := T˜(K ∗ )−1 Φ¯k . Since, in the Dirac bra-ket notation, Pψ = |ψihψ|, the latter equation can be rewritten as (show this) X β(Pψ ) = Vk Pψ Vk∗ . k

Extending this relation, by linearity, to arbitrary density matrices gives (19.16).  Remark 19.27 One can show (see [93], Theorem 4.3) that a positive linear map between two algebras of observables2 one of which is abelian is completely positive.

19.3 Some Proofs First, we give a lengthier but instructive direct proof of Theorem 19.13(b) which we will use below. Let {χi } be an orthonormal basis in the environment space L2 (dy). By the definition of β, (19.15), and the property α(R) = U RU ∗ , we have ∀φ, ψ ∈ L2 (dx) X hφχi , α(ρ ⊗ ρe )ψχi i hφ, β(ρ)ψi = i

=

X i

hU ∗ φχi , ρ ⊗ ρe U ∗ ψχi i.

(19.25)

Taking here φ = ψ, we see the r.h.s is non-negative, provided ρ is non-negative. This gives Theorem 19.13(b).  Proof of Theorem 19.17. To prove (19.16), we write the inner product in L2 (dxdy) as h·i = hh·iE iS , i.e. first as the inner product in L2 (dy) and then in 2

Technically our algebras of observables are C ∗ -algebras. See [45] for the definition of the C ∗ -algebra. A good example to keep in mind is a subset of the space of bounded operators (on some Hilbert space) close under algebraic operations of addition, multiplication and taking adjoints.

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19 Open Quantum Systems

2 HS = L2 (dx). Let χi be an orthonormal basis P in HE =PL (dy) of eigenfunctions of ρe with eigenvalues λj . Then ρe = λj Pχj = λj |χj ihχj |, so that, using (19.25), we obtain, for any φ, ψ ∈ HS , Xp p h λj hχj , U ∗ φχi iE , ρ λj hχj , U ∗ ψχi iE iS hφ, β(ρ)ψi = i,j

X

=

i,j

where Vij∗ φ := Now

hVij∗ φ, ρ0 Vij∗ ψiS ,

(19.26)

p P λj hχj , U ∗ φχi iE , and therefore hφ, β(ρ)ψi = hφ, i,j Vij ρVij∗ ψi.

p hVij φ, ψiS = hφ, Vij∗ ψiS = λj hφ, hχj , U ∗ ψχi iE iS p p = λj hU φχj , ψχi i = h λj hU φχj , χi iE , ψiS .

This implies

Vij φ =

p λj hχi , U φχj iE ,

which, in turn, gives X X p Vij∗ Vij φ = Vij∗ λj hχi , U φχj iE i,j

i,j

=

X ij

=

X j

Since

P

λj hχj , U ∗ hχi , U φχj iE χi iE λj hχj , U ∗

X i

hχi , U φχj iE χi iE .

i hχi , U φχj iE χi

= U φχj , this gives X X X λj hχj , U ∗ U φχj iE = λj hχj , φχj iE . Vij∗ Vij φ = i,j

(19.27)

j

Since hχj , φχj iE = φ and

P

λj = Tr ρe = 1, we have

P

i,j

Vij∗ Vij φ = φ.



The second proof of Theorem 19.21. (Derivation from Theorem 19.24 for arbitrary H.) For φ, ψ ∈ H, we use Stinespring’s Theorem 19.24 to write hφ, β ∗ (A)ψi = hφ, W ∗ (A ⊗ 1K )W ψi = hW φ, (A ⊗ 1K )W ψi for some ancilla space P K. Let {χi } be an orthonormal basis in K. Inserting the partition of unity i Pχi = 1, where recall Pχi ≡ |χi ihχi | is the rank one projection on χi , and writing the inner product in H ⊗ K as h·iH⊗K = hh·iK iH , i.e. first as the inner product in K and then in H, we find X hφ, β ∗ (A)ψiH = hW φ, (A ⊗ Pχi )W ψiH⊗K i

=

X i

hhχi , W φiK , Ahχi , W ψiK iH .

(19.28)

19.4 Communication Channels

265

Now, define the operators Vi on PH by Vi φ := hχi , W φiK , so that the last equation reads hφ, β ∗ (A)ψiH = i hVi φ, AVi ψiH , which is equivalent to X Vi∗ AVi . (19.29) β ∗ (A) = i

Since W is bounded, the operators Vi are bounded as well. By definition (19.18), Eq. (19.29) implies (19.16). Next, if β is trace preserving, then definition (19.18) implies that β ∗ is unital (i.e. β ∗ (1) = 1) and (19.29), with A = 1, becomes (19.17). This proves the theorem. 

19.4 Communication Channels The quantum communication channel is one of the key notions in quantum information theory. In this section we compare classical and quantum channels and describe various ways to process information. We begin by explaining the notion of information. Classical information consists of words constructed from a finite collection of symbols, called an alphabet, with the simplest one being the binary alphabet consisting of two symbols, called bits, say {0, 1}. A more complicated example consists of a written language alphabet. More generally, classical information could be modelled by a probability distribution, p(x), on the set X of wordsPor letters (p : X → R, P the symbol in both the s.t. p(x) ≥ 0 and x p(x) = 1). (We useP R discrete and continuous cases. In the latter case, stands for an integral, .) A classical (communication) channel is a map βcl from probability distributions into probability distributions, satisfying 1) βcl is linear; 2) βcl is positive (i.e. positivity preserving: P p ≥ 0 =⇒Pβcl (p) ≥ 0); 3) βcl preserves the total probability, x βcl (p)(x) = x p(x). A map βcl having the above properties is called a stochastic map. Note ∗ that P we can write 3)∗ as hβcl (p), 1i = hp, 1i ∀p or βcl (1) = 1, where hp, f i = x p(x)f (x), and βcl , defined on bounded functions, is the map dual to βcl defined by ∗ hβcl (p), f i = hp, βcl (f )i. Quantum information is represented by density operators ρ (ρ ≥ 0 and Tr ρ = 1). The simplest physical system – the one with state space C2 – is called the qubit (quantum bit). One can realize a qubit as the lowest two states of an atom or of a Josephson junction between superconductors (with the rest of the spectrum considered as an environment). While the classical binary digit (also called the bit) takes only two values, 0 or 1, the qubit (in a pure state) takes values on the 3dimensional sphere {ψ ∈    1 0 C2 : kψk = 1}. If we take the standard basis, |0i := and |1i := , in 0 1

266

19 Open Quantum Systems

C2 (playing the role of the classical information bits), then a qubit can be in any superposition of the states |0i and |1i : α|0i + β|1i with |α|2 + |β|2 = 1. Clearly, a quantum communication channel should map density matrices into density matrices. Since a transmission of information could be viewed as an evolution process, and since evolution of an open quantum system is given by quantum (dynamical) maps, the quantum channel is, by the definition, a quantum (dynamical) map. Recall that a quantum (dynamical) map is an automorphism on density operators (i.e. a map of density operators into density operators), which is linear, completely positive and trace preserving. Thus depending on the application, the same object is called either a quantum (dynamical) map or a quantum communication channel, or simply quantum channel. One introduces the following definitions: Preparation: The preparation is a linear, bounded, positive map Φ : p → ρ from probability densities, p := {p(x)}, to density operators, ρ (i.e. from classical to quantum information). Given a basis, {ρx }, of density operators ρx (DMB), Φ can be constructed as X Φ(p) = p(x)ρx . x

Reception: The reception is a linear, bounded, positive map R : ρ → p from density matrices to probability densities. Given a collection {Mx } of P positive operators satisfying x Mx = 1 (i.e. {Mx} is a positive operatorvalued resolution of the identity), called positive operator-valued measure or POVM, a reception map can be given by R(ρ)(x) := Tr(Mx ρ). Starting with von Neumann’s axiomatization of the measurement process, one can show that a measurement is a reception map (see [242], Section 3.1). Note that the preparation and reception maps allow us to embed classical channels, βcl , into quantum ones as βcl → Φ ◦ βcl ◦ R. Assume our transmission (computation) is quantum, but the input and/or extraction of information could be either quantum or classical. Further, assume that the originator possesses a DMB, {ρx }, and the receiver a POVM, {Mx }. Thus we can introduce the following refinements: quantum-quantum channel βqq : ρ → ρ′ P classical-quantum channel βclq : p → ρ = px ρx → ρ′ . quantum-classical channel βqcl : ρ → ρ′ →Ppx = T r(Mx ρ′ ). classical-classical channel βclcl : p → ρ = px ρx → ρ′ → px = Tr(Mx ρ′ ).

19.5 Quantum Dynamical Semigroups

267

The last case is the one one expects in most quantum computations. One starts with classical information, converts it into a quantum one, which is manipulated by a quantum computer or quantum channel, and at the end one converts the quantum output into a classical one. Remark 19.28 (Measuring classical and quantum information) Let a source be modelled as a probability distribution {px } (or a random variable X with probability distribution, px = P (X = x)). To measure classical information contained in a probability distribution p(x), Shannon defined the quantity X H(p) = − p(x) log p(x), (19.30) x

which is now called Shannon’s information entropy or just Shannon’s information. It is analogous to the Boltzmann entropy of Statistical Mechanics. The quantum analogue of Shannon’s information is von Neumann’s entropy, which has already appeared in Section 18.3 and will be discussed in Section 19.6.

19.5 Quantum Dynamical Semigroups Now, we address dynamical properties of the reduced evolution (see (19.14)) and restore t in the notation. Definition 19.29 A family βt , ∀t ≥ 0, of quantum maps is called an open quantum evolution. (Thus a reduced evolution is an open quantum evolution.) Note that a single quantum map, β, generates the discrete time flow, β k , k = 0, 1, . . . . Now we may introduce the notion of an an abstract open quantum system as a pair: a space of density operators and an open quantum evolution on it. We further refine the notion of open quantum evolution to the following: Definition 19.30 A quantum dynamical semigroup (qds) is an open quantum evolution which has the Markov property, i.e. it is a semigroup: βt ◦ βs = βt+s ∀t, s ≥ 0.

(19.31)

For a quantum dynamical semigroup βt , we define the generator by L(ρ) := ∂t βt (ρ)|t=0 ,

(19.32)

for those density operators ρ for which βt (ρ) is differentiable at t = 0. Then βt (ρ) satisfies ∂t βt (ρ) = L(βt (ρ)), βt=0 (ρ) = ρ. (19.33)

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19 Open Quantum Systems

Problem 19.31 Show that (19.31) and (19.32) imply (19.33). Problem 19.32 Show that αt given in (19.13) is a qds and find its generator. Theorem 19.33 Under certain technical continuity conditions, the generators of Markov evolutions are of the form ∞ X 1 i (Wj ρWj∗ − {Wj∗ Wj , ρ}), L(ρ) = − [H, ρ] + ~ 2 j=0

(19.34)

where H is a self-adjoint operator (system P ∗ Hamiltonian), {A, B} := AB + BA and Wj are bounded operators s.t. Wj Wj converges strongly.

For a proof of this theorem see [93, 184] and also [7]. Eq (19.33) with (19.34) is called the Lindblad equation. If an open quantum evolution βt has the Markov property, then so has its dual, βt′ . The generator, L′ , of βt′ is the operator dual to L (in the coupling between density operators and observables given by (19.19)). Hence βt′ satisfies the differential equation ∂∂t βt′ = L′ βt′ . Furthermore, for L of the form (19.34), L′ is given by L′ (A) =

X 1 i (Wj∗ AWj − {Wj∗ Wj , A}). [H, A] + ~ 2 i

(19.35)

Write L = L0 + G, where L0 ρ := − ~i [H, ρ] and G := L − L0 . Then, as follows from (19.34), the operator L0 is formally anti-symmetric, while the operator G is formally symmetric (for self-adjoint Wj ’s), in the sense that Tr(AL0 ρ) = −Tr((L0 A)ρ),

Tr(AGρ) = Tr((GA)ρ),

or (abusing notation as primed and not primed operators act on different spaces) L′0 = −L0 and G′ = G. In addition, G satisfies Tr(ρ∗ Gρ) ≤ 0

(19.36)

(and therefore is dissipative in the sense of the following definition: An operator G on density operators is said to be dissipative if for each density operator ρ there exists an observable A, such that Re Tr(AGρ) ≤ 0.) Problem 19.34 Show that Tr(ρ∗ Gρ) ≤ 0 for self-adjoint Wj ’s. Hint: for selfadjoint Wj ’s, Tr(ρ∗ Gρ) = −

∞

1X Tr(([Wj , ρ])∗ ([Wj , ρ])) ≤ 0. 2 j=0

(19.37)

The fact that βt′ has the eigenvalue 1 (with the eigenvector 1) is now translated into the property that L′ has the eigenvalue 0 (with the eigenvector

19.5 Quantum Dynamical Semigroups

269

1), which is apparent from expression (19.35). As above, this suggests that L also has the eigenvalue 0, with the eigenvector being an equilibrium state. In fact, for the Lindblad equation (19.33)-(19.34) derived, non-rigorously, by the 2nd order and Markov approximations, from reduced dynamics (19.14), with the initial state ρe of the environment being the Gibbs state ρT e = e−He /T /ZT e ,

(19.38)

where ZT e := Tre−He /T , at temperature T , see (18.12), we have the following property: - the operator (19.34) satisfies L(AρT ) = (L′ A)ρT , where ρT ≡ ρT s is the Gibbs state of the system, ρT = e−H/T /ZT , with Z(T ) := Tre−H/T . This property is called the detailed balance condition. Taking here A = 1 and using L′ 1 = 0, we see that it implies L(ρT ) = 0, i.e. - the Gibbs state ρT is a stationary state of (19.33)-(19.34). To understand where the latter property comes from, let the Lindblad equation (19.33)-(19.34) be derived as describeded above. Let ρT s and ρT e be the Gibbs states for the system and environment at temperature T . Take ρT s ⊗ ρT e as the initial condition in (19.14). Of course, unless the system and environment are decoupled, ρT s ⊗ ρT e is not a static solution for the total system (19.11)-(19.12). However, it is a static solution in the leading order in the coupling constant, so that ρT s gives a static solution for the approximate reduced dynamics (19.14) generated by L. Given the properties above, we see that in the inner product hA, BiT := Tr(A∗ BρT ) on the algebra of observables, we have that L0 is anti-self-adjoint, while G is self-adjoint, L∗0 = −L0 ,

G∗ = G∗ .

(19.39)

Problem 19.35 Show the relations (19.39). Note that if G = 0, i.e. for the flow generated by L0 , the energy and entropy, as defined in Section 18.3, are conserved. Problem 19.36 Prove this statement. For a general evolution (19.33)-(19.34), this is not true anymore. Instead, as we will show in Section 19.6, if ρT is a static solution, then the free energy and relative entropy are non-increasing functions of time (see Proposition 19.40 and Theorem 19.38). Finally, recall that quantum maps are characterized by either complete positivity, or the explicit form (19.16), which are equivalent by Theorem 19.21 and Homework 19.20. These characterizations extend to quantum dynamical semigroups, since the latter are families of quantum maps. So we expect the generators of these semigroups to be characterized correspondingly. So far, we characterized their generators (L) by the explicit form (19.34) corresponding to (19.16). Thus a natural question is which property of L corresponds to the

270

19 Open Quantum Systems

complete positivity of βt ? The answer is the complete dissipativity, which is defined by lifting the property DL (A, A) ≥ 0,

(19.40)

where DL (A, B) is the dissipation function defined by DL (A, B) := L′ (A∗ B) − L′ (A∗ )B − A∗ L′ (B).

(19.41)

to A ⊗ Mk , for any k ≥ 1, where A is an algebra of observables and Mk is the space of k × k matrices. The form DL has some remarkable properties: - DL (A, A) = 0 if and only if L′ A = ~i [H, A] for some self-adjoint operator H; 3 - DL (c1, c1) = 0; - if DL (A, A) = 0 implies A = c1, then there is a unique stationary state ρ∗ and for every ρ, we have βt (ρ) → ρ∗ , as t → ∞. (If ρ∗ is an equilibrium state, then the last property is called return to equilibrium.) A linear map L′ of a C ∗ -algebra satisfying (19.40) and L′ A∗ = (L′ A)∗ is called a dissipation. It is shown in [178] that dissipations are dissipative in the sense of the definition given before (19.36).

19.6 Irreversibility A quantum (dynamical) map β is said to be irreversible iff β is not invertible (within the class of quantum maps) and reversible, otherwise. An example of a reversible map is β(ρ) = V ρV ∗ ,

(19.42)

where V is an invertible operator. In fact, we have (cf. [242], Section 3.2) Proposition 19.37 Any invertible quantum map β is of form (19.42). Proof. Assume β is invertible. Then β −1 exists and is a quantum map. By Theorem 19.17 (or 19.21), both, β and β −1 , are of the form (19.16) - (19.17). Now, denote the Vn -operators for β −1 by Wn and rewrite the equation β −1 ◦ β = 1 as X ∗ β −1 (β(ρ)) = Wm Vn ρVn∗ Wm = ρ, (19.43) m,n

P ∗ with the bounded operators Vn and Wm satisfying n Vn Vn = 1 and P ∗ m Wm Wm = 1. Taking here ρ = Pψ , for an arbitrary normalized vector 3

In the terminology of C ∗ -algebras, L′ is a derivation. In general, a derivation is a linear map δ of a C ∗ -algebra, satisfying δ(AB) = δ(A)B + Aδ(B), for all A, B in the domain of δ.

19.6 Irreversibility

271

P ψ, we see that (19.43) can be rewritten as m,n PWm Vn ψ = Pψ . Testing this relation on various functions, we see that it holds if and only if every Wm Vn ψ is proportional to ψ: Wm Vn ψ = amn ψ, for some numbers amn and for each pair m, n. OnP the other hand, by the ∗ ∗ completeness relations above, we have Vn∗ Vk = m Vn Wm Wm Vk . Since ∗ ∗ ∗ Vn Wm = (Wm Vn ) , this gives X Vn∗ Vk = a ¯mn amk 1. m

Now, using the polar√decomposition Vk = Uk |Vk |, Pwhere Uk are some unitary operators and |A| := A∗ A, the notation√bnk := m a ¯mn amk , the observation that bkk ≥ 0, ∀k and the relation |Vk | = bkk , we find X p ¯bnk 1 = bnn bkk U ∗ Uk n m

and therefore Un = √bbnkb Uk for each n and k. Hence Vn = bnn Un = cnk Uk , q nn kk where cnk := bnk bbnn , for every n and some fixed k. Remembering (19.16), kk pP 2 this gives (19.42) with V := n |cnk | Vk . Since β is invertible, V should be invertible too.  How to quantify the notion of irreversibility? In classical mechanics we encounter irreversibility when we pass from Newton’s equation to the Boltzmann equation. While Newton’s equation is reversible ((x(t), p(t)) → (x(−t), −p(−t)) is a symmetry of Newton’s equation), the Boltzmann equation is irreversible: the Boltzmann entropy Z H(f ) = − f log f, (19.44) for the particle densities, f (x, v, t), which solve the Boltzmann equation, increases along the evolution (the celebrated Boltzmann H-theorem). In quantum mechanics, while the Schr¨odinger equation is reversible, the reduced evolution, say, ρt = βt (ρ0 ) (or ∂∂t ρt = Kρt , in the Markov case) it leads to when some information is ’integrated out’, is expected to be irreversible. To prove this, we look for an analogue of the Boltzmann entropy, and it is natural to define S(ρ) = −Tr(ρ log ρ),

(19.45)

which is nothing but the von Neumann entropy, defined by von Neumann 20 years earlier, and which was already used in Section 18.3, where the definition of the operator log ρ was discussed. We list properties of S(ρ): 1) S(ρ) = 0 if and only ρ = Pψ is a pure state;

272

19 Open Quantum Systems ∗ 2) S(U PρU ) = S(ρ), P for unitaries U ; P 3) S( λj ρj ) ≥ λj S(ρj ), for λj ≥ 0, λj = 1; 4) For any density operator, ρAB , of a composed system A + B,

S(ρAB ) ≤ S(ρA ) + S(ρB )

(19.46)

where ρA and ρB are the marginals of ρAB : TrB ρAB = ρA and TrA ρAB = ρB . To prove Property P 1), we let λj and φj be eigenfunctions and eigenvalues of ρ so that ρ = − λj Pφj and use that due to the relation if, then X S(ρ) = − λj log λj = H({λj }) (19.47)

(which can be also used to define S(ρ); remember 0 ≤ ρ ≤ 1 and therefore 0 P≤ λj ≤ 1 and S(ρ) ≥ 0). Then the equation S(ρ) = 0 can be rewritten as λj log λj = 0. Since every term in the sum on the l.h.s. is non-positive, it should vanish separately. So λj ’s are either 1 or 0. Next, 2) follows from the cyclicity of the trace and 3), from concavity of the function x log x. Finally, the subadditivity of the entropy, (19.46), was proven in [75, 187]. However, there is no H-theorem for S(ρ), i.e, in general, S(ρ) does not decrease (or increase) under the evolution. We look for a more general object which has monotonicity properties. Such a candidate is the relative entropy defined as S(ρ1 , ρ2 ) := Tr(ρ1 (log ρ1 − log ρ2 )),

(19.48)

if Ranρ1 = Ranρ2 and ∞, otherwise. We have the following result whose proof can be found in [212] and [292], Statement 3.1.12: Theorem 19.38 (Generalized H-theorem (Lindblad)) If β is a quantum map, then S(β(ρ1 ), β(ρ2 )) ≤ S(ρ1 , ρ2 ).

(19.49)

Note that, if β is reversible, then (19.49) holds with equality. This shows that S(βt (ρ1 ), βt (ρ2 )) is monotonically non-increasing, as t increases, and is constant if βt is reversible. It is shown in [229] that for finite-dimensional spaces, any transformation β, which satisfies the equality in (19.49) – i.e., β preserves the relative entropy – is reversible: S(β(ρ1 ), β(ρ2 )) = S(ρ1 , ρ2 )

⇐⇒

β is reversible.

(19.50)

Thus, by this theorem, the strict decrease of the relative entropy implies that the map is irreversible.

19.6 Irreversibility

273

We say a quantum (dynamical) map β is unitary if and only if it is of the form (19.42) with V unitary. 4 If ρ∗ is a stationary state, then Theorem 19.38 implies that for a quantum dynamical semigroup, βt , the relative entropy S(βt (ρ), ρ∗ ) is monotonically non-increasing, while (19.50) says it is constant if and only if βt is unitary. This can be strengthened to the following result ([282]): Theorem 19.39 Let βt be a quantum dynamical semigroup on a finitedimensional space. Assume βt has a stationary state ρ∗ . Then S(βt (ρ), ρ∗ ) is differentiable in t and the entropy production, defined by the relation σ(ρ) := −∂t S(βt (ρ), ρ∗ )|t=0 ,

(19.51)

is non-negative and convex. Moreover, it is equal to 0 if and only if βt is unitary (and consequently, its generator L is a commutation, Lρ = i[H ′ , ρ] for some H ′ ). Recall from Section 18.3 the definition of for the free energy (see (18.13)) FT (ρ) := E(ρ) − T S(ρ).

(19.52)

Proposition 19.40 Let βt be a quantum dynamical semigroup on a finitedimensional space. Assume ρT is a static state of βt . Then the free energy FT is non-increasing under βt , and is constant if and only if βt is unitary. Proof. By definitions (19.38) and (19.48), the free energy (19.52) can be expressed in terms of the relative entropy for ρ and ρT as FT (ρ) = T S(ρ, ρT ) + T ln ZT .

(19.53)

This relation, together with (19.51) (for ρ∗ = ρT ) and the semi-group property of βt , implies ∂t FT (ρt ) = −T σ(ρt ),

(19.54)

where ρt := βt (ρ), which, by Theorem 19.39, yields the result.

4

If V is unitary, then the adjoint quantum map β ∗ is a homomorphism, i.e. it satisfies β ∗ (AB) = β ∗ (A)β ∗ (B).

274

19 Open Quantum Systems

19.7 Decoherence and Thermalization Consider an open quantum system with a state space HS and an evolution βt . Let {ϕj }j≥1 be a preferred orthonormal basis in HS , which will be specified below. The system is said to exhibit (full) decoherence (in the preferred basis) if for any initial condition ρ0 , the off-diagonal matrix elements hϕm , βt (ρ0 )ϕn i of its evolution βt (ρ0 ) vanish in the limit of large time: lim hϕm , βt (ρ0 )ϕn i = 0,

t→∞

(19.55)

whenever m 6= n. Denote ρt := βt (ρ0 ) and [ρt ]m,n := hϕm ρt ϕn i. We claim then that decoherence implies that after some time X Tr(Aρt ) ∼ pm hϕm , Aϕm i, (19.56) m

where pm := [ρt ]m,m P, which is a statistical sum. Indeed, inserting the completeness relation, m |ϕm ihϕm | = 1, for the basis {ϕm } on both sides of A, we find X Tr(Aρt ) = (19.57) [ρt ]m,n hϕn , Aϕm i. m,n

Eqs (19.55) and (19.57) imply the desired relation (19.56). Now assume our open system originates by reduction from a total system which includes an environment. If H ≡ HS is the original hamiltonian of the system of interest (S), then the preferred basis is the one of (generalized) eigenvectors of HS , called the energy basis, and the quantum evolution βt is the reduced one, βt (ρ0 ) = TrE αt (ρ0 ⊗ ρe ), The environment plays a crucial role in the phenomenon of decoherence. Indeed, if the system and environment do not interact, then there is no decoherence. To gain more insight into the role of the environment, let the evolution of the total system this in a basis {ϕm } in L2 (dx), P be given by Ψ = Ψt . Expanding 2 we find Ψ = m ϕm ⊗ αm , for some αm ∈ L (dy). We compute the reduced density operator P for this state. Let {χ Pi } be an orthonormal basis in HE . Then TrE PΨ = hχ , Ψ i hΨ, χ i = i E i E i m,n,i Phχi , αm iE |ϕm ihαn , χi iE hϕn |. Since {χi } is an orthonormal basis, we have i hαn , χi iE hχi , αm iE = hαn , αm iE , which together with the previous relation gives X ρt = TrE PΨ = [ρt ]m,n |ϕm ihϕn |, m,n

where [ρt ]m,n := hϕm , ρt ϕn iS = hαm , αn iE . Hence decoherence means that the environmental dynamics forces hαm , αn iE → 0, whenever m 6= n. According to Subsection 18.3, the equilibrium states of the total system, and environment, at temperature T = 1/β are given by Rβ := e−βHT /Z(β), Z(β) := Tre−βHT , and ρE,β := e−βHE /ZE (β), ZE (β) := Tre−βHE . (The

19.7 Decoherence and Thermalization

275

subindex T for“total”should not be confused with the temperature T = 1/β.) We say that the total system T = S + E has the property of return to equilibrium iff lim TrS+E (αt (A)(ρ0 ⊗ ρE,β )) = TrS+E (ARβ ), t→∞

for all observables (of the joint system) A, and for all initial density matrices ρ0 on HS . If this happens, then the large time limit of the reduced density matrix of such a system is given by ρ∞ := lim ρt = TrE (Rβ ) = ρS,β + O(λ), t→∞

where λ is the coupling constant between in the system S and environment (see (19.11)). In the second relation, we used that, by perturbation theory of equilibrium states, Rβ = ρS,β ⊗ρE,β +O(λ). The leading term of ρ∞ (for small coupling constant λ) is just the Gibbs state of the system (see e.g. [45, 224]). In this sense, the system undergoes the process of thermalization.

20 The Second Quantization

In this chapter we describe a powerful technique used in the analysis of quantum many-body systems – the method of second quantization. In rough terms, it allows one, instead of working with a fixed number of particles, to let the number of particles fluctuate, while keeping the average number fixed. We apply this method to derive a useful mean-field limit for many-body dynamics. The method also provides a natural language for quantum field theory (see Chapters 21 and 22) and can be used to preview, in a much simpler setting, some of the issues arising there.

20.1 Fock Space and Creation and Annihilation Operators Consider a system of n identical particles moving in an external potential W (x) and interacting via pair potentials v(xi − xj ). Its Schr¨odinger operator is given by Hn =

n X

(−

i=1

~2 1X ∆xi + W (xi )) + v(xi − xj ). 2m 2

(20.1)

i6=j

We assume the particles are bosons. In this case Hn acts on the space L2sym (R3n ) of L2 functions on R3n which are symmetric w.r.t permutations of the particle coordinates x1 , ..., xn ∈ R3 . We define the new Hilbert space Fbos := ⊕∞ n=0 Fn ,

(20.2)

of sequences Φ = (Φ0 , Φ1 , . . . ) ≡ ⊕∞ n=0 Φn , Φn ∈ Fn , where F0 = C, and Fn = L2sym (R3n ), n ≥ 1, equipped with the inner product hΨ, Φi =

∞ Z X

Ψn (x1 , ..., xn )Φn (x1 , ..., xn )dn x,

n=0

© Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_20

(20.3)

278

20 The Second Quantization

where Ψn and Φn are the n−th components of Ψ and Φ. The space Fbos is called the bosonic Fock space. On Fbos we define the operator H = ⊕∞ n=0 Hn ,

(20.4)

2

~ where H0 = 0, H1 = − 2m ∆x + W (x) and Hn , n ≥ 2, are as above, ∞ so that HΨ = ⊕n=0 Hn Ψn . The operator H is called the 2nd quantized Schr¨odinger operator. The reason for this name will become clear later. The vector Ω := (1, 0, 0, ...) ∈ Fbos is called the vacuum vector in Fbos . Note that, by construction, HΩ = 0.

Problem 20.1 Assuming the pair potentials, v(xi −xj ), are real and bounded, show that H is self-adjoint. One of the advantages of the 2nd quantization is the representation of operators on the Fock space Fbos in terms of annihilation and creation operators (raising and lowering the number of particles). These are the operator-valued distributions f → a(f ) and f → a∗ (f ), where f ∈ L2 (R3 ) and a(f ) and a∗ (f ) are operators on Fbos , defined as Z √ (20.5) (a(f )Ψ )n = n + 1 f (x)Ψn+1 (x, x1 , ..., xn )dx, and a(f )Ω = 0, and (a∗ (f )Φ)n+1 =

√ n + 1(f sΦn )

(20.6)

S for n ≥ 0. Here f sΦn = Pn+1 (f ⊗ Φn ), with PnS the orthogonal projection 2 3n 2 3n from L (R ) to Lsym (R ):

(PnS f )(x1 , . . . , xn ) :=

1 X f (xπ(1) , . . . , xπ(n) ), n! π∈Sn

where, recall, Sn is the symmetric group of permutations of n indices. The operators a(f ) and a∗ (f ) are unbounded and satisfy ha∗ (f )Φ, Ψ i = hΦ, a(f )Ψ i.

(20.7)

Indeed, we compute, using (20.5), on vectors Φ, Ψ , with finite numbers of components, X hΦ, a(f )Ψ i = hΦn , (a(f )Ψ )n i n

Z X√ = n + 1 dx1 ...dxn+1 Φn (x1 , ...xn )f (xn+1 )Ψn+1 (x1 , ..., xn+1 ). n

It is easy to see by relabeling the variables of integration that Φn (x1 , ...xn )f (xn+1 ) can be replaced by

20.1 Fock Space and Creation and Annihilation Operators

279

S Pn+1 (f (xn+1 )Φn (x1 , ..., xn )) 1 X = f (xj )Φn (x1 , . . . , xj−1 , xj+1 , . . . , xn+1 ) n+1 j

to obtain hΦ, a(f )Ψ i Z X√ S n + 1 dn+1 xPn+1 (Φn (x1 , ..., xn )f (xn+1 ))Ψn+1 (x1 , ..., xn+1 ), = n

which, together with (20.6), implies (20.7). Moreover, a∗ (f ) is adjoint to the operator a(f ) : a∗ (f ) = a(f )∗ . Problem 20.2 Show that [a(f ), a∗ (g)] = hf, gi,

[a(f ), a(g)] = [a∗ (f ), a∗ (g)] = 0.

(20.8)

The operators a(f ), a∗ (f ) are operator-valued distributions and it is convenient to introduce the formal notation a# (x) = a# (δx ), so that, formally, Z Z a(f ) = f (x)a(x)dx, a∗ (f ) = f (x)a∗ (x)dx. (20.9) We consider a(x), a∗ (x) as formal symbols satisfying

[a(x), a∗ (y)] = δ(x − y), [a# (x), a# (y)] = 0.

(20.10)

Representation of vectors in terms of creation operators. Proposition 1. Any Φ = ⊕Φn can be written as n X 1 Z Y √ Φ= Φn (x1 , ..., xn ) a∗ (xj )Ωdn x. n! n j=1

(20.11)

Proof. Using the definition of a∗ (x), we compute (a∗ (yn )...a∗ (y1 )Ω)m =

n Y √ δ(xi − yi ))δn,m . n!PnS (

(20.12)

j=1

Problem 20.3 Show (20.12). Hint: Use induction starting with √ (a∗ (y1 )Ω)n (x1 ) = 1P1S (δ(x1 − y1 ))δn,1 and (a∗ (y2 )a∗ (y1 )Ω)n (x1 , x2 ) =

√

2P2S (δ(x1 − y1 )δ(x2 − y2 ))δn,2 .

The equation (20.12) implies that Z Y 1 √ Φn (y1 , ..., yn ) a∗ (yj )Ωdn y = PnS Φn (x1 , ..., xn ), n! which implies the statement of the proposition. 

(20.13)

280

20 The Second Quantization

20.2 Many-body Hamiltonian One-particle operators. LetR b be an operator on L2 (R3 ). We write it as an integral operator bx f (x) = b(x, y)f (y)dy, where b(x, y) is the integral kernel of R b. We think of bx a(x) as the operator b acting on the parameter x, bx a(x) = b(x, y)a(y)dy. We define an operator B on Fbos by the formula Z B = a∗ (x)bx a(x)dx. (20.14) We call operators of this type one-particle operators. The equation (20.14) is an integral of the product of operator-valued distributions and it is not clear whether it is well-defined to begin with. However, if one thinks about (20.14) as a formal expression and uses formally the definitions of a(x) and a∗ (x), one finds that the operator B defined by (20.14) is equal to B = ⊕∞ n=0 Bn , B0 = 0, Bn :=

n X i=1

bi , n ≥ 1,

(20.15)

where bi ≡ bxi stands for the operator b acting on the variable xi . The latter expression makes a perfect sense. One can demonstrate (20.15) formally as follows. Using the definitions of a(x) = a(δx ) and a∗ (x) = a∗ (δx ) and (20.5) and (20.6) Z √ (BΦ)n = n dxPnS [δ(x1 − x)(bx a(x)Φ)n−1 (x2 , ..., xn )] Z √ √ = n dxPnS [δ(x1 − x)bx nΦn (x, x2 , ..., xn )] = nPnS [bx1 Φn (x1 , ..., xn )] n X bxi Φn (x1 , ..., xn ), = i=1

as claimed. There is a different representation of (20.14) which is well defined directly. Let {fi } be an orthonormal basis in L2 (R3 ) and define the operators ai := a(fi ) and a∗i := a∗ (fi ). Assume fi ∈ D(b) and let bij := hfi , bfj i be the matrix associated with the operator b and the basis {fi }. Consider the expression X a∗i bij ai , (20.16) B= ij

defined on vectors for which the sum on the r.h.s. converges (say on vectors with finite number of components). P Problem 20.4 Show that the operator ij a∗i bij ai acts as (20.15).

20.2 Many-body Hamiltonian

281

Formally, the expression P (20.16) is obtained from (20.14) one by inserting the partition of unity n |fn ihfn | = 1 into the latter expression. Though (20.14) needs an additional interpretation, it is much more convenient to work with and is used commonly. Consider a few examples. The free Schr¨odinger operator: Z ~2 H0 = a∗ (x)(− ∆x )a(x)dx. (20.17) 2m

Pn ~2 The equation (20.15) shows that H0 = ⊕∞ n=0 H0n , where H0n = i=1 (− 2m ∆xi ) is the free n-particle Schr¨odinger operator = 0). R (V ∗ We define the number operator N = a (x)a(x)dx and momentum operaR tor P P = a∗ (x)(−i~∇x )a(x)dx. Then N = ⊕∞ n1 and P = ⊕∞ n=0 n=0 Pn where n Pn = j=1 pj , respectively. We have [H0 , N ] = 0, [H0 , P ] = 0.

(20.18)

These equations imply the conservation of the particle number and total momentum. We prove [H0 , P ] = 0: Z Z ~2 ∗ [H0 , P ] = [ dxa (x)(− ∆x )a(x), dya∗ (y)(−i~∇y )a(y)] 2m Z Z ~2 = dxdya∗ (x)(− ∆x )[a(x), a∗ (y)](−i~∇y )a(y) 2 Z Z + a∗ (y)(−i~∇y )[a(x), a∗ (y)](−i~∆x )a(x) Z ~2 ∆, −i~∇]a(x) = 0, = dxa∗ (x)[− 2m where in the last step we used that [a(x), a∗ (y)] = δ(x − y). Problem 20.5 Show that [a∗ (x)∆x a(x), a(y)] = [a∗ (x), a(y)]∆x a(x). Problem 20.6 Show (20.18). In mathematics, one denotes the operator in (20.15) as dΓ (b). Problem 20.7 Show formally that [dΓ (b1 ), dΓ (b2 )] = dΓ ([b1 , b2 ]). We can generalize (20.17) to a particle moving in an external field W : Z ~2 HW = a∗ (x)(− ∆x + W (x))a(x)dx. (20.19) 2m Two-particle operators. Let Z 1 V = a∗ (x)a∗ (y)v(x − y)a(x)a(y)dxdy. 2

(20.20)

282

20 The Second Quantization

(The integrand on the r.h.s. annihilates particles at x and y, acts with v(x−y), and then creates particles at x and y.) We claim that (V Φ)n =

1X v(xi − xj )Φn (x1 , ..., xn ). 2

(20.21)

i6=j

Q Indeed, consider a(x)a(y) n1 a∗ (xj )Ω. Using this expressionQ and (20.20) and n using a(y)a∗ (z) = a∗ (z)a(y) + δ(y − z), we pull a(y) through j=1 a∗ (xk ) and use a(y)Ω = 0, to obtain a(x)a(y)

n Y

a∗ (xj )Ω = a(x)

i=1

i=1

Then we pull similarly a(x) trough a(x)a(y)

n Y

i=1

n X

a∗ (xj )Ω =

n X i=1

Q

δ(y − xi )

k6=i

δ(y − xi )

Y

a∗ (xi )Ω.

(20.22)

k6=i

a∗ (xk ) and use a(x)Ω = 0 to find

X j6=i

δ(x − xj )

n Y

a∗ (xk )Ω.

(20.23)

k6=i,j

R Qn Let Φn = √1n! Φn (x1 , ..., xn ) 1 a∗ (xj )Ωdn x. Using (20.23) and (20.20) and integrating the delta-functions, we arrive at n n Y 1 XX √ V Φn = a∗ (xk )Ω, v(xi − xj )Φn (x1 , ..., xn ) n! i=1 j6=i i=1

(20.24)

which gives (20.21).  The equations (20.19) and (20.21) show that the Hamiltonian (20.4) introduced at the beginning of this chapter can be written as Z ~2 ∆x + W (x))a(x)dx H = a∗ (x)(− 2m Z (20.25) 1 ∗ ∗ + a (x)a (y)v(x − y)a(x)a(y)dxdy. 2 Problem 20.8 Show that [V, N ] = 0, [V, P ] = 0, [H, N ] = 0, [HW =0 , P ] = 0.

20.3 Evolution of Quantum Fields Consider the Heisenberg evolution of a(x): a(x, t) = e a(x, t) satisfies the Heisenberg equation, i~

∂ a(x, t) = −[H, a(x, t)]. ∂t

iHt

iHt

iHt ~

a(x)e−

iHt ~

. Then

(20.26)

Using that [H, a(x, t)] = e ~ [H, a(x)]e− ~ , and computing the commutator [H, a(x)], one can show that a(x, t) satisfies the differential equation

20.4 Relation to Quantum Harmonic Oscillator

i~

~2 ∂ a(x, t) = (− ∆x + W (x))a(x, t) ∂t Z 2m +

283

(20.27)

dyv(x − y)a∗ (y, t)a(y, t)a(x, t).

Problem 20.9 Show (20.26) and (20.27). Note that if v = 0, then (20.27) is the Schr¨odinger equation but for the operator valued function a(x, t), called a quantum field. This is the origin of the term the second quantization.

20.4 Relation to Quantum Harmonic Oscillator Assume the potential W is positive and confining, so that the operator ~2 ∆x + W (x) has positive, purely discrete spectrum {λj } accumulating − 2m to infinity. Let {fj } be an orthonormal basis of its eigenfunctions. Then applying to the operator HW the formula (20.16), with this basis and using that ~2 the matrix of the operator − 2m ∆x + W (x) in this basis is diagonal with the diagonal elements {λj }, we obtain X λj a∗j aj . (20.28) HW = j

On the other hand, consider the r-dimensional quantum harmonic oscillator, described by the Schr¨odinger operator r

Hho = −

1X ~2 mωi2 x2i ∆+ 2m 2 i=1

(20.29)

on L2 (Rr ). As we know from Section 7.4, it can be rewritten in a form similar to (20.28):   r X 1 ∗ ~ωi ai ai + Hho = , (20.30) 2 i=1

where, recall, ai and a∗i are the harmonic oscillator annihilation and creation operators, aj := p

1 1 (mωj x + ipj ) and a∗j := p (mωj xj − ipj ). 2m~ωj 2m~ωj

(20.31)

(These operators satisfy the commutation relation [ai , a∗j ] = δij .) We see that, Pr modulo the additive constant, 12 i=1 ~ωi , (20.30), with ~ωj = λj , is a finitedimensional approximation to (20.28).

284

20 The Second Quantization

20.5 Scalar Fermions As before, to simplify notation slightly we consider scalar (or spinless) fermions. (For fermions with spin we would have to introduce one more variable - spin- for each particle and require that functions below are antisymmetric with respect to permutations of particle coordinates and spins.) We define the Fock space for scalar fermions as Ff ermi := ⊕∞ n=0 Fn ,

(20.32)

where F0 = C, and Fn = L2asym (R3n ), n ≥ 1, with L2asym (R3n ) the space of functions, Ψ ∈ L2 (R3n ), anti-symmetric w.r.t permutations of coordinates x1 , ..., xn ∈ R3 : Ψ (xπ(1) , . . . , xπ(n) ) = (−1)#(π) Ψ (x1 , . . . , xn ), π ∈ Sn , where, recall Sn is the symmetric group of permutations of n indices and #(π) is the number of transpositions in the permutation π. We equip this space with the inner product (20.3). Again, the vector Ω := (1, 0, 0, ...) ∈ Ff ermi , is called the vacuum vector in Ff ermi . The second quantized hamiltonian H is again defined by (20.4) and HΩ = 0. The creation and annihilation operators are defined now as Z √ (a(f )Ψ )n = n + 1 f (x)Ψn+1 (x, x1 , ..., xn )dx, (20.33) for n ≥ 0, and a(f )Ω = 0, and √ A (f (x1 )Φn (x2 , ..., xn+1 )), (a∗ (f )Φ)n+1 = n + 1Pn+1

(20.34)

where PnA is the orthogonal projection of L2 (R3n ) to L2asym (R3n ). As before, a∗ (f ) is adjoint to the operator a(f ) : a∗ (f ) = a(f )∗ . However, unlike the bosonic annihilation and creation operators, the fermionic ones anti-commute [a(f ), a∗ (g)]+ = hf, gi,

[a(f ), a(g)]+ = [a∗ (f ), a(g)]+ = 0,

(20.35)

where [A, B]+ := AB + BA. A remarkable fact is that the fermionic annihilation and creation operators are bounded. Problem 20.10 Show this. Hint: Use the anti-commutation relation a(f )a∗ (f ) +a∗ (f )a(f ) = kf k2. The second-quantized operator (20.4) is still expressed in the form (20.25) and the derivations above concerning (20.25) remain true.

20.6 Mean Field Regime

285

20.6 Mean Field Regime We introduce the coupling constant g into our Hamiltonian (20.25) by replacing v by gv and consider regime where g ≪ 1 and the number of particles n ≫ 1 but ng = O(1) so that the kinetic and potential energy terms in (20.1) are of the same order. We rescale our Hamiltonian by defining: 1 1 a(f ) = √ ψg (f ) and a∗ (f ) = √ ψg∗ (f ). g g The rescaled creation and annihilation operators are operator-valued distributions obeying commutation relations   ψg (x) , ψg∗ (y) = g δ 3 (x − y), and rest = 0.

ˆ The Hamiltonian H can be written in terms of ψg (x) and ψg∗ (x) as H = g −1 H with Z ~2 ˆ = H ψg∗ (x) (− ∆ + W (x)) ψg (x)d3 x 2m (20.36) Z ∗ ∗ 3 + ψg (x1 ) ψg (x2 ) v(x1 − x2 ) ψg (x1 )ψg (x2 )d x1 dx2 . In the mean-field limit, as g → 0, the rescaled creation and annihilation operators, ψg (x) and ψg∗ (x), commute, and our quantum theory converges to the classical one, which is a Hamiltonian theory with the phase space given by a space of differentiable functionals (classical field observables) A(ψ, ψ), with the Poisson bracket defined (for two functionals A(ψ, ψ) and B(ψ, ψ)) as Z i (∂ψ(x) A∂ψ(x) B − ∂ψ(x) A∂ψ(x) B)dx, {A, B}(ψ, ψ) = (20.37) ~ and the Hamiltonian functional given by Z H (ψ, ψ) := ψ(x) hx ψ(x)d3 x Z ψ(x1 ) ψ(x2 ) v(x1 − x2 ) ψ(x1 )ψ(x2 )d3 x1 d3 x2 , +

(20.38)

2

~ with hx := − 2m ∆ + W (x). With ψ considered as an evaluation functional ψ → ψ(x), the Hamilton equation is given by

∂t ψ = {H(ψ, ψ), ψ},

(20.39)

which has the form similar to (20.26). (For details and discussions see mathematical supplement, Chapter 26, and especially Section 26.6.) Using (20.38) and (20.37) this equation can be written explicitly as

286

20 The Second Quantization

i~∂t ψ = (−

~2 ∆ + W (x) + v ∗ |ψ|2 )ψ, 2m

(20.40)

which is the Hartree equation introduced in Section 14.1. Thus we arrived at the Hartree Hamiltonian system with the Hamiltonian functional H(ψ, ψ) and the Poisson bracket {A, B}(ψ, ψ) and with dynamics given by the Hartree equation (20.40). For more details see Sections 4.7, 26.6 and 26.7. To formalize this heuristic analysis we consider analytic functionals, A(ψ, ψ), i.e. functionals of the form p q Y X Z Y ψ(yi )dxdy, (20.41) ψ(xi )apq (x1 , . . . , xp ; y1 , . . . , yq ) A(ψ, ψ) := pq

1

1

where apq (x1 , ..., xp ; y1 , ..., yq ) are the integral kernels of bounded operators apq , with series converging in an appropriate topology. Now, consider the Wick quantization, i.e. association with classical obsevables A = A(ψ, ψ) their quantum counterparts: A = A(ψ, ψ) −→ Aˆ = A(ψg , ψg∗ ), according to the rule q p Y XZ Z Y ψg (yi )dp xdq y. (20.42) ψg∗ (xi )apq (x1 , . . . , xp ; y1 , . . . , yq ) Aˆ := pq

i=1

i=1

The operator Aˆ is said to be the Wick quantization of the classical field obˆ servable A and A is said to be the Wick symbol of the operator A. Let Φt be the flow generated by the Hartree equation (20.39), or (20.40), i.e. for any reasonable ψ0 , Φt (ψ0 ) is the solution of (20.40) with the initial condition ψ0 . It defined the classical evolution of classical observables according ¯ to αcl t (A) := A ◦ Φt where (A ◦ Φt )(ψ, ψ) = A(Φt (ψ), Φt (ψ)). Furthermore, let ˆ ˆ iHt/g −iHt/g ˆ ˆ αt (A) := e Ae R be the Heisenberg dynamics of quantum observables. Finally, let N := g −1 ψg∗ (x) ψg (x)d3 x be the particle number operator and ~ . One can show (see [124, 16]) that for a certain class of classical τ := 8kvk ∞ observables A and on states, Ψ , satisfying kN Ψ k ≤ c/g, we have for t ≤ τ , as g → 0, cl (A) + O(g). ˆ = α\ αt (A) (20.43) t We sketch a proof of (20.43). Using the Duhamel principle (i.e. writing cl (A))−A cl cl (A)) = α ˆ as the integral of derivative ∂r α−r (α\ ˆ \ α−t (α\ −r [H, αr (A)]− t r
{H,\ αcl r (A)} ), we obtain Z t cl (A) = ˆ − α\ dsαt−s (R(αcl (20.44) αt (A) s (A))), t 0

\ ˆ A] ˆ − {H, where R(A) := gi [H, A}, which gives

20.7 Appendix: the Ideal Bose Gas cl (A)k ˆ − α\ kαt (A) B(FN ≤n ) ≤ t

Z

287

t

0

dskR(A ◦ Φs )kB(FN ≤n) .

(20.45)

Note that the full evolution αt drops out of the estimate. Now, we have to obtain appropriate estimates of the remainder R(A) and the classical observable A ◦ Φs . First, we find a convenient expression for the remainder \ ˆ A] ˆ − {H, ˆ have the symbols, A}. Let the operators Aˆ and B R(A) := gi [H, A(ψ, ψ) and B(ψ, ψ) and assume B(ψ, ψ) is a monomial of degree m in ψ ˆ B], ˆ has the symbol and ψ, separately. Then the commutator operator Cˆ = [A, C(ψ, ψ) given by (see e.g. [13]) C(ψ, ψ) =

m X gk

{A, B}k (ψ, ψ),

(20.46)

(∂ψk A, ∂ψk B − ∂ψk B, ∂ψk A).

(20.47)

k=1

where {A, B}k :=

Z Z

k!

The definition of the Hamiltonian H(ψ, ψ) and Eqns (20.46) and (20.47) imply g2 \ \ ˆ A] ˆ = g {H, A}2 . i[H, A} + {V, 2

(20.48)

\ It is not hard to show that k{V, A}2 k ≤ CkAk∗ with C < ∞ and kAk∗ an appropriate norm of A. This implies (cf. (4.11)) \ ˆ A] ˆ = g {H, i[H, A} + O(g 2 ).

(20.49)

Comparing the relation (20.48) with the expression for the remainder R(A) g \ we see that R(A) = {V, A}2 = O(g) which together with (20.44) implies, in 2 turn, (20.43). Finally, one has to estimate the norm kA ◦ Φs k∗ of the classical evolution and this is the place where our restrictions enter and which we skip here. This completes our sketch. Thus the mean-field regime is nothing else but the quasiclassical regime 1 of the quantum many-body field theory with the reciprocal, = g, of the n number of particles playing role of the quasiclassical parameter. As the number of particle increases a quantum system starts behaving classically. In the opposite direction it is shown in ([124]) that the many-body theory described by the quantum Hamiltonian (20.25) can be obtained by quantizing the classical field theory described by the classical Hamiltonian (20.38) and Poisson brackets (20.37).

20.7 Appendix: the Ideal Bose Gas As an example of an application of the second quantization, we compute the partition function, pressure, and the equation of state for an ideal bose gas (i.e.

288

20 The Second Quantization

a gas with no interparticle interactions), placed in the box Λ ≡ [− L2 , L2 ]d ⊂ Rd . The Schr¨odinger operator for such a system is Z ~2 a∗ (x)(− ∆Λ )a(x)dx, (20.50) HΛ = 2m Λ where ∆Λ is the Laplacian on L2 (Λ) with periodic boundary conditions. The Gibbs equilibrium state with a fixed average number of particles and for the inverse temperature β (grand canonical ensemble) is ρβ,µ := e−β(HΛ −µNΛ ) /ZΛ (β, µ), R where NΛ = Λ a∗ (x)a(x)dx is the number of particles operator in the volume Λ ⊂ R3 and ZΛ (β, µ) = Tr e−β(HΛ −µNΛ ) is the partition function. Here µ is the chemical potential entering as a Lagrange multiplier due to fixing the average number of particles. This state is obtained by maximizing the entropy, while leaving the average energy and average number of particles fixed (hence two Lagrange multipliers, β and µ, appear, see Section 18.3). Note that in our case the one particle configuration space is the flat torus d which is Λ ≡ [− L2 , L2 ]d with opposite sides identified (i.e. Rd / 2π L Z ) and the 2π d corresponding momentum space is the lattice L Z . Using separation of variables, and the result of Section 7.1 for a single particle Q in a box with periodic boundary conditions, or verifying directly that k∈ 2π Zd (a∗ (k))nk Ω are L P eigenvectors of HΛ with the eigenvalues k∈ 2π Zd εk nk , we conclude that the L spectrum of the operator HΛ is X εk nk | nk = 0, 1, . . . ∀k}. σ(HΛ ) = { d k∈ 2π L Z

1 |k|2 and each eigenvalue is of the multiplicity 1. Using equaHere εk = 2m tion (25.50) from the Mathematical Supplement, which expresses the trace of an operator in terms of its eigenvalues, we obtain the following expression for ZΛ (β, µ): X P P ZΛ (β, µ) = z nk e−β εk nk , (20.51) nk

where z = e as

βµ

, is called the fugacity. The latter expression can be transformed ! ∞ Y X Y n −β εk n ZΛ (β, µ) = = (1 − z e−β εk )−1 . (20.52) z e k

n=0

k

Note in passing that if for a self-adjoint operator A Q with a purely discrete ∞ spectrum {λj } accumulating at 1, we define det A := j=1 λj whenever this is finite, then we can rewrite (20.52) as ZΛ (β, µ) = det (1 − z e−β H1,Λ )−1

20.7 Appendix: the Ideal Bose Gas

289

1 where H1,Λ = − 2m ∆Λ is the one particle Hamiltonian acting on L2 (Λ) with periodic boundary conditions. Next, we consider the quantity PV (β, µ) := V1β ln ZΛ (β, µ), where V := vol(Λ), called the pressure. We have

PV (β, µ) = −

1 X ln (1 − z e−β εk ) Vβ k

1 =− Tr ln(1 − z e−β H1,Λ ) . Vβ Using that, as V → ∞, (2π)d V

X

k∈

2π L

Zd

f (k) →

Z

(20.53)

f (k) d k ,

we see that formally PV (β, µ) converges, as V → ∞, to Z 1 P (β, µ) = − ln (1 − z e−β εk ) d k (2π)d β Z (20.54) 1 −εk ln (1 − z e ) d k. =− d+2 (2π)d β 2 √ To get the last integral, we changed variables k → k/ β. We compute average number of particles, n ¯ = Tr(NΛ ρβ,µ ). This definition ∂ ln ZΛ (β, µ) , from which we obtain the expression implies the relation n = z ∂z n/V =

1 V

X

k∈

2π L

Zd

z e−β εk . 1 − z e−β εk

(20.55)

To have n ≥ 0, we should take 0 ≤ z ≤ 1. The terms in the sum on the right hand side are, as can be easily checked, the average numbers of particles having momenta k, nk = Tr (Nk ρβ,µ ) where Nk := a∗ (k)a(k). To show this, one uses that, by (20.51), nk = − β1 ∂εk ZΛ (β, µ) , where ZΛ (β, µ) is considered as a functional of ε = εk , to obtain z e−β εk nk = . 1 − z e−β εk

Equations (20.54) and (20.55) constitute the equation of state of the ideal Bose gas (parameterized by z). More precisely, solving Equation (20.55) for z as a function of the density ρ = n/V , and temperature T = 1/β, and substituting the result into Equation (20.54), we find the pressure P as a function of ρ and T .

290

20 The Second Quantization

However, if we seek the equation of state as a relation between P , E and V (which is, of course, equivalent to the expression involving P , ρ and T ), the answer is much simpler. Indeed, using the definition of E E = E(β, µ, V ) := Tr (HΛ ρβ,µ,Λ ) and considering the partition function, ZΛ (β, z), as a function of β and z (and V ) rather than of β and µ (and V ), and similarly for the pressure, we find E=−

∂ ∂ ln ZΛ (β, z) = − (V β P (β, z)) . ∂β ∂β

(20.56)

Taking into account (20.54) and setting d = 3 we find E=

3 PV . 2

(20.57)

This is the equation of state of the ideal Bose gas. Problem 20.11 It is an instructive exercise to re-derive the results of this section for the ideal Fermi gas. Now, we consider the ideal Bose gas in the domain Λ with a fixed number, n, of particles. Its Hamiltonian is HΛ,n =

n X i=1

−

1 ∆xi 2m

n 2 2 n acting on the space S 1 L (Λ) := Lsym (Λ ) with periodic boundary condi2 n 2 tions. Here S is the symmetric tensor product, and LSym (Λ ) is the L space of functions symmetric with respect to permutations of variables belonging to different factors in the product Λn . Assume we want to compute the canonical partition function, (20.58) ZΛ,n (β) = Tr e−β HΛ,n .

This is not a simple matter (try it!). We show how to derive it from the grand canonical one, ZΛ (β, µ). The considerations below are heuristic, but can be made rigorous. Using expression (20.52) one can show that as a function of z = eβµ , ZΛ (β, µ) is analytic in the disk {|z| ≤ ε}Pfor some ε > 0. Next, by the ∞ definition of ZΛ (β, µ), we have ZΛ (β, µ) = n=0 ZΛ,n (β)z n . Hence ZΛ,n can be computed by the Cauchy formula I dz 1 (20.59) ZΛ (β, µ) n+1 . ZΛ,n (β) = 2πi |z|=ε z By the definition of PΛ (β, µ), we can write ZΛ (β, µ) = eV βPΛ (β,µ) . Writing also z −n = e−n ln z = e−V ν ln z , where ν = n/V , (20.59) becomes

20.7 Appendix: the Ideal Bose Gas

ZΛ,n (β) =

1 2πi

I

eV (βPΛ (β,µ)−ν ln z)

|z|=ε

291

dz . z

Now we take V = |Λ| and n large, while ν = n/V remains fixed. Taking into account the fact that PΛ (β, µ) has a limit as V → ∞, and applying (formally) the method of steepest descent to the integral above, we find ZΛ,n ≈ ce−βµn ZΛ (β, µ),

(20.60)

where | ln c| is uniformly bounded in V , and µ solves the stationary phase equation µ : βz∂z PΛ (β, µ) = n/V, or equivalently (passing from z to µ) µ : ∂µ PΛ (β, µ) = n/V. Define PΛ,n (β) := 1 βV ln ZΛ,n (β). Then relation (20.60) can be rewritten as PΛ,n (β) ≈ (−βµn + PΛ (β, µ))|∂µ PΛ (β,µ)=n/V . That is, PΛ,n (β), as a function of n, is (in the leading order as V → ∞) the Legendre transform of PΛ (β, µ), considered as a function of µ. Similarly, we can pass from ZΛ (β, µ) to ZΛ,n (β) by taking the Legendre transform in the variable ν = n/V . 20.7.1 Bose-Einstein Condensation We analyze formula (20.55) for the average number of particles. From now on we set d = 3. We would like to pass to the thermodynamic limit, V → ∞. The point is that Equation (20.55) is the relation between the average number of particles n (or the average density ρ = n/V ), the temperature T = 1/β, and the chemical potential µ (or fugacity z = eβ µ ). Recall also that 0 ≤ z ≤ 1. As long as ρ and β are such that z < 1, the right hand side of (20.55) converges to the integral Z z e−β εk 1 dk (2π)3 1 − z e−β εk as V → ∞. However, if the solution of Equation (20.55) for z yields, e.g., z = 1 − O(V −1 ), then we have to consider the k = 0 term in the sum on the right hand side of (20.55) separately. In this case we rewrite (20.55) approximately as Z z e−β εk n0 ρ= + dk, (20.61) V 1 − z e−β εk 2

z where we put n0 = 1−z . Now using εk = |k| 2m , changing the variable of inteq β ′ gration as k = 2m k, and passing to spherical coordinates, we obtain

Z

z e−β εk d k = λ−3 g3/2 (z) , 1 − z e−β εk

292

20 The Second Quantization

where λ =

p 2π β/m (the thermal wave length), and g3/2 (z) =

Z

0

∞

2

z e−x x2 dx . 1 − z e−x2

Thus Equation (20.61) can be rewritten as ρ=

n0 + λ−3 g3/2 (z) . V

(20.62)

z Recall that n0 = 1−z , and that this equation connects the density ρ, the thermal wave length λ (or temperature T = 2π/m λ2 ), and the fugacity z (or chemical potential µ = β1 ln z), and is supposed to be valid in the entire range of values, 0 < z < 1, of z. Can z really become very close to 1 (within O(1/V )), or is the precaution we took in the derivation of this equation by isolating the term n0 /V spurious? To answer this question we have to know the behaviour of the function g3/2 (z) for z ∈ (0, 1). One can see immediately that ′ ′ (z) > 0 and g3/2 (1) = ∞ . g3/2 (0) = 0 , g3/2

The function g3/2 (z) is sketched below.

g _3 (1) 2

g _3 (z) 2

1

z

Fig. 20.1. Sketch of g3/2 (z). We see now that if ρ λ3 < g3/2 (1) (with g3/2 (1)−ρ λ3 ≥ a positive number, independent of V ), then the equation ρ λ3 = g3/2 (z) ,

(20.63)

which is obtained from (20.62) by omitting the V -dependent term nV0 , has a unique solution for z which is less than 1, and is independent of V . Conz sequently, taking into account the term n0 /V = V (1−z) would lead to an adjustment of this solution by a term of order O(1/V ), which disappears in the thermodynamic limit V → ∞. However, for ρ λ3 = g3/2 (1), the solution of this equation is, obviously, z = 1, and for ρ λ3 > g3/2 (1), the above equation has no solutions at all. Thus

20.7 Appendix: the Ideal Bose Gas

293

for ρ λ3 ≥ g3/2 (1) we do have to keep the term n0 /V . Moreover, we have an estimate n0 = ρ λ3 − g3/2 (z∗ ) ≥ ρ λ3 − g3/2 (1) V (here z∗ is the solution to Equation (20.62)) which shows that in the case ρλ3 > g3/2 (1), a macroscopically significant (i.e. proportional to the volume or the total number of particles) fraction of the particles is in the single, zero momentum – or condensed – state. This phenomenon is called Bose-Einstein condensation. The critical temperature, Tc , at which this phenomenon takes place can be found by solving the equation ρ λ3 = g3/2 (1) describing the borderline case for λ, and remembering that T = result we have  2/3 2π ρ . Tc = m g3/2 (1)

2π m λ2 .

As a

From Equation (20.62) we can also find the fraction of particles, n0 /V , in the zero momentum (condensed) state as a function of temperature. This dependence is shown in the diagram below.

n 0 /V 1

Tc

T

Fig. 20.2. In this elementary situation, we have stumbled upon one of the central phenomena in macrosystems – the phenomenon of phase transition. Indeed, the states for which all the particles are in the single quantum state corresponding to zero momentum, and those for which the macroscopic fraction of the particles in the quantum state of zero momentum (and consequently in every single quantum state) is zero, can be considered two distinct pure phases of ideal Bose matter (gas). The first pure phase – called the condensate – occurs at T = 0, while the second pure phase takes place for T ≥ Tc . In the interval 0 < T < Tc of temperatures, the Bose matter is in a mixed state in which both phases coexist. Bose-Einstein condensation exhibits a typical property of phase transitions of the second kind: though all the thermodynamic functions and their

294

20 The Second Quantization

first derivatives are continuous at the phase transition, some of the second derivatives are not. Typically one looks at the specific heat CV :=

∂ E(T, µ, V ) , ∂T

the change of heat or energy per unit of temperature. Using Equations (20.54) and (20.56), one can show that while CV is continuous at T = Tc , its derivative with respect to T is not. CV as a function of T is plotted below (see [166], Sect. 12.3]):

Cv/n _3

2

Tc ~~ T 3/2

T

Fig. 20.3. Specific heat of the ideal Bose gas. One can show that the thermodynamic properties (eg., the equation of state – a relation between pressure, temperature and volume) of Bose-Einstein condensation are the same as those of an ordinary gas – liquid condensation. The modern theory relates the phase transitions to superfluid states in liquid helium (He2 ) and to superconducting states in metals and alloys, to the phenomenon of Bose-Einstein condensation. In the mean field description of the phenomena of superfluidity and superconductivity, the wave function of the condensate – the fraction of particles (or pairs of particles, in superconductivity) in the quantum zero momentum state – called the order parameter, is the main object of investigation. Of course, in both cases one deals with interacting particles, and one has to argue that Bose-Einstein condensation persists, at least for weakly interacting Bose matter. Problem 20.12 Extend the above analysis to an arbitrary dimension d.

21 Quantum Electro-Magnetic Field - Photons

To have a theory of emission and absorption of electromagnetic radiation by quantum systems, not only should the particle system be quantized, but the electro-magnetic field as well. Hence we have to quantize Maxwell’s equations. We do this by analogy with the quantization of classical mechanics as we have done this in Section 4.1. This suggests we have to put the classical electromagnetic field theory, which is originally given in terms of Maxwell’s PDEs, into Hamiltonian form. As before we do this in two steps: by introducing the action principle, and performing a Legendre transform. Then we define the quantization map by associating with canonically conjugate classical fields the corresponding operators, and quantizing observables correspondingly. Since Maxwell’s equations are wave equations for vector fields with constraints, to provide the reader with a simpler guide, we first quantize the scalar KleinGordon equation, which gives the wave equation in the limit of vanishing mass. The reader familiar with the quantization of the Klein-Gordon equation can proceed directly to the next section on quantization of the Maxwell equations. In what follows, we work in physical units in which the Planck constant and speed of light are equal to 1: ~ = 1, c = 1.

21.1 Klein-Gordon Classical Field Theory 21.1.1 Principle of minimum action We construct the Hamiltonian formulation of the Klein-Gordon equation. We consider a scalar (real or complex) field φ(x, t) on Rd satisfying the evolution equation ( + m2 )φ = 0, (21.1) where, recall,  := ∂t2 − ∆ is the D’Alembertian operator and the parameter m ≥ 0 is interpreted as mass. For m > 0 this is the Klein-Gordon equation, once proposed to describe relativistic particles, and for m = 0 this is the wave © Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_21

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21 Quantum Electro-Magnetic Field - Photons

equation (assumed to describe massless particles). The corresponding theory is called the Klein-Gordon classical field theory. We write equation (21.1) as a Hamiltonian system. This is done in two steps: introducing the action principle, and performing a Legendre transform. Then we quantize the resulting infinite dimensional Hamiltonian system. To fix ideas, we consider from now on only real fields. We remark on complex fields at the end. As for classical mechanics, we begin with the principle of minimal action (properly, of “stationary” action). Recall that it states that an evolution equation for physical states is an Euler-Lagrange equation for a certain functional called the action. More precisely, one considers a space of functions φ, defined on space-time, called the fields. The equation of motion for φ is given by S ′ (φ) = 0, where S is an action functional on the space of fields. This functional is of the form Z TZ ˙ t))dxdt S(φ) = L(φ(x, t), ∇x φ(x, t), φ(x, (21.2) 0

Rd

for φ : Rdx × Rt → R (for the moment we consider only real fields). Here, L : R × Rd × R → R is the Lagrangian density. The space integral of the Lagrangian density, Z ˙ := ˙ t))dx, L(φ, φ) L(φ(x, t), ∇x φ(x, t), φ(x, (21.3) Rd

is called the Lagrangian functional. Recalling from in Section 4.7 the definitions of critical points and the derivation of the Euler-Lagrange equations (see also Section 26.2 of Mathematical supplement), it is easy to show that critical points of satisfy the Euler-Lagrange equation ˙ + ∂φ L(φ, φ) ˙ = 0. − ∂t (∂φ˙ L(φ, φ))

(21.4)

Now, we turn to the Klein-Gordon equation. Let f : R → R be a differentiable function. Consider the Lagrangian functional  Z  1 ˙2 1 ˙ = |φ| − |∇x φ|2 − f (φ) dx (21.5) L(φ, φ) 2 2 Rd defined on some subspace of H 1 (Rd ) × L2 (Rd ) s.t. f (φ(x)) is integrable. ˙ = 1 |φ| ˙ 2 − 1 |∇x φ|2 − f (φ). The corresponding Lagrangian density is L(φ, φ) 2 2 The critical point equation for the corresponding action functional S(φ) = RT d ˙ L(φ(t), φ(t))d xdt is the (nonlinear) Klein-Gordon equation: 0 φ + f ′ (φ) = 0.

(21.6)

One can generalize the above construction by considering the action RT ˙ S(φ) = 0 L(φ(t), φ(t))dt, defined on the space of paths

21.1 Klein-Gordon Classical Field Theory

297

Pφ0 ,φT = {φ ∈ C 1 ([0, T ]; X) | φ(0) = φ0 , φ(T ) = φT }, for some φ0 , φT ∈ X, where the Lagrangian functional L(φ, η) is defined on X×V . Here X is an open subset of a normed space V (or manifold). X is called the configuration space of the physical system, and its elements are called fields. In an examples above, X is some functional space, say X = H 1 (Rd , R). (If X is a non-linear space, then the Lagrangian functional would be defined on (a subset of) T X, the tangent bundle of X.) The Euler-Lagrange equation in this case is ˙ + ∂φ L(φ, φ) ˙ =0 − ∂t (∂φ˙ L(φ, φ)) (21.7) where ∂φ L and ∂φ˙ L are variational or G˚ ateaux derivative of L with respect ˙ φ and φ, respectively. (See Mathematical Supplement, Section 26.2 for the definition.) 21.1.2 Hamiltonians We generalize the construction we used for classical mechanics. Suppose the dynamics of a system are determined by the minimum action principle with a Lagrangian functional L : X × V → R, which is differentiable. Here V and X are a Banach space and an open subset of V . We pass to the new variables ˙ → (φ, π), where π ∈ V ∗ , as a function of φ and φ, ˙ is given by (φ, φ) ˙ π = ∂φ˙ L(φ, φ).

(21.8)

(Recall that V ∗ is the space dual to V , see Mathematical Supplement, Sec˙ tion 25.1.) We assume that the equation (4.41) has a unique solution for φ. (Typically, L is convex in the second variable.) With this in mind, we define the Hamiltonian functional, H : X × V ∗ → R, as
˙ − L(φ, φ) ˙ |˙ H(φ, π) = hπ, φi . (21.9) ˙ φ:∂ ˙ L(φ,φ)=π φ

As in Classical Mechanics, the space Z := X × V ∗ is called a phase space of the system. ˙ and H(φ, π) are related by (21.8)-(21.9), then the Theorem 21.1 If L(φ, φ) Euler-Lagrange equation (21.4) for the action (21.2) is equivalent to the Hamilton equations φ˙ = ∂π H(φ, π), π˙ = −∂φ H(φ, π). (21.10)

The proof of this theorem is a straightforward generalization of the proof of Theorem 4.8. Problem 21.2 Prove this theorem.

This gives a hamiltonian formulation of CFT. For theR  Klein-Gordon classical field theory, the Lagrange functional is 1 2 2 (|χ| − |∇φ| ) − f (φ) dx, and, consequently, the Klein-Gordon L(φ, χ) = 2 Hamiltonian is

298

21 Quantum Electro-Magnetic Field - Photons

Z 

H(φ, π) =

 1 2 2 (|π| + |∇φ| ) + f (φ) dx. 2

(21.11)

Problem 21.3 Show (21.11). 21.1.3 Hamiltonian System Suppose that Z = X × V ∗ is a space of functions Φ(x) = (φ(x), π(x)) on Rd . The functional on X which maps X ∋ φ 7→ φ(x) is called the evaluation functional (at x), which we denote (with some abuse of notation) as φ(x), and similarly for V . Problem 21.4 Show that ∂φ φ(y) = δy and ∂π π(y) = δy . Now, we recognize that the Hamilton equations equations (21.10) can be written, for all x, as Φ˙ t (x) = {Φ(x), H}(Φt ) (21.12) where {F, G} is the Poisson bracket on Z defined for any pair differentiable functionals F, G on Z as {F, G} = h∂φ F, ∂π Gi − h∂π F, ∂φ Gi Z = {∂π F ∂φ G − ∂φ F ∂π G}dx.

(21.13)

(See Section 4.7 for the notion and another example of the Poisson brackets.) Problem 21.5 Prove this. One can show, formally, that with the Poisson bracket given in (21.13), {π(x), φ(y)} = δ(x − y), {φ(x), φ(y)} = 0, {π(x), π(y)} = 0.

(21.14)

Problem 21.6 Show (21.14). Equation (21.14) says that the evaluation functionals, π, and φ, are canonical coordinates. To have a rigorous interpretation of the first equation in (21.14), we introduce, for f ∈ C0∞ (Rd ), the functionals φ(f ) : φ 7→ hf, φi

and

π(f ) : π 7→ hf, πi.

Then (21.14) means that {π(f ), φ(g)} = hf, gi, etc, for all f, g ∈ C0∞ (Rd ). Note that a path Φt in Z solves (21.12) iff, for all functionals F , d F (Φt ) = {F (Φt ), H}. dt

(21.15)

21.1 Klein-Gordon Classical Field Theory

299

Definition 21.7 A Hamiltonian system is a triple, (Z, {·, ·}, H), a Poisson space (Z, {·, ·}) (a Banach space Z, with a Poisson bracket) together with a Hamiltonian functional H defined on that space. For the Klein-Gordon classical field theory the phase, or state, space is Z = H 1 (Rd ) × H 1 (Rd ), the Hamiltonian is given by (21.11), and the Poisson brackets are defined on it as Z {F, G} = {∂π F ∂φ G − ∂φ F ∂π G}. (21.16) (In principle we could consider a larger phase space, Z = H 1 (Rd ) × L2 (Rd ), but the space above is more convenient.) For a more general situation see Mathematical Supplement, Section 26.6. Remark 21.8 Our definition of a Hamiltonian system differs from the standard one in using the Poisson bracket instead of a symplectic form. The reason for using the Poisson bracket is its direct relation to the commutator. 21.1.4 Complexification of the Klein-Gordon Equation With the view to quantization, it is convenient to pass from the real phase space Z = H 1 (Rd ) × H 1 (Rd ) ≡ H 1 (Rd , R2 ), with the Poisson bracket (21.16), and with the canonical real fields, φ(f ) and π(f ), to the complex one, Z c = H 1 (Rd , C), with the Poisson bracket Z  ∂α(x) F ∂α(x) G − ∂α(x) F ∂α(x) G dx, (21.17) {F, G} = i ¯ ¯

where ∂α(x) := ∂Re α(x) − i∂Im α(x) and ∂α(x) := ∂Re α(x) − i∂Im α(x) , and with ¯ the canonical complex field α(f ), and its complex conjugate α ¯ (f ), defined by
1 α(f ) = √ φ(C −1/2 f ) + iπ(C 1/2 f ) 2
1 α ¯ (f ) = √ φ(C −1/2 f ) − iπ(C 1/2 f ) , 2

(21.18)

where C is an operator on L2 (Rd ). The Poisson brackets (21.16) and (21.17), imply, for f, g ∈ C0∞ , the Poisson brackets {α(f ), α(g)} ¯ = hf¯, gi,

{α(f ), α(g)} = {α ¯ (f ), α ¯ (g)} = 0.

(21.19)

Problem 21.9 Prove the above statement. Thus (21.18) is a canonical transformation. Consider the free classical field theory, i.e. the Hamiltonian (21.11) with

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21 Quantum Electro-Magnetic Field - Photons

F (φ) =

1 2 2 m |φ| , 2

(21.20)

with m > 0. Since −∆ + m2 > 0, we can take C = (−∆ + m2 )−1/2 .

(21.21)

Then the free Klein-Gordon Hamiltonian is expressed in terms of the fields (21.18) as Z H(α, α ¯) = α ¯ C −1 αdx, (21.22)

21.2 Quantization of the Klein-Gordon Equation We are now ready to attempt to quantize the free Klein-Gordon theory, i.e. (21.11) - (21.16), with (21.20). The Klein-Gordon equation (21.1), or the corresponding Hamiltonian system, (21.11) - (21.16), with (21.20), look like a continuum of classical harmonic oscillators. The link becomes especially explicit if pass in the classical fields φ(x, t) and π(x, t) to the Fourier transform ˆ t) and π in the x variable: φ(k, ˆ (k, t). The Fourier transformed fields satisfy 2 the equation (∂t + |k|2 )φˆ = 0, and have the Hamiltonian Z p 1 ˆ 2 }dd k, ˆ π {|ˆ π |2 + ω(k)2 |φ| ω(k) := |k|2 + m2 . (21.23) H(φ, ˆ) = 2 Thus we want to quantize the free Klein-Gordon classical field as a continuum of harmonic oscillators. In finite dimensional case we can quantize the corresponding hamiltonian system in any set of canonically conjugate variables. In the infinite dimensional case, the choice of such variables makes a difference between a meaningful and meaningless quantum theory. Recall, that we obtained a representation (7.22) of theP harmonic oscillator, which can be rewritten, with m = 1 and ~ = 1, as H = rj=1 ωj a∗j aj on the space −1

L2 (Rr , dµc ), dµc (x) := [det(2πc)]−1/2 e−hx,c xi dr x, Pr Qr 2 r where c := diag(~/ωi ), so that hx, c−1 xi = i=1 ωi xi , d x = i=1 dxi is the Lebesgue measure on Rr , and the operators aj and their adjoints a∗j , are defined on the space L2 (Rr , dµc ) by s s p 1 1 ∂xj , a∗j = − ∂xj + 2ωj xj . aj := (21.24) 2ωj 2ωj dµc is Gaussian measure of mean zero and covariance c. The point here is that unlike the standard representation, where the harmonic oscillator Hamiltonian acts on the space L2 (Rr , dr x), which has no r → ∞ limit, the new

21.2 Quantization of the Klein-Gordon Equation

301

representation is on the space L2 (Rr , dµc ), which can be considered as a finitedimensional approximation of the Gaussian space L2 (Qs , dµC ). Here Qs = {f | (−∆ + |x|2 + 1)−s/2 f ∈ L2 (Rd )}, for s sufficiently large, and dµC is the Gaussian measure on Qs of mean 0, and covariance operator C which acts on L2 (Rd ) and has matrix c as a finite– dimensional approximation. Elements of the space L2 (Qs , dµC ) are functionals F (φ) on Qs such that Z |F (φ)|2 dµC (φ) < ∞. Qs

We explain this in more detail later, in Subsection 21.3 below. The representation (21.22) can be thought of as coming from quantization of the classical hamiltonian system with the hamiltonian h(α, α) ¯ := hα, c−1 αi, where α := (αj ). Motivated by this representation, we associate with the complex fields α(f ), and α ¯ (f ), the annihilation operator a(f ), and its adjoint, the creation operator a∗ (f ), acting on the space L2 (Qs , dµC ), with some covariance operator C, for s sufficiently large, according to the relations a(f ) = hC 1/2 f, ∂φ i, a∗ (f ) = −hC 1/2 f ∂φ i + hC −1/2 f φi.

(21.25)

(More precisely, a(f )F = hC 1/2 f, ∂φ F i = ∂λ |λ=0 F (φ + λC 1/2 f ), etc.) These operators satisfy the commutation relations [a(f ), a∗ (g)] = hf¯, gi, [a(f ), a(g)] = [a∗ (f ), a∗ (g)] = 0, where f, g ∈ C0∞ . (For representation of canonical commutation relation, see [45].) It is convenient to introduce the operator-valued distributions, a(x) and R R a∗ (x) by a(f ) = a(x)f¯(x)dx and a∗ (f ) = a∗ (x)f (x)dx. Then a(x) = C 1/2 ∂φ(x) , a∗ (x) = −C 1/2 ∂φ(x) + C −1/2 φ(x).

With this correspondence, the Hamiltonian functional H(α, α) ¯ = mapped into the quantum Hamiltonian operator Z Hf = a(x)∗ C −1 a(x)dx,

(21.26)

R

α ¯ C −1 α is

(21.27)

acting on the space L2 (Qs , dµC ). (The subindex f stands for the ”field”.) Here we put the creation operator a(x)∗ on the left of the annihilation operator a(x) for a reason, to be explained later. Recall, that for the Klein-Gordon CFT, the covariance operator is given by (21.21). In this case it is convenient to

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21 Quantum Electro-Magnetic Field - Photons

pass to the Fourier representation, a(k) = (2π)−d/2 the Plancherel theorem, to obtain Z Hf = ω(k)a∗ (k)a(k)dk,

R

e−ik·x a(x)dx, and use

(21.28)

p where ω(k) = |k|2 + m2 . (The meaning of the integral on the r.h.s. is explained in Section 20.2.) Define the particle number and the momentum operators (cf. Section 26.8 of Mathematical Supplements and Problem 21.13) Z Z N = a∗ (x)a(x)dx and Pf = a∗ (x)(−i∇x )a(x)dx. (21.29) Problem 21.10 Show (formally) that the operators P , H, and N commute.

21.3 The Gaussian Spaces We sketch a definition of the Gaussian measure, dµC (φ), on Qs , of mean 0 and covariance operator C, acting on L2 (Rd ). (For detailed exposition see [135, 270, 271].) One way to describe dµC is through finite-dimensional approximations, X, of Q∗s = Q−s given in terms of cylinder sets. For each f ∈ Q∗s , we define the linear functional on Qs , Z f (φ) := f φdx

R where the integral on the r.h.s. is understood as (−∆ + |x|2 + 1)s/2 f · (−∆ + |x|2 + 1)−s/2 φd3 x. Let n = dim X. Given n linearly independent vectors f1 , . . . , fn from X, we associate with each Borel set B in Rn , the X− cylinder set CB := {φ | (f1 (φ), . . . , fk (φ)) ∈ B}.

With every measurable function g on Rn we associate the functional G(φ) = g(φ(f1 ), . . . , φ(fn )) on Q∗s , called a X−cyliner function. We define dµC (φ) by giving its integrals of X−cyliner functions over X−cylinder sets Z Z g(x1 , . . . , xn )dµX (21.30) G(φ)dµC (φ) = C (x) CB

B

for every B and g and for increasing sequence of finite-dimensional spaces X, whose limit is Q∗s . Here dµX C is the finite-dimensional Gaussian measure on Rn , −1 −1/2 −hx,CX xi/2 n dµX e d x C (φ) = (det 2πCX ) where dn x is the usual Lebesgue measure on Rn (here we display the dimension of the measure) and CX is the matrix (hfi , Cfj i). dµC (φ) can be extended

21.3 The Gaussian Spaces

303

to a measure on the σ−algebra generated by cylinder sets which leads to the notion of the integral of measurable w.r.to this algebra complex functionals. Next, as usual we define L2 (Qs , dµC ) Rto be the space of measurable complex functionals F (φ) on Qs such that Qs |F (φ)|2 dµC (φ) < ∞. The basic examples of square integrable functionals are F : φ ∈ Qs 7→ p(φ(f1 ), . . . , φ(fn )) where p(t1 , . . . , tn ) are polynomials in t1 , . . . , tn and f1 , . . . , fn ∈ Q∗s . In fact, one can show that, by our construction, the span of vectors of the form

n Y 1

φ(fj )Ω, n ≥ 1, is dense in L2 (Qs , dµC ).

(21.31) Proving this latter fact requires some general considerations from the theory of functional spaces. (See however the corresponding proof for the harmonic oscillator.) We introduce the expectation of a functional F with respect to dµC : Z E(F ) := F (φ)dµC (φ). The terminology “mean 0” and “covariance C” corresponds to the properties E(φ(f )) = 0

and

E(φ(f )φ(g)) = hf, Cgi,

E(φ(x)) = 0

and

E(φ(x)φ(y)) = C(x, y)

or, formally,

where C(x, y) is the integral kernel of C. These formulae can be easily proven by using finite dimensional approximations (21.30). What should s be? Compute the expectation ofRthe square of the norm functional φ → kφk2Qs := hφ, (−∆ + |x|2 + 1)−s φi = φ(x)Ks (x, y)φ(y)dxdy, where Ks (x, y) is the integral kernel of the operator (−∆ + |x|2 + 1)−s . We have formally Z Z E(kφk2Qs ) = δ(x − y)Ks (x, y)E(φ(x)φ(y))dxdy Z Z = δ(x − y)Ks (x, y)C(x, y)dxdy. Since C(x, y) = C(y, x), this gives E(kφk2Qs ) = Tr((−∆y + |x|2 + 1)−s C). Now, the operator C is bounded and the operator (−∆y + |x|2 + 1)−s is trace class if s > d = 3. So in this case E(kφk2Qs ) < ∞. If we know more about C, we can relax the condition s > d = 3. For example, for the Klein-Gordon theory, C = (−∆ + m2 )−1/2 and we have s > d − 1 for dimension d ≥ 2.

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21 Quantum Electro-Magnetic Field - Photons

The integration by parts formula in our space is Z Z ¯ (−i∂φ + iC −1 φ)F GdµC (φ). F (−i∂φ G)dµC (φ) = Qs

To prove (21.32) we use Z Z ¯ F (−i∂φ G)dµC (φ) = Qs

(21.32)

Qs

(−i∂φ F )GdµC (φ) + i

Qs

Z

F¯ G∂φ dµC (φ)

Qs

and the relation ∂φ dµC (φ) = −C −1 φdµC (φ).

To see the latter relation, think formally about the Gaussian measure as being dµC (φ) = const e−hφ,C

−1

φi/2

Dφ.

The integration by parts formula implies that −i∂φ is not symmetric on L2 (Qs , dµC ): (−i∂φ )∗ = −i∂φ + iC −1 φ. However, the above implies that the operator π := −i∂φ + 12 iC −1 φ is symmetric. In fact, it is self-adjoint: π ∗ = π.

21.4 Wick Quantization We now describe a systematic way to quantize observables, called Wick quantization. To see why we need a special procedure let us quantize naively the free Klein-Gordon Hamiltonian (21.11), with (21.20), which we rewrite R as H(φcl , π cl ) = 21 {(π cl )2 + (C −1 φcl )2 }dx, where C is given in (21.21) and where we denote for the moment the classical coordinate and momentum field by φcl and π cl , respectively. First we observe that the quantum fields corresponding to the classical fields φcl and π cl are 1 1 φ(x) = √ C 1/2 (a(x) + a∗ (x)) and π(x) = √ C −1/2 (a(x) − a∗ (x)). (21.33) 2 i 2 Here, remember, φ(x) is considered as an operator of multiplication by the evaluation functional φ(x). The second equation gives 1 π := −i∂φ + iC −1 φ. 2 We have shown in Subsection 21.3 that this operator is symmetric. Using this expression, one can show that i[π(x), φ(y)] = δ(x − y) i[π(x), π(y)] = i[φ(x), φ(y)] = 0.

(21.34)

21.4 Wick Quantization

305

Problem 21.11 Derive formally the commutation relations above. Now in the expression (21.11), with (21.20), we replace the classical fields by the quantum ones above to obtain Z 1 H(φ, π) = {π 2 + (C −1 φ)2 }dx 2 Z 1 i 1 = {[ C −1/2 (a∗ − a)]2 + [ C −1/2 (a∗ + a)]2 }dx 2 2 2 Z 1 {C −1/2 a∗ C −1/2 a + C −1/2 aC −1/2 a∗ }dx = 4 Z Z 1 1 = a∗ C −1 a + (C −1 δx )dx 2 4 where we have used the commutation relation for a and a∗ , and the selfadjointness of C −1/2 . The first term on the right hand side is non-negative, and the second is infinite (recall (21.21), which gives C −1 = (−∆ + m2 )1/2 ), and therefore the r.h.s. is infinite. We can make this argument rigorous as follows. First, we move to momentum space via the Fourier transform: a(x) 7→ a(k), Ra∗ (x) 7→ a∗ (k) (we omit the ’hats’ over a(k) and a∗ (k)). We wish to show that ω(k)a(k)a∗ (k)dk = ∞, where ω(k) = (|k|2 + m2 )1/2 (the dispersion law). Let Rd = ∪α∈Zd Bα (a disjoint union), where Bα is the unit box with center at a lattice point in Zd , R aα = Bα a, and ωα = mink∈Bα ω(k). Then [aα , a∗β ] = δα,β

so that

Z

On the other hand, X

ωaa∗ ≥

ωα aα a∗α =

X

X

ωα aα a∗α .

α∈Z3

ωα a∗α aα +

X

ωα = ∞.

Now we describe the quantization procedure which avoids the above problem. Fix the covariance C. Recall that we denote for the moment the classical coordinate and momentum fields by φcl and π cl . Classical observables are functionals A(φcl , π cl ) of φcl and π cl . We express A(φcl , π cl ) in terms of the functions α and α, ¯ using 1 1 ¯ ) and π cl = √ C −1/2 (α − α ¯) φcl = √ C 1/2 (α + α 2 i 2

(see (21.18)): A(φcl , π cl ) = B(α, α). ¯ We consider functionals B(α, α) ¯ : L2 (Rd )× 2 d L (R ) → C, analytic in α and α, ¯ of the sense of the convergent power series,

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21 Quantum Electro-Magnetic Field - Photons

B(α, α) ¯ =

XZ

Bm,n (x1 , . . . , xm+n )

m,n

m Y

α(x ¯ i)

α(xi )d(n+m) x.

i=m+1

i=1 cl

m+n Y

cl

We associate with classical observables, A(φ , π ), quantum observables, i.e. operators A on the state space L2 (Qs , µC ), according to the rule A(φcl , π cl ) 7→ A :=: A(φ, π) : ≡ : B(a∗ , a) :

(21.35)

where : B(a∗ , a) : is a Wick ordered operator defined by : B(a∗ ,a) : =

XZ

Bmn (x1 , . . . , xm+n )

m,n

m Y

a∗ (xi )

i=1

m+n Y

a(xi )d(n+m) x. (21.36)

i=m+1

Here are some examples of Wick ordering: Example 21.12 : φ2 : = : [C −1/2 (a + a∗ )]2 : = : (C −1/2 a)2 + (C −1/2 a∗ )2 + C −1/2 a∗ C −1/2 a + C −1/2 aC −1/2 a∗ : = (C −1/2 a)2 + (C −1/2 a∗ )2 + 2C −1/2 a∗ C −1/2 a. Using above, we compute Z Z 1 1 Hf = : {π 2 + |∇φ|2 + m2 φ2 } : dx = a∗ C −1 adx. 2 2 Problem 21.13 Show that the momentum operator Pf = ×a(x)dx can be written as (cf. (26.26)) Z Pf = : π(x)∇x φ(x) : dx.

R

(21.37)

a∗ (x)(−i∇x )

Problem 21.14 (see [135]) Let c = hf, Cf i = E(φ(f )2 ). Show

1. : φ(f )n := cn/2 Pn (c−1/2 φ(f )) where Pn is the nth Hermite polynomial 2. : eφ(f ) := eφ(f )−c/2 .

21.5 Fock Space Using the definitions (21.26) of the annihilation operator a(x), we see that the only solution to the equation aΩ = 0 is Ω = const. We thus set Ω ≡ 1, and call it the vacuum. Now we show that any element of L2 (Q, dµC ) can be obtained by applying creation operators to the vacuum. From now on, we omit the subindex s in Qs . Let S denote the symmetrized tensor product and L2sym (Rnd ), the subspace of L2 (Rnd ) consisting of functions which are symmetric with respect to permutations of the n variables xj ∈ Rd .

21.5 Fock Space

307

Theorem 21.15 Any vector F ∈ L2 (Q, dµC ) can be written uniquely as F =

Z ∞ X 1 √ Fn (x1 , . . . , xn )a∗ (x1 ) · · · a∗ (xn )Ωdx1 · · · dxn n! n=0

(21.38)

n 2 d where Fn ∈ L2sym (Rnd ) = S1 L (R ).

P Proof. For simplicity, we will denote the right hand side in (21.38) by n √1n! R Fn (a∗ )n Ω. We first remark that for any Gn ∈ L2 (Rnd ), Z Z Gn (a∗ )n Ω = Gsym (a∗ )n Ω (21.39) n where Gsym (x1 , . . . , xn ) =

1 X G(xπ(1) , . . . , xπ(n) ) ∈ L2sym (Rnd ) n! π∈Sn

is the symmetrization of φ. Here Sn is the group of permutations of the n variables, and for π ∈ Sn , π(1, . . . , n) = (π(1), . . . , π(n)). Next, we have by straightforward computation  Z Z 0 n 6= m h Fn (a∗ )n Ω, Gm (a∗ )m Ωi = (21.40) n!hFn , Gn i n = m. Problem 21.16 Show (21.40). Thus, for two vectors, F and G, of the form (21.38), we have hF, Gi :=

∞ X

hFn , Gn i.

(21.41)

n=1

Qn Now we use the fact that the span of vectors of the form 1 φ(fj )Ω, n ≥ 1, is dense in L2 (see (21.31)). By the relation (21.33), we have n Y 1

n Y 1 √ (a(C 1/2 fj ) + a∗ (C 1/2 fj )). φ(fj ) = 2 1

Using the commutation relation, a(f )a(f )∗ = Q a(f )∗ a(f ) + hf, gi, to pull a’s to ∗ the a ’s,Pwe transform the product (a + a∗ ) to the normal form R Qn right of ∗ ∗ k l (a + a ) = Akl (a ) a . But (a∗ )k al Ω = 0 unless l = 0, so R P k+l≤n Qn1 Ak0 (a∗ )k Ω. By (21.39), this can be rewritten as k≤n 1 φ(fj )Ω = n Y 1

φ(fj )Ω =

XZ

k≤n

1 √ Fk (a∗ )k Ω n!

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21 Quantum Electro-Magnetic Field - Photons

k 2 d where Fk ∈ L2sym (Rkd ) = S1 sym L (R ). Thus vectors of the form X 1 Z Fn (a∗ )n Ω n!

are dense in L2 . On the other hand, by (21.41), the subspace ( ) X 1 Z √ F = Fn (a∗ )n Ω | Fn ∈ L2sym (Rnd ) n! n

(21.42)

is closed. Since by above, it contains a dense set, it is the whole L2 space. Definition 21.17 The (bosonic) Fock space is F :=

∞ n M 2 d [ S L (R )]

with

n=0 1

hF, Gi :=

∞ X

hFn , Gn i.

n=1

n 2 d We call Fn := S1 L (R ) the n-particle sector. By convention, F0 = C.

The previous theorem provides a unitary isomorphism L2 (Q, dµC ) ≃ F given by ∞ X 1 Z M ∗ n √ φn (a ) Ω ↔ φn ≡ (φ0 , φ1 , φ2 , . . . ) . n! n=0

Moreover it is easily checked that Z E(F ) := F dµC = hΩ, F Ωi Q

and on F ,

a(f ) : φn ∈ Fn 7→

and

a∗ (f ) : φn ∈ Fn 7→

√ nhf, φn i ∈ Fn−1

√

n + 1f S φn ∈ Fn+1 .

Proposition 2. In the Fock space representation, the Hamiltonian, Hf R R = a∗RC −1 a, the momentum operator Pf = a∗ (−i∇)a and particle number, N = a∗ a, operators (see (21.27) and (21.29)) are of the form Hf φ ↔

Nφ ↔

n ∞ X M n=0 ∞ M n=0

Cx−1 φn j

1


nφn ,


,

Pf φ ↔

n ∞ X M n=0

1


pxj φn ,

where the subscript xj in Cxj indicates that this operator acts on the variable xj .

21.6 Quantization of Maxwell’s Equations

309

This is simply a matter of using the commutation relations. We leave the proof as an exercise. Thus we have obtained a very simple realization of our state space L2 (Q, dµC ) which is independent of C, and in which the Klein-Gordon Hamiltonian acts as a direct sum of simple operators in a finite but increasing number of variables: n p X Hf = ⊕ ∞ −∆xi + m2 . n=0 1

In particular, the spectrum of Hf is

σ(Hf ) = {0} ∪ {∪n≥1 [nm, ∞)} where the zero-eigenfunction is the vacuum, Ω. Physically, this theory describes non-interacting particles (bosons) of mass m. Problem 21.18 Find the spectrum of the momentum operator Z Pf = a∗ (x)(−i∇x )a(x)dx. Remark 21.19 For connections with stochastic fields and with infinite dimensional pseudodifferential calculus, see [304] and Berezin [38], respectively. There is also a relation to the Wiener chaos expansion used in stochastic differential equations.

21.6 Quantization of Maxwell’s Equations Maxwell’s equations as a hamiltonian system. As a prelude to quantizing Maxwell’s equations we have first to write them in Hamiltonian form. Recall that we work in physical units in which the Planck constant and speed of light are equal to 1: ~ = 1, c = 1. The Maxwell equations for vector fields E : R3+1 → R3 (the electric field) and B : R3+1 → R3 (the magnetic field) in vacuum are ∂E (21.43) ∇·E = 0 ∇×B = ∂t ∂B ∇×E =− ∇ · B = 0. (21.44) ∂t The third and fourth equations are actually constraints. They can be used to reduce number of unknowns by introducing the potentials U : R3+1 → R and A : R3+1 → R3 such that B = ∇ × A,

E = −∂A/∂t − ∇U.

(21.45)

Then the last the equations (21.44) are satisfied automatically. There is still a redundancy in the choice of A and U . Specifically, any gauge transformation

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21 Quantum Electro-Magnetic Field - Photons

A 7→ A + ∇χ,

U 7→ U − ∂χ/∂t

(21.46)

for χ : R3+1 7→ R, results in new potentials A and U which yield the same fields E and B. By appropriate choice of χ, we may take ∇·A= 0 which is called the Coulomb gauge. From now on, we work in this gauge. The equation for U is ∆U = 0, so we can take U = 0, which gives E = −∂A/∂t,

B = ∇ × A.

Now, the first, third and fourth equations in (21.43) - (21.44) are automatically satisfied while the second equation results in A = 0,

∇ · A = 0.

(21.47)

Thus the vector potential A is a transverse vector field satisfying a wave equation. (A vector field f : R3 → R3 is called transverse (or incompressible or divergence free) if ∇ · f = 0.) Similarly as for the Klein-Gordon equation, one shows that Equation (21.47) is the Euler-Lagrange equation for the action Z Z 1 T ˙ 2 − |∇ × A|2 }dxdt S(A) = {|A| (21.48) 2 0 R3 where the variation is among transverse vector fields. Again repeating the steps we went through in the Klein-Gordon case, we arrive, for Maxwell’s equations, at 1. the phase space, which is the direct sum of two Sobolev spaces of transverse vector fields (i.e. the derivatives are considered in distributional sense), Z = H 1,trans (R3 ; R3 ) ⊕ H 1,trans (R3 ; R3 );

2. the Hamiltonian functional (the Legendre transform of the Lagrangian R ˙ := 3 {|A| ˙ 2 − |∇ × A|2 }), L(A, A) R Z 1 H(A, E) = {|E|2 + |∇ × A|2 }dx 2 Z (21.49) 1 {|E|2 + |B|2 }dx = 2 ˙ = A˙ = −E, where −E is the canonically conjugate field to A : ∂A˙ L(A, A) and ∇ · E = 0; 3. the Poisson bracket {F, G} := h∂A F, T ∂E Gi − h∂E F, T ∂A Gi

(21.50)

(on Z), where T be the projection operator of vector fields onto transverse vector fields: T F := F − (∆)−1 ∇(∇ · F );

21.6 Quantization of Maxwell’s Equations

311

4. Canonically conjugate fields are A and E. If Tij (x − y) is the matrix integral kernel of the operator T , then {Ei (x), Aj (y)} = Tij (x − y). Note that by the definition, ∂A F = T ∂A F and similarly for ∂E F . The Hamilton equations for the above system are   0 −T Φ˙ = JT ∂Φ H(Φ), JT := , Φ = (A, E). (21.51) T 0 Note that the first Hamilton equation yields, as expected, A˙ = T ∂E H(A, E) = −T E = −E. Problem 21.20 Check that ∇ · (T F ) = 0. Problem 21.21 Check that Maxwell’s equations are equivalent to the Hamilton equations (21.51). (Cf. Problem 21.2.) Quantization of the EM field. We quantize the Maxwell hamiltonian system in the same way as the Klein-Gordon one, but replacing scalar generalized operator-functions, say a(x), by generalized transverse operator-vectorfields, say a(x) = (a1 (x), a2 (x), a3 (x)) with ∇ · a(x) = 0, and using the projection T onto the be the transverse vector fields, when needed. Here ∇ · a(x) is understood as a distributional divergence of the operator-valued distribution, Z Z ¯ ∇ · a(x)f (x)dx = − a(x) · ∇f¯(x)dx = −a(∇f ), i.e. a symbolic way to represent the operator-valued functional −a(∇f ). The resulting theory can be summarized as follows: 1. The quantum state space is L2 (Qtrans , dµC ), where, for s sufficiently large, Qtrans = {A : R3 → R3 | (−∆ + |x|2 + id)−s/2 A ∈ L2 , divA = 0} and dµC is the Gaussian measure of Qtrans with the covariance operator C = (−∆)−1/2 . 2. The quantized A(x) and E(x) are operator-valued transverse vector fields, acting on the space L2 (Qtrans , dµC ) as the multiplication operator by A(x) and E(x) := −i∂A(x) + 21 iC −1 A(x), respectively. They give canonically conjugate fields with the non-trivial commutation relation i[Ek (x), Al (y)] = Tkl (x − y)1 where T (x − y) is the integral kernel of the projection operator, T , onto the transverse vector fields (recall our convention ~ = 1).

312

21 Quantum Electro-Magnetic Field - Photons

3. A(x) and E(x) are expressed in terms of the annihilation and creation operators, a(x) and a∗ (x), as 1 1 A(x) = √ C 1/2 (a(x)+a∗ (x)), E(x) = √ C −1/2 (a(x)−a∗ (x)). (21.52) 2 i 2 Here a(x) and a∗ (x) are operator-valued transverse vector-fields, a(x) = (a1 (x), a2 (x), a3 (x)) with ∇ · a(x) = 0, satisfying the commutation relations [ai (x), a∗j (y)] = Tkl (x − y)1,

[ai (x), aj (y)] = 0 = [a∗i (x), a∗j (y)]. (21.53)

4. The quantum Hamiltonian operator acts on the space L2 (Qtrans , dµC ) and is given by Z Z √ 1 2 2 (21.54) Hf = : |E| + |∇ × A| : dx = a(x)∗ · −∆a(x)dx, 2 √ √ P where a(x)∗ · −∆a(x) = i ai (x)∗ · −∆ai (x). (Here again the subindex f stands for the ”field” and the precise definition of the integral on the r.h.s can be found in Section 21.2.) Note an essential difference in the EM case: there is no mass (m = 0). That is, the covariance operator is C = (−∆)−1/2 . The equations (21.52) - (21.54) give the full description of the quantized electro-magnetic field in vacuum. For the Maxwell theory, the Fock space is built on the one-particle space L2transv (R3 , R3 ) := {f ∈ L2 (R3 , R3 )|divf = 0}, instead of L2 (Rd ) ≡ L2 (Rd , C). The Maxwell Hamiltonian acts on it as Hf = ⊕ ∞ n=0

n X p −∆xi . 1

This representation can be used to obtain the spectrum of H: σ(Hf ) = {0} ∪ {∪n≥1 [0, ∞)} where the zero-eigenfunction is the vacuum, Ω. Physically, this theory describes non-interacting massless particles (bosons), called photons. The Fourier transforms, a(k) and a∗ (k), of the operators a(x) and a∗ (x) satisfy a(k) = (a1 (k), a2 (k), a3 (k)) with k · a(k) = 0. The commutation relations for ai (k), a∗i (k) are [ai (k), a∗j (k)] = (δij −

ki kj )δ(k − k ′ ), |k|2

[ai (k), aj (k ′ )] = 0 = [a∗i (k), a∗j (k ′ )]. (21.55)

21.6 Quantization of Maxwell’s Equations

313

In these terms, the Maxwell quantum Hamiltonian is of the form Z Hf = ω(k)a∗ (k) · a(k)dk, with ω(k) = |k|, and the quantized vector potential, A(x), is given by Z dk A(x) = {eix·k a(k) + e−ix·k a∗ (k)} p . (21.56) ω(k)

In the Fourier representation, we can choose an orthonormal basis eλ (k) ≡ e(k, λ), λ ∈ {−1, 1} in k ⊥ ⊂ R3 , satisfying k · eλ (k) = 0. The vectors eλ (k) ≡ e(k, λ), λ ∈ {−1, 1} are called polarization vectors. We can write the operatorvalued transverse vector fields a# (k) as X eλ (k)a# a# (k) := λ (k), λ∈{−1,1}

# # where a# λ (k) ≡ a (k, λ) := eλ (k) · a (k), the components of the creation and # annihilation operators a (k) in the direction transverse to k; they satisfy the commutation relations  #    ′ aλ (k) , a# aλ (k) , a∗λ′ (k ′ ) = δλ,λ′ δ 3 (k − k ′ ). (21.57) λ′ (k ) = 0,

Now the Hamiltonian Hf and the quantized vector potential, A(x), can be written as XZ Hf = ω(k)a∗λ (k)aλ (k)dk λ

and

A(x) =

XZ λ

dk . {eix·k aλ (k) + e−ix·k a∗λ (k)}eλ (k) p ω(k)

(21.58)

Elements of Fn can be written as ψn (k1 , λ1 , . . . , kn , λn ), where λj ∈ {−1, 1} are the polarization variables. Roughly in the case of photons, compared with scalar bosons, one replaces the variable k by the pair (k, λ) and adds to the integrals in k also the sums over λ.

22 Standard Model of Non-relativistic Matter and Radiation

In this chapter we introduce and discuss the standard model of non-relativistic quantum electrodynamics (QED). Non-relativistic QED was proposed in the early days of Quantum Mechanics (it was used by Fermi ([98]) in 1932 in his review of theory of radiation). It describes quantum-mechanical particle systems coupled to a quantized electromagnetic field, and appears as a quantization of the system of non-relativistic classical particles interacting with a classical electromagnetic field (coupled Newton’s and Maxwell’s equations). It is the most general quantum theory obtained by quantizing a classical system.

22.1 Classical Particle System Interacting with an Electro-magnetic Field We consider a system of n classical particles of masses mi and electric charges ei interacting with electro-magnetic field. Recall that we work in physical units in which the speed of light is equal to 1: c = 1. The coupled Newton’s and Maxwell’s equations, describing interacting particles and electromagnetic field are mi x¨i (t) = ei [E + x˙ i ∧ B](xi (t), t), (22.1) ∇ · E = ρ,

∇×B =

∂E + j, ∂t

(22.2)

∂B , ∇ · B = 0. (22.3) ∂t P Here ρ and j are P the charge and current densities: ρ(x, t) = i ei δ(x − xi (t)) and j(x, t) = i ei x˙ i (t)δ(x − xi (t)). The first equation is Newton’s equation with the Lorentz force. The last four equations are Maxwell’s equations in presence of charges and currents. To find a hamiltonian formulation of these equation we first present the minimum action principle for this system. As before, we express the electric and magnetic fields, E and B, in terms of the vector potential, A : R3+1 7→ R3 , ∇×E =−

© Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_22

316

22 Standard Hamiltonian of Non-relativistic Matter and Radiation

and the scalar potential, U : R3+1 7→ R via (21.45). The action functional which gives (22.1) is given by S(γ|U, A) =

Z

T 0

 |γ˙i |2 − ei U (γi ) + ei γ˙ i · A(γi ) .

X  mi 2

i

(22.4)

Indeed, let us compute the Euler-Lagrange equation for functional (see P this mi 2 (4.40)). Using the Lagrangian function L(γ, γ|U, ˙ A) = | i 2 γ˙i | − eU (γi ) −
eγ˙ i · A(γi ) , we compute ∂γ˙ L(γ, γ|U, ˙ A) = mγ˙ + eA(γ), ∂γ L(γ, γ|U, ˙ A) = e∇U (γ) + e∇γ˙ · A(γ).

d Plug this into (4.40) and use the relations dt A(γ) = ∂t A(γ)+(γ·∇)A(γ), ˙ ∇(γ˙ · A) − (γ˙ · ∇)A(γ) = γ˙ ∧ curlA, and (21.45) to obtain (22.1). To (22.4) we add the action of electromagnetic field found in the previous chapter, but with a non-zero scalar potential,

Sem (U, A) =

1 2

Z

Z

T


|A˙ + ∇U |2 − |∇ × A|2 (x, t)dxdt.

R3

0

(We do not have R to assume the Coulomb gauge, divA =R0, here.) Using that ˙ · A(x(t), t) = R3 j(x, t) · A(x, t)dx, eU (x(t), t) = R3 ρ(x, t)U (x, t)dx and ex(t) to rewrite the last two terms in (22.4), we obtain the action functional of the coupled system S(γ, U, A) = Sp (γ) + Sem (U, A) + Sint (γ, U, A),

(22.5)

R T P mi 2 where Sp (γ) = 0 i 2 |γ˙i | dt is the action of the free particle, familiar to us from Section 4.7, Sem (U, A) is the action of a free electro-magnetic field given above, and Sint (φ, U, A) is the interaction action, coupling them, Sint (γ, U, A) =

Z

0

T

Z


− ρ(x, t)U (x, t) + j(x, t) · A(x, t) dxdt.

R3

(22.6)

It is easy to check that the Euler-Lagrange equation for the action (22.5) gives the Newton - Maxwell system (22.1) - (22.3). Gauge invariance. The fields E and B are not changed under gauge transformations (21.46) of A and U . Hence we would like to make sure that the action (22.6) gives the same equations of motion for different A and U , connected by a gauge transformation (21.46). Under (21.46), the action (22.6) changes as S(γ, U, A) → S(γ, U, A) + Λ(γ, χ), where Λ(γ, χ) :=

Z

0

T

Z

R3


ρ∂t χ + j(x, t) · ∇χ dxdt

(22.7)

22.1 Classical Particle System Interacting with an Electro-magnetic Field

=

Z

T 0

Z

R3


∂t (ρχ) + ∇ · (jχ) − (∇ · j + ∂t ρ) dxdt.

317

(22.8)

Hence the gauge transformation leads to an equivalent action/Lagrangian, i.e. to the action which gives the same Euler-Lagrange equation. Covariant formulation. Geometrically, the electric field is a one-form, while the magnetic field is a two-form. One can form the field tensor F := −E ∧ dx0 + B, as a two-form on the Minkowskii space M 3+1 , with coordinates (x0 , x1 , x2 , x3 ), where x0 = t, and the metric g = diag(1, −1, −1, −1). By the last two Maxwell’s equations, F is a closed form, dF = 0, and therefore it is exact, i.e. there is a one-form A, s.t. F = dA. (A can be thought of as a connection, and F as a curvature on a U (1)−bundle.) Then R the action2 functional 1 can be written as a ’Dirichlet’ integral, S(A) = − 16π M 3+1 (kdAk + hJ , Ai), where J := (ρ, j) on the Minkowskii space M 3+1 , with the norm and inner product related to the Minkowskii metrics, g = diag(1, −1, −1, −1). This gives the first two of Maxwell’s equations as d∗ F = J , where d∗ is the operator adjoint to d. Elimination of scalar potential. Note that the time derivative of scalar potential U does not enter this Lagrangian. This indicates that U is not a dynamical variable and we can eliminate it from the action. We do this in the Coulomb gauge, divA = 0, as follows. Varying the action S(γ, U, A) w.r. to U , we obtain the first of Maxwell’s equations − ∆U = ρ.

(22.9)

R Furthermore, divA = 0 implies, after integration by parts, that R3 |A˙ +
R R R ˙ 2 +|∇U |2 dx. Using this and 3 |∇U |2 dx = 3 U (−∆)U dx ∇UR|2 dx = R3 |A| R RR = R3 U ρdx, we see that the terms involving U add to − 12 R3 U ρdx. Now, solving the Poisson equation (22.9) for U as U = (−∆)−1 ρ, we obtain Z Z ρU dx = ρ(−∆)−1 ρdx = Vcoul (x), R3

R3

where x = (xP 1 , . . . , xn ), plus the infinite Coulomb self-energy term. Here e e Vcoul (x) := 21 i6=j |xi i−xj j | . With this and dropping the Coulomb self-energy term, we can write the action as coul S coul (γ, A) = Spcoul (γ) + Sf (A) + Sint (γ, A),

(22.10)

R T P mi
RT R ˙ t)|2 − where Spcoul (γ) = 0 |γ˙i |2 − Vcoul dt, Sf (A) = 21 0 R3 |A(x, i 2
|∇ × A(x, t)|2 dxdt is the free action of electromagnetic field, encountered in the previous chapter, and coul Sint (φ, A)

=

Z

T 0

Z

R3

j(x, t) · A(x, t)dxdt.

(22.11)

318

22 Standard Hamiltonian of Non-relativistic Matter and Radiation

Hamiltonian formulation. Now, we pass to the hamiltonian formulation. To this end we impose the Coulomb gauge, divA = 0 and use the Lagrangian given by the above action. The generalized particle momenta ˙ the same are ki = mi x˙ i + eA(xi ), while the field momentum is −E = A, as in the free case. The classical Hamiltonian functional is H(x, k, A, E) = , which gives k · v − L(x, v, A, E)|mi x˙ i =ki −eA(xi ),A=−E ˙ H(x, k, A, E) =

X 1 (ki − ei A(xi ))2 + Vcoul (x) + Hf (A, E) 2m i i

where k = (k1 , . . . , kn ) and Hf (A, E) is the Hamiltonian functional of the free electro-magnetic field. Defining the Poisson bracket as the sum of classical mechanics one, (4.47), and electro-magnetic one, (21.50), we arrive at the hamiltonian formulation for a system of n particles of masses m1 , . . . , mn and charges e1 , . . . , en interacting with electromagnetic field, (E, B).

22.2 Quantum Hamiltonian of Non-relativistic QED According to our general quantization procedure, we replace the classical cl cl cl canonical variables xcl i and ki and classical fields A (x), and E (x) by the quantum canonical operators xi , pi , A(x), and E(x) (see (4.4) and (21.52)), acting on the state space L2 (R3n ) ⊗ L2 (Qtrans , dµC ) ≃ L2 (R3n ) ⊗ F ≡ Hpart ⊗ Hf .

(22.12)

In the units in which the Planck constant divided by 2π and the speed of light are equal to 1, ~ = 1 and c = 1, the resulting Schr¨odinger operator, acting on Hpart ⊗ Hf , is n X 1 H= (pi − ei A(xi ))2 + Vcoul (x) + Hf , 2m i i=1

(22.13)

where, recall, mi and ei are the mass (in fact, the ’bare’ mass, see below) and charge of the i-th particle. Recall the the quantized vector potential A(x) and quantum Hamiltonian P annihilation and creP Hf are given in terms of the ation operators a(k) = λ eλ (k)aλ (k) and a∗ (k) = λ eλ (k)a∗λ (k) (obeying canonical commutation relations (21.57)) by Z d3 k A(x) = {eix·k a(k) + e−ix·k a∗ (k)} p (22.14) ω(k) and

Hf =

Z

ω(k)a(k)∗ · a(k)dk.

(22.15)

22.2 Quantum Hamiltonian of Non-relativistic QED

319

Note that in the units we have chosen, various physical quantities entering the quantum hamiltonian are not dimensionless. We choose these units in order to keep track of the particle mass. Also note that we omitted the identities 1part and 1f on Hpart and Hf , respectively, and a careful notation would have Vcoul (x) ⊗ 1f and 1part ⊗ Hf , instead of Vcoul (x) and Hf . In what follows, as a rule, we do not display these identities. Gauge principle (minimal coupling). We see that the full, interacting Hamiltonian, is obtained by replacing the particle momenta pj by the covariant momenta pA,j = pj − ej A(xj ), and adding to the result the field Hamiltonian, Hf , responsible for the dynamics of A(x). This procedure is called the minimal coupling and it is justified by a gauge principle requiring the global gauge symmetry of the particle system also to be the local one. Ultra-violet cut-off. The quantum Hamiltonian H we obtained above is illdefined: its domain of definition contains no non-zero vectors. The problem is in the interaction (p − eA(x))2 − p2 = −2eA(x)p + e2 A(x)2 (written here in the one-particle case). The A(x) is too rough an operator-valued function. It has the empty domain. E. g. A(x)(f (x) ⊗ Ω) = f (x)A(x)Ω Z d3 k = f (x) e−ix·k a∗ (k)Ω p ∈ / L2 (R3 ) ⊗ F . ω(k)

To remedy this we institute an ultraviolet cut-off. It consists of replacing A(x) in (22.14) by the operator Aχ (x) := χ ˇ ∗ A(x) where χ ˇ is a smooth well-localized around 0 (i.e. sufficiently fast decaying away from 0) function: A(x) 7→ Aχ (x) := χ ˇ ∗ A(x).

(22.16)

Recall that χ ˇ denotes the inverse Fourier transform of a function χ. We choose χ ˇ to be a positive function whose integral is equal to 1 (a smoothed-out δfunction). In fact, the specific shape of χ ˇ is not important for us. Now, Aχ (x) is of the form Z χ(k) dk. (22.17) Aχ (x) = (eik·x a(k) + h.c.) p ω(k)

Assuming that the ultra-violet cut-off χ decays on the scale κ, we arrive, as a result, at the Hamiltonian of non-relativistic matter interacting with radiation: Heκ =

n X j=1

1 (pAχ ,j )2 + e2j Vcoul (x) + Hf 2mj

(22.18)

acting on Hpart ⊗ Hf , where pAχ ,j = pj − ej Aχ (xj ), and pj = −i∇xj (note that the parameter e enters the definition of the operator pAχ ,j ). This is the

320

22 Standard Hamiltonian of Non-relativistic Matter and Radiation

standard model of non-relativistic QED. We show below that Heκ is welldefined and is self-adjoint. To sum up, we arrived at a physical system with the state space H = Hp ⊗ Hf and the quantum Hamiltonian Heκ . Its dynamics is given by the time-dependent Schr¨odinger equation with i∂t ψ = Heκ ψ, where ψ is a differentiable path in the Hilbert space H = Hp ⊗ Hf . A few comments are now in order. Choice of κ. We assume that our matter consists of electrons and the nuclei and that the nuclei are infinitely heavy and therefore are manifested through the interactions only). We reintroduce the Planck constant, ~ and speed of light, c, for a moment. The electron mass is denoted by mel . Since we assume that χ(k) decays on the scale κ, in order to correctly describe the phenomena of interest, such as emission and absorption of electromagnetic radiation, i.e. for optical and rf modes, we have to assume that the cut-off energy, ~cκ, is much greater than the characteristic energies of the particle motion. The latter motion takes place on the energy scale of the order of the 1 e2 ≈ 137 is the ionization energy, i.e. of the order α2 mel c2 , where α = 4π~c 2 2 fine-structure constant. Thus we have to assume α mel c ≪ ~cκ. On the other hand, for energies higher than the rest energy of the the electron (mel c2 ) the relativistic effects, such as electron-positron pair creation, take place. Thus it makes sense to assume that ~cκ ≪ mel c2 . Combining the last two conditions we arrive at the restriction α2 mel c2 ≪ ~cκ ≪ mel c2 or α2 mel c/~ ≪ κ ≪ mel c/~. In our units (~ = 1, c = 1) this becomes α2 mel ≪ κ ≪ mel . Free parameters. We will see later that the physical mass, mel , is not the same as the parameter m ≡ mj (the ’bare’ mass) entering (22.18), but depends on m and κ. Inverting this relation, we can think of m as a function of mel and κ. If we fix and the particle potential V (x) (e.g. taking it to be the total Coulomb potential), and mel and e, then the Hamiltonian (22.18) depends on one free parameter, the bare electron mass m (or the ultraviolet cut-off scale, κ). Gauge equivalence. We quantized the system in the Coulomb gauge. Assume we quantized the system in a different gauge, say the Lorentz gauge, how is the latter Hamiltonian related to the former one? As we saw above, classically, different gauges give equivalent descriptions of a classical system. Do they lead to equivalent descriptions of the corresponding quantum system? The answer is yes. One can show that quantum Hamiltonians coming out of different gauges, as well as other observables, are related by unitary (canonical) transformations.

22.2 Quantum Hamiltonian of Non-relativistic QED

321

22.2.1 Translation invariance The system of particles interacting with the quantized electromagnetic fields is invariant under translations of the particle coordinates, x → x + y, where y = (y, . . . , y) (n− tuple) and the fields, A(x) → A(x − y), i.e. Heκ commutes with the translations Ty : Ψ (x, A) → Ψ (x + y, ty A),

(22.19)

where (ty A)(x) = A(x−y). (We say that Heχ is translation invariant.) Indeed, we use (22.19) and the definitions of the operators A(x) and E(x), to obtain Ty A(x) = (ty A)(x)Ty and Ty E(x) = (ty E)(x)Ty , which, due to the definition of Heχ , give Ty Heκ = Heκ Ty , In the Fock space representation (22.19) becomes Ty : ⊕n Ψn (x, k1 , . . . , kn ) → ⊕n eiy·(k1 +...kn ) Ψn (x + y, k1 , . . . , kn ) and therefore can be rewritten as Ty : Ψ (x) → eiy·Pf Ψ (x + y), where Pf is the momentum operator associated to the quantized radiation field, XZ Pf = dk k a∗λ (k)aλ (k). λ

As we know from Section 3.4, typically, symmetries of a physical system lead to conservation laws. (In the classical case, this is the content of the Noether theorem.) In our case, of a particular importance is space-time translational symmetry, which leads to conservation laws of the energy, Heχ , and the total momentum. We check this for the spatial translations. It is straightforward to show that Ty are unitary operators and that they satisfy the relations Tx+y = Tx Ty (and therefore y → Ty is a unitary Abelian representation of R3 ). Finally, we observe that the group Ty is generated by the total momentum operator, Ptot , of the electrons and the photon field: Ty = eiy·Ptot . Here Ptot is the selfadjoint operator on H, given by X pi ⊗ 1f + 1part ⊗ Pf (22.20) Ptot := i

where, as above, pj := −i∇xj , the momentum of the j−th electron and Pf is the field momentum given above. Hence [Heκ , Ptot ] = 0. 22.2.2 Fiber decomposition with respect to total momentum Since the Hamiltonian Heκ commutes with translations, Ty , it has a fiber decomposition w.r.to the generator Ptot of the translations (cf. Section 6.6). We R⊕ construction this decomposition. Let H be the direct integral H = R3 HP dP,

322

22 Standard Hamiltonian of Non-relativistic Matter and Radiation

P with the fibers HP := L2 (X) ⊗ F , where X := {x ∈ R3n | i mi xi = 0} ≃ R3(n−1) , (this means that H = L2 (R3 , dP ; L2 (X) ⊗ F )) and define U : Hel ⊗ Hf → H on smooth functions, compactly supported in x, by the formula Z ei(P −Pf )·xcm Ψ (x′ + xcm )dy, (22.21) (U Ψ )P (x′ ) = R3

′

where x are the coordinates of the n particles in the center-of-mass frame and P xcm = (xcm , . . . , xcm ) (n− tuple), with xcm = P 1mi i mi xi , the center-ofi mass coordinate, so that x = x′ + xcm . Then U extends uniquely to a unitary operator (see below). Its converse is written, for ΦP (x′ ) ∈ L2 (X) ⊗ F , as Z (U −1 Φ)(x) = e−ixcm ·(P −Pf ) ΦP (x′ )dP. (22.22) R3

The functions ΦP (x′ ) = (U Ψ )P (x′ ) are called fibers of Ψ . Lemma 1. The operations (23.8) and (22.22) define unitary maps L2 (R3n )⊗ F → H and H → L2 (R3n ) ⊗ F , and are mutual inverses. Proof. By density, we may assume that Ψ is a C0∞ in x. Then, it follows from standard arguments in Fourier analysis that Z Z −1 −i(P −Pf )xcm (U U Ψ )(x) = dP e dy ei(P −Pf )·y Ψ (x′ + y) Z Z = dy dP e−iP ·(xcm −y) eiPf ·(xcm −y) Ψ (x′ + y) Z = dy δ(xcm − y) eiPf ·(xcm −y) Ψ (x′ + y) = Ψ (x) .

(22.23)

On the other hand, for Φ ∈ H, Z Z (U U −1 Φ)P (x′ ) = dy ei(P −Pf )y dq ei(q−Pf )y Φq (x′ ) Z Z = dq dy ei(p−q)y Φq (x′ ) Z = dq δ(P − q) Φq (x′ ) = ΦP (x′ ) .

(22.24)

From the density of C0∞ in x functions, we infer that (22.23) and (22.24) define bounded maps which are mutual inverses. Unitarity can be checked easily.  Since Heκ commutes with Ptot , it follows that Heκ admits the fiber decomposition Z U Heκ U −1 =

⊕

R3

Heκ (P )dP,

(22.25)

22.3 Rescaling and decoupling scalar and vector potentials

323

where the fiber operators Heκ (P ), P ∈ R3 , are self-adjoint operators on the space fibers HP . The latter means that U Heκ U −1 ΦP = Heκ (P )ΦP for R⊕ ΦP (x′ ) ∈ R3 HP dP . We compute Heκ (P ). Using a(k)e−iy·Pf = e−iy·(Pf +k) a(k) and a∗ (k)e−iy·Pf = e−iy·(Pf −k) a∗ (k), we find ∇y eiy·(P −Pf ) Aχ (x′ +y)eiy·(P −Pf ) = 0 and therefore Aχ (x)eiy·(P −Pf ) = eiy·(P −Pf ) Aχ (x − y).

(22.26)

Using this and (22.22), we compute Z −1 Heκ (U Φ)(x) = eixcm ·(P −Pf ) Heκ (P )Φ(P )dP, R3

where the fiber Hamiltonians Heκ (P ) are given explicitly by Heκ (P ) =

X 1
2 P − Pf − i∇x′j − ei Aχ (x′j ) + Vcoul (x′ ) + Hf 2mi j

(22.27)

where x′i = xi −xcm is the coordinate of the i−th particle in the center-of-mass frame.

22.3 Rescaling and decoupling scalar and vector potentials In order to simplify the notation and some of the analysis from now on we assume that our matter consists of electrons and the nuclei and that the nuclei are fixed (the Born-Oppenheimer approximation in the case of molecules, see Section 10.1) and therefore are manifested through the interactions only. Recall that we work in physical units in which the Planck constant and speed 1 , the of light are equal to 1: ~ = 1 and c = 1, so that e2 /4π = α ≈ 137 2 e fine-structure constant. In the original units, α = 4π~c . In the Hamiltonian (22.18) the coupling constant - the electron charge e enters in two places - into the particle system itself (the Coulomb potential) and into coupling the particle systems to the quantized electro-magnetic field and thus has two different effects onto the total system. To decouple these two effects, we rescale our Hamiltonian as follows: x→

1 x, α

k → α2 k ,

Under this rescaling, the Hamiltonian Heκ is mapped into the Hamiltonian α2 H(ε), where H(ε) =

n X 1 (i∇xj + εAχ (xj ))2 + V (x) + Hf , 2m j=1

(22.28)

324

22 Standard Hamiltonian of Non-relativistic Matter and Radiation

√ where ε = 2 πα3/2 , Aχ (x) = Aχ (αx) with χ(k) replaced by χ(α2 k), and V (x) = Vcoul (x, αR). We write out the operator Aχ (x): Z χ(α2 k) 3 χ d k. (22.29) A (x) = (eik·αx a(k) + h.c.) p ω(k) The Hamiltonian H(ε) is, of course, equivalent to our original Hamiltonian Heκ . After the rescaling performed above the new UV cut-off momentum scale, κ′ = α−2 κ, satisfies mel ≪ κ′ ≪ α−2 mel ,

which is easily accommodated by our estimates (e.g. we can have κ′ = O(α−1/3 mel )). From now on we fix κ′ and we do not display α in the coupling function in (22.29) (the presence of α multiplying x above only helps our arguments). At this point we forget about the origin of the potential V (x), but rather assume it to be a general real function satisfying standard assumptions, say (V) V ∈ L2 (R3N ) + L∞ (R3N )ǫ ,

i.e. V can be written as a sum of L2 − and L∞ −functions, where the second component can be taken arbitrary small. It is shown in Section 22.3.1 that a sum of Coulomb potentials satisfies this condition and it is shown in Section 22.3.1 that under conditions (V), the operator Hpart := −

n X 1 ∆x + V (x) 2m j j=1

Pn is self-adjoint on the domain D( j=1 key object of our analysis.

1 2m ∆xj ).

The Hamiltonian H(ε) is a

22.3.1 Self-adjointness of H(ε) The key properties of a quantum Hamiltonian are boundedness below and self-adjointness. One way to approach proving these properties is to use the gaussian space representation (22.12) and the Kato inequality (see [73], cf. Section 10.1). This would give the result for for all coupling constants ε. (See also a proof of the self-adjointness of H(ε) for an arbitrary ε, using path integrals by [160].) We take a different approach which demonstrates some of estimates we need later, but proves these properties only for ε sufficiently small. Theorem 22.1 Assume ε is sufficiently small. Then the Hamiltonian H(ε) is defined on D(H(0)) and is self adjoint and bounded below.

22.3 Rescaling and decoupling scalar and vector potentials

325

Proof. To keep the notation simple we consider one-particle systems (n = 1). We decompose the Hamiltonian (22.28) into unperturbed part and perturbation as H(ε) = H(0) + I(ε). (22.30) In this proof we omit the super-index χ and write A(x) for Aχ (x) in (22.28). Using (22.28), we find 1 I(ε) = −εp · A(x) + ε2 |A(x)|2 . 2

(22.31)

Now the self-adjointness of H(ε) follows from the self-adjointness of H(0) = Hpart ⊗ 1f + 1part ⊗ Hf , Proposition 22.2 below, the smallness of ε and an abstract result, Theorem 2.11 of Section 2.2 .  The next statement shows that Iε is a relatively bounded perturbation in the sense of forms. Proposition 22.2 There is an absolute constant c > 0, s. t.
hψ, I(ε)ψi ≤ cε hψ, H(0)ψi + kψk2 .

(22.32)

Proof. Let k · kF stand for the norm in the Fock space F . First we prove the bounds Z |f |2 1/2 ka(f )ψk2F ≤ kHf ψk2F (22.33) ω and Z Z |f |2 1/2 ∗ 2 2 ka (f )ψkF ≤ kHf ψkF + |f |2 kψk2F . (22.34) ω The first inequality follows from the relations Z ka(f )ψkF ≤ |f |kaψkF

(22.35) 1/2 Z 1/2 |f |2 2 . ωkaψkF ≤ ω R (due to the Schwarz inequality) and ωkaψk2 = hψ, ωa∗ aψi = hψ, Hf ψi. The second inequality follows from the first and the relation Z ∗ 2 ∗ 2 ka(f ) ψkF = hψ, a(f )a(f ) ψiF = ka(f )ψkF + |f |2 kψk2F . (22.36) Z

The inequalities (22.33) and (22.34) yield the following bound kA(x)ψkF ≤
1/2 c kHf ψkF + kψkF , which implies that kA(x)(Hf + 1)−1/2 k ≤ c .

(22.37)

Now, chose c′ s.t. Hpart + c′ 1 ≥ 1. Using the equation H(0) = Hpart ⊗ 1f + 1 1part ⊗ Hf and using repeatedly that for any positive operator B, kB 2 ψk2 =

326

22 Standard Hamiltonian of Non-relativistic Matter and Radiation

hψ, Bψi and that hψ, Hpart ψi ≤ hψ, H(0)ψi (since Hf ≥ 0) and hψ, Hf ψi ≤ hψ, (H(0) + c′ 1)ψi, we find that 1

1

k(Hf + 1) 2 (H(0) + c′ 1)− 2 k ≤ 1, 1

1

kk(Hpart + c′ 1) 2 (H(0) + c′ 1)− 2 k ≤ 1.

1

These inequalities together with the estimates kpψk ≤ ck(Hpart + c′ 1) 2 ψk, and (22.37) give, k(Hpart + Hf + c′ 1)−1/2 pA(x)(Hpart + Hf + c′ 1)−1/2 k ≤ c.

(22.38)

Similarly, (22.37) implies that k(Hpart + Hf + c′ 1)−1/2 A(x)2 (Hpart + Hf + c′ 1)−1/2 k ≤ c.

(22.39)

The estimates (22.39) and (22.38) together with the relation (22.31) imply (22.32).  One can also prove a bound on I(ε) itself, not just on its form hψ, I(ε)ψi, so that the self-adjointness of H(ε) would follow from the Kato-Rellich theorem 2.9. This is done in Appendix 23.8.

22.4 Mass Renormalization In this section we study electron mass renormalization. First we analyze the definition of (inertial) mass in Classical Mechanics. Consider a classical particle with the Hamiltonian h(x, k) := K(k)+V (x), where K(k) is some function describing the kinetic energy of the particle. To find the particle mass in this case we have to determine the relation between the force and acceleration at very low velocities. The Hamilton equations give x˙ = ∇k K and k˙ = F , where F = −∂x V is the force acting on the particle. Assuming that K has a minimum at k = 0, we expand ∇k K(k) around 0 to obtain x˙ = K ′′ (0)k, where K ′′ (0) is the hessian of K at k = 0. Differentiating the above relation w.r. to time and using the second Hamilton equation, we obtain x ¨ = K ′′ (0)F (x). This ′′ suggests to define the mass of the particle as m = K (0)−1 , i.e. as the inverse of the Hessian of the energy, in the absence of external forces, as a function of momentum, at 0. (K(k) is called the dispersion relation.) We adopt this as a general definition: the (effective) mass of a particle interacting with fields is the inverse of the Hessian of the energy of the total system as a function of of the total momentum at 0. Now, we consider a single non-relativistic electron coupled to quantized electromagnetic field. Recall that the charge of electron is denoted by −e and its bare mass is m. The corresponding Hamiltonian in our units is (1) Heκ :=

1 (− i∇x ⊗ 1f + Aχ (x) )2 + 1part ⊗ Hf , 2m

(22.40)

22.4 Mass Renormalization

327

acting on the space L2 (R3 ) ⊗ F ≡ Hpart ⊗ Hf . It is the generator for the dynamics of a single non-relativistic electron, and of the electromagnetic radiation field, which interact via minimal coupling. Here recall Aχ (x) and Hf are the quantized electromagnetic vector potential with ultraviolet cutoff and the field Hamiltonian, defined in (22.17) and (22.15) (1)

The system considered is translationally invariant in the sense that Heκ commutes with the translations, Ty , (1) (1) Ty Heκ = Heκ Ty ,

which in the present case take the form Ty : Ψ (x, A) → Ψ (x + y, ty A),

(22.41) (1)

where (ty A)(x, A) = A(x − y). This as before leads to Heκ commuting with the total momentum operator, Ptot := Ppart ⊗ 1f + 1part ⊗ Pf ,

(22.42)

(1)

of theP electron and the photon field: [Heκ , Ptot ] = 0. Here Pel := −i∇x and R Pf = λ dk k a∗λ (k)aλ (k), the electron and field momenta. Again as in Section 22.2.2, this leads to the fiber decomposition Z ⊕ (1) −1 (1) U Heκ U = Heκ (P )dP, (22.43) R3

(1)

where the fiber operators Heκ (P ), P ∈ R3 , are self-adjoint operators on F . (1) The computation of the operator Heκ (P ) is the same as of thje corresponding fiber operator in Section 22.2.2. Specifying (22.27) to the present case (i.e. taking x′ = 0), we obtain (1) Heκ (P ) =

1 P − Pf − eAχ )2 + Hf 2m

where Aχ := Aχ (0). Explicitly, Aχ is given by XZ κ(|k|) χ A = dk 1/2 eλ (k) { aλ (k) + a∗λ (k) }. |k|

(22.44)

(22.45)

λ

(1)

Consider the infimum, E(p) := inf σ(Heκ (P )), of the spectrum of the (1) fiber Hamiltonian Heκ (P ). Note that for e = 0, E(P )|e=0 =: E0 (P ) is the
2 (1) 1 ground state energy of H0 (P ) := Heκ (P )|e=0 = 2m P − Pf + Hf with the 2

| ground state Ω and is E0 (P ) = |P 2m . Moreover, it is easy to show that E(p) is spherically symmetric and has a minimum at P = 0 and is twice differentiable

328

22 Standard Hamiltonian of Non-relativistic Matter and Radiation

P = 0. Following the heuristic discussion at the beginning of this section, we define the renormalized electron mass as mren := E ′′ (0)−1 , where, recall, E ′′ (0) is the Hessian of E(P ) at the critical point P = 0. A straightforward computation gives 
Pi Pj  ∂|P | E(P ) Pi Pj 2 Hess E(P ) ij = δij − + ∂ E(P ) |P |2 |P | |P |2 |P |

(22.46)

Since E(P ) is spherically symmetric and C 2 at P = 0, and satisfies ∂|P | E(0) =
P P 2 0, we find Hess E(0) ij = |Pi |2j ∂|P | E(0) and therefore as mren :=

1 2 E(0) . ∂|P |

Recalling our discussion at the beginning of this section, the kinematic meaning of this expression is as follows. The ground state energy E(P ) can be considered as an effective Hamiltonian of the electron in the ground state. (The propagator exp(−itE(P )) determines the propagation properties of a wave packet formed of dressed one-particle states with a wave function supported near P = 0 – which exist as long as there is an infrared regularization.) The first Hamilton equation gives the expression for the electron velocity as v = ∂P E(P ) . Expanding the right hand side in P we find v = Hess E(0)P + O(P 2 ). Since E(P ) is spherically symmetric, and C 2 in |P | near P = 0, this becomes 2 2 v = ∂|P | E(0, σ) P + O(P ) . 2 −1 This suggests taking (∂|P as the renormalized electron mass. | E(0)) (1)

It is shown in [28, 59, 61] that the infimum, E(P ) = inf spec(Heκ (P )), of (1) the spectrum of Heκ (P ) is twice differentiable and P = 0 and the expression for mren is computed to the order e3 .

22.5 Appendix: Relative bound on I(ε) and Pull-through Formulae Proposition 22.3 There is an absolute constant c > 0, s. t.
kI(ε)ψk ≤ cε kH(0)ψk + kψk .

(22.47)

22.5 Appendix: Relative bound on I(ε) and Pull-through Formulae

329

Proof. We use the pull-through formula a(k)f (Hf ) = f (Hf + ω(k))a(k), valid for any piecewise continuous, bounded function, f , on R, proven in Appendix 1 23.8. Applying it to the function f (λ) := (λ + 1)− 2 gives 1

1

a(k)(Hf + 1)− 2 = (Hf + ω(k) + 1)− 2 a(k).

(22.48)

Using this relation we obtain Z ka(g)a(f )ψk ≤ |f |ka(g)a(k)ψk Z 1 1 = |f |ka(g)(Hf + ω(k) + 1)− 2 a(k)(Hf + 1) 2 ψk Z 1 1 ≤ |f |ka(g)(Hf + ω(k) + 1)− 2 kka(k)(Hf + 1) 2 ψk. 1/2

Applying now the bounds (22.33) and (22.35) and using that kHf (Hf + 1/2

1

1

ω(k) + 1)− 2 ψk ≤ kψk and kHf (Hf + 1) 2 ψk ≤ k(Hf + 1)ψk, we obtain ka(g)a(f )ψk ≤

Z

|f |2 ω

1/2 Z

|g|2 ω

1/2

k(Hf + 1)ψk.

Now, using this relation together with (22.33), (22.35) and (22.36), we find Z Z |g|2 |f |2 k(Hf + 1)ψk2 ka∗ (g)a(f )ψk2 ≤ ω ω Z Z |f |2 1/2 |g|2 kHf ψk2 , + ω and 1/2 Z 1/2 |g|2 |f |2 k(Hf + 1)ψk ω ω Z Z |f |2 1/2 +3 |g|2 kHf ψk2 ω Z Z 2 + 3 |f | |g|2 kψk2 .

ka∗ (g)a∗ (f )ψk ≤ 2

Z

The above estimates imply
kA(x)A(x)ψk ≤ c kHf ψk + kψk .

(22.49) 1

Finally, using the estimates (22.37) and kpψk ≤ ck(Hpart + c′ 1) 2 ψk, for c′ s.t. Hpart + c′ 1 ≥ 1, and the fact that Hpart and A(x) commute, we find 1

1

1

kpA(x)ψk ≤ ckA(x)(Hpart +c′ 1) 2 ψk ≤ ck(Hf +1) 2 (Hpart +c′ 1) 2 ψk. (22.50)

330

22 Standard Hamiltonian of Non-relativistic Matter and Radiation

Using the equation H(0) = Hpart ⊗ 1f + 1part ⊗ Hf and using repeatedly 1 that for any positive operator B, kB 2 ψk2 = hψ, Bψi and that hψ, Hpart ψi ≤ 1 hψ, H(0)ψi and hψ, Hf ψi ≤ hψ, (H(0)+c′ 1)ψi, we find that k(Hf +1) 2 (Hpart + 1 c′ 1) 2 ψk ≤ k(H(0) + c′ 1)ψk. This inequality together with (22.50) gives kpA(x)ψk ≤ ck(H(0) + c′ 1)ψk.

(22.51)

The relation (22.31) and estimates (22.49) and (22.51) imply (22.47). Now we prove the very useful “pull-through” formulae (see [135, 28]) used above: a(k)f (Hf ) = f (Hf + ω(k))a(k) (22.52) and f (Hf )a∗ (k) = a∗ (k)f (Hf + ω(k)),

(22.53)

valid for any piecewise continuous, bounded function, f , on R. Problem 22.4 Using the commutation relations above, prove relations (22.52)¯ +. (22.53) for f (H) = (Hf − z)−1 , z ∈ C/R Using the Stone-Weierstrass theorem, one can extend (22.52)- (22.53) from ¯ + , to the class of functions functions of the form f (λ) = (λ − z)−1 , z ∈ C\R mentioned above. Alternatively, (22.52)- (22.53) follow from the relation f (Hf )

N Y

j=1

N N Y X a∗ (kj )Ω. ω(ki )) a∗ (kj )Ω = f ( i=1

j=1

Problem 22.5 Prove this last relation, and derive (22.52)- (22.53) from it.

23 Theory of Radiation

Emission and absorption of electromagnetic radiation by systems of nonrelativistic particles such as atoms and molecules is a key physical phenomenon, central to the existence of the world as we know it. Attempts to understand it led, at the beginning of the twentieth century, to the birth of quantum physics. In this chapter we outline the theory of this phenomenon. It addresses the following fundamental physical facts: (a) a system of matter, say an atom or a molecule, in its lowest energy state is stable and well localized in space, while (b) the same system placed in the vacuum in an excited state, after awhile, spontaneously emits photons and descends to its lowest energy state. The starting point of theory of radiation is the Schr¨odinger equation i

∂ψ = H(ε)ψ, ∂t

describing quantum particles interacting amongst themselves, and with quantized electro-magnetic field. Here ψ is a path in the state space Hpart ⊗ Hf and the quantum Hamiltonian operator H(ε) entering it acts on Hpart ⊗ Hf and is given by (see (22.28)): H(ε) =

n X 1 (i∇xj + εAχ (xj ))2 + V (x) + Hf , 2m j=1

(23.1)

with the notation explained in the previous chapter. (Recall that we use the units ~ = 1 and c = 1 and, as a rule, we do not display these identities 1part and 1f on Hpart and Hf , respectively. A careful notation would have Vcoul (x) ⊗ 1f and 1part ⊗ Hf , instead of Vcoul (x) and Hf .) The mathematical manifestation of the fact (a) is that H(ε) has a ground state, which is well localized in the particle coordinates, while the statement (b), rendered in mathematical terms, says that the system in question has no stable states in a neighbourhood of the excited states of the particle system, but ‘metastable’ ones. © Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_23

332

23 Theory of Radiation

23.1 Spectrum of the Uncoupled System To understand the spectral properties of the operator H(ε) we first examine a system consisting of matter and radiation not coupled to each other. Such a system is described by the Hamiltonian H(0) = Hpart ⊗ 1f + 1part ⊗ Hf ,

(23.2)

which is obtained by setting the parameter ε in (22.28) to zero. The corresponding time-dependent Schr¨odinger equation is i

∂ψ = H(0)ψ. ∂t

(23.3)

The dynamics of a system of quantum matter (atom, molecule, etc., with fixed nuclei) is described by the Schr¨odinger operator Hpart =

n X 1 2 p + V (x) 2m j j=1

acting on Hpart = L2 (R3n ) (or a subspace of this space of a definite symmetry type). Recall the spectral structure of the Schr¨odinger operator Hpart . By HVZ theorem (see Section 13.4), we have σ(Hpart ) = {negative EV’s, Ej } ∪ { continuum [0, ∞) }.

(23.4)

Here j = 0, 1, . . . and we assume E0 < E1 ≤ . . . . The eigenfunction, ψ0part , corresponding to the smallest eigenvalue, E0 is called the ground state, while the eigenfunctions ψjpart for the higher eigenvalues Ej with j ≥ 1, are called the excited states. The generalized eigenfunctions of the essential spectrum are identified with the scattering states (see Figure 23.1). E0 E 1 . . . x x x bound states

0 scattering states

Fig. 23.1. Spectrum of Hpart For the field, σess (Hf ) = [0, ∞) and σd (Hf ) = {0}. The eigenvalue 0 is non-degenerate, and corresponds to the vacuum vector: Hf Ω = 0. Using separation of variables, we obtain the eigenfunctions and the generalized eigenfunctions, ψjpart ⊗ Ω,

and

ψjpart ⊗

s Y

i=1

a∗ (ki )Ω,

(23.5)

23.2 Complex Deformations and Resonances

333

for various s ≥ 1 and k1 , . . . , ks in R3 , of H(0), corresponding P to the eigens values Ej , j = 0, 1, . . ., and the spectral points Ej (k) = Ej + i=1 ω(ki ) ∈ [Ej , ∞), respectively. This leads to the following stationary solutions e−iEj t (ψjpart ⊗ Ω)

and

e−iEj (k) ψjpart ⊗

s Y

i=1


a∗ (ki )Ω ,

of the time-dependent Schr¨odinger equation (23.3). The first of these states describes the particle system in the state ψjpart with no photons around, while the second one corresponds to the system in the state ψjpart and s photons with momenta k1 , . . . , ks . Both states are stationary in time. In the absence of coupling between matter and radiation, the system of matter and radiation placed in one of these states remains in the same state forever. Radiation is neither absorbed nor emitted by this system. Note finally that various eigenfunctions and generalized eigenfunctions above lead to different branches of the spectrum of H(0), which, as a set, is of the form [
σ H(0) = {EV′ s Ej } {continuum [Ej , ∞)} ∪ continuum [0, ∞) (23.6) j≥0

(see Section 25.13 of the Mathematical Supplements). The spectrum of H(0) is pictured in Figure 23.2. branches of the continuous spectrum x x x E0 E 1 . . .

0

bound states

Fig. 23.2. Spectrum of H(0). The question we want to address is: how does this picture change as an interaction between the matter and radiation is switched on? This is the main problem of the mathematical theory of radiation.

23.2 Complex Deformations and Resonances In this section we discuss a key notion of the (quantum) resonance. It gives a clear-cut mathematical description of processes of emission and absorption of the electro-magnetic radiation. In this description, the process of radiation corresponds to formation of resonances out of the excited states of particle

334

23 Theory of Radiation

systems. The most effective way to define resonances is to use complex transformation of the Hamiltonian under consideration (see Section 17.1) which we proceed to describe in the present setting. Define the dilatation transformation, Uf θ , on the Fock space, Hf ≡ F , as the unitary group of operators, given by Y Y Uf θ Ω = Ω and Uf θ a∗ (fj )Ω = a∗ (uθ fj )Ω. (23.8)

Here uθ is the rescaling transformation acting on L2 (R3 ), given by (uθ f )(k) = e−3θ/2 f (e−θ k).

(23.9)

(Uf θ is the Rsecond quantization of uθ .) This definition implies Uf θ := eiT θ where T := a∗ (k)ta(k)dk and t is the generator of the group uθ (see Chapter R 20 for the careful definition of the integral a∗ (k)ta(k)dk) and # Uf θ a# (f )Uf−1 θ = a (uθ f )

(23.10)

where a# (f ) stands for either a(f ) or a∗ (f ). Applying this transformation to the photon Hamiltonian Hf , we find Hf θ = e−θ Hf . Denote by Upθ the 3n standard dilation group on the particle space: Upθ : ψ(x) → e 2 θ ψ(eθ x) where, recall, n is the number of particles (cf. Section 17.1). We define the dilation transformation on the total space H = Hp ⊗ Hf by Uθ = Upθ ⊗ Uf θ .

(23.11)

For θ ∈ R the above operators are unitary and map the domains of the operators H(ε) into themselves. Consequently, we can define the family of Hamiltonians originating from the Hamiltonian H(ε) as Hε,θ := Uθ H(ε)Uθ−1 .

(23.12)

We would like to extend this family analytically into complex θ’s. To this end we impose the following condition (in addition to condition (V)): (A) Vθ (x) := V (eθ x), as a multiplication operator from D(∆x ) to Hpart , has an analytic continuation, Vθ , in θ from R into a complex neighbourhood of θ = 0, which is bounded from D(∆x ) to Hpart . Furthermore, we fix the ultra-violet cut-off χ(k) from now on so that χθ (k) := e−3θ/2 χ(e−θ k) is an analytic function in a neighbourhood of θ = 0, vanishing 2 sufficiently fast at ∞, e.g. by taking χ(k) = e−|k| . Under Condition (A), there is a family H(ε, θ) of operators Type-A analytic ([176, 247, 162]) in the domain |Imθ| < θ0 , which is equal to (23.12) for θ ∈ R and s.t. H ∗ (ε, θ) = H(ε, θ) and −1 H(ε, θ) = UReθ H(ε, iImθ)UReθ .

(23.13)

23.2 Complex Deformations and Resonances

335

Indeed, using (22.28), we decompose Pn H(ε) = Hpart + Hf + I(ε), where I(ε) is defined by this relation: I(ε) = j=1 [i∇xj · εAχ (xj ) + 12 ε2 Aχ (xj )2 ], where we used that divAχ = 0. Using this decomposition, we write for θ ∈ R H(ε, θ) = Hp (θ) ⊗ 1f + e−θ 1p ⊗ Hf + I(ε, θ) ,

(23.14)

Furthermore, using (23.10) and the definitions of the interaction I(ε), we see 3θ that I(ε, θ) is obtained from I(ε) by the replacement a# (k) → e− 2 a# (k) and, in the coupling functions only, k → e−θ k

and

xj → eθ xj .

(23.15)

This gives the required analytic continuation of (23.12). We call H(ε, θ) with Imθ > 0 the complex deformation of H(ε). One can show show that: 1) The essential spectrum of moves (e.g. the spectrum of the deformation Hf θ is σ(Hf θ ) = {0} ∪ e− Im θ [0, ∞)); 2) The real eigenvalues of H(ε, θ), Imθ > 0, coincide with eigenvalues of H(ε) and that complex eigenvalues of H(ε, θ), Imθ > 0, lie in the complex half-plane C− ; 3) The complex eigenvalues of H(ε, θ), Imθ > 0, are locally independent of θ. Let Ψθ = Uθ Ψ , etc., for θ ∈ R and z ∈ C+ . Use the unitarity of Uθ for real θ, to obtain (the Combes argument) hΨ, (H(ε) − z)−1 Φi = hΨθ¯, (H(ε, θ) − z)−1 Φθ i.

(23.16)

Assume now that for a dense set of Ψ ’s and Φ’s (say, D, defined below), Ψθ and Φθ have analytic continuations into a complex neighbourhood of θ = 0 and continue the r.h.s analytically first in θ into the upper half-plane and then in z across the continuous spectrum. • •

The real eigenvalues of H(ε, θ) give real poles of the r.h.s. of (23.16) and therefore they are the eigenvalues of H(ε). The complex eigenvalues of H(ε, θ) are poles of the meromorphic continuation of the l.h.s. of (23.16) across the spectrum of H onto the second Riemann sheet.

The poles manifest themselves physically as bumps in the scattering crosssection or poles in the scattering matrix. An example of the dense set D for which the r.h.s. of (23.16) has an analytic continuation into a complex neighbourhood of θ = 0, is [
D := Ran χN ≤n χ|T |≤a . (23.17) n>0,a>0

R

Here N = d3 ka∗ (k)a(k) be the photon number operator and T be the selfadjoint generator of the one-parameter group Uθ , θ ∈ R. It is easy to show that the set D is dense.

336

23 Theory of Radiation

We define the the resonances of H(ε) as the complex eigenvalues of H(ε, θ) with Imθ > 0. Thus to find resonances (and eigenvalues) of H(ε) we have to locate complex (and real) eigenvalues of H(ε, θ) for some θ with Imθ > 0.

23.3 Results The rigorous answer to the question of how the spectral properties of H(0) change as the interaction is switched on is given in the theorem below. This theorem refers to the notion of resonance described in Section 23.2 (see also Section 17.1). We have Theorem 23.1 Let ε 6= 0 be sufficiently small. For statements (ii) and (iii), we assume, besides (V), Condition (A). Then (i) H(ε) has a unique ground state, ψε . This state converges to ψ0part ⊗ Ω as ε → 0, and is exponentially localized in the particle coordinates: i.e, keδ|x| ψε k < ∞ for some δ > 0. (ii) H(ε) has no other bound states. In particular, the excited states of Hpart (i.e. ψjpart ⊗ Ω, j ≥ 1) are unstable. (iii) The excited states of Hpart turn into resonances of H(ε), ε 6= 0 (see Fig. 23.3). Statement (ii) uses in addition to (V) and (A) a technical condition called (positivity of) the Fermi Golden Rule, which is satisfied except in a few “degenerate” cases).

Fig. 23.3. Bifurcation of eigenvalues of H(0)(the second Riemann sheet). This theorem gives mathematical content to the physical picture based on formal calculations performed with the help of perturbation theory. Statement (i) says that a system of matter, say an atom or a molecule, in its lowest energy state is stable and well localized in space, and according to statements (ii) and (iii), the excited states of the particle system disappear and give rise to longliving, metastable states of the total system. The latter are solutions of the time-dependent Schr¨odinger equation i

∂ψ = H(ε)ψ ∂t

which are localized for long intervals of time, but eventually disintegrate. (Recall that the metastable state is another term for a resonance.) These

23.4 Idea of the proof of Theorem 23.1

337

metastable states are responsible for the phenomena of emission and absorption of radiation and their life-times tell us how long, on average, we have to wait until a particle system emits (or absorbs) radiation. The real parts of the resonance eigenvalues – the resonance energies – produce the Lamb shift, first experimentally measured by Lamb and Retherford (Lamb was awarded the Nobel prize for this discovery). The imaginary parts of the resonance eigenvalues – the decay probabilities – are given by the Fermi Golden Rule (see, eg, [169]). In the rest of this chapter we outline main steps of the proof of Theorem 23.1, with more machinery given in the next chapter.

23.4 Idea of the proof of Theorem 23.1 A complete proof of statements (i)-(iii) can be found in [265] (see also [29, 27]). We will describe the proof of (i) in this chapter and in Chapter 24. The proofs of (ii) and (iii) are similar. Namely, techniques developed in the proof of (i) are applied to the family H(ε, θ), Im θ > 0, instead of H(ε). Since we are dealing only with the ground state, we will not need the analyticity Condition (A) above. The proof of statement (i) of Theorem 23.1 – existence of the ground state – is done in two steps. On the first step, after performing a convenient canonical transformation, we map the family H(ε) − z1 into a family H0 (ε, z), acting on the subspace of the Fock space Hf ≡ F corresponding to the photon energies ≤ ρ for some ε ≪ ρ ≤ E1part − E0part , where E0part and E1part are the ground state and the first excited state energies of Hpart . On the second step, done in Chapter 24, we apply to H0 (ε, z) a spectral renormalization group which brings it isospectrally into a very simple form which can be analyzed easily. We explain some intuition underlying the proof. To understand the problem we are facing we look at the spectrum of H(ε = 0), i.e., when the interaction is turned off (see Figure 23.4). We see that the unperturbed ground state energy is at the bottom of continuous spectrum. This suggests that only the ground state ψ0part of Hpart and the low energy states of Hf are essential.

region of interest x E0

x E1 . . .

Fig. 23.4. Region of interest w.r.t. specH(0). Hence - and this is the key idea - we would like to project out the inessential parts of the spectrum without distorting the essential ones. But how do we do this? Let us try the first idea that comes to mind:

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23 Theory of Radiation

H → Qρ HQρ ,

(23.18)

where Pρ is the spectral projection for H(0) associated with the interval [E0 , E0 + ρ]. The operator Qρ can be written explicitly as Qρ = P0part ⊗ E[0,ρ] (Hf ) .

(23.19)

Here P0part is the orthogonal projection onto ψ0part and E[0,ρ] (Hf ) is the spectral projection of Hf corresponding to the L interval [0, ρ], defined as follows. L∞ ∞ First, recall that Hf = n=0 Hf,n on F = n=0 Fn , where HP f,0 is the opern ator of multiplication by 0 on F0 := C, and for n ≥ 1, Hf,n = i=1 ω(ki ) are n 2 3 multiplication operators on Fn := S 1 L (R ). Next, let χ∆ be the characteristic function of the set ∆ ⊂ R. Now we define E∆ (Hf ) = ⊕∞ n=0 E∆ (Hf,n ) where P for n ≥ 1, E∆ (Hf,n ) is the operator of multiplication by the function χ∆ ( nk=1 ω(ki )), acting on Fn , and E∆ (Hf,0 ) is the operator of multiplication by χ∆ (0) on F0 . The operator E[0,ρ] (Hf ) “cuts-off” the energy states of Hf with energy above ρ. The new operator Qρ H(ε)Qρ acts on the subspace L2 (R3 ) ⊗ F which consists of states of the form ψ0part ⊗ φ , φ ∈ RanE[0,ρ] (Hf ) (see Section 12.7). This is exactly what we want. However, the low energy spectrum of the operator Qρ H(ε)Qρ is different from that of H(ε). So we have lost the spectral information we are after. In Section 11.1, we learned how to project operators to smaller subspaces without losing the spectral information of interest. (We will refine this procedure in Sections 23.6 and 24.3.) But, as usual, there is a trade-off involved. While the map (23.18) acting on operators H is linear, the new map we introduce is not. This new map is called the decimation map (or Feshbach-Schur map). The result of application of this map to the family H(ε) − z1 is a family Hz (ε) of operators which act on the subspace RanE[0ρ] (Hf ) of the Fock space F and which is a small perturbation of the operator Hf plus a constant. In fact, for technical reasons it is convenient to use not the Feshbach-Schur map, but its extension - the smooth Feshbach-Schur map. (The latter is more difficult to define but much easier to use.) This is done below. In the next chapter we apply a renormalization group, based on the smooth Feshbach-Schur map, which brings isospectrally the operators Hz (ε) arbitrarily close to operators of the form E · 1 + wHf , with E ∈ R and w > 0.

23.5 Generalized Pauli-Fierz Transformation In order to improve the infra-red behaviour of the Hamiltonian H(ε), in this section we apply to it a canonical transformation. To simplify notation we

23.5 Generalized Pauli-Fierz Transformation

339

only consider the case n = 1 and m = 1. (The generalizations to an arbitrary number of particles is straightforward.) In this case we have H(ε) =

1 (i∇x + εAχ (x))2 + V (x) + Hf , 2

(23.20)

where x ∈ R3 . The vector potential operator Aχ (x) is still given by (22.29). Now Hpart = − 21 ∆ + V (x) acts on Hpart = L2 (R3 ), so that H(ε) acts on L2 (R3 ) ⊗ Hf . Now, we introduce the generalized Pauli-Fierz transformation Hε := e−iεF (x) H(ε)eiεF (x) ,

(23.21)

where F (x) is the self-adjoint operator on the state space H given by XZ d3 k (23.22) F (x) = (f¯x,λ (k)aλ (k) + fx,λ (k)a∗λ (k)) p , |k| λ

with the coupling function fx,λ (k) chosen as (recall that we do not display α in the coupling function in (22.29)) 1 χ(k) fx,λ (k) := e−ik·x p ϕ(|k| 2 eλ (k) · x). |k|

(23.23)

The function ϕ is assumed to be C 2 , bounded, with bounded second derivative, and satisfying ϕ′ (0) = 1. We assume also that ϕ has a bounded analytic continuation into the wedge {z ∈ C| | arg(z)| < θ0 }. We call the resulting Hamiltonian, Hε , the generalized Pauli-Fierz Hamiltonian. We compute 1 (p − εA1 (x))2 + Vε (x) + Hf + εG(x) (23.24) 2 R 2 P 2 3 where A1 (x) = Aχ (x) − ∇F (x), Vε (x) := V (x) + ε2 λ |k||fx,λ (k)| d k and Hε =

G(x) := −i

XZ λ

d3 k |k|(f¯x,λ (k)aλ (k) − fx,λ (k)a∗λ (k)) p . |k|

(23.25)

(The terms εG and Vε − V come from the commutator expansion e−iεF (x) Hf 2 ×eiεF (x) = −iε[F, Hf ] − ε2 [F, [F, Hf ]].) Observe that the operator-family A1 (x) is of the form XZ d3 k A1 (x) = (χx,λ (k)aλ (k) + χx,λ (k)a∗λ (k)) p , (23.26) |k| λ where the coupling function χλ,x (k) is defined as follows

χλ,x (k) := eλ (k)e−ikx χ(k) − ∇x fx,λ (k).

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23 Theory of Radiation

It satisfies the estimates |χλ,x (k)| ≤ const min(1, with hxi := (1 + |x|2 )1/2 , and Z

p |k|hxi),

d3 k |χλ,x (k)|2 < ∞. |k|

(23.27)

(23.28)

The fact that the operators A1 and G have better infra-red behavior than the original vector potential A, is used in proving, with a help of a renormalization group, the existence of the ground state and resonances for the Hamiltonian Hε and therefore for H(ε). We note that for the standard Pauli-Fierz transformation, the function fx,λ (k) is chosen to be χ(k)eλ (k) · x, which results in the operator G (which in this case is proportional to (the electric field at x = 0) · x) growing as x. This transformation can be used if our system is placed in an external confining potential W (x) satisfying the estimate W (x) ≥ c|x|2 , for |x| ≥ R, for some c > 0 and R > 0. For further reference, we mention that the operator (23.24) can be written as Hε = H0ε + Iε , (23.29) where H0ε = H0 +

Z ε2 X |χλ,x (k)|2 3 )d k, (|fx,λ (k)|2 + 2 2|k|

(23.30)

λ

with H0 := H(ε = 0) = Hpart + Hf (see (23.20)), and Iε is defined by this relation. Note that the operator Iε contains linear and quadratic terms in creation- and annihilation operators, with coupling functions (form-factors) in the linear terms satisfying estimate (23.27) and with coupling functions in the quadratic terms satisfying a similar estimate. Moreover, the operator H0ε is of the form H0ε = Hεpart + Hf where Hεpart := Hpart +

Z ε2 X (|fx,λ (k)|2 + |χλ,x (k)|2 2|k|)d3 k, 2

(23.31)

λ

where, recall, Hpart = − 12 ∆ + V (x). Note that similarly to Proposition 22.3 one can show the following Proposition 23.2 We have
1/2 kG(x)ψk, kA(x)ψk ≤ c kHf ψk + kψk .

(23.32)

This theorem implies that Iε is a relatively bounded perturbation of H0ε .

23.6 Elimination of Particle and High Photon Energy Degrees of Freedom

341

23.6 Elimination of Particle and High Photon Energy Degrees of Freedom Since we are looking at a vicinity of the ground state energy of H(ε), the degrees of freedom connected to the excited particle states and to high photon energies should not be essential. So we eliminate them isospectrally using the smooth Feshbach-Schur (decimation) map. In this section we construct this map and use it to pass isospectrally from the family Hε − z · 1 to a family H0 (ε, z) of operators which act non-trivially only on the subspace RanE[0,ρ] (Hf ), where recall E[0,ρ] (λ) is the characteristic function of the interval [0, ρ], of the Fock space Hf ≡ F . The advantage of this family is that it is even smaller perturbation of Hf plus a constant and that it can be treated by the renormalization group approach developed in the next chapter. Passing from Hε − z · 1 to H0 (ε, z) will be referred to as elimination of the particle and high photon energy (actually photon energy ≥ ρ) degrees of freedom. As was already mentioned, the smooth Feshbach-Schur map is a generalization of the Feshbach-Schur map, discussed in Section 11.1 and it arises when the projections P and P¯ := 1−P are replaced by more general operators P and P¯ forming a partition of unity P 2 + P¯ 2 = 1.

(23.33)

Here we give a quick definition of the smooth Feshbach-Schur map. For more details see Section 24.3. Consider operators H on a Hilbert space H with specified decompositions H = H0 + I. We define HP¯ := H0 + P¯ I P¯ .

(23.34)

IP and P I extend to bounded operators on H

(23.35)

Assume now that

and that z is s.t. HP¯ − z is (bounded) invertible on Ran P¯ .

(23.36)

We define smooth Feshbach-Schur map, FPsmooth , as FPsmooth (H − z1) := H0 − z1 + P IP − P I P¯ (HP¯ − z)−1 P¯ IP .

(23.37)

Definition 23.3 We say that operators A and B are isospectral at a point z ∈ C iff (a) z ∈ σ(A) ⇔ z ∈ σ(B), (b) Aψ = zψ ⇔ Bφ = zφ

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23 Theory of Radiation

where ψ and φ are related by φ = P ψ and ψ = Qφ for some bounded operators P and Q. We have the following theorem : Theorem 23.4 Assume H − z1 is in the domain of FPsmooth . Then the operators H and FPsmooth (H − z1) + z1 are isospectral at z, in the sense of the definition above, with the operator P , the same as in FPsmooth and the operator Q defined by QP (H − z1) := P − P¯ (HP¯ − z)−1 P¯ IP. (23.38) This theorem generalizes Theorem 11.1 of Section 11.1; its proof is similar and is sketched in Section 24.3. Now we adapt these notions to the families of operators (23.24). In this case, a partition (P, P¯ ) (see (23.33) ) is defined as follows. Let χρ ≡ χρ (Hf ) := χHf /ρ≤1 and χ ¯ρ ≡ χ ¯ρ (Hf ) := χHf /ρ≥1 , where χλ≤1 and χλ≥1 are smooth functions satisfying the relations  1 if λ ≤ 45 2 2 χλ≤1 + χλ≥1 = 1, 0 ≤ χλ≤1 , χλ≥1 ≤ 1, χλ≤1 = 0 if λ > 1 (and the corresponding relation for χλ≥1 ). Thus χρ (Hf ) = χHf ≤ρ is an almost the spectral projection for the operator Hf onto energies ≤ ρ. With the definition (23.31), let furthermore P0part = orthogonal projection onto the ground state eigenspace of Hεpart , (23.39) and P¯0part := 1 − P0part . (Recall that Hεpart is defined in (23.31).) Assume the ground state of Hεpart is simple and the corresponding eigenspace is spanned by a function ψ0part . Then, in the Dirac notation, P0part = |ψ0part ihψ0part |. We introduce a pair of almost projections Pρ := P0part ⊗ χρ (Hf )

(23.40) 2

and P ρ := P¯0part +P0part ⊗χ ¯ρ (Hf ), which form a partition of unity Pρ2 +P ρ = 1. Note that Pρ and Pρ commute with H0ε . Let E0 and E1 be the ground state energy and the first excited state energy of Hεpart and set 1 Ωρ := {z ∈ C | Rez ≤ E0 + ρ}. 4

(23.41)

Now we have Theorem 23.5 Let |ε| ≪ ρ ≤ E1 − E0 . Then FPsmooth is defined on for the ρ families of operators, Hε − z · 1, z ∈ Ωρ , where Hε is given by (23.24), with the decomposition (23.29), and is isospectral (in the sense of Theorem 23.4). Proof. The first part of the theorem follows from the definition of the domain of the map FPsmooth given above and Propositions 23.9 and 23.10 given in Appendix 23.10 below. The isospectrality follows from the first part of the theorem and Theorem 23.4. 

23.7 The Hamiltonian H0 (ε, z)

343

Let, as above, E[0,ρ] (λ) be the characteristic function of the interval [0, ρ]. We define the subspace Xρ := ψ0part ⊗ E[0,ρ] (Hf )Hf ≈ E[0,ρ] (Hf )Hf of Hpart ⊗ Hf . Here ≈ stands for the isomorphic isomorphism. The subspace Xρ and its orthogonal complement, Xρ⊥ , are invariant under FPsmooth (Hε − z1) ρ and span Hpart ⊗ Hf . Let ∆E = E1 − E0 . We choose ρ0 such that ε ≪ ρ0 < min(1, ∆E). We apply FPsmooth to the families of operators (23.24) with the decomposition ρ0 Hε = H0ε + Iε , as given in (23.29) - (23.30), to obtain the new Hamiltonian H0 (ε, z) := FPsmooth (Hε − z1)|Xρ0 . ρ0

(23.42)

We claim that for Rez ≤ E0 + 14 ρ0 and 0 < ρ0 ≤ E1 − E0 , Hε − z1

and

H0 (ε, z)

are isospectral at

0.

(23.43)

FPsmooth (Hε − z1)|Xρ⊥ = (H0ε − z1)|RanXρ⊥ . ρ

(23.44)

Indeed, since Pρ = 0 on Xρ⊥ , we have that

Therefore, for Rez ≤ E0 + 14 ρ and 0 < ρ ≤ E1 − E0 , FPsmooth (Hε − z1)|Xρ⊥ ≥ ρ

3 ρ>0 4

(23.45)

and, in view of Theorem 23.4, (23.43) holds. Observe that since the projection P0part has rank 1, the operator H0 (ε, z) is of the form   H0 (ε, z) = hψ0part , FPsmooth (Hε − z1)ψ0part iHpart |RanE[0,ρ0 ] (Hf ) . (23.46) ρ Since we are interested in the part of the spectrum which lies in the set {z ∈ R | Re z ≤ E0 + 14 ρ0 }, we can study H0 (ε, z), which acts on the space Xρ0 ≈ RanE[0,ρ0 ] (Hf ), instead of Hε . Thus we have passed from the operator Hε acting on Hpart ⊗ Hf to the operator H0 (ε, z) acting on the subspace, RanE[0,ρ] (Hf ), of the Fock space Hf ≡ F , which is isospectral (in the sense of Definition 23.3) to Hε − z1 at 0, provided z is in the set (23.41). We eliminated, in an isospectral way, the degrees of freedom corresponding to the particle and to the photon energies ≥ ρ0 ; i.e., we projected out the part Ran (1 − P0part) of the particle space Hpart , and the part Ran(1 − E[0,ρ0 ] (Hf )) of the Fock space Hf ≡ F . The parameter ρ0 is called the photon energy scale.

23.7 The Hamiltonian H0 (ε, z) Key properties of H0 (ε, z) are summarized in the next theorem.

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23 Theory of Radiation

Theorem 23.6 Let z ∈ Ωρ0 and ε ≪ ρ0 ≤ ∆E (ρ0 is the scale parameter entering the definition of H0 (ε, z)). The operator H0 (ε, z) has a generalized normal form, X H0 (ε, z) = H0,00 + χρ0 H0,rs χρ0 , H0,00 := h0,00 (Hf ) r+s≥1

H0,rs :=

Z

r Y

j=1

r+s Y
a(ki )dr+s k, a∗ (kj ) h0,rs (Hf , k1 . . . kr+s )

(23.47)

i=r+1

with coupling functions h0,rs which are analytic in z ∈ Ω, and satisfy the estimates |∂µn h0,rs (µ, k)| ≤ (const)ρ0

r+s X

1

ω(ki ) 2

i=1

r+s Y j=1


− 21 const · ε · ρ−1 , (23.48) 0 · ω(kj )

for any i, 1 ≤ i ≤ r + s, µ ∈ [0, 1], and n = 0, 1,where the product is absent in the case r = s = 0, and |h0,00 (0)| ≤ (const)|E0 − z + O(ε2 )|,

|∂µ h0,00 (µ) − 1| ≤ (const)ε2 /ρ20 , (23.49)

for µ ∈ [0, ∞).

P 1 2 We note that the crucial for the future analysis term r+s i=1 ω(ki ) is due to the estimate (23.27) gained in the generalized Pauli-Fierz Hamiltonian (23.24). The proof of this theorem is simple but lengthy. It can be found in [27, 121]. Below we sketch its main ideas. Sketch of proof of Theorem 23.6. Using the definition of the smooth Feshbach-Schur map FPsmooth (Hε − z1), we write the operator H0 (ε, z) in the ρ0 form H0 (ε, z) = Hf + χρ0 W χρ0 , where with the same definitions as in the proof of Proposition 23.10, ¯ ρ0 (z)Iε ψ part iHpart , W = Iε − hψ0part , Iε R 0 ¯ ρ (z) = P¯ρ (H0ε + Pρ0 Iε Pρ0 − z)−1 P¯ρ . Now we use estimates obtained with R in the proof of Proposition 23.10 of Appendix 23.10 below, to expand the resolvent (H0ε +Pρ0 Iε Pρ0 −z)−1 in a Neumann series, which with the notation
−1 I = Iε , P = Pρ0 , P¯ = P¯ρ0 , and R0,P¯ = P¯ H0ε − z P¯ can be written as W =

∞ X

hψ0part , I(−R0,P¯ I)n ψ0part i .

(23.50)

n=0

Next we bring each term on the right hand side of (23.50) to the generalized normal form. To this end we observe that the terms on the right hand side of (23.50), with n ≥ 1, consist of sums of products of five operators: R0,P¯ ,

23.8 Estimates on the operator H0 (ε, z)

345

P¯ , p, a(k) and a∗ (k). We do not touch the operators R0,P¯ , P¯ and p, while we move the operators a(k) to the extreme right and the operators a∗ (k) to the extreme left. In doing this we use the following rules: 1. a(k) is pulled through a∗ (k) according to the relation a(k)a∗ (k ′ ) = a∗ (k ′ )a(k) + δ(k − k ′ ) 2. a(k) and a∗ (k) are pulled through R0,P¯ according to the relations ω(k)

a(k)R0,P¯ = R0,P¯ a(k) ω(k)

R0,P¯ a∗ (k) = a∗ (k)R0,P¯ , ω(k) where R0,P¯ = R0,P¯ H

, and similarly for pulling a(k) and a∗ (k) through other functions of Hf , such as P¯ (see the equations (22.52) of Appendix 23.8). f →Hf +ω(k)

The procedure above brings the operator H0 (ε, z) to generalized normal form (23.47), at least formally. It remains to estimate the coupling functions h0,rs entering (23.47). A direct estimate produces large combinatorial factors which we must avoid. So we use a special technique which amounts to a partial resummation of the series, in order to take advantage of cancelations. This can be found in [27, 121]. 

23.8 Estimates on the operator H0 (ε, z) Though the operator H0 (ε, z) looks rather complicated, we show now that the complicated part of it gives a very small contribution. To this end we use Proposition 23.11 of Appendix 23.11 and the estimates (23.48) and  Z Z 2! n  n Y Jj Jj (kj )2 1 Y 3n d k≤ Pn ω(kj ) n! j=1 ω 1 ω(kj )≤ρ j=1 (valid for any functions Ji and used for Ji = ω −1/2 or = 1), to obtain for r + s ≥ 1, √ −1/2 3/2 kχρ0 H0,rs (ε, z)χρ0 k ≤ (r!s!)−1/2 ( 4περ0 )r+s ρ0 . (23.51) With choosing, say, ρ0 = ε9/5 , these estimates and the relation h0,00 (Hf ) − h0,00 (0) − Hf =

Z1 0


(h′0,00 sHf ) − 1 ds Hf

imply that the operator H0 (ε, z) is of the form

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23 Theory of Radiation 9
H0 (ε, z) = (E0 + ∆0 E − z) · 1 + w(Hf )Hf + O ε 5 ,

1
where E0 + ∆0 E − z = h0,00 (0) and w(Hf ) = 1 + O ε 5 . Here ∆0 E is a explicitly computable energy shift of the order O(ε2 ). Since the spectrum of the operator Hf + E0 + ∆0 E fills in the semi-axis [E0 + ∆0 E, ∞), the isospectrality Theorem 23.4 implies that spec H(ε) = spec Hε ⊂ [E0 + ∆0 E + 9
O ε 5 , ∞), which implies the following intermediary result:

Theorem 23.7 Assume ε 6= 0 is sufficiently small and that the ground state of the particle Hamiltonian, Hpart , is non-degenerate. Define E0 (ε) = inf σ(H(ε)), the ground state energy of H(ε). Then 9
E0 (ε) = E0 + ∆0 E + O ε 5 , with explicitly computable energy shift ∆0 E of the order O(ε2 ).

Applying the renormalization transformation iteratively (see the next chapter) to quantum Hamiltonians of the form (23.47) - (23.49), given in Theorem 23.6, 1 we find energies E (n) = E0 + O(ε2 ) and numbers w(n) = 1 + O(ε 5 ), such that for ρ = 0(ε) ≪ 1 and any n ≥ 1, Hε (or H(ε)) is isospectral to E (n) + w(n) Hf + O(ε2 ρn ) in the disk D(E (n) , ρn ). This will give us a much more precise information about the spectral properties of the Hamiltonian H(ε). The above procedure is at the heart of the renormalization group approach. Remark 23.8 The property that the interaction vanishes under renormalization transformations, i.e. when we go closer and closer to the ground energy (or farther and farther from the particle system) is called infrared asymptotic freedom. Note that the spectrum of the operator Hf +E0 +∆0 E contains the eigenvalue E0 + ∆0 E, with eigenfunction Ω, and the continuum [E0 + ∆0 E, ∞), with generalized eigenfunctions ψ0part ⊗ Πa∗ (kj )Ω. Hence, extending the results of Theorem 23.4, QPρ0 (Hε − z)Ω, where QP (H − z1) is given by (23.38), gives an approximate eigenfunction of Hε with approximate eigenvalue, E0 + ∆0 E and similarly for the continuum.

23.9 Ground state of H(ε) Let Uf θ be a unitary group of operators, given by (23.8). We define the rescaling transformation, Sρ , on operators on the Fock space F by Sρ (H) := Uf θ HUf−1 θ , In particular, we have (see (23.10))

θ = − ln ρ.

(23.52)

23.9 Ground state of H(ε)

Sρ (a# (k)) = ρ−3/2 a# (ρ−1 k),

347

(23.53)

so that Sρ (Hf ) = ρHf

and Sρ (χρ ) = χ1 . (23.54)
H0 (ε, z) , where, recall, H0 (ε, z) is defined in (ε, z) is of the form

ρ−1 0 Sρ0 (0)

(0)

Define H (ε, z) := (23.42). The operator H X H = H00 + χ1 Hrs χ1 , H00 := h00 (Hf ) r+s≥1

Hrs :=

Z Y r

a∗ (kj ) hrs (Hf , k1 . . . kr+s )

r+s Y

(23.55) a(ki )dr+s k,

i=r+1

j=1

acting on acting on the space E[0,1] (Hf )F , where recall E[0,ρ] (λ) denotes the characteristic function of the interval [0, ρ], and with coupling functions hrs satisfying the estimates |∂µn hrs (µ, k)| ≤ (const)ρn0

r+s X i=1

1

ω(ki ) 2

r+s Y j=1

1
ξ0 · ε · ω(kj )− 2 ,

(23.56)

for any i, 1 ≤ i ≤ r + s, µ ∈ [0, 1], ξ0 > 0, and n = 0, 1,where the product is absent in the case r = s = 0, and const 2 const |E0 − z + O(ε2 )|, |∂µ h00 (µ) − 1| ≤ ε , (23.57) |h00 (0)| ≤ ρ0 ρ20 for µ ∈ [0, ∞). To prove this we use (23.53). Then Theorem 23.4 implies that

z ∈ σ# Hε ↔ 0 ∈ σ# H (0) (ε, z) (23.58)

as long as z is in the set Ωρ0 := {z ∈ R | z ≤ E0 + 14 ρ0 }. Next, let S := {w ∈ C|Rew ≥ 0, |Imw| ≤ 13 Rew}. Theorem 23.6 above implies that for ε sufficiently small, the operators H (0) (ε, z) (more precisely, H (0) (ε, z) − hH (0) (ε, z)iΩ ) satisfy the assumptions of Theorem 24.1 of the next chapter, which implies, in particular, that • •

H (0) (ε, z) has
a simple eigenvalue λ(0) (ε, z) ∈ D(0, cε2 ); σ# H (0) (ε, z) ⊂ λ(0) (ε, z) + S.

With some extra work (in which the analyticity of λ(0) (ε, z) in z ∈ Ωρ0 plays an important role) we show that the equation λ(0) (ε, z) = 0 for z has a unique solution. Let E0 (ε) solve this equation.
By (23.58), this gives that E0 (ε) is a simple eigenvalue of Hε and σ# H(ε) ⊂ S. This together with (23.58) shows that E0 (ε) is the ground state energy of the operator Hε . Finally, if φ is the eigenfunction of H (0) (ε, E0 (ε)) corresponding to the eigenvalue 0, then QPρ (Hε − E0 (ε)1)φ, where QP (H − z1) is given by (23.38), is an eigenfunction of Hε corresponding
to the eigenvalue E0 (ε).
Finally, we use the relation σ# Hε = σ# H(ε) with a simple relation between the eigenfunctions to transfer the spectral information from Hε to H(ε). 

348

23 Theory of Radiation

23.10 Appendix: Estimates on Iε and HP¯ρ (ε) Recall that we consider the Hamiltonian Hε , given in (23.24), whose decomposition into unperturbed part and perturbation is given in (23.29) - (23.30). In this section we omit the subindex 1 in A1 (x), entering (23.24), so that the operator-family A(x) is given by XZ d3 k (23.59) A(x) = (χx,λ (k)aλ (k) + χx,λ (k)a∗λ (k)) p , |k| λ

with the coupling function χλ,x (k) satisfying the estimates (23.27) - (23.28). Using (23.29)-(23.30), we find 1 Iε = −p · A(x) + ε|A(x)|2 + G(x). 2 Proposition 23.9 kIε Pρ k ≤ C|ε| .

(23.60)

Proof. We write P , P¯ for Pρ , P¯ρ , respectively. Since p is bounded relative to Hpart , we have kP · pk ≤ C . Finally, since ρ ≤ c, the bound (23.32) implies that kP G(x)k, kP A(x)k ≤ C. Collecting the last four estimates and using the fact that Re z ≤ E0 + ρ, we arrive at (23.60).  In the present context, the operator HP¯ , introduced in (23.34), is given by HP¯ρ (ε) := H0ε + P¯ρ Iε P¯ρ .

(23.61)

Proposition 23.10 Assume |ε| ≪ ρ. Then Ωρ ⊂ ρ(HP¯ρ (ε)) and, for z ∈ Ωρ , the inverse of HP¯ρ (ε) − z satisfies the estimate 1

kP¯ρ (HP¯ρ (ε) − z)−1 k ≤ Cρ 2 .

(23.62)

Proof. To simplify notation we assume z is real. If we omit the subindexes ρ and ε and denote P = Pρ , P¯ = P¯ρ and I = Iε (e.g. IP¯ stands for P¯ρ Iε P¯ρ ). Recall the definition (23.39) (P = P0part ⊗ χρ (Hf )) and the definitions before and after this equation. We have ¯ρ (Hf ) . P¯ = P¯0part ⊗ 1f + P0part ⊗ χ

(23.63)

¯ρ (Hf ) ≥ Now using Hεpart P¯0part ≥ E1 P¯0part , Hεpart ≥ E0 and Hf ≥ 0, Hf χ ρχ ¯ρ (Hf ) , where recall E0 and E1 are the ground state energy and the first excited state of Hεpart , and using that H0ε = Hεpart ⊗ 1f + 1part ⊗ Hf , we estimate

23.10 Appendix: Estimates on Iε and HP¯ρ (ε)

349

P¯ H0ε P¯ ≥ (E1 + Hf )P¯0part ⊗ 1f 2 1 + (E0 + ρ + Hf )1part ⊗ χ ¯ρ (Hf ). 3 3 Setting δ := 3 min(E1 − E0 − 14 ρ, 31 ρ) ≥ ρ. Since for z ∈ Ωρ , z ≤ E0 + 14 ρ , we conclude that 1 P¯ (H0ε − z)P¯ ≥ (δ + Hf )P¯ 2 . (23.64) 3 Due to (23.64) and the fact that H0ε commutes with P¯ , we can define, for any α real α, the invertible, positive operator R0, := P¯ (H0ε −z)−α = (H0ε −z)−α P¯ , P¯ −α α 2 satisfying R0,P¯ R0,P¯ = P¯ so that the following identity holds: −1/2 −1/2 P¯ (HP¯ (ε) − z)P¯ = R0,P¯ [1 + K]R0,P¯ , 1/2

(23.65)

1/2

where K = R0,P¯ IR0,P¯ . Next we show that kKk ≤ const ε. Using the definition of I, we find 1/2

1/2

kKk ≤ εkpR0,P¯ kkA(x)R0,P¯ k

ε2 1/2 1/2 kR0,P¯ A(x)kkA(x)R0,P¯ k 2 1/2 1/2 + ε2 kR0,P¯ kkG(x)R0,P¯ k . +

(23.66)

The relative bound on A(x) proven above implies that 1/2 1/2 1/2 1/2
kA(x)R0,P¯ k ≤ c kHf R0,P¯ k + kR0,P¯ k .

(23.67)

1/2

Now let u = R0,P¯ v . Then the estimate (23.64) can be rewritten as 1/2

k(δ + Hf )1/2 R0,P¯ uk ≤

√ 3kP¯ uk .

This gives, in particular, that, for z ∈ Ω ∩ R, √ 1/2 1/2 kHf R0,P¯ k ≤ 3 and

1/2

kR0,P¯ k ≤

√

3δ −1/2 .

(23.68)

(23.69) 1/2

Next, using that kpvk2 = hv, p2 vi ≤ hv, (H0ε + C)vi and taking v := R0,P¯ u, we obtain 1/2 kpR0,P¯ k ≤ Cδ −1/2 . (23.70) Estimates (23.66) - (23.70) imply kKk ≤ Cερ−1 , if z ∈ Ωρ . Pick now |ε| ≪ δ so that kKk ≤ 1/2 and therefore the right hand side of (23.65) is invertible and satisfies (23.62). 

350

23 Theory of Radiation

23.11 Appendix: Key Bound Recall that E[0,ρ] (λ) denotes the characteristic function of the interval [0, ρ]. Proposition 23.11 (Key bound) Let Hrs be an rs-monomial of the form Z

r Y

j=1

r+s Y
a(ki )dr+s k , a∗ (kj ) hrs (Hf , k1 . . . kr+s )

(23.71)

i=r+1

where hrs (µ, k), k = (k1 . . . kr+s ), are measurable functions on [0, 1] × R3(r+s), r P ω(kj ) ≤ called coupling functions, and let Ωrs (ρ) := {k ∈ R3(r+s) | j=1

ρ,

r+s P

j=r+1

ω(kj ) ≤ ρ}. Then we have the following bound:

kE[0,ρ] (Hf )Hrs E[0,ρ] (Hf )k2 ≤ ρr+s

Z

Ωrs (ρ)

supµ |hrs (µ, k)|2 r+s d k r+s Q ω(kj )

(23.72)

j=1

Proof. In this proof we denote Eρ := E[0,ρ] (Hf ). Using the form (23.71) of Hrs , taking the norm under the integral sign, and using the norm inequality for the product of operators, as well as kA∗ k = kAk, we have Z Z kEρ Hrs Eρ k ≤ kar Eρ kkhrs kkas Eρ k . (23.73) Here khrs k is the operator norm of hrs (Hf , k). So khrs k = supµ |hrs (µ, k)|. Let f be a positive, continuous function on R3n . We will prove the estimate Z Z f 2
1/2 f kan Eρ k ≤ ρn , (23.74) n Q P ω j ωj ≤ρ 1

where ωj = ω(kj ), which will imply (23.72). First, we prove the estimate for n = 1. By the pull-through formula (22.52), we have a(k)Eρ = Eρ (Hf + ω(k))a(k)Eρ .

Since Hf ≥ 0, this implies ka(k)Eρ k ≤ χω(k)≤ρ ka(k)Eρ k. Proceeding as in the proof of the relative bound on a(k) (see (22.33)), we obtain

Z

f ka(k)Eρ k ≤ ≤

23.11 Appendix: Key Bound

Z

ω≤ρ

351

f ka(k)Eρ k

Z

f2 ω

ω≤ρ

 Z ≤ ρ

ω≤ρ

1/2

f2 ω

1/2

kHf Eρ k

(23.75)

1/2

which implies (23.74) for n = 1. Now we prove (23.74) for arbitrary n ≥ 1. First of all, applying the pull-through formula (22.52) n times, we find n Y

n X

a(kj )Eρ (Hf ) = Eρ (Hf +

1

ω(kj ))

1

Hence

Z

n

f ka Eρ k ≤

Z

Pn 1

ωj ≤ρ

n Y

a(kj )Eρ .

1

f kan Eρ k.

(23.76)

Secondly, applying the pull-through formula (22.52) n times again, we obtain n Y

a(kj )Eρ =

1

n Y

−1/2

a(kj )Hf

1/2

Hf Eρ

1

= (Hf +

n X

ω(kj ))−1/2

1

n Y

1/2

a(kj )Hf Eρ .

1

This formula and inequalities (23.75) and (23.76) give Z

n

f ka ψk ≤

Z

Z

P

ωj ≤ρ

f2 dkn ω

!1/2

1/2

kan−1 Hf Eρ kdn−1 k.

Proceeding in the same fashion we arrive at (23.74). R Applying the bound (23.74) with n = r to the integral kar Eρ kkhrs kdr k, we bound the r.h.s. of (23.73) by Z

s

ka Eρ k ρ

r

Z

Pr

1

ωj ≤ρ

khrs k2 r Qr d k 1 ωj

!1/2

ds k.

Applying bound (23.74) with n = s to the outer integral gives (23.72). 

24 Renormalization Group

In this chapter we investigate spectral properties of quantum Hamiltonians of the form (23.55) - (23.57). To this end we develop a spectral renormalization group method. It consists of the following steps (see Sections 24.2-24.5): • • •

Pass from a single operator H (= Hε ) to a Banach space, B, of Hamiltoniantype operators (of the form (23.55) - (23.57)); Construct a map, Rρ , on B, with the following properties: (a) Rρ is isospectral in the sense of Definition 23.3; (b) Rρ removes the photon degrees of freedom related to energies ≥ ρ. Relate the dynamics of semi-flow, Rnρ , n ≥ 1, to spectral properties of individual operators in B. The map Rρ is called the renormalization map. It is of the form Rρ = ρ−1 Sρ ◦ Fρ

where Fρ is the smooth Feshbach - Schur (decimation) map, and Sρ is a simple rescaling map (see Sections 23.9 and 24.4). Fρ maps operators which act non-trivially only on the subspace RanE[0,1] (Hf ) consisting of states with photon energies ≤ 1, to operators which act non-trivially on states with photon energies ≤ ρ. The rescaling Sρ brings us back to the subspace RanE[0,1] (Hf ). By design, the renormalization map Rρ is isospectral in the sense that the operators K and Rρ (K) have the same spectrum near 0, modulo rescaling. The renormalization map gives rise to an isospectral (semi-)flow Rnρ , n ≥ 1, (called renormalization group (RG)) on the Banach space B. We will see that orbits of this flow, with appropriate initial conditions approach the operators of the form ωHf (for some ω ∈ C) as n → ∞. In fact, CHf is a line of fixed points of the flow (Rρ (ωHf ) = ωHf ). By studying the behaviour of the flow near this line of fixed points, we can relate the spectrum of an initial condition, H, near 0 to that of Hf , which we know well. This is the basic idea behind the proof of Theorem 23.1.

© Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_24

354

24 Renormalization Group

24.1 Main Result In this section we formulate the main result of this chapter. It concerns the spectra of quantum Hamiltonians of the form (23.55) - (23.57), X H = H00 + χ1 Hrs χ1 , H00 := h00 (Hf ), (24.1) r+s≥1

Hrs :=

Z Y r

a∗ (kj ) hrs (Hf , k1 . . . kr+s )

r+s Y

a(ki )dr+s k,

(24.2)

i=r+1

j=1

acting on the space Hred := E[0,1] (Hf )F , where recall E[0,ρ] (λ) denotes the characteristic function of the interval [0, ρ]. Let B r denote the unit ball in R3r and I := [0, 1]. Above, χρ := χρ (Hf ) are the operators defined in Section 23.6 (the shorthand we use from now on) and the coupling functions, hrs : I × B r+s → C, satisfy the estimates |∂µn hrs (µ, k)| ≤ γ0

r+s X i=1

1

ω(ki ) 2

r+s Y j=1

1
ξ0 ω(kj )− 2 ,

(24.3)

for any 1 ≤ i ≤ r + s, µ ∈ [0, 1], ξ0 > 0, and n = 0, 1, where the product is absent in the case r = s = 0, and |h00 (0)| ≤ α0 ,

|∂µ h00 (µ) − 1| ≤ β0 ,

(24.4)

for µ ∈ [0, ∞). Here, recall, k = (k1 , . . . , kr+s ). Note that, in order to be able to apply our theory to the analysis of resonances of Hε , the space of operators H should include non-selfadjoint. Let D(0, r) stand for the disc in C of the radius r and centered at 0. We denote by Ds the set of operators of the form (24.1) - (24.4) with h00 (0) = 0. We define a subset S of the complex plane by S := {w ∈ C| Rew ≥ 0, |Imw| ≤

1 Rew}. 3

(24.5)

Recall that a complex function f on an open set A in a complex Banach space W is said to be analytic if ∀H ∈ A and ∀ξ ∈ W, f (H + τ ξ) is analytic in the complex variable τ for |τ | sufficiently small (see [40]). We are now prepared to state the main result of this chapter. Theorem 24.1 Assume that β0 and γ0 are sufficiently small. Then there is an analytic map e : Ds → D(0, cγ02 ) such that for H ∈ Ds the number e(H) is a simple eigenvalue of the operator H and σ(H) ⊂ e(H) + S. Moreover, e(H) ∈ R, for H = H ∗ . Note that our approach also provides an effective way to compute the eigenvalue e(H) and the corresponding eigenvector.

24.2 A Banach Space of Operators

355

Operators on the Fock space of the form (24.1) - (24.4) will be said to be in generalized normal (or Wick) form. Operators of the form (24.2) will be called (rs)-monomials. Though Hf can be expressed in the standard Wick R form, Hf = ωa∗ a , the corresponding coupling function, ω(k1 )δ(k1 − k2 ), is more singular than we allow. But even if this coupling function were smooth, finding the Wick form of operators like χρ (Hf ), or the hrs , is not an easy matter. In what follows we manipulate the operators a∗ (k), a(k) and Hf as if they were independent, using only the commutation relations [a(k), Hf ] = ω(k)a(k) , etc.

24.2 A Banach Space of Operators In this section we define the Banach space of operators on which the renormalization group acts. We consider formal expressions of the form (24.1) (24.4) acting on RanE[0,1] (Hf ). For ξ, ν > 0, we define the Banach space, Bξν , consisting of formal expressions, (24.1) - (24.2)acting on the space Hred := E[0,1] (Hf )F and satisfying kHkξν < ∞, where X kHkξν := ξ −(r+s) kHrs kν (24.6) r+s≥0

with kHrs kν := khrs kν and khrs kν := (n)

1 X

n=0

(r+s)

max sup[Ωj j

(n)

where Ωj = 1 for n = 0, and Ωj Similarly to (23.51) we obtain

µ,k

|∂µn hrs |] < ∞ ,

= ω(kj )−ν

Qn

i=1

(24.7)

p ω(ki ) for r + s ≥ 1.

Proposition 24.2 Let Hrs be as above. Then for any ν ≥ 0, √ (24.8) kχρ Hrs χρ k ≤ (r!s!)−1/2 ( 4πρ)r+s ρν kHrs kν . P Thus r+s≥1 χρ Hrs χρ converges in norm. PR ∗ (a )hrs as is oneNext we state without proof that the map {hrs } 7→ to-one (see [27, Thm. III.3]). Here the {hrs } satisfy (24.7). Hence the normed space Bξν is indeed a Banach space. For the future analysis it is important to know how different operators behave under rescaling, more precisely, under the action of the rescaling map ρ−1 Sρ , defined in (23.52), as ρ → 0 (or of (ρ−1 Sρ )n as n → ∞, which in the present case is the same). We have for r + s ≥ 1

356

24 Renormalization Group

ρ

−1

Sρ (Hrs ) =

Z

s (a∗ )r h(ρ) rs a ,

(24.9)

where 3/2(r+s)−1 h(ρ) hrs (ρHf , ρk) . (24.10) rs (Hf , k) = ρ Pr+s Q 1 r+s If Hrs ∈ Bνξ , then hrs behaves for small |kj |’s like i=1 ω(ki )ν j=1 ω(kj )− 2 . Since by (23.53), Sρ (a# (k)) = ρ−3/2 a# (ρ−1 k), we have for r + s ≥ 1

ρ−1 Sρ (Hrs ) ∼ ρr+s−1+ν Hrs .

(24.11)

In the r + s = 0 case, we have to specify the behaviour of the function h00 (µ) at µ = 0. In our case, h00 (µ) − h00 (0) ∼ µ and therefore ρ−1 Sρ (H00 − hH00 iΩ ) ∼ H00 − hH00 iΩ ),

ρ−1 Sρ (hH00 iΩ ) = ρ−1 hH00 iΩ .

(24.12)

where we used the notation hAiΩ := hΩ, AΩi. Hence, Hrs , r +s ≥ 1, contract, H00 −hH00 iΩ are roughly invariant, while hH00 iΩ expand under our rescaling. This suggests to decompose the Banach space Bξν into the direct sum Bξν = C · 1 + T + W ,

(24.13)

of the subspaces which expand, are roughly invariant or contract under our rescaling: T = {T (Hf ) | T : [0, ∞) → C is C 1 with T (0) = 0} , and W := {

X

r+s≥1

χ1 Hrs χ1 ∈ Bξν }.

(24.14) (24.15)

This decomposition will plat an important role below. Remark 24.3 The subspace T can be further decomposed into the invariant substace C · Hf and and contracting one, Ts = {T (Hf ) | T : [0, ∞) → C is C 1 with T (0) = T ′ (0) = 0} .

(24.16)

The decomposition of the Banach space Bξν into the subspaces C · 1, C · Hf and Ts + W is related to the spectral decomposition of the map ρ−1 Sρ .

24.3 The Decimation Map In this section we construct a map which projects out the degrees of freedom corresponding to high photon energies. We use the smooth Feshbach-Schur map, which we define here in a greater generality than in Section 23.6, and formulate its important isospectral property. Let χ, χ be a partition of unity

24.3 The Decimation Map

357

on a separable Hilbert space H, i.e. χ and χ are operators on H whose norms are bounded by one, 0 ≤ χ, χ ≤ 1, and χ2 + χ2 = 1. We assume that χ and χ are nonzero. Let τ be a (linear) projection acting on closed operators on H with the property that operators in its image commute with χ and χ. We also assume that τ (1) = 1. Let τ := 1 − τ and define Hτ,χ# := τ (H) + χ# τ (H)χ# ,

(24.17)

where χ# stands for either χ or χ. Given χ and τ as above, we denote by Dτ,χ the space of closed operators, H, on H which belong to the domain of τ and satisfy the following three conditions: D(τ (H)) = D(H) and χD(H) ⊂ D(H),

(24.18)

Hτ,χ is (bounded) invertible on Ran χ,

(24.19)

τ (H)χ and χτ (H) extend to bounded operators on H.

(24.20)

(For more general conditions see [27, 139].) The smooth Feshbach-Schur map (SFM) maps operators on H belonging to Dτ,χ to operators on H by H 7→ Fτ,χ (H), where −1 χW χ. Fτ,χ (H) := H0 + χW χ − χW χHτ,χ

(24.21)

Here H0 := τ (H) and W := τ (H). Note that H0 and W are closed operators on H with coinciding domains, D(H0 ) = D(W ) = D(H), and H = H0 + W . We remark that the domains of χW χ, χW χ, Hτ,χ , and Hτ,χ all contain D(H). Define operator −1 χW χ. Qτ,χ (H) := χ − χ Hτ,χ

(24.22)

The following result ([27]) generalizes Theorem 11.1 of Section 11.4. Theorem 24.4 (Isospectrality of SFM) Let 0 ≤ χ ≤ 1 and H ∈ Dτ,χ be an operator on a separable Hilbert space H. Then we have the following results: (i) Hψ = 0 =⇒ Fτ,χ (H) ϕ = 0, ϕ := χψ ∈ Ran χ. (ii) Fτ,χ (H) ϕ = 0 =⇒ Hψ = 0, ψ := Qτ,χ (H)ϕ ∈ H. (iii) dim NullH = dim NullFτ,χ (H). (iv) H is bounded invertible on H if and only if Fτ,χ (H) is bounded invertible on Ran χ. In this case H −1 = Qτ,χ (H) Fτ,χ (H)−1 Qτ,χ (H)# + χ Hχ−1 χ, −1

Fτ,χ (H)

= χH

−1

−1

χ + χ τ (H)

χ.

(24.23) (24.24)

358

24 Renormalization Group

We also mention the following useful property of Fτ,χ : H is self-adjoint

⇒

Fτ,χ (H) is self-adjoint.

(24.25)

The proof of this theorem is similar to the one of Theorem 11.1 of Section 11.4. We demonstrate only the proof of the statement (ii) which we use extensively below and refer for the rest of the proof to [27]. The statement (ii) follows from the relation H Qτ,χ (H) = χ Fτ,χ (H). (24.26) Now we prove the latter relation, using the shorthand Q ≡ Qτ,χ (H), Hχ ≡ H0 + χ W χ, Hχ ≡ H0 + χ W χ: H Q = Hχ − Hχ Hχ−1 χW χ = χHχ
+ χ2 W χ − χHχ + χ2 W χ Hχ−1 χW χ . (24.27)
Since χHχ Hχ−1 χW χ = χH0 + χ2 W χ Hχ−1 χW χ = χ2 W χ, the r.h.s. gives χ Fτ,χ (H). The Feshbach-Schur map is a special case of the smooth Feshbach-Schur 2 map and is obtained from the latter when χ is a projection, χ
= χ, by taking −1 τ = 0. Then Fτ,χ becomes Fχ (H) := χ H − HχHχ χH χ, Hχ := H0χ = χHχ, as defined in Section 11.1. P Hrs considered in Sections 24.1For Hamiltonians of the form H = r+s≥0

24.2, we define the decimation map as

Fρ ≡ Fτ,χ ,

(24.28)

where Fτ,χ is the smooth Feshbach-Schur map and the operators τ and χ are chosen as τ (H) = H00 := h00 (Hf ) and χ := χρ = χρ (Hf ) ≡ χρ−1 Hf ≤1 ,

(24.29)

where the cut-off function χλ≤1 is defined in SectionP 23.6. To isolate a set on χ1 Hrs χ1 ∈ Bξν as which the map Rρ is defined we write H = H00 + r+s≥1

H = E + T + χ1 W χ1 ,

where E := hΩ, HΩi = h00 (0), T := H00 − E and W := - (24.15)) and define the following polydisc in Bξν : n Dξν (α, β, γ) := H ∈ Bξν |h0,0 [0]| ≤ α,

(24.30)

Hrs (cf. (24.13)

r+s≥1

sup |h′0,0 [r] − 1| ≤ β, kW kξν

r∈[0,∞)

P

(24.31) o ≤γ .

The following lemma shows that the domain of the decimation map Fρ contains Dξν (α, β, γ), for appropriate numbers α, β, γ > 0.

24.4 The Renormalization Map

359

Lemma 24.5 Fix 0 < ρ < 1, ν ≤ 1/2, and 0 < ξ ≤ (4π)−1/2 . Then it follows that the polydisc Dξν (ρ/8, 1/8, ρ/8) is in the domain of the decimation map Fρ . Proof. Let H ∈ Dξν (ρ/8, 1/8, ρ/8). We observe that χ1 W χ1 := H − h0,0 defines a bounded operator on F , and we only need to check the invertibility of Hτ χρ on Ran χρ . Now the operator h0,0 is invertible on Ran χρ since for all r ∈ [3ρ/4, ∞) Re h0,0 [r] ≥ r − |h0,0 [r] − r|


≥ r 1 − sup |h′0,0 [r] − 1| − |h0,0 [0]| r

3ρ ρ ρ ≥ (1 − 1/8) − ≥ . (24.32) 4 8 2

Furthermore, by (24.8), χ1 W χ1 k ≤ kW kξν ≤ γ = ρ/8. Hence Re(h0,0 + χ1 W χ1 ) ≥ ρ3 on Ran χρ , i.e. Hτ,χρ is invertible on Ran χρ . 

Note that the decimation map, Fρ maps isospectrally operators which act nontrivially on Ran χ1 into those which nontrivially on Ran χρ .

24.4 The Renormalization Map Using that the subspace RanE[0,1] (Hf ) is invariant under the composition map Sρ ◦ Fρ , we define the renormalization map as a composition of a decimation map and a rescaling map on the domain of the decimation map Fρ (see (24.28)) as Rρ := ρ−1 Sρ ◦ Fρ |RanE[0,1] (Hf ) (24.33) where Sρ is the rescaling map defined in (23.52). By Lemma 24.5, its domain contains the polydiscs Dξν (α, β, γ), with α, β, γ ≤ ρ8 . Note that the map Sρ ◦Fρ acts on the orthogonal complement, RanE[0,1] (Hf )⊥ , trivially: ρ−1 Sρ ◦ Fρ = ρ−1 Sρ ◦ τ

on

RanE[0,1] (Hf )⊥

(24.34)

where the map τ is defined in (24.29). Note that while the standard FeshbachSchur maps have the semigroup property (see Supplement 24.6): Rρ2 ◦ Rρ1 = Rρ2 ρ1 , the smooth ones have this property only under certain conditions on ρ1 and ρ2 . Next, we list some elementary properties of the map Rρ which follow readily from the definitions 1. Rρ is isospectral in the sense that ρ · Rρ (H) and H are isospectral at 0 in the sense of Theorem 24.4, 2. Rρ (wHf + z1) = wHf + 1ρ z1 ∀w, z ∈ C.

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24 Renormalization Group

In particular, CHf is a complex line of fixed points of Rρ : Rρ (wHf ) = wHf ∀w ∈ C, and C · 1 is (a part of) the unstable manifold. The first property follows from the relations (24.34). Problem 24.6 Prove the statements above. Describing the range of the renormalization map Rρ on polydiscs Dξν (α, β, γ) is considerably harder. The following result, proven in [121], shows that contraction is actually a key property of along ’stable’ directions. Theorem 24.7 Let ǫ0 : H → hΩ, HΩi, and µ := ν > 0. Then, for any σ ≥ 1, 0 < ρ < 1/2, α, β ≤ ρ8 , and γ ≪ 1, we have that Rρ − ρ−1 ǫ0 : Dξν (α, β, γ) → Dξν (rρ (α, β, γ)). continuously, with ξ :=

1 4

(24.35)

and, for an absolute constant c,

rρ (α, β, γ) := (cγ 2 /ρ, β + cγ 2 /ρ, cρν γ).

(24.36)

Moreover, Rρ (H) and H are isospectral (modulo rescaling) at 0. Remark 24.8 Subtracting the term ρ−1 ǫ0 from Rρ allows us to control the expanding direction during the iteration of the map Rρ . In [27], such a control was achieved by changing the spectral parameter λ, which controls hΩ, HΩi. The proof of this theorem is similar to the proof of Theorem 23.6 which we outlined above: expand Rρ (H) in the perturbation W := H − H00 , reduce the resulting series to generalized normal form, and estimate the obtained coupling functions. To bring the operators we deal with into generalized normal form, we use the “pull-through” formulae (22.52) and (22.53). To explain the result above, we, as usual in the study of nonlinear dynamics, consider the linearization (variational derivative) of ∂Rρ (w · Hf ) in order to understand the dynamics of the map Rρ near its near its fixed points w · Hf , w ∈ C. The variational derivative of the map Rρ at a point H0 is defined as ∂ Rρ (H0 + sξ) s=0 ∂Rρ (H0 )ξ := ∂s for any ξ ∈ Bξν . Thus ∂Rρ (H0 ) is a linear operator on Bξν . By the definition of the map Fρ we have the relation Fρ (w · Hf + sξ) = w · Hf + sτ (ξ) + O(s2 ), which in turn implies (informally) that ∂Dρ (w·Hf )ξ = τ (ξ)+χρ τ¯(ξ)χρ . Using this and the relation Sρ (χρ ) = χ1 , we obtain ∂Rρ (w · Hf )ξ = ρ−1 Sρ (ξ) . (24.37) R Scaling properties of Hrs = (a∗ )r hrs as are given in (24.11) - (24.12). They suggest that for ν > 0, the termsHrs , r + s ≥ 1 contract, the terms H00 − hH00 iΩ are essentially invariant, and the terms hH00 iΩ expand. In the physics terminology and for ν > 0,

24.5 Dynamics of RG and Spectra of Hamiltonians

r+s≥1

H00 − hH00 iΩ hH00 iΩ

↔

↔ ↔

361

irrelevant terms, marginal terms, relevant terms.

where we used the notation hAiΩ := hΩ, AΩi. The following equation follows from Eqs (24.9) - (24.10) : kρ−1 Sρ (Hrs )kξν = ρ(1+ν)(r+s)−1 kHrs kξν = ρ−1 kHrs kρ−1−ν ξ,ν . (24.38) P Hrs , we find Applying these equalities to operators of the form W = r+s≥1

kρ−1 Sρ (W )kξν ≤ kW kρ−ν ξ,ν ≤ ρν kW kξν .

(24.39)

−1 (Recall that ν ≥ 0.) Let Rlin Sρ and ρ := ∂Rρ (w · Hf ) = ρ

rρlin (α, β, γ) := (0, β, ρν γ).

(24.40)

The estimate (24.39), together with the relations ρ−1 Sρ (E1) = ρ−1 1 and ρ−1 Sρ (wHf ) = wHf , implies that −1 Rlin ǫ0 : Dξν (α, β, γ) → Dξν (rρlin (α, β, γ)). ρ −ρ

(24.41)

This is the linearization of the estimate (24.35). Theorem 24.7 deals essentially with controlling the nonlinear part of the map Rρ . In the next section we address the dynamics of Rnρ as n → ∞ in a vicinity of the fixed point manifold Mf p ⊇ CHf , and connect this dynamics with spectral properties of operators of interest.

24.5 Dynamics of RG and Spectra of Hamiltonians To describe the dynamics of Rkρ we need some definitions. Consider an initial set of operators to be D := Dξν (α0 , β0 , γ0 ), with α0 , β0 , γ0 ≪ 1. We let Ds := Dξν (0, β0 , γ0 ) (the subindex s stands for ’stable’). We fix the scale ρ so that 1 α0 , β0 , γ0 ≪ ρ ≤ . (24.42) 2 Below, we will use the n−th iteration of the numbers α0 , β0 , γ0 under the map (24.36): ∀n ≥ 1 (αn , βn , γn ) := rρn (α0 , β0 , γ0 ). For H ∈ D we denote Hu := hHiΩ and Hs := H − hHiΩ 1 (the unstableand stable-central-space components of H, respectively). Note that Hs ∈ Ds . It is shown in [121] that the following objects are defined inductively in n ≥ 1 (with e0 (Hs ) = 0 ∀Hs ∈ Ds )

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24 Renormalization Group

Vn := {H ∈ D| |Hu − en−1 (Hs )| ≤

1 n+1 ρ }; 12


En (H) := Rnρ (H) u , H ∈ Vn ;

en (Hs ) is the unique zero of the function λ → En (Hs − λ1) 1 in the disc D(en−1 (Hs ), ρn+1 ), 12 and have the following properties:

(24.43) (24.44) (24.45)

Vn ⊂ Vn−1 ⊂ D(Rnρ );

(24.46)

Rnρ (Vn ) ⊂ Dµ,ξ (ρ/8, βn , γn );

(24.47)

4 ∂λ En (λ) ≤ − ρ−n ; 5 |en (Hs ) − en−1 (Hs )| ≤ 2αn ρn ; en (Hs ) ∈ R,

if

∗

H=H .

(24.48) (24.49) (24.50)

Proof (Proof of Theorem 24.1). Now we prove the first statement of Theorem 24.1. By (24.49), the limit e(Hs ) := limj→∞ ej (Hs ) exists pointwise for H ∈ D. Iterating Eqn (24.49) and using that (cρµ )2 ρ ≤ 1/2, we find the estimate |en (Hs ) − e(Hs )| ≤ 2αn+1 ρn+1 , n ≥ 0.

(24.51)

Since e0 (Hs ) = 0, for n = 0 this estimate gives |e(Hs )| ≤ 2α1 ρ = cγ02 , which shows that e : Ds → D(0, 2α1 ρ). Moreover, (24.50) shows that e(Hs ) ∈ R, if H = H ∗ . We skip the proof of analyticity of e(Hs ) in Hs . Next, we prove that e(Hs ) is a simple eigenvalue. T We omit the reference (0) to H and set e ≡ e(H ). Let H := H − e1 ∈ s n Vn . (24.46) implies that T s k V ⊂ D(R ), ∀k. Hence we can define a sequence of operators (H (n) )∞ ρ n=0 n n
in Bµ,s ⊆ B(Hred ) by H (n) := Rnρ H (0) . Recall the definitions of Sρ and Uθ in (23.52) and (23.8) and let Γρ := Uθ , ρ = e−θ . Then the definition (24.33) of Rρ implies that, for all integers n ≥ 0, H (n) =


 ∗ 1 Γρ , Γρ Fρ H (n−1) ρ

(24.52)

where, recall, Fρ is defined in (24.28). We will use the operators Qτ χ defined in (24.22), and the identity (24.26) (HQτ χ = χFτ χ (H)) which they satisfy. Let
(24.53) Q(n) := Qτ χ H (n) , with τ and χ = χρ given in (24.29). Then the equation H (n) Q(n) = χρ Fρ (H (n) ) together with (24.52), implies the intertwining property

24.5 Dynamics of RG and Spectra of Hamiltonians

H (n−1) Q(n−1) Γρ∗ = ρ Γρ∗ χ1 H (n) .

363

(24.54)

Eq. (24.54) is the key identity for the proof of the existence of an eigenvector with the eigenvalue e. For the construction of this eigenvector, we define, for non-negative integers β, vectors Ψk in H by setting Ψ0 := Ω and Ψk := Q(0) Γρ∗ Q(1) Γρ∗ · · · Q(k−1) Ω .

(24.55)

We first show that this sequence is convergent, as k → ∞. To this end, we observe that Ω = Γρ∗ χρ Ω and hence
Ψk+1 − Ψk = Q(0) Γρ∗ Q(1) Γρ∗ · · · Q(k−1) Γρ∗ Q(k) − χρ Ω . (24.56)

Since kχρ k ≤ 1, this implies that

Yn



β−1

o

Ψk+1 − Ψk ≤ Q(k) − χρ 1 + Q(j) − χρ .

(24.57)

j=0

To estimate the terms on the r.h.s. we consider the j-th step Hamiltonian H (j) . By (24.30) and (24.47), we can write H (j) as H (j) = Ej · 1 + Tj + Wj , with |Ej | ≤

(24.58)

ρ ρ and kWj k ≤ γj ≤ . 8 16

(24.59)

Recalling the definition (24.53) of Q(j) , we have χρ − Q(j) = χρ Ej + Tj + χρ Wj χρ


−1

χρ Wj χρ .

(24.60)

By (24.59), for all j ∈ N, we may estimate ρ −1 16 γj kχρ − Q(j) k ≤ − kWj k . (24.61) kWj k ≤ 8 ρ P∞ Q∞ Inserting this estimate into (24.57) and using that j=0 (1 + λj ) ≤ e j=0 λj , for λj ≥ 0, we obtain k−1 Yn

16 γj o

Ψk+1 − Ψk ≤ 16 γk 1+ ρ j=0 ρ

≤

  16 γk exp 32 γ0 ρ−1 , ρ

(24.62)

P∞ where we have used that j=0 γj ≤ 2γ0 (recall the definition of γj after Eqn P∞ (24.42)). Since j=0 γj < ∞, we see that the sequence (Ψk )k∈N0 of vectors in H is convergent, and its limit Ψ∞ := limk→∞ Ψk , satisfies the estimate

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24 Renormalization Group



 

Ψ∞ − Ω = Ψ∞ − Ψ0 ≤ 32 γ0 exp 32 γ0 ρ−1 , ρ

(24.63)

which guarantee that Ψ(∞) 6= 0. The vector Ψ∞ constructed above is an element of the kernel of H (0) , as we will now demonstrate. Observe that, thanks to (24.54),

H (0) Ψk = H (0) Q(0) Γρ∗ Q(1) Γρ∗ · · · Q(k−1) Ω

= ρ Γρ∗ χ1 H (1) Q(1) Γρ∗ Q(2) Γρ∗ · · · Q(k−1) Ω .. .
k = ρk Γρ∗ χ1 H (k) Ω . (24.64)

Eq (24.58) together with the estimate (24.59) and the relation Tk Ω = 0 implies that

(k)

H Ω = (Wk + Ek ) Ω (24.65) ≤ γk + 8α2k ≤ 2γk .

Summarizing (24.64)–(24.65) and using that the operator norm of Γρ∗ χ1 is bounded by 1, we arrive at

(0)

H Ψk ≤ 2γk → 0 (24.66) as k → ∞. Since H (0) ∈ B(H) is continuous, (24.66) implies that
H (0) Ψ∞ = lim H (0) Ψk = 0. k→∞

(24.67)

Thus 0 is an eigenvalue of the operator H (0) := H − e1, i.e. e is an eigenvalue of the operator H, with the eigenfunction Ψ∞ . Finally, we prove the second statement of Theorem 24.1. To simplify exposition we restrict ourself to self-adjoint operators. We omit the reference to Hs and set e ≡ e(Hs ) and en ≡ en (Hs ). Let H (n) (λ) := Rnρ (Hs − λ) and, recall, En (λ) := H (n) (λ)u . Using the equation En (en ) = 0, the mean value theorem and the estimate (24.48), we obtain that En (λ) ≥ − 54 ρ−n (λ − en ), provided λ ≤ en . Hence, if λ ≤ en − θn , with θn ≫ γn ρn and θn → 0 as n → ∞, then H (n) (λ) ≥ 45 ρ−n θn − O(γn ) ≥ 12 γn . This implies 0 ∈ ρ(H (n) (λ)) and therefore, by Theorem 24.4, 0 ∈ ρ(Hs − λ) or λ ∈ ρ(Hs ). Since en → e and θn → 0 as n → ∞, this implies that σ(Hs ) ⊂ [e, ∞), which is the second statement of the theorem for self-adjoint operators.  We discuss the geometrical meaning of the results obtained above. Let H ∈ Vn ⊂ D(Rn+1 ). According to (24.30), H (n) := Rnρ (H) can be written as ρ

24.6 Supplement: Group Property of Rρ

H (n) = En 1 + Tn + Wn ,

365

(24.68)

where Tn ≡ Tn (Hf ) with Tn (r) ∈ C 1 and Tn (0) = 0. By (24.47) we have s ≤ γn . Hence the function τn (r) := Tn (r)/r = |∂r Tn (r)−1| ≤ βn and kWn kWop R1 ′ satisfies |τn (r)−1| ≤ βn . One can also show that 0 T (sr)ds is continuous andT τn → τ = (constant) for H ∈ n Vn . By the definition of Vn , |En −en−1 (Hs )| ≤ 1 n+1 . Hence 12 ρ Rnρ (H) → e(Hs ) + τ Hf in the norm of Bξν ,

(24.69) T

where e(Hs ) := limj→∞ ej (Hs ), as n → ∞. In other words Ms := n Vn is the (local) stable manifold for the invariant manifold Mf p := CHf of fixed points. (Formally, a local stable manifold can be defined as a manifold invariant under Rρ and such that Ms = {H ∈ Bξν Rnρ (H) → Mf p as n → ∞}.) By the definition of Vn , it is the graph of the map e : Ds → Vu : Ms = {H ∈ D

| hHiΩ = e(Hs )}.

(24.70)

One can show that Ms is invariant under Rρ . Consider the invariant manifolds Mu , Mf p and Ms . Since Ms is of the codimension one and contains the manifold CHf of fixed points, while Mu := C1 is invariant and expanding under Rρ , we see CHf =: Mf p and Mu := C1 are fixed point and unstable manifolds, respectively. The subspaces Vu := C · 1, Vc := C · Hf and Vc + Vs := T + W spanning the Banach space Bξν (see Section 24.2) are tangent spaces to the manifolds Mu , Mf p and Ms at Mf p . C .1 Mu = C . Hf +C . 1 Mfp= C . Hf

Ms . H−z1

Fig. 24.1. RG flow.

24.6 Supplement: Group Property of Rρ The Feshbach-Schur maps has the semi-group property: Proposition 24.9 (semigroup property) Assume the projections P1 and P2 commute. Then FP1 ◦ FP2 = FP1 P2 Proof. Assume for simplicity that H and FP2 (H) are invertible. Then the statement follows by applying equation (11.36) twice. 

25 Mathematical Supplement: Elements of Operator Theory

We have seen already in the first chapter that the space of quantummechanical states of a system is a vector space with an inner-product (in fact a Hilbert space). We saw also that an operator (a Schr¨odinger operator) on this space enters the basic equation (the Schr¨odinger equation) governing the evolution of states. In fact, the theory of operators on a Hilbert space provides the basic mathematical framework of quantum mechanics. This chapter describes some aspects of operator theory and spectral theory that are essential to a study of quantum mechanics. To make this chapter more self-contained, we repeat some of the definitions and statements from the chapters in the main text.

25.1 Spaces In this section we review briefly some background material related to linear (or vector) spaces. We introduce the simplest and most commonly used spaces, Banach and Hilbert spaces, and describe the most important examples. We begin with the basic definitions. Vector spaces. A vector space V is a collection of elements (here denoted u, v, w, ...) for which the operations of addition, (u, v) → u + v and multiplication by a (real or complex) number, (α, u) → αu, are defined in such a way that u + v = v + u (commutativity) u + (v + w) = (u + v) + w (associativity), u + 0 = 0 + u = u (existence of zero vector), α(βu) = (αβ)u, (α + β)u = αu + βu, α(u + v) = αu + αv, © Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_25

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25 Mathematical Supplement: Elements of Operator Theory

0v = 0,

1v = v.

We also denote −v := (−1)v. Elements of a vector space are called vectors. Here are some examples of vector spaces: (a) Rn = {x = (x1 , ..., xn )| − ∞ < xj < ∞ ∀ j}– the Euclidean space of dimension n; (b) C(Ω) – the space of continuous real (or complex) functions on Ω, where Ω is Rn , or a subset of Rn ; (b) C k (Ω) – the space of k times continuously differentiable real (or complex) functions on Ω, k = 1, 2, 3, . . . . The addition and multiplication by real/complex numbers in these spaces is defined in the pointwise way: (x + y)j = xj + yj and (αx)j = αxj ∀ j and (f + g)(x) := f (x) + g(x) and (αf )(x) := αf (x) ∀x ∈ Ω. Problem 25.1 Show that Rn , C(Ω) and C k (Ω) are vector spaces. Norms. To measure the size of vectors, one uses the notion of norm. A norm on a vector space V is defined to be a map, V ∋ u → ||u|| ∈ [0, ∞), which has the following properties: (a) kuk = 0 ⇐⇒ u = 0; (b) kαuk = |α|kuk; (c) ku + vk ≤ kuk + kvk. The last inequality is called the triangle inequality. We give some examples of norms: P (a) kxk := |x| = ( i x2i )1/2 in Rn ; (b) kf k∞ = supx∈Ω |f (x)| in C(Ω) (also denoted kf kC ); dj k (c) kf kC k = max0≤j≤k supx∈R | dx j f (x)| in C (R); R p 1/p in C(Ω) (d) kf kp := ( Ω |f (x)| dx)

For examples (b) and (d), clearly, ||f ||p = 0 ⇔ f = 0, ||αf ||p = |α| ||f ||p , ∀α ∈ C, and for p = 1, ∞, ||f + g||p ≤ ||f ||p + ||g||p . We will prove the triangle inequality for 1 < p < ∞ later.

A vector space equipped with a norm is called a normed vector space. Here are some examples of normed vector spaces (a) Rn with the norm kxk = |x|; (b) The subspace Cb (Ω) of C(Ω) consisting of all continuous, bounded functions on Ω ⊂ Rn , equipped with the norm kf k∞ ;

25.1 Spaces

369

(c) The space C k (Ω), for Ω = [0, 1], equipped with the norm kf kC k or the norm kf kp . Problem 25.2 Show that Cb (Ω) is a normed vector space. Banach spaces. A normed vector space is said to be complete, if Cauchy sequences in it converge; that is, if uj ∈ V , j = 1, 2, . . . is such that limj,k→∞ kuj − uk k = 0, then there exists u ∈ V such that limj→∞ kuj − uk = 0. A normed vector space which is also complete is called a Banach space. Examples of Banach spaces include (a) Rn with the norm kxk = |x|; (b) Cb (Ω) with the norm kf k∞ ; (c) C k (Ω), for Ω = [0, 1] with the norm kf kC k ; (d) For 1 ≤ p < ∞, the Lp -space Z p L (Ω) := { f : Ω → C | |f (x)|p dx < ∞ } Ω

with the norm kf kp . Dual spaces. Next we define the important notion of dual space. 1. A bounded linear functional on a vector space V , with a norm k · k, is a map l : V → C (or → R if V is a real, rather than a complex vector space) such that l(αξ + βη) = αl(ξ) + βl(η) for all ξ, η ∈ V , and α, β ∈ C (or R), and there is C < ∞ such that |l(ξ)| ≤ Ckξk for all ξ ∈ V . 2. The dual space of a normed vector space V , is the space V ∗ of all bounded linear functionals on V . Note that on a finite dimensional space all linear functionals are bounded. The dual space V ∗ is also a normed vector space, under the norm klkV ∗ :=

sup ξ∈V,kξkV =1

|l(ξ)|.

If V is a Banach space, then so is V ∗ , under this norm. One often denotes the action of l ∈ V ∗ on ξ ∈ V by hl, ξi := l(ξ). Inner products and Hilbert spaces. Now let H be a (complex) vector space. We assume H is endowed with an inner product, h·, ·i. This means the map h , i:H×H→C satisfies the properties

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25 Mathematical Supplement: Elements of Operator Theory

1. linearity (in the second argument): hv, αw + βzi = αhv, wi + βhv, zi 2. conjugate symmetry: hw, vi = hv, wi 3. positive definiteness: hv, vi > 0 for v 6= 0

for any v, w, z ∈ H and α, β ∈ C. It follows that the map k · k : H → [0, ∞) given by kvk := hv, vi1/2

is a norm on H. If H is also complete in this norm – that is, if it is a Banach space – then H is called a Hilbert space. Our main example of a Hilbert space is the space of square-integrable functions, the L2 -space (the state space of a a quantum system): Z |ψ|2 < ∞} L2 (Rd ) := {ψ : Rd → C | Rd

with the inner-product hψ, φi :=

Z

¯ ψφ. Rd

R R Here R and throughout, we will often use the simplified notations f or Rd f for Rd f (x)dx. Another important example of a Hilbert space is the Sobolev space of order n, n = 1, 2, 3, . . .: H n (Rd ) := {ψ ∈ L2 (Rd ) | ∂ α ψ ∈ L2 (Rd ) ∀ α, |α| ≤ n}. Here α is a multi-index: α = (α1 , . . . , αd ), αj non-negative integers, and |α| := Pd α α1 αd j=1 αj . The expression ∂ ψ denotes the partial derivative ∂x1 · · · ∂xd ψ of order |α|. In other words, H n (Rd ) is the space of functions all of whose derivatives up to order n lie in L2 (Rd ). The inner-product that makes H n (Rd ) into a Hilbert space is X h∂ α ψ, ∂ α φi hψ, φiH n := 0≤|α|≤n

where h·, ·i is the L2 inner-product defined above. The Fourier transform – see Section 25.14 – provides a very convenient characterization of Sobolev spaces: Z 2 n d ˆ ψ ∈ H (R ) ⇐⇒ (1 + |k|2n )|ψ(k)| dk < ∞. (25.1) Rd

We recall here two frequently used facts about Hilbert spaces (see, eg., [106] or [244] for proofs). Proposition 25.3 (Cauchy-Schwarz inequality) For v, w ∈ H, a Hilbert space, |hv, wi| ≤ kvkkwk.

25.2 Operators on Hilbert Spaces

371

A set {vn } ⊂ H, n = 1, 2, . . . is called orthonormal if kvn k = 1 for all n and hvn , vm i = 0 for n 6= m. It is a complete orthonormal set (or basis) if the collection of finite linear combinations of the vn ’s is dense in H. Recall that for a subset D ⊂ H to be dense in H means that given any v ∈ H and ǫ > 0, there exists w ∈ D such that kv − wk < ǫ. A Banach space which has a countable dense subset is said to be separable. A Hilbert space is separable if and only if it has a a countable orthonormal basis. Proposition 25.4 (Parseval relation) Suppose {vn } ⊂ H is a complete orthonormal set. Then for any w ∈ H, X kwk2 = |hw, vn i|2 . n

If H is a Hilbert space, then we can identify its dual, H∗ , with H itself, via the map H ∋ u → lu ∈ H∗ with lu v := hu, vi for v ∈ H (here the notation h·, ·i indicates the Hilbert space inner-product). The fact that this map is an isomorphism between H and H∗ is known as the Riesz representation theorem.

25.2 Operators on Hilbert Spaces In this section we explain the notion of a linear operator on a Hilbert space H (often just called an operator), which abstracts some of the key properties of the Schr¨odinger operator introduced in Chapter 1. Operators are maps, A, from H to itself, satisfying the linearity property A(αv + βw) = αAv + βAw for v, w ∈ H, α, β ∈ C. Actually, we only require an operator A to be defined on a domain D(A) ⊂ H which is dense in H: A : D(A) → H. An example of a dense subset of L2 (Rd ) is C0∞ (Rd ), the infinitelydifferentiable functions with compact support. (Recall, the support of a function f is the closure of the set where it is non-zero: supp(f ) := {x ∈ Rd | f (x) 6= 0}. Thus a function with compact support vanishes outside of some ball in Rd . For Ω ⊂ Rd , C0∞ (Ω) denotes the infinitely differentiable functions with support contained in Ω.) Here are some simple examples of linear operators, A, acting on the Hilbert space L2 (Rd ). In each case, we can simply choose D(A) to be the obvious domain D(A) := {ψ ∈ L2 (Rd ) | Aψ ∈ L2 (Rd )}.

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25 Mathematical Supplement: Elements of Operator Theory

1. The identity map 1 : ψ 7→ ψ 2. Multiplication by a coordinate xj : ψ 7→ xj ψ (i.e. (xj ψ)(x) = xj ψ(x)) 3. Multiplication by a continuous function V : Rd → C V : ψ 7→ V ψ (again meaning (V ψ)(x) = V (x)ψ(x)). 4. The momentum operators (differentiation) pj : ψ 7→ −i~∂j ψ 5. The Laplacian ∆ : ψ 7→

d X

∂j2 ψ

j=1

6. A Schr¨odinger operator H : ψ 7→ −

~2 ∆ψ + V ψ 2m

7. An integral operator K : ψ 7→

Z

K(·, y)ψ(y)dy

R (i.e. (Kψ)(x) = K(x, y)ψ(y)dy). The function K : Rd × Rd → C is called the integral kernel of the operator K.

The domain of the first example is obviously the whole space L2 (Rd ). The domain of the last example depends on the form of the integral kernel, K. The domains of the other examples are easily seen to be dense, since they contain C0∞ (Rd ) (assuming V (x) is a locally bounded function). If the potential function V (x) is bounded, then the largest domain on which the Schr¨odinger operator, H, is defined, namely D(H) := {ψ ∈ L2 (Rd ) | Hψ ∈ L2 (Rd )}, is the Sobolev space of order two, H 2 (Rd ). Remark 25.5 If the kernel K is allowed to be a distribution (a generalized function), then the last example above contains all the previous ones as special cases. It is useful in operator theory to single out those operators with the property of boundedness (which is equivalent to continuity).

25.2 Operators on Hilbert Spaces

373

Definition 25.6 An operator A on H is bounded if kAk :=

sup {ψ∈H | kψk=1}

kAψk < ∞.

(25.2)

In fact, the expression (25.2) defines a norm which makes the space B(H) of bounded operators on H into a complete normed vector space (a Banach space). As we will see, bounded operators are, in some respects, much easier to deal with than unbounded operators. However, since some of the most important operators in quantum mechanics are unbounded, we will need to study both. Problem 25.7 Which of the operators in Examples 1-7 above are bounded? ~2 In particular, show that the operators pj := −i~∂j and H0 := − 2m ∆ are 2 d unbounded on L (R ). Often we can prove a uniform bound for an operator A on a dense domain, D. The next lemma shows that in this case, A can be extended to a bounded operator. Lemma 25.8 If an operator A satisfies kAψk ≤ Ckψk (with C independent of ψ) for ψ in a dense domain D ⊂ H, then it extends to a bounded operator (also denoted A) on all H, satisfying the same bound: kAψk ≤ Ckψk for ψ ∈ H. Proof. For any u ∈ H, there is a sequence {un } ⊂ D such that un → u as n → ∞ (by the density of D). Then the relation kAun − Aum k ≤ Ckun − um k shows that {Aun } is a Cauchy sequence, so Aun → v, for some v ∈ H (by completeness of H), and we set Au := v. This extends A to a bounded operator on all of H (with the same bound, C).  A converse statement – that an operator defined on all of H must be bounded – holds for certain important classes of operators. One such class is closed operators: Definition 25.9 An operator A on H is called closed if whenever {uj }∞ j=1 ⊂ D(A) is a sequence with uj → u and Auj → v as j → ∞, u ∈ D(A) and Au = v. Another way to say this is that the graph of A, {(u, A(u)) | u ∈ D(A)} ⊂ H × H, is closed. A second such class is symmetric operators – i.e., those satisfying hu, Avi = hAu, vi

(25.3)

for all u, v ∈ D(A). As we have seen, the operators of most importance in quantum mechanics are symmetric (indeed, self-adjoint). Symmetric and selfadjoint operators will be discussed in more detail below.

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Theorem 25.10 A closed or symmetric operator, defined on the entire Hilbert space, is bounded. Proof. For a closed operator, this is the “Closed Graph Theorem” of functional analysis. For a symmetric operator, this is the “Hellinger-Toeplitz” theorem – see, eg., [244]. We conclude this section with a useful definition. Definition 25.11 The commutator, [A, B], of two bounded operators A and B is the operator defined by [A, B] := AB − BA. Defining the commutator of two operators when one of them is unbounded requires caution, due to domain considerations. Given this warning, we will often deal with commutators of unbounded operators formally without giving them a second thought.

25.3 Integral Operators Let K be an integral operator on L2 (Rd ): Z (Kψ)(x) := K(x, y)ψ(y)dy Rd

where K : Rd × Rd → C is the integral kernel of the operator K. Examples include 1. K = g(−i~∇) for which the kernel is K(x, y) = (2π~)−d/2 gˇ(x − y)

(25.4)

(here gˇ denotes the inverse Fourier Transform of g – see Section 25.14). 2. K = V (multiplication operator) for which the kernel is K(x, y) = V (x)δ(x − y). The following statement identifies the kernel of the composition of integral operators. The proof is left for the reader. Proposition 25.12 If K1 and K2 are integral operators (with kernels K1 and K2 ), then the integral kernel of K := K1 K2 is Z K1 (x, z)K2 (z, y)dz. K(x, y) = Rd

Problem 25.13 Prove this.

25.4 Inverses and their Estimates

375

Proposition 25.14 Let K be an integral operator with kernel, K(x, y), which lies in L2 of the product space: K(x, y) ∈ L2 (Rd × Rd ). Then K is a bounded operator on L2 (Rd ), and Z kKk2 ≤ |K(x, y)|2 dxdy. (25.5) Rd ×Rd

Proof. To show that the operator K is bounded, we estimate by the CauchySchwarz inequality, Z Z 1/2 Z 1/2 2 2 K(x, y)u(y)dy ≤ |K(x, y)| dy |u(y)| dy .

This implies

2

kKuk ≤ which in turn yields (25.5).

Z

2

|K(x, y)| dxdy

Z

|u(y)|2 dy



25.4 Inverses and their Estimates A key notion of theory of operators is that of the inverse operator. Given an operator A on a Hilbert space H, an operator B is called the inverse of A if D(B) = Ran(A), D(A) = Ran(B), and BA = 1|Ran(B) ,

AB = 1|Ran(A) .

Here Ran(A) denotes the range of A: Ran(A) := {Au | u ∈ D(A)}. It follows from this definition that there can be at most one inverse of an operator A. The inverse of A is denoted A−1 . Put differently, finding the inverse of an operator A is equivalent to solving the equation Au = f for all f ∈ Ran(A). A convenient criterion for an operator A to have an inverse is that A be one-to-one: that is, Au = 0 =⇒ u = 0, or equivalently that it has trivial kernel or nullspace: Null(A − z) := {u ∈ D(A) | (A − z)u = 0} = { 0 }.

(25.6)

The operator A is said to be invertible if A has a bounded inverse. Since by definition a bounded operator is defined on all of H, an invertible operator A, in addition to being one-to-one, must also be onto: that is, Ran(A) = H.

(25.7)

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Remark 25.15 Conditions (25.6) and (25.7) ensure that A−1 exists and is defined on all H. In fact, they are enough to ensure that A is actually invertible (i.e. that A−1 is also bounded) in some important cases, namely: • •

if A is a closed operator, since by Definition 25.9 then so is A−1 , and so by Theorem 25.10, A−1 is bounded; if A is symmetric, since then so is A−1 , and so by Theorem 25.10, A−1 is bounded.

Problem 25.16 Show that if operators A and C are invertible, and C is bounded, then the operator CA is defined on D(CA) = D(A), and is invertible, with (CA)−1 = A−1 C −1 . The following result provides a widely used criterion for establishing the invertibility of an operator. Theorem 25.17 Assume the operator A is invertible, and the operator B is bounded and satisfies kBA−1 k < 1. Then the operator A + B, defined on D(A + B) = D(A), is invertible. This theorem follows from the relation A + B = (1 + BA−1 )A, Problem 25.16 above, and Problem 25.18 below. Problem 25.18 Suppose P∞ an operator K is bounded, and satisfies P∞ kKk < 1. Show that the series n=0 (−K)n is absolutely convergent (i.e. n=0 k(−K)n k < ∞) and provides the inverse of the operator 1 + K. In other words, if kKk < 1, then the operator 1 + K has an inverse given by (1 + K)−1 =

∞ X

(−K)n .

(25.8)

n=0

The series (25.8) is called a Neumann series, and is used often in this book.

25.5 Self-adjointness To make this section more self-contained, we repeat the definitions of symmetric and self-adjoint, as well as some basic results for self-adjoint operators from Section 2.2 of the main text. Recall, 1. A linear operator A acting on a Hilbert space H is symmetric if hu, Avi = hAu, vi

(25.9)

for all u, v ∈ D(A). 2. A linear operator A acting on a Hilbert space H is self-adjoint if it is symmetric, and Ran(A ± i) = H.

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377

Note that the condition Ran(A ± i) = H is equivalent to the fact that the equations (A ± i)ψ = f (25.10) have solutions for all f ∈ H. Example 25.19 On H = L2 (Rd ), the operators xj , pj := −i~∂xj , H0 := ~2 − 2m ∆ (on their natural domains), f (x) and f (p) for f real and bounded, R and integral operators Kf (x) = K(x, y)f (y) dy with K(x, y) = K(y, x) and K ∈ L2 (Rd × Rd ), are all self-adjoint. (See Section 25.14 for the definition of f (p) using the Fourier transform.) d Proof. As an example, we show this for the operator p = −i~ dx on L2 (R) 1 with domain D(p) = H (R). This operator is symmetric, so we compute Ran(−i~∂x + i). For f ∈ L2 (R), solve

(−i~∂x + i)ψ = f, which, using the Fourier transform (see Section 25.14), is equivalent to ˆ (k + i)ψ(k) = fˆ(k), and therefore fˆ(k) ˆ , ψ(k) = k+i

−1/2

ψ(x) = (2π~)

Z

eikx/~

fˆ(k) dk. k+i

ˆ 2 = |fˆ|2 , so since f ∈ L2 (and hence fˆ ∈ L2 ), by (25.1), ψ ∈ Notice (1+|k|2 )|ψ| 1 H (R) = D(p), and therefore Ran(−i~∂x + i) = L2 . Similarly Ran(−i~∂x − i) = L2 .  2

~ Problem 25.20 Show that on L2 (Rd ), xj , − 2m ∆ (on their natural domains), d and f (x) and f (p) for f : R → R bounded are self-adjoint (the last two are bounded operators, and so have domain all of L2 (Rd )).

The next result provides important information about the invertibility of self-adjoint operators. Lemma 25.21 Let A be a symmetric operator. If Ran(A − z) = H for some z with Im z > 0, then it is true for every z with Im z > 0. The same holds for Im z < 0. Moreover, if A is self-adjoint, then A − z is invertible for every z with Im z 6= 0 and satisfies the estimate k(A − z)−1 k ≤

1 . | Im z|

(25.11)

Proof. Write z = λ + iµ with λ, µ ∈ R. Then, since A is symmetric, we have k(A−z)uk2 = h(A−z)u, (A−z)ui = k(A−λ)uk2 +kµuk2 ≥ |µ|2 kuk2. (25.12)

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Hence, if Im z > 0 (or Im z < 0), then Null(A − z) = { 0 }. If also Ran(A − z) = H, then (A − z)−1 is defined on all of H. Further, (25.12) implies that (A − z)−1 is bounded, with bound (25.11) (if one defines v := (A − z)u), and so in particular (A − z) is invertible. Then, by Theorem 25.17, A − z ′ = (A − z) + (z − z ′ ) is invertible for |z ′ − z| < k(A − z)−1 k−1 . Therefore, A − z ′ is invertible if |z ′ − z| < | Im z|, so we can extend invertibility of A − z ′ to all Im z ′ > 0 (or Im z ′ < 0). if A is self-adjoint, then Ran(A ± i) = H and therefore A − z is invertible and satisfies (25.11) for every z in C/R.  This lemma shows that if A is self-adjoint, then αA + β is self-adjoint for any real α 6= 0 and β, and also that A is self-adjoint ⇒ (A − z)ψ = f has a unique solution ∀ Im z 6= 0. (25.13) The next theorem shows that for bounded operators, self-adjointness is easy to check. Theorem 25.22 If A is symmetric and bounded, then A is self-adjoint. Proof. By Lemma 25.21, it suffices to show that Ran(A + iλ) = H provided |λ| is sufficiently large. This is equivalent to solving the equation (A + iλ)ψ = f

(25.14)

for all f ∈ H and such a λ. Now, divide this equation by iλ to obtain ψ + K(λ)ψ = g, where K(λ) = (iλ)−1 A and g = (iλ)−1 f . Let |λ| > kV k. Then kK(λ)k = 1 |λ| kAk < 1 and we conclude that 1 + K(λ) is invertible, as shown in Problem 25.18 above.  As an example, we consider an integral operator K with kernel, K(x, y), which lies in L2 of the product space: K(x, y) ∈ L2 (Rd × Rd ), and satisfies K(x, y) = K(y, x). Then the integral operator K is symmetric (see Problem 2.3). By Proposition 25.14 it is bounded, and therefore by the theorem above, it is a self-adjoint operator on L2 (Rd ). The property of self-adjointness can also be described in terms of the general notion of adjoint of an operator. Definition 25.23 The adjoint of an operator A on a Hilbert space H, is the operator A∗ satisfying hA∗ ψ, φi = hψ, Aφi (25.15) for all φ ∈ D(A), for ψ in the domain D(A∗ ) := {ψ ∈ H | |hψ, Aφi| ≤ Cψ kφk for some constant Cψ (independent of φ), ∀φ ∈ D(A)}.

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It is left as an exercise to show this definition makes sense. Problem 25.24 Show that equation (25.15) defines a unique linear operator A∗ on D(A∗ ) (hint: use the “Riesz lemma” – see, eg., [244]). The subtleties surrounding domains in the definition above are absent for bounded operators: if A is bounded, then by Lemma 25.8 we may assume D(A) = H. Since |hψ, Aφi| ≤ kψkkAkkφk,

for ψ, φ ∈ H, we have D(A∗ ) = H. Not surprisingly, one can show that an operator A is self-adjoint according to our definition above, if and only if A = A∗ (that is, A is symmetric, and D(A∗ ) = D(A)). We conclude this section with a useful definition. Definition 25.25 A self-adjoint operator A is called positive (denoted A > 0) if hψ, Aψi > 0 for all ψ ∈ D(A), ψ 6= 0. Similarly, we may define non-negative, negative, and non-positive operators. Problem 25.26 Show that the operator −∆ on L2 (Rd ) is positive (take D(−∆) = H 2 (Rd )). Hint: integrate by parts (equivalently, use the divergence theorem) assuming that ψ ∈ Dβ := {ψ ∈ C 2 (Rd ) | |∂ α ψ(x)| ≤ Cα (1 + |x|)−β ∀ α, |α| ≤ 2} for some β > d/2, Then use the fact that Dβ is dense in H 2 (Rd ) to extend the inequality to all ψ ∈ H 2 (Rd ).

25.6 Exponential of an Operator In this section we construct the exponential e−itA for a self-adjoint operator A, which allows us to solve the abstract Schr¨odinger equation i

∂ψ = Aψ ∂t

(25.16)

where ψ : t → ψ(t) is a path in a Hilbert space H and A is a self-adjoint operator on H. In our applications, ~A is a Schr¨odinger operator. As before, we supplement equation (25.16) with the initial condition ψ|t=0 = ψ0

(25.17)

where ψ0 ∈ H. Our goal is to prove Theorem 2.16 of Section 2.3 which shows that self-adjointness of A implies the existence of dynamics. We restate this theorem here in terms of an operator A:

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Theorem 25.27 If A is a self-adjoint operator, then there is a unique family of bounded operators, U (t) := e−itA , having the following properties, for t, s ∈ R, ∂ i U (t) = AU (t) = U (t)A, (25.18) ∂t U (0) = 1 and U (t)ψ → ψ, as t → 0, (25.19) U (t)U (s) = U (t + s),

(25.20)

kU (t)ψk = kψk.

(25.21)

Furthermore, the initial value problem (25.18)-(25.19) has a unique solution. Proof. We will define the exponential eiA for an unbounded self-adjoint operator A, by approximating A by bounded operators, and then using the power series definition of the exponential for bounded operators: eA :=

∞ X An n! n=0

(25.22)

which converges absolutely since ∞ ∞ X kAn k X kAkn ≤ = ekAk < ∞. n! n! n=0 n=0

We have already shown in Section 2.3 that Theorem 25.27 holds for bounded, self-adjoint operators A. (The self-adjointness is only needed for (25.21).) Now, we extend it to unbounded operators. By Lemma 25.21, the operators Aλ :=

1 2 λ [(A + iλ)−1 + (A − iλ)−1 ] 2

are well-defined and bounded for λ > 0. The operators Aλ approximate A in the sense that Aλ ψ → Aψ as λ → ∞ for all ψ ∈ D(A). To see this, note first that Aλ = Bλ A,

Bλ :=

and 1 − Bλ =

1 iλ[(A + iλ)−1 − (A − iλ)−1 ] 2

1 [(A + iλ)−1 + (A − iλ)−1 ]A. 2

So using the estimate

1 , λ which is established in Lemma 25.21, we find, for any φ ∈ D(A), k(A ± iλ)−1 k ≤

1 k(1 − Bλ )φk = k [(A + iλ)−1 + (A − iλ)−1 ]Aφk 2 1 ≤ kAφk → 0 as λ → ∞. λ

(25.23)

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381

And since D(A) is dense and kBλ k ≤ 1, we have Bλ φ → φ as λ → ∞ for any φ ∈ H. Finally, taking φ = Aψ for any ψ ∈ D(A), we conclude by (25.23), that Aλ ψ → Aψ as λ → ∞ for ψ ∈ D(A), (25.24)

as required. Since Aλ is bounded, we can define the exponential eiAλ by power series. We will show now that the family {eiAλ , λ > 0} is a Cauchy family, in the sense that
k eiAλ′ − eiAλ ψk → 0 (25.25)

as λ, λ′ → ∞ for all ψ ∈ D(A). To prove this fact, we represent the operator inside the norm as an integral of a derivative: Z 1 ∂ isAλ′ i(1−s)Aλ e e ds. (25.26) eiAλ′ − eiAλ = ∂s 0 Since Aλ is symmetric and bounded, it is self-adjoint (Theorem 25.22). Using (25.18) and (25.21) in (25.26), we find (noting that Aλ and Aλ′ commute) Z 1 k(eiAλ′ − eiAλ )ψk = k eisAλ′ ei(1−s)Aλ i(Aλ′ − Aλ )ψdsk ≤ =

Z

0 1

keisAλ′ ei(1−s)Aλ i(Aλ′ − Aλ )ψkds

0

Z

0

(25.27)

1

k(Aλ − Aλ′ )ψkds = k(Aλ′ − Aλ )ψk.

(The inequality used in the first step – the Minkowski inequality – can be proved by writing the integral as a limit of Riemann sums and using the triangle inequality – see [106]). Due to (25.24), relation (25.25) follows. The Cauchy property (25.25) shows that for any ψ ∈ D(A), the vectors eiAλ ψ converge to some element of the Hilbert space as λ → ∞. Thus we can define eiA ψ := lim eiAλ ψ (25.28) λ→∞

for ψ ∈ D(A). Since we have already shown that the theorem holds for bounded operators, we have that keiA ψk ≤ kψk for all ψ in D(A), which is dense in H. Thus, as in Lemma 25.8, we can extend this definition of eiA to all ψ ∈ H. This defines the exponential eiA function for any self-adjoint operator A. Now we prove (25.18). We use the definition (25.28) and the fact Aλ is a bounded operator and therefore e−itAλ satisfies (25.18). Formally bringing the differentiation into the limit, we obtain, for φ ∈ D(H), i~

∂ ∂ ∂ hφ, e−itA ψ0 i = i lim hφ, e−itA ψ0 i = i lim hφ, e−itA ψ0 i λ→∞ ∂t ∂t λ→∞ ∂t = lim hφ, Aλ e−iAλ t ψ0 i = lim hAλ φ, e−iAλ t ψ0 i λ→∞

= hAφ, e−iAt ψ0 i.

λ→∞

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This exchange of limits is readily justified (the reader is invited to supply the details). Furthermore, if ψ0 ∈ D(A), one can show that e−iAt ψ0 ∈ D(A), and ∂ −itA therefore i ∂t e ψ0 = Ae−itA ψ0 . So ψ(t) := e−itAλ ψ0 satisfies (25.16) and therefore (25.18) holds. Next, clearly U (0) = 1. Moreover, for any ψ0 ∈ D(A), Z t U (s)Aψ0 ds → 0 as t → 0. U (t)ψ0 − ψ0 = i 0

Hence (25.19) holds. To prove (25.21), we observe that by the self-adjointness of A, the derivative of the square of the l.h.s. of (25.21) w.r.to t is 0. Hence, it is independent of t and therefore (25.21) follows. This shows that the initial value problem (25.18)-(25.19) has a unique solution. Indeed, if Ui (t), i = 1, 2, satisfy (25.18)-(3.11), then U (t) := U1 (t) − U2 (t) satisfies (25.18) with the initial condition U (t = 0) = 0, which by (25.21) implies that U (t) = 0. To prove that U (t) has the group property (25.20), we notice that both sides of equation (25.20) satisfy the same differential equation (in t) with the same initial condition. Hence, by the uniqueness, they are equal. Another way to prove (25.20) is to use representation (25.28). By Problem 2.17, e−iAλ t has the group property (25.20). Given this result, for any ψ, φ ∈ H, hψ, U (t)U (s)φi = hU (t)∗ ψ, U (s)φi = lim heiAλ t ψ, e−iAλ s/~ φi λ→∞

= lim hψ, e−iAλ (t+s) φi = hψ, U (t + s)φi λ→∞

which proves (25.20).  The theorem above has the following corollary Corollary 25.28 If A is self-adjoint, then the Cauchy problem (25.16)(25.17) has a unique solution which conserves probability. Indeed, the family ψ(t) := U (t)ψ0 is a solution of the Cauchy problem (25.16)(25.17), which conserves the probability. It is the unique solution of (25.16)˜ solves (25.16) (25.17), since, if there are two solutions, then their difference, ψ, ˜ with ψ|t=0 = 0, and therefore by conservation of probability (a consequence ˜ ˜ of symmetry of A), kψ(t)k = kψ(0)k = 0 for all t and hence ψ˜ ≡ 0. −itA The operator family U (t) := e is called the propagator or evolution operator generated by A, or for the equation (25.16). The properties recorded in the equations (25.20) and (25.21) are called the group and isometry properties. The operator U (t) = e−itA is invertible (since U (t)U (−t) = 1) and is isometry (i.e. kU (t)ψk = kψk). Such operators are called unitary. More precisely, Definition 25.29 An operator U is called unitary if it preserves the inner product: hU ψ, U φi = hψ, φi, for all ψ, φ ∈ H, or what is the same, U U ∗ = U ∗ U = 1.

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383

To show that U (t) = e−itA is unitary, we observe that kU (t)ψk = kψk implies hψ, ψi = hU (t)ψ, U (t)ψi = hψ, U ∗ (t)U (t)ψi, from which U ∗ (t)U (t) = 1 follows. Similarly, U (t)U ∗ (t) = 1. So U ∗ (t) = U (−t) = (U (t))−1 . This if A is self-adjoint, the operator U (t) := e−iAt exists and is unitary for all t ∈ R. The following very simple example illustrates the connection between unitarity and self-adjointness. Example 25.30 If φ : Rd → R is continuous, then the bounded operator U : ψ 7→ eiφ ψ is easily checked to be unitary on L2 (Rd ) (just note that U ∗ is multiplication by e−iφ ). Now φ is bounded as a multiplication operator iff it is a bounded function. Note, however, that U is well-defined (and unitary) even if φ is unbounded. Remark 25.31 If A is a positive operator, then we can define the operator e−A in a way similar to our definition of eiA above. We take e−A := limλ→∞ e−Aλ where Aλ = (A + λ)−1 λA is a family of bounded operators. To conclude this section we describe an important extension of Theorem 25.27 to the situation when the operator A is time-dependent, A = A(t). Our goal now is to solve the abstract Schr¨odinger equation i

∂ψ = A(t)ψ, ∂t

ψ|t=0 = ψ0 ,

(25.29)

where ψ : t → ψ(t) is a path in a Hilbert space H, for every t, A(t) is a self-adjoint operator on H and ψ0 ∈ H. We have the following result Theorem 25.32 Let A(t) = A + W (t), where A and W are self-adjoint operators and, in addition, W is bounded. Then there is a unique two-parameter family of bounded operators, U (t, s) with the following properties, for t, s ∈ R, ∂ U (t, s) = A(t)U (t, s), ∂t U (s, s) = 1 and U (t, s)ψ → ψ, as t → s, U (t, s)U (s, r) = U (t, r),

i

kU (t, s)ψk = kψk.

(25.30) (25.31) (25.32) (25.33)

Proof. We construct the two-parameter family of bounded operators, U (t, s) satisfying (25.30) - (25.31). First, we pass to the interaction representation by introducing the two-parameter family V (t, s) := ei(t−s)A U (t, s). This family satisfies the differential equation i

∂ V (t, s) = W (t, s)V (t, s), ∂t

V (s, s) = 1,

(25.34)

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where W (t, s) := ei(t−s)A W (t)e−i(t−s)A . Now, we integrate (25.34) in t from s to t and use the fundamental theorem of calculus to obtain Z t V (t, s) = 1 + i W (r, s)V (r, s)dr. s

We construct a unique solution to this equation. Iterating this equation, we arrive at the series (Dyson series) U (t, s) = 1 +

∞ X

Un (t, s).

(25.35)

n=1

R where Un (t, s) := in ∆(s,t) dn rW (r1 , s) . . . W (rn , s), with ∆(s, t) := {(r1 , r2 , . . . rn ) : s ≤ rn ≤ · · · ≤ r2 ≤ r1 ≤ t}. Now, using that W (r, s) are bounded (in fact, kW (r, s)k ≤ kW (r)k), we estimate kUn (t, s)k ≤

Z

dn r

∆(s,t)

n Y 1

kW (rj )k.

Next, the relation Z

∆(s,t)

dn r

n Y 1

kW (rj )k = =

1 n!

Z

dr1 . . . drn

[0,t]n

Z t 1 ( drkW (r)k)n n! 0

n Y 1

kW (rj )k

shows that the series above converges in norm and represents a family of bounded operators. Representation (25.35) shows that it satisfies (25.30) (25.31). Similarly, one shows that U (t, s) satisfies also the equation i

∂ U (t, s) = U (t, s)A(s). ∂s

(25.36)

Using (25.30) and (25.36), one checks easily that the derivative of the l.h.s. of (25.32) w.r.to s is zero. Hence, it is independent of s and therefore (25.32) follows. Finally, differentiating the square of the l.h.s. of (25.33) w.r.to t, we see that it is independent of t and therefore (25.33) follows.  A two-parameter family of bounded operators, U (t, s), satisfying (25.30) (25.33) is called the propagator, or evolution, generated by A(t). (Recall that, if A is t-independent then the same term is applied to U (t) = U (t, 0) = e−iAt .

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385

Problem 25.33 Let A(t) = A + W (t), where A and W are self-adjoint operators on a Hilbert space H and, in addition, W is bounded. Prove directly (i.e without using Theorem 25.32, but using the ideas of its proof) that the abstract Schr¨odinger initialvalue problem i

∂ ψ(t) = A(t)ψ(t), ψ(t = 0) = ψ0 , ∂t

(25.37)

where ψ(·) : R → H is a differentiable vector function, has a unique solution for every ψ0 ∈ H. Show that the solution satisfies kψ(t)k = kψ0 k.

25.7 Projections Let H be a Hilbert space. A bounded operator P on H is called a projection operator (or simply a projection) if it satisfies P 2 = P. This relation implies kP k ≤ kP k2 , and so kP k ≥ 1 provided P 6= 0. We have v ∈ RanP ⇐⇒ P v = v

and v ∈ (RanP )⊥ ⇐⇒ P ∗ v = 0.

(25.38)

Indeed, if v ∈ RanP , then there is a u ∈ H s.t. v = P u, so P v = P 2 u = P u = v; the second statement is left as an exercise. Problem 25.34 Prove that (a) P ∗ v = 0 if and only if v ⊥ RanP , (b) RanP is closed and (c) P ∗ is also a projection. Example 25.35 The following are projection operators: 1. let H = L2 (Rd ) and let E be a subset of Rd . Then χx∈E : f (x) 7→ χE (x)f (x) where χE (x) :=



1x∈E 0x∈ /E

is a projection. 2. again let H = L2 (Rd ) and let E be a subset of Rd . Then χp∈E = F −1 χk∈E F : u(x) 7→ (χE (k)ˆ u(k))ˇ(x) is a projection. 3. let H be any Hilbert space, and ϕ, ψ ∈ H satisfying hϕ, ψi = 1. Then f 7→ hϕ, f iψ is a projection.

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25 Mathematical Supplement: Elements of Operator Theory

4. as in 3), but now let {ψi }N i=1 be an orthonormal set (i.e. hψi , ψj i = δij ). Then N X f 7→ hψi , f iψi 1

is a projection.

Definition 25.36 1. A projection P is said to be of rank r < ∞ if dim RanP = r. 2. A projection P is called an orthogonal projection if it is self-adjoint, i.e. if P = P ∗ . If P be an orthogonal projection, then (25.38) implies that v ⊥ RanP ⇐⇒ P v = 0, i.e. NullP = (RanP )⊥ .

(25.39)

The projections in Examples 1), 2) and 4) above are orthogonal. The projection in Example 3) is orthogonal if and only if ϕ = ψ. Problem 25.37 Let P be an orthogonal projection. Show that 1. kP k ≤ 1, and therefore ||P || = 1 if P 6= 0 (Hint: Use (25.38)); 2. 1 − P is also an orthogonal projection, Ran(1 − P )⊥RanP , and Null(1 − P ) = RanP ; 3. H = RanP ⊕ NullP . Remark 25.38 Orthogonal projections on H are in one-to-one correspondence with closed subspaces of a Hilbert space H. This correspondence is obtained as follows. Let V = RanP . Then V is a closed subspace of X. To show that V is closed, let {vn } ⊂ V , and vn → v ∈ X, and show that v ∈ V . Since P is a projection, we have vn = P vn , so kv−P vk = kv−vn −P (v−vn)k ≤ kv − vn k + kP k kv − vn k → 0, as n → ∞. Therefore v = P v, so v ∈ V , and V is closed. Conversely, given a closed subspace V , define a projection operator P by P u = v, where u = v + v ⊥ ∈ V ⊕ V ⊥ . (25.40) Problem 25.39 Show that P defined in (25.40) is an orthogonal projection with RanP = V . For any given V , show that there is only one orthogonal projection (the one given in (25.40)) such that RanP = V .

25.8 The Spectrum of an Operator Again to keep this section self-contained we repeat some definitions and results from Section 6.1 of the main text. Definition 25.40 The spectrum of an operator A on a Hilbert space H is the subset of C given by σ(A) := {λ ∈ C | A − λ is not invertible (has no bounded inverse)}

25.8 The Spectrum of an Operator

387

(here and below, A − λ denotes A − λ1). The complement of the spectrum of A in C is called the resolvent set of A: ρ(A) := C\σ(A). For λ ∈ ρ(A), the operator (A − λ)−1 , called the resolvent of A, is well-defined. The following exercise asks for the spectrum of our favourite operators. Problem 25.41 Prove that as operators on L2 (Rd ) (with their natural domains), σ(1) = {1}. σ(pj ) = R. σ(xj ) = R. σ(V ) = range(V ), where V is the multiplication operator on L2 (Rd ) by a continuous function V (x) : Rd → C. 5. σ(−∆) = [0, ∞). 6. σ(f (p)) = range(f ), where f (p) := F −1 f F with f (k), the multiplication operator on L2 (Rd ) by a continuous function f (k) : Rd → C. 1. 2. 3. 4.

Theorem 25.42 The spectrum σ(A) is a closed set. Proof. We show that the complement of the spectrum, ρ(A) := C/σ(A), called the resolvent set, is an open set. Indeed, let z ∈ ρ(A). Then A − z is invertible (has a bounded inverse) and therefore by Theorem 25.17, so is A − z ′ = (A − z)[1 + (z − z ′ )(A − z)−1 ], if |z ′ − z| < k(A − z)−1 k−1 .  We observe that self-adjoint operators have real spectrum. Theorem 25.43 If A = A∗ , then σ(A) ⊂ R. Proof. This follows immediately from Lemma 25.21.  One familiar reason for A − λ not to be invertible is that (A − λ)ψ = 0 has a non-zero solution ψ ∈ D(A) ⊂ H. In this case we say that λ is an eigenvalue of A and ψ is called a corresponding eigenvector. Definition 25.44 The discrete spectrum of an operator A is σd (A) = {λ ∈ C | λ is an isolated eigenvalue of A with finite multiplicity} (isolated meaning some neighbourhood of λ is disjoint from the rest of σ(A)). Here the multiplicity of an eigenvalue λ is the dimension of the eigenspace Null(A − λ) := {v ∈ H | (A − λ)v = 0}. Problem 25.45 1. Show Null(A − λ) is a vector space. 2. Show that if A is self-adjoint, eigenvectors of A corresponding to different eigenvalues are orthogonal.

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The rest of the spectrum is called the essential spectrum of the operator A: σess (A) := σ(A)\σd (A).

Remark 25.46 Some authors may use the terms “point spectrum” and “continuous spectrum” rather than (respectively) “discrete spectrum” and “essential spectrum’. Problem 25.47 For the operators xj and pj on L2 (Rd ) show that 1. σess (pj ) = σ(pj ) = R; 2. σess (xj ) = σ(xj ) = R; 3. σess (−∆) = σ(−∆) = [0, ∞). Hint: Show that these operators do not have discrete spectrum. Problem 25.48 Show that if U : H → H is unitary, then σ(U ∗ AU ) = σ(A), σd (U ∗ AU ) = σd (A), and σess (U ∗ AU ) = σess (A). Problem 25.49 Let A be an operator on H. If λ is an accumulation point of σ(A), then λ ∈ σess (A). Hint: use the definition of the essential spectrum and Theorem 25.42. For a self-adjoint operator A the sets {span of eigenfunctions of A} and {span of eigenfunctions of A}⊥ , where W ⊥ := {ψ ∈ H | hψ, wi = 0 ∀ w ∈ W }, are invariant under A in the sense of the definition Definition 25.50 A subspace W ⊂ H of a Hilbert space H is invariant under an operator A if Aw ∈ W whenever w ∈ W ∩ D(A). Problem 25.51 Assume A is a self-adjoint operator. Show that 1. If W is invariant under A, then so is W ⊥ ; 2. The span of the eigenfunctions of A and its orthogonal complement are invariant under A;

25.9 Functions of Operators and the Spectral Mapping Theorem Our goal in this section is to define functions f (A) of a self-adjoint operator A. We do this in the special case where A is bounded, and f is a function analytic in a neighbourhood of σ(A).

25.9 Functions of Operators and the Spectral Mapping Theorem

389

Problem 25.52 Let A be a bounded self-adjoint operator. Show σ(A) ⊂ [−kAk, kAk]. Hint: use that if |z| > kAk, then k(A − z)uk ≥ (|z| − kAk)kuk. Suppose f (λ) is analytic in a complex disk of radius R, {λ C | |λ| < R}, P∈ ∞ with R > kAk. So f has a power series expansion, f (λ) = n=0 an λn , which converges for |λ| < R. Define the operator f (A) by the convergent series f (A) :=

∞ X

an An .

(25.41)

n=0

We have already encountered an example of this definition: the exponential eA discussed in Section 2.3. As another example, consider the function f (λ) = (λ − z)−1 , for |z| > kAk, which is analytic in a disk of radius R, with kAk < R < |z|, and has power-series expansion ∞

1 1X 1 (λ/z)j . =− f (λ) = − z 1 − λ/z z j=0 The corresponding operator defined by (25.41) is the resolvent (A − z)−1 = −

∞

1 X −j j z A . z j=0

(25.42)

(Recall that the series in (25.42) is a Neumann series.) Of course, the resolvent is defined for any z in the resolvent set ρ(A) = C\σ(A). In fact, (A − z)−1 is an analytic (operator-valued) function of z ∈ ρ(A). To see this, we start with the relation (A − z)−1 = (A − z0 )−1 − (z0 − z)(A − z0 )−1 (A − z)−1 for z, z0 ∈ ρ(A), which the reader is invited to verify (this relation is called the first resolvent identity). Thus (A − z)−1 = [1 − (z − z0 )(A − z0 )−1 ]−1 (A − z0 )−1 . If |z − z0 | < (k(A − z0 )−1 k)−1 , the first inverse on the right hand side can be expanded in a Neumann series, yielding (A − z)−1 =

∞ X j=0

(z − z0 )j (A − z0 )−j−1 .

Thus (A − z)−1 is analytic in a neighbourhood of any z0 ∈ ρ(A). The following useful result relates eigenvalues and eigenfunctions of A to those of f (A). Theorem 25.53 (Spectral mapping theorem) Let A be a bounded operator, and f a function analytic on a disk of radius > kAk. If Aφ = λφ, then f (A)φ = f (λ)φ.

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Proof. If {aj } are the coefficients of the power series for f , then f (A)φ =

∞ X j=0



∞ X aj λj )φ = f (λ)φ. aj Aj φ = ( j=0

We conclude with a brief discussion of alternatives to, and extensions of, the above definition. Suppose f is analytic in a complex neighbourhood of σ(A). We can replace definition (25.41) by the contour integral (called the Riesz integral) I 1 f (A) := f (z)(A − z)−1 dz (25.43) 2πi Γ where Γ is a contour in C encircling σ(A). The integral here can be understood in the following sense: for any ψ, φ ∈ H, I 1 f (z)hψ, (A − z)−1 φidz hψ, f (A)φi := 2πi Γ (knowledge of hψ, f (A)φi for all ψ, φ determines the operator f (A) uniquely).

Problem 25.54 Show that the definition (25.43) agrees with (25.41) when f (λ) is analytic on {|λ| < R} with R > kAk. Hint: by the Cauchy theorem, and analyticity of the resolvent, the contour Γ can be replaced by {|z| = R0 }, kAk < R0 < R. On this contour, (A − z)−1 can be expressed as the Neumann series (25.42). A similar formula can be used for unbounded operators A to define certain functions f (A) (see [162]). If A is an unbounded self-adjoint operator, and f is a continuous, bounded function, the bounded operator f (A) can still be defined. One example of this is the definition of eiA in Section 25.6. The definition of f (i∇) using the Fourier transform (Section 25.14) provides another example. More generally, the Fourier transform, together with Theorem 25.27, allows one to define functions of self-adjoint operators as follows: Definition 25.55 Assume A is a self-adjoint operator R and f (λ) is a function whose inverse Fourier transform, fˇ is integrable, |fˇ(t)|dt < ∞. Then the operator Z ∞ −1/2 f (A) := (2π~) fˇ(t)eiAt/~ dt (25.44) −∞

is well-defined, bounded, and is self-adjoint if f is real.

We present without justification a formula connecting the equations (25.43) and (25.44) (i.e. connecting the propagator and the resolvent): Z 1 ∞ −iAt dλf (λ)e−iλt Im(A − λ − i0)−1 . (25.45) e f (A) = π −∞ The reader is referred to [244] for the general theory.

25.10 Weyl Sequences and Weyl Spectrum

391

25.10 Weyl Sequences and Weyl Spectrum We now want to address the question of how to characterize the essential spectrum of a self-adjoint operator A. Is there a characterization of σess (A) similar to that of σd (A) in terms of some kind of eigenvalue problem? To answer this question, we observe that there is another reason for A−λ not to be invertible, besides λ being an eigenvalue of A, namely (A − λ)ψ = 0 “almost” having a non-zero solution. More precisely, there a sequence {ψn } ⊂ H s.t. 1. kψn k = 1 for all n 2. k(A − λ)ψn k → 0 as n → ∞ 3. ψn → 0 weakly as n → ∞ (this means hφ, ψn i → 0 for all φ ∈ H). We say {ψn } ⊂ H is a Weyl sequence for A and λ if these statements hold. Definition 25.56 The Weyl spectrum of an operator A is σw (A) = {λ | there is a Weyl sequence for A and λ}. The following result says that when A is self-adjoint, the sets σd (A) and σw (A) are disjoint, and comprise the whole spectrum: Theorem 25.57 (Weyl) If A is self-adjoint, then σess (A) = σw (A), and therefore the spectrum of A is the disjoint union of the discrete spectrum of A and the Weyl spectrum of A: σ(A) = σd (A) ∪ σw (A). Proof. Suppose first that λ ∈ σess (A). Then inf kψk=1,ψ∈D(A) k(A − λ)ψk = 0, for otherwise A − λ would be invertible. Hence there is a sequence ψn ∈ D(A) such that kψn k = 1 and k(A − λ)ψn k → 0 as n → ∞. By the Banach-Alaoglu theorem (see, eg., [244]), there is a subsequence {ψn′ } ⊂ {ψn } and an element ψ0 ∈ H such that ψn′ → ψ0 weakly as n′ → ∞ (we drop the prime in n′ henceforth). This implies that for all f ∈ D(A), h(A − λ)f, ψ0 i = lim h(A − λ)f, ψn i = lim hf, (A − λ)ψn i = 0. n→∞

n→∞

Hence ψ0 ∈ D(A) (since D(A) = D(A∗ ) = {ψ ∈ H | |hAf, ψi| ≤ Ckf k ∀f ∈ D(A)} and |hAf, ψ0 i| = |λhf, ψ0 i| ≤ |λ|kψ0 kkf k) and Aψ0 = λψ0 . If ψ0 = 0, then {ψn } is a Weyl sequence for A and λ, and so λ ∈ σw (A). If ψ0 6= 0, this implies that λ is an eigenvalue of A. So λ must therefore have infinite multiplicity or be non-isolated. If λ is an eigenvalue of infinite multiplicity, then an orthonormal basis of Null(A − λ) yields a Weyl sequence for A and λ, and therefore λ ∈ σw (A). If λ is not isolated, then consider a sequence λj ∈ σ(A)\{λ} with λj → λ. If there is a subsequence consisting of distinct eigenvalues, a corresponding sequence of normalized eigenvectors is orthonormal, and so converges weakly to 0 – hence it is a Weyl sequence for A and λ.

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On the other hand, if the sequence λj consists (eventually) of non-eigenvalues, then, arguing as above for each λj , one can construct a diagonal sequence which is Weyl for A and λ. So we conclude λ ∈ σw (A), and we have shown that σess (A) ⊂ σw (A). Now suppose λ ∈ σw (A), and let ψn be a corresponding Weyl sequence. Then certainly λ ∈ σ(A), otherwise kψn k = k(A − λ)−1 (A − λ)ψn k ≤ k(A − λ)−1 kk(A − λ)ψn k → 0, a contradiction. Suppose λ is an isolated eigenvalue of finite multiplicity. For simplicity, suppose the multiplicity is one (the argument is straightforward to generalize), and let ψ0 be a normalized eigenvector. Write ψn = cn ψ0 + ψ˜n with cn = hψ0 , ψn i and hψ˜n , ψ0 i = 0. Since ψn → 0 weakly, cn → 0, and so kψ˜n k → 1. Also (A − λn )ψ˜n → 0. Because λ is isolated in the spectrum, (A − ζ)−1 is uniformly bounded on (ψ0 )⊥ for ζ near λ, and so kψ˜n k = k(A − λn )−1 (A − λn )ψ˜n k ≤ (const)k(A − λn )ψ˜n k → 0, a contradiction. Thus λ ∈ σess (A), showing σw (A) ⊂ σess (A) and completing the proof of Theorem 25.57. As an application of the Weyl theorem we consider a Schr¨odinger operator on a bounded domain, with Dirichlet boundary conditions. Theorem 25.58 Let Λ be a cube in Rd , and V a continuous function on Λ. Then the Schr¨odinger operator H = −∆ + V , acting on the space L2 (Λ) with Dirichlet boundary conditions, has purely discrete spectrum, accumulating at +∞. To be precise, the operator “H on L2 (Λ) with Dirichlet boundary conditions” should be understood as the unique self-adjoint extension of H from C0∞ (Λ). Proof. Suppose Λ = [0, L]d . Consider the normalized eigenfunctions of the operator −∆ on L2 (Λ) with Dirichlet boundary conditions: φk (x) =

 d/2 Y d 2 sin(kj xj ), L j=1

k∈

π + d (Z ) L

(see Section 7.1), so that − ∆φk = |k|2 φk .

(25.46)

π Now we recall that the eigenfunctions φk , k ∈ L (Z+ )d , form an orthonormal 2 basis for L (Λ): X ψ= hφk , ψiφk π k∈ L (Z+ )d

for any ψ ∈ L2 (Λ) (this is a special case of a general phenomenon valid for self-adjoint operators).

25.10 Weyl Sequences and Weyl Spectrum

393

We show now that the operator H has no essential spectrum. Assume on the contrary that λ ∈ σess (H), and let un be a corresponding Weyl sequence; i.e. kun k = 1, un → 0 weakly, and k(H −λ)un k → 0. By the triangle inequality k(H − λ)un k ≥ k(−∆ − λ)un k − kV un k ≥ k(−∆ − λ)un k − max |V | (25.47) since kun k = 1. Writing

un =

X

akn φk

k

where

akn

= hφk , un i, and using (25.46), we compute X (−∆ − λ)un = (|k|2 − λ)akn φk k

and so by the Parseval relation (Proposition 25.4) X k(−∆ − λ)un k2 = (|k|2 − λ)2 |akn |2 .

(25.48)

k

The Parseval relation also gives 1 = kun k2 =

X k

|akn |2 .

(25.49)

√ Now choose K such that |k|2 − λ ≥ 2(max |V | + 1) for |k| ≥ K. Then by (25.48) and (25.49), X X |akn |2 ). |akn |2 = 2(max |V |+1)2 (1− k(−∆−λ)un k2 ≥ 2(max |V |+1)2 |k|≥K

Since un → 0 weakly,

|k|N

!2

2

|hφn , ψj i|hφn , ρφn i1/2  1/2

|hφn , ψj i|hφn , ρφn i

!2

.

Applying the Cauchy-Schwarz inequality, we obtain    X X hψj , ρψj i ≤ 2  hφn , ρφn i |hφn , ψj i|2   n≤N

+2

X

n>N

|hφn , ψj i|

2

!

n≤N

X

n>N

!

hφn , ρφn i .

Since ρ is trace class, given any ǫ > 0, there is N (ǫ) such that X hφn , ρφn i ≤ ǫ. n>N (ǫ)

Since ψj converges weakly to zero, for any ǫ > 0 and N > 0, there is J(ǫ, N ) such that X |hφn , ψj i|2 ≤ ǫ for all j ≥ J(ǫ, N ). n≤N

The last three inequalities imply that for all j ≥ N (ǫ, N (ǫ)), hψj , ρψj i ≤ 2ǫ Tr ρ + 2kψj k2 ǫ, where we used

X

n≤N

and

X

n>N

hφn , ρφn i ≤

|hφn , ψj i|2 ≤

(25.52)

X hφn , ρφn i = Tr ρ n

X n

|hφn , ψj i|2 = kψj k2 .

Since kψj k ≤ M , the inequality (25.52) completes the proof of the lemma. 

25.11 The Trace, and Trace Class Operators

401

Lemma 25.67 If ρ ≥ 0 is trace class, then so is ρ2 . Proof. Since hφ, ρ2 φi = hρφ, ρφi = kρφk2

= kρ1/2 ρ1/2 φk2 ≤ kρ1/2 k2 kρ1/2 φk2 = kρ1/2 k2 hφ, ρφi,

for any orthonormal basis {φj }, X X hφ, ρφi < ∞. hφj , ρ2 φj i ≤ kρ1/2 k2 j

j

Hence ρ2 is trace class. Now we are ready to prove our spectral statement. Let {ψj } be a sequence with ψj → 0 weakly, and kψj k = 1. Then by Lemmas 25.66 and 25.67, k(ρ − λ)ψj k2 = hψj , (ρ2 − 2λρ + λ2 )ψj i

= hψj , ρ2 ψj i − 2λhψj , ρψj i + λ2 → λ2 .

Hence ρ and λ 6= 0 have no Weyl sequence. Thus by Theorem 25.57, λ 6= 0 is not a point of the essential spectrum of ρ. Thus σess (ρ) ⊂ {0}.  Theorem 25.68 Let ρ be a self-adjoint trace class operator on a Hilbert space H. Then the normalized eigenvectors of ρ form a basis in H. Proof. The normalized eigenvectors of ρ are independent, since ρ is selfadjoint. Let V denote the span of the normalized eigenvectors of ρ, and define ρ⊥ to be the operator ρ restricted to V ⊥ . Then ρ⊥ is a self-adjoint trace-class operator on V ⊥ . It cannot have non-zero eigenvalues, since all such eigenvectors would lie in the space V . Hence σ(ρ⊥ ) ⊂ {0}. Since ρ⊥ is self-adjoint and non-negative, we apply Theorem 8.2 to ρ⊥ and −ρ⊥ to conclude that hψ, ρ⊥ ψi = 0 for all ψ ∈ V ⊥ , and so ρ⊥ = 0. That means V ⊥ must consist of zero-eigenvectors of ρ, and hence must be empty. Thus V = H.  A self-adjoint trace-class operator ρ can be written in the form X pj Pψj ρ=

(25.53)

j

where {ψj } are orthonormal eigenfunctions and pj are the corresponding eigenvalues, ρψj = pj ψj . Indeed, since by PTheorem 25.68, {ψj } forms a basis in H, any φ ∈ H can be written as φ = j hψj , φiψj , and so X X pj Pψj φ. pj hψj , φiψj = ρφ = j

j

P P Since TrPψ = P 1 and Tr( j Aj ) = j Tr Aj , the last relation implies the relation Trρ = j pj .

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25.12 Operator Determinants In this section we discuss determinants of differential operators, which appear in the stationary phase expansion of the path integrals developed in Chapter 16. For a square matrix A, the determinant function has the properties 1. 2. 3. 4. 5.

A is invertible iff det A 6= 0 A = A∗ ⇒ det A ∈ R det(AB) = det(A) det(B) A > 0 ⇒Qdet A = eTr(ln A) det A = λ ev of A λ

We would like to define the determinant of a Schr¨odinger operator. Example 25.69 Let H = −∆ + V on [0, L] with zero boundary conditions (assume V is bounded and continuous). Then using the fact that for n ∈ Z, sin(πnx/L) is an eigenfunction of −∆ with eigenvalue (πn/L)2 , we obtain k(H−(πn/L)2 ) sin(πnx/L)k = kV (x) sin(πnx/L)k ≤ (max |V |)k sin(πnx/L)k, and spectral theory tells us that H hasp an eigenvalue in the interval [(πn/L)2 − 2 max |V |, (πn/L) + max |V |]. Since { 2/L sin(πnx/L) | n = 1, 2, . . .} is an orthonormal basis in L2 [0, L], we have σ(H) = {(πn/L)2 + O(1) | n ∈ Z}. Q So trying to compute the determinant directly, we get λ ev H λ = ∞. P For a positive matrix, A, we can define ζA (s) := TrA−s = λ ev of A λ−s . ′

Problem 25.70 Show in this case that det(A) = e−ζA (0) .

Now for H = −∆ + V on [0, L]d with zero boundary conditions, X ζH (s) = tr H −s := λ−s λ ev of H exists for Re(s) > 1/2 (see Example 25.69 for d = 1). If ζH has an analytic continuation into a neighbourhood of s = 0, then we define ′

det H := e−ζH (0) . So defined, det H enjoys Properties 1-4 above, but not Property 5. It turns out that for H = −∆ + V , ζH does have an analytic continuation to a neighbourhood of 0, and this definition applies. It is difficult, however, to compute a determinant from this definition. In what follows, we describe some useful techniques for computation of determinants.

25.12 Operator Determinants

403

Using the formula 1 = Γ (s)

−s

λ

for each λn ∈ σ(H) leads to ζH (s) =

1 Γ (s)

Z

Z

∞

ts−1 e−tλ dt

0

∞

ts−1

0

X

e−tλn dt.

n

Now λ is an eigenvalue of H iff e−tλ is an eigenvalue of e−tH (this is an example of the spectral mapping theorem – see Section 25.9), and so e−tλn is the n-th eigenvalue of e−tH . Thus Z ∞ 1 ts−1 Tre−tH dt. ζH (s) = Γ (s) 0 formula can be useful, as it may be easier to deal with Tr(e−tH ) = RThis −tH e (x, x)dx than Tr(H −s ).

Example 25.71 We consider H = −∆ in a box B = [−L/2, L/2]d with periodic boundary conditions. In this case e−tH (x, y) ≈ (2πt)−d/2 e−|x−y|

2

/2t

in B × B, if B is very large, and so Z Z Tre−tH = e−tH (x, x)dx ≈ (2πt)−d/2 = (2πt)−d/2 vol(B). B×B

B

But calculation of det H by this method is still a problem. Remark 25.72 Often (and in all cases we consider here), we have to compute det A a ratio, det B , of determinants of two operators A and B, such that det A and det B must be defined through a regularization procedure such as the one described above, but the determinant of the ratio AB −1 can be defined det A −1 directly. Since det ), we can make sense of the ratio on the left B = det(AB hand side without going to regularization. The most useful calculational technique for us is as follows. Let A and B be Schr¨odinger operators on L2 ([0, T ]; Rd) with Dirichlet boundary conditions. Denote by JA the solution to AJA = 0 with JA (0) = 0, J˙A (0) = 1 (J a d × d matrix valued function on [0, T ]). Then one can show (see, eg., [195, 180]) det JA (T ) det A = . det B det JB (T )

(25.54)

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¯ for a critical path φ, ¯ then JA is the Jacobi Remark 25.73 If A = S ′′ (φ) ¯ matrix along φ (see Section 26.4). Problem 25.74 Let A(T ) be the operator −∂t2 +q(t) defined on L2 ([0, T ]; Rd) with Dirichlet boundary conditions, and let JA be the corresponding Jacobi matrix. Show that the functions det A(T ) and det J(T ) have the same zeros of the same multiplicities (t0 is a zero of f (t) of multiplicity n if ∂ k /∂tk f (t0 ) = 0 for k = 0, . . . , n − 1 and ∂ n /∂tn f (t0 ) 6= 0).

25.13 Tensor Products We collect here a few facts about tensor products of Hilbert spaces, and tensor products of operators and their spectra (see [244] for details and proofs). Let H1 and H2 be two separable Hilbert spaces. The tensor product of H1 and H2 is a Hilbert space H1 ⊗ H2 constructed as follows. To ψ1 ∈ H1 and ψ2 ∈ H2 , we associate a map ψ1 ⊗ ψ2 : H1 × H2 → C

(f1 , f2 ) 7→ hf1 , ψ1 iH1 hf2 , ψ2 iH2

which is conjugate linear in each component (ψ1 ⊗ ψ2 (αf1 , f2 ) = α ¯ ψ1 ⊗ ψ2 (f1 , f2 ), ψ1 ⊗ ψ2 (f1 + g1 , f2 ) = ψ1 ⊗ ψ2 (f1 , f2 ) + ψ1 ⊗ ψ2 (g1 , f2 ), and the same for the second component). On the vector space, V , of all finite linear combinations of such conjugate bilinear maps, we define an inner-product by setting hψ1 ⊗ ψ2 , φ1 ⊗ φ2 i := hψ1 , φ1 iH1 hψ2 , φ2 iH2 (25.55) and extending by linearity (it is straightforward to check that this is welldefined). Then H1 ⊗ H2 is defined to be the completion of V in the innerproduct determined by (25.55). A simple example, which appears in Section 13.1, is L2 (Rm ) ⊗ L2 (Rn ) ≈ L2 (Rm+n ) for positive integers m, n. This Hilbert space isomorphism is determined by the map f ⊗ g 7→ f (x)g(y) (see, eg., [244] for details). Given bounded operators A and B acting on H1 and H2 , the operator A ⊗ B, which acts on H1 ⊗ H2 , is defined by setting A ⊗ B(ψ1 ⊗ ψ2 ) = Aψ1 ⊗ Bψ2 , extending by linearity to all finite linear combinations of elements of this form, and then by density of these finite linear combinations, to H1 ⊗ H2 . This produces a well-defined operator.

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405

This construction can be extended to unbounded self-adjoint operators A and B, yielding a self-adjoint operator A ⊗ B (see, eg., [244]). Of particular interest for us are operators of the form A ⊗ 1 + 1 ⊗ B, acting on H = H1 ⊗ H2 , where A and B are operators acting on H1 and H2 respectively. This is an abstract version of the “separation of variables” situation of differential equations. It is intuitively clear that we should be able to reconstruct characteristics of such operators from the corresponding characteristics of the operators A and B. As an example, we have the following important (and simple) description of the spectrum of A ⊗ 1 + 1 ⊗ B under certain conditions on A and B, and, in particular, for A and B self-adjoint: σ(A ⊗ 1 + 1 ⊗ B) = σ(A) + σ(B)

σd (A ⊗ 1 + 1 ⊗ B) ⊂ σd (A) + σd (B) ⊂ { ev’s of A ⊗ 1 + 1 ⊗ B} σess (A ⊗ 1 + 1 ⊗ B) = σess (A) + σess (B) ∪ [σess (A) + σd (B)] ∪ [σd (A) + σess (B)].

Rather than prove any such statements (an involved task, requiring further assumptions), let us just do a simple, suggestive computation. Suppose Aψ1 = λ1 ψ1 and Bψ2 = λ2 ψ2 . Then note that (A ⊗ 1 + 1 ⊗ B)ψ1 ⊗ ψ2 = Aψ1 ⊗ ψ2 + ψ1 ⊗ Bψ2 = λ1 ψ1 ⊗ ψ2 + ψ1 ⊗ (λ2 ψ2 )

= λ1 (ψ1 ⊗ ψ2 ) + λ2 (ψ1 ⊗ ψ2 ) = (λ1 + λ2 )ψ1 ⊗ ψ2 ,

which shows, in particular, that σd (A) + σd (B) ⊂ { ev’s of A ⊗ 1 + 1 ⊗ B}.

25.14 The Fourier Transform The Fourier transform is a useful tool in many areas of mathematics and physics. The purpose of the present section is to review the properties of the Fourier transform, and to discuss the important role it plays in quantum mechanics. The Fourier transform is a map, F , which sends a function ψ : Rd → C to another function ψˆ : Rd → C where for k ∈ Rd , Z −d/2 ˆ ψ(k) := (2π~) e−ik·x/~ ψ(x)dx Rd

(it is convenient for quantum mechanics to introduce Planck’s constant, ~, into the Fourier transform). We first observe thatR ψˆ = F ψ is well-defined if ψ is an integrable function (ψ ∈ L1 (Rd ), meaning Rd |ψ(x)|dx < ∞), and that F acts as a linear operator on such functions.

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In the following exercise, the reader is asked to compute a few Fourier transforms. Problem 25.75 Show that under F |x|2

a|k|2

1. e− 2a~2 7→ (~a)d/2 e− 2 (Re(a) > 0). Hint: try d = 1 first – complete the square in the exponent and move the contour of integration in the complex plane. −1 1 1 2. e− 2~2 x·A x 7→ ~d/2 (det A)1/2 e− 2 k·Ak (A a positive d × d matrix). Hint: diagonalize and use the previous result. p π e−√b/~2 |x| 3. 7→ (|k|2 + b)−1 (b > 0, d = 3). Hint: use spherical coordi2~ |x| nates. Alternatively, see Problem 25.77 below. In the first example, if Re(a) > 0 then the function on the left is in L1 (Rd ), and the Fourier transform is well-defined. However, we can extend this result to Re(a) = 0, in which case the integral is convergent, but not absolutely convergent. Properties of the Fourier Transform. The utility of the Fourier transform derives from the following properties. 1. The Plancherel theorem: F is a unitary map from L2 (Rd ) to itself (note that initially the Fourier transform is defined only for integrable (L1 (Rd )) functions – the statement here is that the Fourier transform extends from L1 (Rd ) ∩ L2 (Rd ) to a unitary map on L2 (Rd )). 2. The inversion formula: the adjoint F ∗ of F on L2 (Rd ) is given by the map ψ 7→ ψˇ where Z −d/2 ˇ ψ(x) := (2π~) eix·k/~ ψ(k)dk Rd

3. 4. 5. 6.

(and by the Plancherel theorem, this is also the inverse, F −1 ). For the next four statements, suppose ψ, φ ∈ C0∞ (Rd ). ˆ [ −i~∇ x ψ(k) = k ψ(k). c ˆ xψ(k) = i~∇k ψ(k). c = (2π~)−d/2 φˆ ∗ ψ. ˆ φψ d/2 ˆ ˆ [ φ ∗ ψ = (2π~) φψ.

Here

(f ∗ g)(x) :=

Z

Rd

f (y)g(x − y)dy

(25.56)

is the convolution of f and g. The last four properties can be loosely summarized by saying that the Fourier transform exchanges differentiation and coordinate multiplication, and products and convolutions. Proof. The proof of Property 1 is somewhat technical and we just sketch it here (see, eg, [106] for details). In particular, we will show that kf k = kfˆk. Suppose f ∈ C0∞ , and let Cǫ be the cube of side length 2/ǫ centred at the

25.14 The Fourier Transform

407

origin. Choose ǫ small enough so that the support of f is contained in Cǫ . One can show that {Ek := (ǫ/2)d/2 eik·x/~ | k ∈ ǫ~πZd }

is an orthonormal basis of the Hilbert space of functions in L2 (Cǫ ) satisfying periodic boundary conditions. Thus by the Parseval equation (Proposition 25.4), Z Z X |f |2 = |f |2 = |hEk , f i|2 Rd

Cǫ

k∈ǫ~πZd

d

= (πǫ~)

X

k∈ǫ~πZd

|fˆ(k)|2 →

Z

Rd

|fˆ|2

as ǫ → 0. Problem 25.76 Show that {Ek } is an orthonormal set. We will prove Property 3, and we leave the proofs of the other properties as exercises. Integrating by parts, we have Z −d/2 d −i~∇ψ(k) = (2π~) e−ix·k/~ · (−i~∇)ψ(x)dx Z ˆ = k · (2π~)−d/2 e−ix·k/~ ψ(x)dx = k ψ(k).  Problem 25.77 1. Show that for b > 0 and d = 3, under F −1 , √ r − b/~2 |x| π e (|k|2 + b)−1 7→ 2~ |x| (hint: use spherical coordinates, then contour deformation and residue theory). 2. Show that under F −1 , δ(k − a) 7→ (2π~)−d/2 eia·x/~ . Here δ is the Dirac delta functionR – not really a function, but a distribution – characterized by the property f (x)δ(x − a)dx = f (a). The exponential function on the right hand side is called a plane wave. Functions of the derivative. As an application, we show how the Fourier transform can be used to define functions of the derivative operator. Recall our notation p := −i~∇. Motivated by Property 3 of the Fourier transform, we define an operator g(p) (for “sufficiently nice” functions g) on L2 (Rd ) as follows.

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\ ˆ Definition 25.78 g(p)ψ(k) := g(k)ψ(k) or, equivalently, g(p)ψ := (2π~)−d/2 gˇ ∗ ψ. Let us look at a few examples. Example 25.79 1. If g(k) = k, then by Property 3 of the Fourier transform, the above definition gives us back g(p) = p (so at least our definition makes some sense). \ ˆ 2. Now suppose g(k) = |k|2 . Then g(p)ψ(k) = |k|2 ψ.

d = |k|2 ψ. ˆ Problem 25.80 Show that −~2 ∆ψ

Thus we have |p|2 = −~2 ∆. Extending this example, we can define g(p) when g is a polynomial “with our bare hands”. It is easy to see that this definition coincides with the one above. 1 3. Let g(k) = ( 2m |k|2 + λ)−1 , λ > 0, and d = 3. Then due to Problem 25.77, we have m ((H0 + λ)−1 ψ)(x) = 2π~2 where we have denoted H0 :=

Z

1 2 2m |p|

R3

e−

√

2mλ |x−y| ~

|x − y|

ψ(y)dy

(25.57)

2

~ = − 2m ∆.

Problem 25.81 Let yP∈ Rd be fixed. Find how the operator eiy·p acts on functions (here y · p = dj=1 yj pj , pj = −i~∂xj ).

26 Mathematical Supplement: The Calculus of Variations

The calculus of variations, an extensive mathematical theory in its own right, plays a fundamental role throughout physics. This supplement contains an overview of some of the basic aspects of the variational calculus. This material will be used throughout the book, and in particular in Chapters 16 and 17 to obtain useful quantitative results about quantum systems in the regime close to the classical one.

26.1 Functionals The basic objects of study in the calculus of variations are functionals, which are just functions defined on Banach spaces (usually spaces of functions). (Recall that a Banach space is a complete normed vector space.) If we specify a space, X, then functionals on X are just maps S : X → R (or into C). In the calculus of variations, one often uses spaces other than L2 (Rd ), and which are not necessarily Hilbert spaces. Among the most frequently encountered spaces are the Sobolev spaces H s (Rd ), s = 1, 2, . . ., introduced in Section 25.1. Recall that the Sobolev spaces are Hilbert spaces. Spaces of continuously differentiable functions also arise frequently. For k ∈ {0, 1, 2, . . .}, and an open set Ω ⊂ Rd , we define C k (Ω; Rm ) to be the set of all functions φ : Ω → Rm such that ∂ α φ is continuous in Ω for all |α| ≤ k, and for which the norm X sup |∂ α φ(x)| kφkC k (Ω;Rm ) := |α|≤k

x∈Ω

is finite. Equipped with this norm, C k (Ω; Rm ) is a Banach space. Example 26.1 Here are some common examples of functionals, S, and spaces, X, on which they are defined.

© Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_26

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1. X = L2 ([a, b]; R), f ∈ X ∗ = X is a fixed function (X ∗ denotes the dual space to X, as explained in Section 25.1), and S : φ 7→

Z

b

f (x)φ(x)dx.

a

Note that S(φ) is well-defined, by the Cauchy-Schwarz inequality. 2. Evaluation functional: X = C([a, b]), x0 ∈ (a, b) fixed, and S : φ 7→ φ(x0 ) Compare this with the first example by taking f (x) = δ(x − x0 ) (∈ X ∗ ; in this case X ∗ 6= X). 3. Let V : Rm → R be continuous. Set X = {φ : Rd → Rm | V (φ) ∈ L1 (Rd )}, and take Z S : φ 7→

V (φ(x))dx.

Rd

4. Dirichlet functional: X = H 1 (Rd ), and Z 1 S : φ 7→ |∇φ|2 dx. 2 Rd

5. Classical action: fix x, y ∈ Rm , set X = Pxy := {φ ∈ C 1 ([0, T ]; Rm ) | φ(0) = x, φ(T ) = y}, and define S : φ 7→

Z

T 0



 1 ˙ 2 − V (φ) dt. m|φ| 2

m

6. Classical action: Let L : R × Rm → R be a twice differentiable function (a Lagrangian function), and S : φ 7→

Z

T

˙ L(φ, φ)dt.

0

Here, X = Pxy is as in Example 5. 7. Action of a classical field theory: fix f, g, ∈ H 1 (Rd ; Rm ), set X = {φ ∈ H 1 (Rd × [0, T ]; Rm) | φ(x, 0) = f (x), φ(x, T ) = g(x)}, (26.1) and define S : φ 7→

Z

0

T

  1 1 2 2 − |∂t φ| + |∇x φ| + f (φ) dxdt 2 2 Rd

Z

where f : Rm → R is a differentiable function.

26.2 The First Variation and Critical Points

411

8. Lagrangian functional: Suppose L : Rm × Rd+1 → R (a Lagrangian density) is a twice differentiable function, and set ZZ S : φ 7→ L(φ(x, t), ∇x,t φ(x, t))dxdt (here X is an appropriate space of vector functions from space-time Rdx ×Rt to Rm , whose specific definition depends on the form of L). We encountered many of these functionals when we considered the problems of quantization and of (quasi-)classical limit in quantum mechanics and quantum field theory.

26.2 The First Variation and Critical Points The notion of a critical point of a functional is a central one. It is a direct extension of the usual notion of a critical point of a function of finitely many variables (i.e., a place where the derivative vanishes). The solutions of many physical equations are critical points of certain functionals, such as action or energy functionals. In what follows, the spaces X on which our functionals are defined will generally be linear (i.e. vector) spaces or affine spaces. By an affine space, we mean a space of the form X = {φ0 + φ | φ ∈ X0 }, where φ0 is a fixed element of X, and X0 is a vector space. We will encounter examples of functionals defined on non-linear spaces when we study constrained variational problems in Section 26.5. Let X be a Banach space, or else an affine space based on a Banach space X0 . (Recall that the notion of Banach space is defined in Section 25.1 of the previous mathematical supplement.) Definition 26.2 A path, φλ , in X is a differentiable function I ∋ λ 7→ φλ from an interval I ⊂ R containing 0, into X. Definition 26.3 The tangent space, Tφ X, to X at φ ∈ X is the space of all “velocity vectors” at φ:   ∂φλ Tφ X := | φλ is a path in X, φ0 = φ . ∂λ λ=0

If X is a Banach space, then Tφ X = X (and if X is an affine space based on the Banach space X0 , Tφ X = X0 ). To see this, just note that for any ξ ∈ X, φλ := φ + λξ is a path in X, satisfying φ0 = φ and ∂φλ /∂λ = ξ (the reader is invited to check the corresponding statement for affine spaces). So the tangent space to a linear space is not very interesting. The notion of a tangent space is useful when working in non-linear spaces (we will see an example of this shortly).

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Definition 26.4 A variation of φ ∈ X along ξ ∈ Tφ X is a path, φλ in X, such that φ0 = φ and ∂φλ /∂λ|λ=0 = ξ. Example 26.5 One of our main examples is the classical action, Example 6 above (and (4.36)), defined on Pxy = {φ ∈ C 1 ([0, T ], X) | φ(0) = x, φ(T ) = y} (an affine space). Then Tφ Pxy = P00 (see Fig. 26.1), and an example of a variation of φ ∈ Pxy in the direction ξ ∈ P00 is φλ = φ + λξ ∈ Xxy . φ+λξ

ξ

y 00

φ

xy

x

Fig. 26.1. Variations of a path. We wish to define a notion of differentiation of functionals which is a direct extension of usual differentiation of functions of a finite number of variables. To do so, we use the concepts of the dual space X ∗ to X and the notation h·, ·i for the coupling between the X ∗ to X, described in Section 25.1.

Definition 26.6 Let S : X → R be a functional on a real Banach space X, and let φ ∈ X. We say that S is differentiable at φ if there is a linear functional, ∂S(φ) ∈ X ∗ , such that d S(φλ )|λ=0 = h∂S(φ), ξi dλ

(26.2)

for any variation φλ of φ along ξ ∈ X. The functional ∂S(φ) is the (variational) derivative of S at φ. Remark 26.7 1. The notion of differentiability introduced here is often called Gˆ ateaux differentiability. A stronger notion of differentiability, called Fr´echet differentiability, demands (for linear spaces) that S(φ + ξ) = S(φ) + h∂S(φ), ξi + o(kξkX ) as kξkX → 0. The reader can check that if S is continuously (Gˆ ateaux) differentiable at φ0 ∈ X (i.e. ∂S(φ) exists in a neighbourhood of φ0 , and is a continuous map from this neighbourhood into X ∗ ), then S is Fr´echet differentiable at φ0 . 2. We will sometimes use the notation S ′ (φ) for the variational derivative ∂S(φ). 3. Recall that if X is a Hilbert space, then we can identify its dual, X ∗ , with X itself, via the map X ∋ φ 7→ lφ ∈ X ∗ with lφ ξ := hφ, ξi for ξ ∈ X (here the notation h·, ·i indicates the Hilbert space inner-product).

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413

(The fact that this map is an isomorphism between X and X ∗ is known as the Riesz representation theorem.) Thus when X is a Hilbert space, we can identify ∂S(φ) with an element of X. This element is called the gradient of S at φ (in the inner product of X), and is sometimes denoted by gradX S(φ). Example 26.8 We compute the derivatives of some of the functionals in Example 26.1. We suppose that whenever X is a space of functions, it is a subspace of an L2 -space. Then the variational derivative can be identified with a function (or distribution), using integration by parts where necessary. This is related to the L2 gradient as discussed above. Rb 1. For the functional S(φ) = a f φdx, we compute Z b d ∂ S(φλ )|λ=0 = f φλ dx|λ=0 dλ ∂λ a Z b = f ξdx = hf, ξi.

h∂S(φ), ξi =

a

Thus we identify ∂S(φ) = f . R 3. For S(φ) = Rd V (φ), we compute

Z d d ∂ S(φλ )|λ=0 = V (φλ )|λ=0 dx dλ dλ Rd ∂λ Z ∇V (φ) · ξdx = h∇V (φ), ξi =

h∂S(φ), ξi =

Rd

and so we identify ∂S(φ) = ∇V (φ). R 4. For S(φ) = 12 Rd |∇φ|2 , we compute

Z 1 ∂ d S(φλ )|λ=0 = |∇φλ |2 |λ=0 dx h∂S(φ), ξi = dλ 2 Rd ∂λ Z Z (−∆φ)ξdx = h−∆φ, ξi ∇φ · ∇ξdx = = Rd

Rd

where we integrated by parts (Gauss theorem), and used the fact that the functions decay at ∞. Thus we identify ∂S(φ) = −∆φ. We leave the remaining examples as an exercise. Problem 26.9 Compute the variational derivatives for the remaining functionals in Example 26.1. You should find 5. 6. 7. 8.

∂S(φ) = −mφ¨ − ∇V (φ) d ∂S(φ) = − dt (∂φ˙ L) + ∂φ L ∂S(φ) = φ + ∇f (φ) where  := ∂t2 − ∆ is the D’Alembertian operator ∂S(φ) = −∇x,t · (∂∇φ L) + ∂φ L

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As in the finite-dimensional case, a critical point is a place where the derivative vanishes. Definition 26.10 An element φ ∈ X is a critical point (CP) of a functional S : X → R if ∂S(φ) = 0. In fact, many physical equations are critical point equations for certain functionals. Example 26.11 Continuing with the same list of examples of functionals above, we can write down some of the equations describing their critical points: 4. 5. 6. 7. 8.

∆φ = 0 (Laplace equation, φ a harmonic function) mφ¨ = −∇V (φ) (Newton’s equation) d dt (∂φ˙ L) = (∂φ L) (Euler-Lagrange equation) φ + ∇f (φ) = 0 (nonlinear wave/Klein-Gordon equation) −∇x,t · (∂∇φ L) + ∂φ L = 0 (classical field equation)

The following connection between critical points and minima (or maxima) is familiar from multi-variable calculus. Theorem 26.12 If φ locally minimizes or maximizes a differentiable functional S : X → R, then φ is a critical point of S. (We say φ is a local minimizer ˜ ≥ S(φ) (resp. (resp. maximizer) of S if there is some δ > 0 such that S(φ) ˜ ˜ ˜ S(φ) ≤ S(φ)) for all φ with kφ − φkX < δ.) Problem 26.13 Prove this (hint: it is similar to the finite-dimensional case). Recall that a function f (v) is called strictly convex if f (sv + (1 − s)v ′ ) < sf (v) + (1 − s)f (v ′ ), ∀s ∈ (0, 1), ∀x ∈ X, v, v ′ ∈ V . This condition holds if f is twice differentiable and has positive Hessian, d2v f (v) > 0, ∀v ∈ V . A function f (v) is called strictly concave iff −f (v) is strictly convex. Theorem 26.14 Assume f (v) is a differentiable and strictly convex/concave function on a finite-dimensional space V . Then it has a unique critical point, and this critical point minimizes/maximizes f . We sketch a proof of this theorem. To fix ideas we consider only the convex case. Assume for simplicity that f (v) is twice differentiable. Then as was mentioned above d2v f (v) > 0, ∀v ∈ V . Hence every critical point is a (local) minimum. Assume there are two critical points, v1 and v2 . Then the function f (sv1 + (1 − s)v2 ) would have a maximum for some s ∈ (0, 1), a contradiction. Hence f (v) has at most one critical point. One can show furthermore that f (v) → ∞, as kvk → ∞ and therefore f (v) has at least one minimizing point.  To extend this theorem to the infinite-dimensional case one would have to make some additional assumptions, i.e. that f (v) is weakly lower semicontinuous and V is reflexive (see e.g. [244]).

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415

26.3 The Second Variation In multi-variable calculus, if one wishes to know if a critical point is actually a minimum (or maximum), one looks at the second derivative. For the same reason, we need to define the second derivative of a functional. Definition 26.15 Let η, ξ ∈ Tφ X. A variation of φ along η and ξ is a two∂ parameter family, φλ,µ ∈ X, such that φ0,0 = φ, ∂λ φλ,µ |λ=µ=0 = ξ, and ∂ φ | = η. ∂µ λ,µ λ=µ=0 Definition 26.16 Let S : X → R be a functional. We say S is twice differentiable is there is a bounded linear map ∂ 2 S(φ) : Tφ X → (Tφ X)∗ (called the Hessian or second variation of S at φ) such that ∂2 S(φλ,µ )|λ=µ=0 = h∂ 2 S(φ)η, ξi ∂λ∂µ

(26.3)

for all ξ, η ∈ Tφ X and all variations φλ,µ of φ along ξ and η. Remark 26.17 1. The Hessian ∂ 2 S(φ) can also be defined as the second derivative of S(φ), i.e., ∂ 2 S(φ) = ∂ · ∂S(φ). That is, we consider the map ∂ φ 7→ ∂S(φ) and define, for η ∈ Tφ X, ∂ 2 S(φ)η := ∂λ ∂S(φλ ), where φλ is a variation of φ along η. 2. We will often use the notation S ′′ (φ) to denote ∂ 2 S(φ). Computations of the second derivatives of the functionals in our list of examples are left as an exercise (again, we suppose where appropriate that the action of the dual space is just given by integration). Problem 26.18 Continuing with our list of examples of functionals above, show that 3. S ′′ (φ) = D2 V (φ) (a matrix multiplication operator). 4. S ′′ (φ) = −∆ (the Laplacian). 5. S ′′ (φ) = −m∂t2 − D2 V (φ) (a Schr¨odinger operator) acting on functions satisfying Dirichlet boundary conditions: ξ(0) = ξ(T ) = 0. 6.   d 2 d 2 d ′′ ∂ L + ∂φ2 L, (26.4) S (φ) = − (∂φ˙ L) − dt dt dt φφ˙ with Dirichlet boundary conditions. (The first term on the r.h.s. is a product of three operators while the second one is the time-derivative of ∂φ2φ˙ L. 7. S ′′ (φ) =  + V ′′ (φ), with Dirichlet boundary conditions: ξ(x, 0) = ξ(x, T ) = 0. 8.  2    ∂ L ∂2L ∂2L S ′′ (φ) = −∇x,t + ∇ − ∇ . x,t x,t 2 ∂(∇φ) ∂(∇φ)∂φ ∂φ2

The following criterion for a critical point to be a minimizer is similar to the finite-dimensional version, and the proof is left as an exercise.

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Theorem 26.19 Let φ be a critical point of a twice continuously differentiable functional S : X → R.

1. If φ locally minimizes S, then S ′′ (φ) ≥ 0 (meaning hS ′′ (φ)ξ, ξi ≥ 0 for all ξ ∈ Tφ X). 2. If S ′′ (φ) > c, for some constant c > 0 (i.e. hS ′′ (φ)ξ, ξi ≥ ckξk2X for all ξ ∈ Tφ X), then φ is a local minimizer of S.

Problem 26.20 Prove this. Let us now pursue the question of whether or not a critical point of the classical action functional Z T ˙ S(φ) = L(φ(s), φ(s))ds 0

(which is a solution of the Euler-Lagrange equation – i.e., a classical path) minimizes the action. As we have seen, the Hessian S ′′ (φ) is given by (26.4). We call ∂φ2˙ L the generalized mass. Theorem 26.21 Suppose ∂φ2˙2 L > 0. Suppose further that ∂φ2 L is a bounded function. Then there is a T0 > 0, such that S ′′ (φ) > 0 for T ≤ T0 . ′′ 2 2 ′′ ˙2 Proof for L = m 2 φ −V (φ). In this case S (φ) = −md /ds −V (φ), acting on L2 ([0, T ]) with Dirichlet boundary conditions. Since inf σ(−d2 /ds2 ) = (π/T )2 , we have, by Theorem 8.2, −d2 /ds2 ≥ (π/T )2 . So S ′′ (φ) ≥ m(π/T )2 − max |V ′′ |, which is positive for T sufficiently small. 

Corollary 26.22 For T sufficiently small, a critical point of S (i.e., a classical path) locally minimizes the action, S.

26.4 Conjugate Points and Jacobi Fields In this section we study the classical action functional and its critical points (classical paths) in some detail. While such a study is of obvious importance in classical mechanics, it is also useful in the quasi-classical analysis of quantum systems that we undertook in Chapters 16 and 17. Thus we consider the action functional Z t ˙ S(φ) = L(φ(s), φ(s))ds. 0

We have shown above that if t is sufficiently small, then S ′′ (φ) > 0, provided (∂ 2 L/∂ φ˙ 2 ) > 0. So in this case, if φ¯ is a critical path, it minimizes S(φ). On the other hand, Theorem 26.19 implies that if φ¯ is a critical path such that ¯ has negative spectrum, then φ¯ is not a minimizer. We will show later S ′′ (φ) ¯ decrease monotonically as t increases. So the point that eigenvalues of S ′′ (φ)

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¯ becomes zero, separates the t’s for t0 when the smallest eigenvalue of S ′′ (φ) ¯ which φ is a minimizer, from those for which φ¯ has lost this property. The ¯ becomes zero play a special points at which one of the eigenvalues of S ′′ (φ) role in the analysis of classical paths. They are considered in this section. ¯ is In this discussion we have used implicitly the fact that because S ′′ (φ) 2 a Schr¨odinger operator defined on L ([0, t]) with Dirichlet (zero) boundary conditions, it has a purely discrete spectrum running off to +∞. We denote ¯ this spectrum by {λn (t)}∞ 1 with λn (t) → +∞ as n → ∞. Note that if φ is a ¯ ¯ critical point of S on [0, t], then for τ < t, φτ := φ|[0,τ ] is a critical point of S ¯ on [0, τ ] on [0, τ ]. Thus for τ ≤ t, {λn (τ )} is the spectrum of S ′′ (φ¯τ ) = S ′′ (φ) with zero boundary conditions. We specialize now to the classical action functional Z t m ˙ 2 S(φ) = { |φ(s)| − V (φ(s))}ds 2 0 on the space X = {φ ∈ C 1 ([0, t]; Rd ) | φ(0) = x, φ(t) = y}, and continue to ¯ a critical point of this functional (classical path). denote by φ, Theorem 26.23 The eigenvalues λn (τ ) are monotonically decreasing in τ . Sketch of proof. Consider λ1 (τ ), and let its normalized eigenfunction be ψ1 . Define ψ˜1 to be ψ1 extended to [0, τ + ǫ] by 0. So by the spectral variational principle Theorem 8.1, ¯ ψ˜1 i = λ1 (τ ). λ1 (τ + ǫ) ≤ hψ˜1 , S ′′ (φ) Further, equality here is impossible by uniqueness for the Cauchy problem for ordinary differential equations, which states the following: if a solution of a linear, homogeneous, second-order equation is zero at some point, and its derivative is also zero at that point, then the solution is everywhere zero. To extend the proof to higher eigenvalues, one can use the min-max principle.  ¯ 0 ) such that λn (τ0 ) = 0 for some n is called a Definition 26.24 A point φ(τ ¯ ¯ conjugate point to φ(0) = x along φ. ¯ 0 ) is a conjugate point to x, then S ′′ (φ) ¯ on [0, τ0 ] has a 0 So if c = φ(τ 2 eigenvalue. That is, there is some non-zero ξ ∈ L ([0, τ0 ]) with ξ(0) = ξ(τ0 ) = 0 such that ¯ = 0. S ′′ (φ)ξ (26.5) This is the Jacobi equation. A solution of this equation with ξ(0) = 0 will be called a Jacobi vector field. ¯ is the number of negative eigenvalues Definition 26.25 The index of S ′′ (φ) 2 it has (counting multiplicity) on L ([0, t]) with zero boundary conditions. ¯ has no zero eigenvalues on [0, τ ] (TheoWe recall that for τ small, S ′′ (φ) rem 26.21). Combining this fact with Theorem 26.23 gives the following result.

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¯ is equal to the number of Theorem 26.26 (Morse) The index of S ′′ (φ) ¯ ¯ points conjugate to φ(0) along φ, counting multiplicity (see Fig. 26.2).

spec[S"(φ t )]

t

Fig. 26.2. Index = # of conjugate points. The picture that has emerged is as follows. For sufficiently small times, a ¯ classical path φ(0) locally minimizes the action. As time increases, the path might lose this property. This happens if there is a point in the path conjugate ¯ to φ(0). Example 26.27 An example of a conjugate point is a turning point in a one-dimensional potential (see Fig. 26.3, and remember that we are working with the functional of Example 26.1, no. 5).

V(x) φ

x a

b

Fig. 26.3. A turning point. The classical path φ starts at a, and turns back after hitting b at time τ . Now S ′′ (φ) = −m∂s2 − V ′′ (φ) and it is easy to check that S ′′ (φ)φ˙ = 0 (just differentiate Newton’s equation). ˙ ˙ ) = 0 (the velocity at a turning point is zero), b is conjugate Since φ(0) = φ(τ to a. We return to the Jacobi equation (26.5), and consider its fundamental solution, J(s). J(s) is the d × d matrix satisfying ¯ =0 S ′′ (φ)J

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419

with the initial conditions J(0) = 0

and

˙ J(0) = 1.

J is called the Jacobi matrix. Proposition 26.28 The Jacobi matrix has the following properties 1. For any h ∈ Rd , Jh is a Jacobi field. Conversely, any Jacobi field is of the form Jh for some h ∈ Rd . ¯ 0 ) is a conjugate point to φ(0) ¯ 2. φ(τ iff J(τ0 ) has a zero-eigenvalue, i.e. det J(τ0 ) = 0. Proof. 1. The first part is obvious. To prove the second part let ξ be a ˙ Jacobi field, and let h = ξ(0). Then ξ˜ := Jh satisfies the same differential ˜ and therefore equation as ξ with the same initial conditions. Hence ξ = ξ, ξ = Jh. ¯ 0 ) is a conjugate point iff there is a Jacobi 2. We have shown above that φ(τ field ξ such that ξ(τ0 ) = 0. By the previous statement, there is h 6= 0 such that ξ = Jh, which implies J(τ0 )h = 0. So J(τ0 ) has a zero eigenvalue (with eigenvector h), and det J(τ0 ) = 0.  Now we give the defining geometric/dynamic interpretation of J. Consider ¯ a family of critical paths φv (s) starting at φ(0) with various initial velocities ˙ d ¯ v ∈ R . Denote v¯ = φ(0). Then J(s) =

∂φv (s) |v=¯v ∂v

¯ Indeed, φv satisfies the equation ∂S(φv ) = is the Jacobi matrix (along φ). 0. Differentiating this equation with respect to v, and using that S ′′ (φ) = ∂φ ∂S(φ), we find ∂ ∂φv 0= ∂φ S(φv ) = S ′′ (φv ) . ∂v ∂v Thus, ∂φv /∂v|v=¯v satisfies the Jacobi equation. Next, ∂ ¯ ∂ φv (0) = φ(0) = 0 ∂v ∂v and

which completes the proof.

∂ ∂ ˙ φv (0) = v=1 ∂v ∂v

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26.5 Constrained Variational Problems Let S and C be continuously differentiable functionals on a real Banach space X. We consider the problem of minimizing the functional S(φ), subject to the constraint C(φ) = 0. That means we would like minimize S(φ) for φ in the (non-linear) space M := {φ ∈ X | C(φ) = 0}.

We assume that C ′ (φ) 6= 0 for φ ∈ M (here, and below, C ′ (φ) and S ′ (φ) denote the variational derivatives of the respective functionals considered as functionals on all of X, rather than just M ). If C is a C 2 (twice continuously differentiable) functional, then (see eg. [122]) Tφ M = {ξ ∈ X | hC ′ (φ), ξi = 0}.

To see this, suppose φλ is a variation of φ in M . Differentiating the relation C(φλ ) = 0 with respect to λ at λ = 0 yields hC ′ (φ), ξi = 0, where ξ = ∂ ′ ∂λ φλ |λ=0 . Thus Tφ M ⊂ {ξ ∈ X | hC (φ), ξi = 0}. Conversely, given ξ ∈ X ′ such that hC (φ), ξi = 0, one can show (using the “implicit function theorem”) ∂ φλ |λ=0 = ξ. So ξ ∈ Tφ M . that there is a path φλ ∈ M satisfying φ0 = φ and ∂λ Concerning the constrained variational problem, we have the following result: Theorem 26.29 (Lagrange multipliers) Let S be a C 1 and C a C 2 functional on a Banach (or affine) space X. Suppose φ¯ locally minimizes S(φ) subject to the constraint C(φ) = 0 (i.e. φ¯ locally minimizes S on the space ¯ 6= 0. Then φ¯ is a critical point of the functional S − λC on M ) and C ′ (φ) the space X, for some λ ∈ R (called a Lagrange multiplier). In other words, φ¯ satisfies the equations ¯ = λC ′ (φ) ¯ S ′ (φ)

and

¯ =0 C(φ)

(the first equation is as linear functionals on X). Proof. The fact that φ¯ minimizes S over M implies that φ¯ is a critical point ¯ = 0 on T ¯M . of S considered as a functional on M . This means that S ′ (φ) φ Recall ¯ ξi = 0}. Tφ¯M = {ξ ∈ X | hC ′ (φ), ¯ ρi 6= 0. Then for all ξ ∈ X, Let ρ ∈ X be such that hC ′ (φ), η := ξ −

¯ ξi hC ′ (φ), ¯ ρi ρ ∈ Tφ¯M. hC ′ (φ),

Hence ¯ ηi = hS ′ (φ), ¯ ξi − λhC ′ (φ), ¯ ξi, 0 = hS ′ (φ), ¯ is a multiple of C ′ (φ).  Thus S ′ (φ)

λ :=

¯ ρi hS ′ (φ), ′ ¯ ρi . hC (φ),

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Example 26.30 Quadratic form: let B be a a self-adjoint operator on a Hilbert space, X = D(B), and set S : φ 7→

1 hφ, Bφi 2

(26.6)

and C(φ) := 21 (kφk2 − 1). Easy computations show ∂S(φ) = Bφ and ∂C(φ) = φ. Hence by the result above, any critical point of S(φ) subject to the constraint C(φ) = 0 (kφk = 1) satisfies the equation Bφ = λφ

(26.7)

for some Lagrange multiplier λ ∈ R. This is an eigenvalue equation for B with the eigenvalue being the Lagrange multiplier.

26.6 Legendre Transform and Poisson Bracket Passing from the lagrangian to hamiltonian constitutes a Legendre transform, defined formally as: g(π) := sup (hπ, ui − f (u)). (26.8) u∈X

Here f is a function (or functional) on a normed vector space, V , while g is defined on the dual space V ∗ , and hπ, ui is a coupling between V and V ∗ (See the previous mathematical supplement, Section 25.1 for definitions of the dual space and the coupling hπ, ui.) Of course, for the supremum in (26.8) to exist we have to make some assumptions on on the class of functions on which the Legendre transform is defined. In the finite-dimensional case, it suffices to assume that f is differentiable and strictly convex, i.e. ∀v, v ′ ∈ V, f (sv + (1 − s)v ′ ) < sf (v) + (1 − s)f (v ′ ), ∀s ∈ (0, 1). In the infinite-dimensional case, we have to make extra assumptions. In order not to complicate the exposition, we make an assumption which is much stronger than needed, but which suffices for our needs. Namely, we assume that f is of the form f (u) := 12 hu, Lui where the linear operator L : V → V ∗ is invertible and satisfies hv, Lwi = hLv, wi and hv, Lvi ≥ δkvk2 , δ > 0. In this case, like in the finite-dimensional one, the function hπ, ui − f (u) has a unique critical point satisfying is ∂f (u) = π, (in this case, u = L−1 π), and this point is a maximum point. Hence g(π) = (hπ, ui − f (u))|u:∂f (u)=π . Problem 26.31 Show that g is differentiable, convex and that (Legendre transform)2 = 1. Hint: use the fact that ∂g(π) = u(π) where u(π) solves ∂f (u(π)) = π.

(26.9)

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ProblemR26.32 Show that the Legendre transform maps the functional R f (ψ) = 12 |ψ|2 on X = L2 (Rd ) into g(π) = 12 |π|2 .

The equation (26.9) shows that the classical and Klein-Gordon Hamiltonians, (4.45) and  Z  1 2 1 2 H(φ, π) = |π| + |∇φ| + f (φ) dx (26.10) 2 2 Rd

(see (21.11)), are the Legendre transforms of the classical and Klein-Gordon Lagrange functionals, (4.37) and  Z  1 2 1 2 L(φ, χ) = |χ| − |∇φ| − f (φ) dx 2 2 in the second variables, h(x, k) := sup (hk, vi − L(x, v)).

(26.11)

H(φ, π) := sup (hπ, ηi − L(φ, η)).

(26.12)

v∈V

and η∈V

Now we consider the Poisson brackets, which were defined in Section 4.7. If Z is a real inner-product space, on which there is a linear invertible operator J : Z ∗ → Z such that J ∗ = −J (J is called a symplectic operator), then we can define a Poisson bracket of functions (or functionals) F and G as {F, G} := h∂F, J∂Gi.

(26.13)

For one-particle Classical Mechanics with the phase space Z = R3 × R3 , the symplectic operator is   0 1 J= , (26.14) −1 0 yielding the Poisson bracket (4.47). For the phase space Z = H 1 (Rd , Rm ) × H 1 (Rd , Rm ), the symplectic operator is (26.14) (but defined on a different space), yielding the Poisson bracket Z {F, G} = {∂π F · ∂φ G − ∂φ F · ∂π G}dx. (26.15) Suppose that Z = X × V ∗ is a space of functions Φ(x) = (φ(x), π(x)) on R . Recall that the functional on X which maps X ∋ φ 7→ φ(x) is called the evaluation functional (at x), which we denote (with some abuse of notation) as φ(x), and similarly for V . Consider the Hamilton equations d

Φ˙ t (x) = {Φ(x), H}(Φt )

(26.16)

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423

where {F, G} is a Poisson bracket on Z. If this Poisson R structure is given by a symplectic operator J as in (26.13) with hφ, πi := φ(x) · π(x)dx, then, by Problem 21.4, Z {Φ(x), H}(Φ) = (∂Φ(x))(y)J∂H(Φ)(y)dy Z = δx (y)J∂H(Φ)(y) = J∂H(Φ)(x) which gives {Φ, H}(Φ) = J∂H(Φ),

which leads to Hamilton’s equation

Φ˙ = J∂H(Φ).

(26.17)

Example 26.33 The Klein-Gordon Hamiltonian theory: the phase space is X = H 1 (Rd , Rm ) × H 1 (Rd , Rm ), the Poisson bracket (26.13), with J given in (26.14), and the Hamiltonian given by (26.10). So for Φ(t) = (φ(t), π(t)) a path in H 1 (Rn ) × H 1 (Rn ), Equation (26.17) is     −∆φ + ∇f (φ) φ˙ =J , π π˙ and we recover the Klein-Gordon equation φ¨ = ∆φ + ∇f (φ). For a general Hamiltonian system (Z, {·, ·}, H) Hamilton’s equation can be written as Φ˙ = {Φ, H}, (26.18)

where we identified the derivation F → {F (Φ, t), H} with the vector field XH on Z, determined by this derivation. The map t → Φt (Φ0 ), where Φt (Φ0 ) is the solution to (26.18) with the initial condition Φ0 is called the flow generated by (26.18). Consider the equation d F (Φ, t) = {F (Φ, t), H}, dt

(26.19)

with an initial condition F (Φ, 0) = F0 (Φ), where F0 (Φ) is a smooth functional on Z. The solution of this equation for various F0 (Φ) defines the flow Φt (Φ0 ) on Z, by the equation F0 (Φt (Φ0 )) = F (Φ0 , t). This is the flow for the vector field XH on Z. We conclude this section with brief remarks about an important classical field theory which does not fit the above framework, the Schr¨odinger CFT. Assume V = W ⊕ W and define the Lagrangian on V ⊗ V by L(ψ, χ) = hJ −1 ψ, χi − E(ψ), where J : V ∗ → V is a symplectic operator as above (i.e. a linear invertible operator satisfying J ∗ = −J, e.g. (26.14) w.r.to the decomposition V = W ⊕

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W ) and E(ψ) is a functional on V . The generalized momentum now is π = J −1 ψ. Then using the equations H(ψ, π) = (hπ, χi − L(ψ, χ))|χ:∂χ L(ψ,χ)=π

(26.20)

and ∂χ L(ψ, χ) = J −1 ψ we see that hπ, χi and hJ −1 ψ, χi cancel and we obtain H(ψ) = E(ψ).

(26.21)

The phase space here is V and the Poisson bracket is given by (26.13) (where the derivatives are understood to be w.r.to ψ).

26.7 Complex Hamiltonian Systems In this section we sketch the Lagrangian and Hamiltonian formalism on complex Banach spaces. We begin with a specific complex Banach space and discuss an abstract case briefly at the end of this section. We consider the space X = {φ ∈ H 1 (Rd × [0, T ]; C) | φ(x, 0) = h(x), φ(x, T ) = g(x)}, where h, g, ∈ H 1 (Rd ; C) are fixed functions. We identify X with the real space (26.1) with m = 2 (i.e. with Re X ⊕ Im X), as → − φ ⇐⇒ φ := (φ1 , φ2 ), φ1 := Re φ, φ2 := Im φ.

With this identification, we can define the variational (or Gˆateaux or Fr´echet) − S(φ), for any functional S(φ) on X, and differentiability and derivative, ∂→ φ specifically partial derivatives, ∂φ1 S(φ) and ∂φ2 S(φ), with respect the real, φ1 , and imaginary, φ2 , parts of the field φ. After that we introduce the derivatives with respect φ and φ¯ as follows ∂φ S(φ) := ∂φ1 S(φ) − i∂φ2 S(φ)

∂φ¯S(φ) := ∂φ1 S(φ) + i∂φ2 S(φ). (26.22)

To fix ideas, we consider the complex Klein-Gordon and Schr¨odinger field theories. Complex Klein-Gordon CFT. Here the action is given as in the item (7) of Example 26.1, i.e. by S(φ) :=

Z

T 0

Z

Rd

 |∂t φ|2 − |∇x φ|2 − f (φ) dxdt,

where f : C → R is a continuous function. The equation ∂φ¯S(φ) = 0 for critical points gives the complex non-linear Klein-Gordon equation, as follows from

26.7 Complex Hamiltonian Systems

425

∂φ¯ S(φ) = −∂t2 φ + ∆x φ − ∂φ¯ f (φ). Here ∂φ¯ f (φ) is the regular complex derivative of a function. With the La R  ˙ := d |∂t φ|2 − |∇x φ|2 − f (φ) , the generalized momengrangian L(φ, φ) R ˙ = ∂t φ. ¯ This gives the same Hamiltonian as in (26.10). tum is π = ∂φ˙ L(φ, φ) The Legendre transform is defined as above, (26.8). The Poisson bracket is given now by again by (26.15), but with complex derivatives w.r.to φ and π. Schr¨ odinger CFT. The action for the nonlinear Schr¨odinger classical field theory, with the self-interaction described by a differentiable function f : C → R, is given by Z TZ  ¯ t ψ) − |∇x ψ|2 − V |ψ|2 − f (ψ) dxdt, S(ψ) := − Im(ψ∂ 0

Rd

RT R ¯ t ψ)dxdt, modulo an on the same space as above. (We think of 0 Rd Im(ψ∂ R T R −1 RT R 1 ¯ t ψdxdt or ¯ additive constant, as either 0 Rd i ψ∂ ∂ ψψdxdt.) The 0 Rd i t critical point equation ∂φ¯S(ψ) = 0 gives the non-linear Schr¨odinger equation. Indeed, we compute

∂ψ¯ S(ψ) = i∂t ψ − (−∆x + V )ψ − ∂ψ¯ f (ψ). R  ¯ t ψ) − |∇x ψ|2 − V |ψ|2 − f (ψ) , ˙ := d − Im(ψ∂ With the Lagrangian L(ψ, ψ) R ˙ = ψ. ¯ This gives the Hamiltonian the generalized momentum is π = ∂ψ˙ L(ψ, ψ) Z  ¯ := H(ψ, ψ) |∇x ψ|2 + V |ψ|2 + f (ψ) dx. Rd

The Legendre transform is as above and it gives the Hamiltonian above. In this case, the phase space is the complex Hilbert space, H 1 (Rd , C) and the Poisson bracket is given now by Z {F, G} = {∂ψ F ∂ψ¯ G − ∂ψ¯ F ∂ψ G}dx. (26.23)

The symplectic operator is J = multiplication by − i, so that {F, G} := Reh∂F, (−i)∂Gi. Finally the Hamilton equation for this system is ψ˙ = {ψ, H}, or in detail, i∂t ψ = (−∆x + V )ψ + ∂ψ¯ f (ψ). (26.24)

Now we describe briefly an abstract construction. Let X be a complex Banach space, or else an affine space based on a complex Banach space X0 . Assume there is an antilinear map, C, on X satisfying C 2 = 1, so that one 1 (1 − C)X, can define a real part, X1 := 12 (1 + C)X, and imaginary, X2 := 2i 1 1 part of X ( 2 (1 + C) and 2 (1 − C) are projection operators). Then we can identify X with the real space X1 ⊕ X2 , as → − 1 1 φ ⇐⇒ φ := (φ1 , φ2 ), φ1 := (1 + C)φ, φ2 := (1 − C)φ. 2 2i With this identification, we can define the variational (or Gˆateaux or Fr´echet) differentiability and derivatives with respect to φ and φ¯ as in (26.22)

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26.8 Conservation Laws As in classical mechanics, an observable F (Φ) is conserved or is a constant of motion, i.e. F (Φt ), where Φt is a solution of the Hamilton equation, is independent of t, if and only if its Poisson bracket with the Hamiltonian, H, vanishes, {F, H} = 0. This follows from the equation (21.15), d F (Φt ) = {F (Φt ), H}. dt

(26.25)

As in classical mechanics, conservation laws often arise from symmetries. Assume a system has a symmetry in the sense that there exists a one-parameter group τs , s ∈ R, of bounded operators on the space V , s.t. Ts H = H, ∀s ∈ R, where ′ Ts H(φ, π) := H(τs φ, τ−s π),

with τs′ being the dual group action of τs on V ∗ , defined by hτs′ π, φi = hπ, τs φi (recall that h·, ·i is the coupling between X and X ∗ ). Let A be the generator of the group τs , ∂s τs = Aτs . Then the classical observable Q(φ, π) := hπ, Aφi has vanishing Poisson bracket (”commutes”) with the Hamiltonian, ′ {H, Q} = ∂s H(τs φ, τ−s π)|s=0 = 0,

and consequently is conserved under evolution: Q(φt , πt ) is a constant in t where Φt = (φt .πt ) is a solution to (26.18). In analogy with quantum mechanics we formulate this as Us is a symmetry of (26.18) → hπ, Aφi is conserved For the Schr¨odinger CFT, the one-parameter group τs , s ∈ R, can be chosen to be unitary and the dual group is given by τs′ = Cτ−s C, where C is the ¯ we have τ ′ π = τs ψ. Hence the complex conjugation, so that, since π = ψ, −s operator Ts acts on H as Ts H(ψ, ψ) := H(τs ψ, τs ψ). The conserved classical ¯ := hψ, iAψi where A is the (anti-self-adjoint) observable is defined as Q(ψ, ψ) generator of the group τs , ∂s τs = Aτs , and hψ, φi is the standard scalar product on L2 (Rd , C). As above, it has vanishing Poisson bracket (”commutes”) with the Hamiltonian, {H, Q} = ∂s H(τs φ, τs ψ)|s=0 = 0, and consequently is conserved under evolution: Q(ψt , ψ¯t ) is a constant in t where ψt is a solution to (26.24). We list some examples of symmetries and corresponding conservation laws for a CFT with the phase space Z = H 1 (Rd , C) × H 1 (Rd , C) • •

Time translation invariance ((τs ψ)(x, t) := ψ(x, t + s), s ∈ R) → conser¯ vation of energy, H(ψ, ψ); Space translation invariance ((τs ψ)(x, t) := ψ(x + s, t), s ∈ Rd ) → conserR ¯ ¯ vation of momentum P (ψ, ψ) := ψ(−i∇)ψdx;

26.8 Conservation Laws

• •

427

−1 Space rotation invariance ((τR ψ)(x, t) := ψ(RP x, t), R ∈ SO(Rd )) → ¯ ¯ ∧ −i∇)ψdx; conservation of angular momentum L(ψ, ψ) := ψ(x iγ Gauge invariance ((τγ ψ)(x, t)R := e ψ(x, t), γ ∈ R) → conservation of ‘charge/number of particles’ |ψ|2 dx.

Note that, except for the first case, the families above are multi-parameter groups, but as mentioned in Section 4.7, they can be reduced to one-parameter ones. As an example we also give the momentum field on the phase space Z = H 1 (Rd , R) × H 1 (Rd , R): Z P (φ, π) := π∇φdx. (26.26) Sometimes real theories have complex representations and these complex theories have gauge symmetries which lead to the conservation of the number of particles, which is not obvious in the original real representation. We encountered an example of this phenomenon in Section 21.2, where we found that the real Klein-Gordon theory is equivalent to a complex theory on the space H 1 (Rd , C) with the latter theory having gauge symmetry, resulting in the conservation of the number of particles.

27 Comments on Literature, and Further Reading

General references There is an extensive literature on quantum mechanics. Standard books include [36, 186, 226, 261]. More advanced treatments can be found in [257, 41, 81]. For rigorous treatments of quantum mechanics see [39, 291, 94, 285, 287, 288, 285, 79]. Mathematical background is developed in the following texts: [17, 42, 73, 106, 162, 204, 244, 245]. Further mathematical developments and open problems are reviewed in [95, 169, 201, 208, 273]. Chapter 2 The definition of self-adjointness given in this section and in Section 25.5 of Mathematical Supplement is different from the one commonly used, but is equivalent to it. It allows for a straightforward verification of this property, avoiding an involved argument. For more on the semi-classical approximation see e.g. [250]. Chapter 4 For a discussion of the relation between quantization and pseudodifferential operators and semi-classical asymptotics, see [39, 49, 50, 51, 78, 95, 158, 164, 250, 263]. Detailed discussions of identical particles can be found in most of the books on quantum mechanics. For a particularly nice discussion of the relation between spin and statistics see [115]. The orthogonal, unitary and symmetric groups and their representations and algebras (in the first two cases) are discussed in most of the books on the group theory, of which there is a considerable number. © Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3_27

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Chapter 5 Extensive development and discussion of coherent states can be found in [70, 150] For more discussion about the relation between the uncertainty principle and the stability of atoms, see [95, 199]. For stability of bulk matter see [95, 199, 201] and the book [208] (the papers [199, 201] can be found in [203]). Chapter 9 The wave operators for short-range interactions were introduced in [228], and for long-range interactions, in [82, 52]. Scattering theory is presented in the textbooks mentioned above, as well as in [136, 221, 232]. Modern mathematical treatment of scattering theory can be found in [169, 77, 302]. Chapter 10 Certain aspects of the theory of Coulomb systems (atoms, molecules, and aggregates thereof) are reviewed in [95, 199, 202, 203, 264, 291]. The result on existence of bound states of atoms is due to G.M. Zhislin. Existence of the hydrogen molecule was proven in [249]. Chapter 11 The Feshbach projection method was introduced by H. Feshbach in connection with perturbation problems in Nuclear Physics (see [226]) and by I. Schur in theory of matrices. A similar method, called the method of Lifshitz matrix, was developed independently in systems theory and in linear partial differential equations, where it is called the Grushin problem. See [27, 139] for further extensions and historical remarks The abstract results presented in this chapter follow the work [29], which introduced the notion of Feshbach-Schur map FP : H → FP (H) and used it as a basis of a renormalization group (RG) approach to spectral problems (which is presented in Chapter 24). There is an extensive physics literature on Zeeman effect, and on perturbation of atoms by time-periodic electric fields (see e.g. [186, 226]). The first rigorous results on the Zeeman problem were obtained in [24] and the time-periodic one, in [303, 165]. See [73] and [168] for further discussion and references. A rigorous approach to the Fermi Golden Rule was proposed by Simon ([270]). For more on perturbation theory, see [35, 176, 248, 247] and the recent comprehensive review [275].

27 Comments on Literature, and Further Reading

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Chapter 12 The Born-Oppenheimer approximation plays an important role in quantum chemistry, and there is an enormous literature on the subject. We mention here only rigorous works: [69, 148, 149, 151, 179, 218, 237, 280, 289]. This approximation in conjunction with the Feshbach - Schur method is used in [15] to give a rigorous derivation of the van der Waals forces between atoms. The results (12.40) and (12.41) should be compared with those in [230]. The Born-Oppenheimer approximation for a molecule coupled to the quantized electro-magnetic field (the non-relativistic quantum electrodynamics), including a computation of the probability of spontaneous emission, was developed in [290]. For adiabatic theory and geometrical phases, see excellent review [20] and books [262, 289]. Book [289] covers also application of adiabatic theory to the Born-Oppenheimer approximation and many related topics. For adiabatic theory in relation to thermodynamics, Lindblad dynamics and quantum resonances, see [3, 4], [22] and [5], respectively. For the relation of the adiabatic theory to the integer quantum hall effect and to charge transport and for the space-time adiabatic theory, see [20, 25] and [237], respectively. For the adiabatic theory without the spectral gap condition, see [21, 289] For different aspects of the Born-Oppenheimer approximation, adiabatic theory and geometrical phases, see [104, 256, 84]. Chapter 13 This chapter is somewhat more advanced than the chapters preceding it. Recommended reading for this chapter includes [73, 169, 9]. The key result of quantum many-body theory - the HVZ theorem - is due to W. Hunziker, C. van Winter, and G.M. Zhislin. Recently, mathematical many-body theory, especially scattering theory for many-particle systems, has undergone rapid and radical development. It is covered in [169, 77, 132, 76]. Chapter 14 For the rigorous derivation of Hartree and Hartree-Fock equations see [159, 279, 133, 34, 33, 86, 87, 90, 91, 89, 117, 123, 124, 1, 253, 238, 239, 182, 14]. The Gross-Pitaevski equation was proposed independently by Gross and Pitaevski in connection with the problem of many bosons (boson gas) which we consider here. It was derived by [90] with the derivation simplified in [238, 239]. (For additional important aspects see [181, 63, 64].) Earlier, [209] have shown that the bosonic many-body ground state looks asymptotically (in the Gross-Petaevskii regime, in which the number of particles n → ∞ and the scattering length a → 0, so that na =: λ/(4π) is fixed) as the product

432

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of the ground states of the Gross-Pitaevski equation. For detailed discussion and references see [210]. For the first result on the Hartree-Fock theory see the paper [211], with further mathematical developments in [214]. (See [203, 191] for more references.) For asymptotic stability of the Gross-Pitaevski, nonlinear Schr¨odinger and Hartree equations see [277, 126, 293, 127, 128, 145, 72] and references therein. Other interesting properties (in particular, the blow-up) and generalizations are described in [185, 192, 193]. A small but very effective modification of the Hartree-Fock equations leads to the Kohn-Sham equation, which is at the foundation of the density functional theory, see [55, 65, 85] for reviews and references. See [286] and [58] for background and results on the nonlinear Schr¨odinger equations. A rigorous proof of BEC in the Gross-Pitaevski limit is given in [207] (see also [209]).

Chapter 15 Standard references on path integrals are [101, 180, 259, 254]. Rigorous results can be found in [272]. Important original papers are [188, 68, 195]. Chapter 16 An extensive rigorous treatment of semi-classical asymptotics can be found in [49, 158, 170, 219, 220, 250, 251, 252]. Chapter 17 The mathematical theory of resonances started with the paper [270], whose impetus came from the work [10, 32]. A classic, influential review of the theory of resonances is [271]. See [162] for earlier rigorous references and [48, 71, 225, 276], for some recent ones. Our analysis of the resonance free energy is close to [269], its physical predecessors are [188, 68, 54, 8, 189, 190]. For complex classical trajectories, see, eg., [49, 50, 51, 297, 298]. A more careful treatment of positive temperatures would involve coupling of a quantum system to a thermal reservoir, or, as an intermediate step, replacing the commutator with the Hamiltonian by Lindblad generator, discussed in Chapter 18. Resonances (metastable states) at zero and positive temperatures are a subject of intensive study in condensed matter physics, quantum chemistry, nuclear physics, and cosmology. See [54, 44, 156, 296, 269, 80] for reviews and further references. For formal treatment of tunneling of extended objects see [236].

27 Comments on Literature, and Further Reading

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Moreover, [247] gives an excellent exposition of the perturbation theory of resonances. Chapter 18 More material on density matrices can be found in [46, 157, 292, 76]. For a rigorous semi-classical analysis in terms of the Wigner functions, see [102, 103, 215, 243, 278]. There is an extensive literature on the Hartree-Fock and Kohn-Sham equations, for relevant discussion and references see [65]. The generalized Hartree-Fock equations were introduced in [65]. Chapter 19 For an abstract definition and discussion of the non-commutative conditional expectation see [143, 175, 76]. For the relation of quantum dynamical maps to completely positive maps see [292], Statement 3.1.4, and [45], Remarks to Section 5.3.1. The form of generators L was found by Lindblad and Gorini, Kossakowski and Sudarshan ([213, 137], see [66] for the most general result and [76], for a book discussion). For the detailed balance condition see e.g. [227, 183, 108] and references therein. Inequality (19.36) was shown in [281]. For a discussion of dissipation function (19.41) and dissipation see [76], Lecture 12. For far a reaching generalization, see [174]. The dissipation function DL was introduced in [213]. Theorem 19.38 was proven in [212]. For important generalizations of the subadditivity of the entropy, (19.46), see [205, 206]. For more on quantum entropy, see book [235] and reviews [56, 255]. For rigorous results on the decoherence - [222, 223, 224] (these papers contain also a very brief review of the relevant physics literature) and on relation between the decoherence and measurement, [107]. The approach to equilibrium is one of the central subjects of theoretical and mathematical physics. In the last 20 or so years its mathematical underpinning underwent a vigorous development on the level of microscopic theory, with robust and elegant theory mainly due to works of V. Jakˇsi´c and C.-A. Pillet, see some reviews in [19, 173] For the relation of the topics covered to quantum information theory, see [116, 157, 177, 233, 242, 295, 301]. Chapter 20 A standard reference for mathematical treatment of the second quantization is [38].

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The result on the comparison of the quantum and Hartree (mean-field) dynamics was obtained in [124]. The sketch of its proof follows [16]. The expansion (20.46) - (20.47) was derived in [13] it follows from the symbol composition formula of [38]. It was shown in [124] that the quantum many-body theory with the quantum Hamiltonian (20.25) can be obtained by quantizing the classical field theory described by the classical Hamiltonian (20.38) and Poisson brackets (20.37). Chapter 22 The renormalization of the electron mass and one-particle states were studied in [109, 110, 240, 241, 152, 161, 28, 59, 60, 61, 62]. See [26] for the review and references. Chapter 23 The results on the theory of radiation, as well as the renormalization group approach, are taken from [265, 121], which builds upon [29, 30, 27]. (Theorem 23.1 was established for systems with the coupling to photons regularized at spacial infinity in [29] - [30]. It would have been applicable to confined systems, i.e. systems in external potentials, growing at infinity sufficiently fast.) These works assume that the fine-structure constant is sufficiently small. The method in these papers also provides an effective computational technique to any order in the electron charge, something the conventional perturbation theory fails to do. For other approaches and references see the book [283] and a recent reviews [26, 266]. The existence of the ground state for the physical range of the parameters was proved in [140]. The notion of resonance in the non-relativistic QED was introduced in [29, 30]. The generalized Pauli-Fierz transformation was introduced in [265]. Theorem 23.4 was proven in [27]. Chapter 24 The spectral renormalization group method, described in this chapter is due to [29], with further development due to [27, 121, 154, 155]. We follow [121] It shares its philosophy with the standard renormalization group due to Wilson and others which can be found in any standard book on quantum field theory or statistical mechanics (for some papers see e.g. [130, 299]) and which is treated rigorously ina large number of papers, see [37, 100, 47, 97, 96, 31, 100, 129, 258] for a review of rigorous results. The Feshbach-Schur map could be also used for the problem at hand. In fact, the Feshbach-Schur maps were introduced in [29] for exactly this purpose.

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However, for technical reasons, it is convenient to use its present generalization. Chapters 25 and 26 These chapters cover standard material though many of the proofs are simpler or require less advanced material than those found in standard books. For a discussion of determinants, see also [53, 45, 180, 195, 254, 259, 260] and references therein. For elementary facts on the calculus of variations, one can consult [131, 43].

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Index

C k (Ω), 409 C0∞ (Ω), 371 C0∞ (Rd ), 371 H s (Rd ), 409 L1 (Rd ), 121, 406 L2 -space, 370 L2 (Rd ), 5, 370 Lp space, 369 n-particle sector, 308 action, 6, 196, 213, 296, 410, 416 adiabatic partition function, 225 adjoint, 85, 378 affine space, 411 analytic continuation, 223, 231 angular momentum, 25, 34, 74, 81 annihilation operator, 85, 301, 306 asymptotic completeness, 118, 119, 122, 184 asymptotic stability, 192 average value, 237 Balmer series, 84 Banach space, 369, 373, 409 basis, 371 Birman-Schwinger operator, 115 principle, 115 Bohr-Sommerfeld quantization, 211, 232, 235 Born-Oppenheimer approximation, 128, 151, 152 Bose-Einstein condensation, 192, 293 boson, 41, 45, 46, 287, 309, 312

bound state, 64, 110, 117 bounded operator, 373 canonical variables, 33 commutation relations, 34 coordinates, 298 canonically conjugate operators, 34 Cauchy problem, 11 Cauchy-Schwarz inequality, 371 causality, 5 centre-of-mass motion, 172, 173 channel evolution, 183 chemical potential, 288 classical action, 416 field theory, 410 observables, 50 path, 209, 416 closed graph theorem, 374 closed operator, 373 cluster decomposition, 175, 183 coherent states, 54 commutator, 23, 34, 374 complete, 369 complex classical field theory, 424 complex deformation, 219, 225 condensate, 293 configuration space, 297 confining potential, 68 conjugate point, 417, 418 conservation of energy, 27, 48

© Springer-Verlag GmbH Germany, part of Springer Nature 2020 S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext, https://doi.org/10.1007/978-3-030-59562-3

452

Index

conservation of probability, 11 constraint, 420 convolution, 406 correspondence principle, 5 Coulomb gauge, 39, 310 Coulomb potential, 14, 83, 111, 182 covariance operator, 301, 302, 311 creation operator, 85, 301, 306 critical point, 187, 202, 205, 411, 414 critical temperature, 293 decimation map, 338, 358 delta function, 395, 407 dense set, 371 density matrix, 237, 238, 254 density operator, 237 determinant of an operator, 213, 402 dilation, 110, 220 Dirac delta function, 407 direct sum, 174 Dirichlet boundary conditions, 80 Dirichlet functional, 410 discrete spectrum, 62, 387 dispersion , 53 law, 305 distribution, 407 domain of an operator, 371 dual space, 48, 369 Dyson series , 384 eigenspace, 62, 387 eigenvalue, 61, 387 eigenvector, 387 electric field, 38, 51, 309, 315 embedded eigenvalue, 144 entropy, 242 equation of state, 290 equilibrium state, 242 ergodic mean convergence, 64 essential spectrum, 62, 388 Euler-Lagrange equation, 296, 414 evaluation functional, 298, 422 evolution operator, 16, 195, 382 excited state, 332 existence of dynamics, 16 expected value, 303 exponential

decay, 182 of an operator, 17, 380 Fermi golden rule, 146, 337 fermion, 41, 45, 46 Feshbach map, 338 Feshbach-Schur method, 135 Feynman path integral, 197 Feynman-Kac theorem, 212, 243 field, 296 first resolvent identity, 389 fixed point, 353 flow, 423 Fock space, 308 Fourier inversion formula, 406 Fourier transform, 405 Fr´echet derivative, 412 free evolution, 118, 119 Hamiltonian, 19 propagator, 19, 205 free energy , 212, 224, 243 Helmholtz, 243 fugacity, 288 functional , 409 linear, 369 Gˆ ateaux derivative, 412 gauge invariance, 39, 316, 320 transformation, 39, 309 Gaussian, 19 generalized eigenfunction, 120 generalized normal form, 355 generator of dilations, 110 Gibbs entropy, 242 Gibbs state, 242 gradient of a functional, 413 grand canonical ensemble, 288 Green’s function, 206, 208, 232 Gross-Pitaevski equation, 187 ground state, 332 ground state energy, 211, 213 group property, 16, 382 Hamilton’s equations, 25, 26, 33, 49, 50, 298, 422

Index Hamilton-Jacobi equation, 8, 208, 214, 216 Hamiltonian function, 33, 214 functional, 49, 297 operator, 25, 34, 308 system, 50, 299 hamiltonian formulation, 318 harmonic function, 414 harmonic oscillator, 84, 213, 214 Hartree equation, 186, 189 Hartree-Fock equation, 189 Heisenberg equation, 26, 34 representation, 26 uncertainty principle, 53 Hellinger-Toeplitz theorem, 374 Hessian , 198 of a functional, 415 Hilbert space, 370, 409 Hilbert-Schmidt operator, 398 HVZ theorem, 177, 180, 332 hydrogen atom, 56, 83, 111 ideal Bose gas, 287, 290 Ideal quantum gas, 97 identical particle symmetry, 44 index, 417 initial condition, 11, 63, 379 initial value problem, 11 inner product, 369, 370 instanton, 227, 228 integrable function, 405 integral kernel, 113, 120, 195, 254, 372, 374 integral operator, 372, 374, 398 intercluster distance, 183 intercluster potential, 176, 183 internal energy, 242 internal motion, 174 intertwining relations, 119 intracluster potential, 176 invariant subspace, 63, 388 inverse of an operator, 375 invertible operator, 375 ionization threshold, 68 isometry, 16, 118, 382 isospectral map, 359

453

Jacobi equation, 208, 417 identity, 50 matrix, 404, 419 vector field, 417 Josephson junction, 228 Kato’s inequality, 38 kernel of an integral operator, 372 of an operator, 375 Klein-Gordon equation, 296, 414 Lagrange multiplier, 420 Lagrangian, 47, 296 Lagrangian , 199, 411 density, 296 functional, 296 Lamb shift, 337 Landau levels, 89 Landau-von Neumann equation, 238 Laplace method, 212 Laplace-Beltrami operator, 77, 82 Laplacian, 6, 372, 408, 415 lattice translations, 71 Lebesgue space, 369 Legendre function, 76 Legendre transform, 207, 215, 310, 421, 422, 425 Lie algebra, 50 Lieb-Thirring inequalities, 100 linear functional, 369 linear operator, 371 linearization, 360 Lippmann-Schwinger equation, 113 long-range interaction, 120 potential, 181, 184 Lyapunov stability, 192 magnetic field, 38, 51, 309, 315 Maxwell’s equations, 295, 309, 315 mean value, 23 mean-field theory, 186 metastable state, 219, 226 min-max principle, 106, 111, 182 minimal coupling, 319 minimizer of a functional, 414, 416 Minkowski inequality, 381

454

Index

mixed state, 239 momentum operator, 24, 372, 407 Morse theorem, 418 multi-index, 117, 370 multiplication operator, 372 multiplicity of an eigenvalue, 62, 387 Neumann series, 344, 376, 389 Newton’s equation, 25, 414 nonlinear Schr¨ odinger equation, 187 norm, 368 normal form, 85 normed vector space, 368 nullspace of an operator, 375 observable, 23, 24, 237, 305 one-parameter unitary group, 17, 219 open system, 255 operator , 371 adjoint, 85, 378 angular momentum, 25, 34 bounded, 373 closed, 373 evolution, 195 Hamiltonian, 25, 34, 308 Hilbert-Schmidt, 398 integral, 372, 374, 398 Laplacian, 372, 408 linear, 371 momentum, 24, 372, 407 multiplication, 372 particle number, 308 position, 23 positive, 238, 379 projection, 385 Schr¨ odinger, 7, 11, 24, 109, 195, 372, 379, 415 self-adjoint, 379 symmetric, 11 trace class, 238, 397 unitary, 27, 382 orbital stability, 192 order parameter, 294 orthogonal projection, 136, 175, 238, 386 orthonormal set, 371 Parseval relation, 371 particle number operator, 85, 288, 308

partition function, 212, 224, 242, 288 partition of unity, 178 path integral, 197, 204, 212, 226, 232 Penrose-Onsager criterion, 193 periodic boundary conditions, 80 permutation, 307 perturbation theory, 135 phase space, 33, 297 phase transition, 293 photon, 333 Plancherel theorem, 406 Planck’s constant, 7 plane wave, 396, 407 Poisson bracket, 25, 33, 50, 299, 310 polarization identity, 12 position operator, 23 positive operator, 238, 379 potential, 6 pressure, 289 principle of maximum entropy, 242 of minimal action, 47, 296 probability distribution, 23 projection , 167, 385 operator, 385 orthogonal, 136, 175, 238, 386 rank-one, 238, 255 spectral, 338 propagator, 16, 195, 382 propagator , 384 free, 19 pseudodifferential operators, 38 pure state, 239 quadratic form, 421 quantization, 33 quantization of Maxwell’s equations, 311 quantum statistics, 237, 253 range of an operator, 375 rank of a projection, 386 rank-one projection, 238, 255 reduced mass, 173 reflection coefficient, 125 relative boundedness, 135 relative motion, 173 renormalization group, 353

Index renormalization map, 353, 359 repulsive potential, 110 resolvent of an operator, 59, 387, 389 set, 59, 112, 387 resonance , 126, 147, 219, 220, 235, 336, 337 decay probability, 225 eigenvalue, 225 energy, 126, 223, 225 free energy, 225 lifetime, 223, 225 width, 225 Riesz integral, 390 Ritz variational principle, 103 Ruelle theorem, 64 Rydberg states, 132 scalar potential, 38, 51, 316 scaling dimension, 355 scattering channel, 183 eigenfunction, 120 operator, 119 state, 64, 117, 118, 219, 332 Schr¨ odinger eigenvalue equation, 80 equation, 5, 7, 11, 63, 117, 195, 379, 383 operator, 7, 11, 24, 109, 195, 372, 379, 415 representation, 26 second law of thermodynamics, 242 second variation of a functional, 415 self-adjoint, 376 self-adjoint operator, 379 self-adjointness, 12 semi-classical analysis, 208, 213, 226 semi-classical approximation, 20 semi-group, 359, 365 separation of variables, 332, 405 short-range interaction, 118 potential, 181, 183 Sobolev space, 370, 409 spaces, 367 specific heat, 294 spectral projection, 338 spectrum

455

discrete, 62, 387 essential, 62, 388 of an operator, 59, 309, 312, 387 Weyl, 391 spherical coordinates, 82 spherical harmonics, 76 spin, 40 spreading sequence, 66 stability, 191 stable manifold, 365 standard model of non-relativistic QED, 320 state space, 5, 33, 34, 370 state vector, 4 stationary phase method, 202, 204, 207, 212, 232 stationary state, 241 statistical mechanics, 237 strong operator topology, 200 superposition principle, 5 symbols, 38 symmetric operator, 11 symmetrization, 307 symplectic form, 50, 299 operator, 422 tangent space, 411 temperature, 242, 288 tensor product of Hilbert spaces, 404 of operators, 404 test function, 103 thermal equilibrium, 242 thermodynamic limit, 291 threshold , 180, 181 set, 180 two-cluster, 181 trace of an operator, 115, 397 trace class operator, 238, 397 translation invariance, 321, 327 transmission coefficient, 126 transverse vector-field, 310 triangle inequality, 67, 368 Trotter product formula, 195, 199 tunneling, 223, 228, 232

456

Index

two-cluster threshold, 181 ultraviolet cut-off, 319 uncertainty principle, 53, 112 unitary operator, 27, 382 unstable manifold, 365 vacuum, 306, 332 variation, 412 variational derivative, 48, 360, 412 vector potential, 38, 51, 199, 316 vector space, 367 virial relation, 110

theorem, 231 wave equation , 310 nonlinear, 414 wave function, 4, 23, 237 wave operator, 118, 119, 121, 183 weak convergence, 391 Weyl criterion, 391 sequence, 391 spectrum, 391 Wick ordering, 304 quantization, 304 Zeeman effect, 140, 141