Quantum Mechanics of Charged Particle Beam Optics: Understanding Devices from Electron Microscopes to Particle Accelerators 1138035920, 9781138035928

Table of contents :
Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
Authors
Chapter 1 Introduction
Chapter 2 An Introductory Review of Classical Mechanics
2.1 Single Particle Dynamics
2.1.1 Lagrangian Formalism
2.1.1.1 Basic Theory
2.1.1.2 Example: Motion of a Charged Particle in an Electromagnetic Field
2.1.2 Hamiltonian Formalism
2.1.2.1 Basic Theory
2.1.2.2 Example: Motion of a Charged Particle in an Electromagnetic Field
2.1.3 Hamiltonian Formalism in Terms of the Poisson Brackets
2.1.3.1 Basic Theory
2.1.3.2 Example: Dynamics of a Charged Particle in a Constant Magnetic Field
2.1.4 Changing the Independent Variable
2.1.4.1 Basic Theory
2.1.4.2 Example: Dynamics of a Charged Particle in a Constant Magnetic Field
2.1.5 Canonical Transformations
2.1.5.1 Basic Theory
2.1.5.2 Optical Hamiltonian of a Charged Particle Moving Through an Electromagnetic Optical Element with a Straight Axis
2.1.6 Symplecticity of Canonical Transformations
2.1.6.1 Time-Independent Canonical Transformations
2.1.6.2 Time-Dependent Canonical Transformations: Hamiltonian Evolution
2.1.6.3 Canonical Invariants: Poisson Brackets
2.2 Dynamics of a System of Particles
Chapter 3 An Introductory Review of Quantum Mechanics
3.1 Introduction
3.2 General Formalism of Quantum Mechanics
3.2.1 Single Particle Quantum Mechanics: Foundational Principles
3.2.1.1 Quantum Kinematics
3.2.1.2 Quantum Dynamics
3.2.1.3 Different Pictures of Quantum Dynamics
3.2.1.4 Ehrenfest’s Theorem
3.2.1.5 Spin
3.3 Nonrelativistic Quantum Mechanics
3.3.1 Nonrelativistic Single Particle Quantum Mechanics
3.3.1.1 Free Particle
3.3.1.2 Linear Harmonic Oscillator
3.3.1.3 Two-Dimensional Isotropic Harmonic Oscillator
3.3.1.4 Charged Particle in a Constant Magnetic Field
3.3.1.5 Scattering States
3.3.1.6 Approximation Methods, Time-Dependent Systems, and the Interaction Picture
3.3.1.7 Schrödinger–Pauli Equation for the Electron
3.3.2 Quantum Mechanics of a System of Identical Particles
3.3.3 Pure and Mixed States: Density Operator
3.4 Relativistic Quantum Mechanics
3.4.1 Klein–Gordon Equation
3.4.1.1 Free-Particle Equation and Difficulties in Interpretation
3.4.1.2 Feshbach–Villars Representation
3.4.1.3 Charged Klein–Gordon Particle in a Constant Magnetic Field
3.4.2 Dirac Equation
3.4.2.1 Free-Particle Equation
3.4.2.2 Zitterbewegung
3.4.2.3 Spin and Helicity of the Dirac Particle
3.4.2.4 Spin Magnetic Moment of the Electron and the Dirac–Pauli Equation
3.4.2.5 Electron in a Constant Magnetic Field
3.4.3 Foldy–Wouthuysen Transformation
3.4.3.1 Foldy–Wouthuysen Representation of the Dirac Equation
3.4.3.2 Foldy–Wouthuysen Representation of the Feshbach–Villars form of the Klein–Gordon Equation
3.5 Appendix: The Magnus Formula for the Exponential Solution of a Linear Differential Equation
Chapter 4 An Introduction to Classical Charged Particle Beam Optics
4.1 Introduction: Relativistic Classical Charged Particle Beam Optics
4.2 Free Propagation
4.3 Optical Elements with Straight Optic Axis
4.3.1 Axially Symmetric Magnetic Lens: Imaging in Electron Microscopy
4.3.2 Normal Magnetic Quadrupole
4.3.3 Skew Magnetic Quadrupole
4.3.4 Axially Symmetric Electrostatic Lens
4.3.5 Electrostatic Quadrupole
4.4 Bending Magnet: An Optical Element with a Curved Optic Axis
4.5 Nonrelativistic Classical Charged Particle Beam Optics
Chapter 5 Quantum Charged Particle Beam Optics: Scalar Theory for Spin-0 and Spinless Particles
5.1 General Formalism of Quantum Charged Particle Beam Optics
5.2 Relativistic Quantum Charged Particle Beam Optics Based on the Klein–Gordon Equation
5.2.1 General Formalism
5.2.2 Free Propagation: Diffraction
5.2.3 Axially Symmetric Magnetic Lens: Electron Optical Imaging
5.2.3.1 Paraxial Approximation: Point-to-Point Imaging
5.2.3.2 Going Beyond the Paraxial Approximation: Aberrations
5.2.3.3 Quantum Corrections to the Classical Results
5.2.4 Normal Magnetic Quadrupole
5.2.5 Skew Magnetic Quadrupole
5.2.6 Axially Symmetric Electrostatic Lens
5.2.7 Electrostatic Quadrupole Lens
5.2.8 Bending Magnet
5.3 Effect of Quantum Uncertainties on Aberrations in Electron Microscopy and Nonlinearities in Accelerator Optics
5.4 Nonrelativistic Quantum Charged Particle Beam Optics: Spin-0 and Spinless Particles
5.5 Appendix: Propagator for a System with Time-Dependent Quadratic Hamiltonian
Chapter 6 Quantum Charged Particle Beam Optics: Spinor Theory for Spin-1/2 Particles
6.1 Relativistic Quantum Charged Particle Beam Optics Based on the Dirac–Pauli Equation
6.1.1 General Formalism
6.1.1.1 Free Propagation: Diffraction
6.1.1.2 Axially Symmetric Magnetic Lens
6.1.1.3 Bending Magnet
6.1.2 Beam Optics of the Dirac Particle with Anomalous Magnetic Moment
6.1.2.1 General Formalism
6.1.2.2 Lorentz and Stern–Gerlach Forces, and the Thomas–Frenkel–BMT Equation for Spin Dynamics
6.1.2.3 Phase Space and Spin Transfer Maps for a Normal Magnetic Quadrupole
6.1.2.4 Phase Space and Spin Transfer Maps for a Skew Magnetic Quadrupole
6.2 Nonrelativistic Quantum Charged Particle Beam Optics: Spin-1/2 Particles
Chapter 7 Concluding Remarks and Outlook on Further Development of Quantum Charged Particle Beam Optics
Bibliography
Index

Citation preview

Quantum Mechanics of Charged Particle Beam Optics

Multidisciplinary and Applied Optics Series Editors: Vasudevan Lakshminarayanan, University of Waterloo, Ontario, Canada Hassen Ghalila, University Tunis El Manar, Tunisia Ahmed Ammar, University Tunis El Manar, Tunisia L. Srinivasa Varadharajan, University of Waterloo, Ontario, Canada Understanding Optics with Python Vasudevan Lakshminarayanan, Hassen Ghalila, Ahmed Ammar, L. Srinivasa Varadharajan

For more information about this series, please visit: https://www.crcpress.com/Multidisciplinary-and-Applied-Optics/book-series/ CRCMULAPPOPT

Quantum Mechanics of Charged Particle Beam Optics Understanding Devices from Electron Microscopes to Particle Accelerators Ramaswamy Jagannathan and Sameen Ahmed Khan

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 2019 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business ©

No claim to original U.S. Government works Printed on acid-free paper International Standard Book Number-13: 978-1-138-03592-8 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-7508400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Dedication To the memory of my parents Thirali Aravamudha Ramaswamy Iyengar & Andal Ramaswamy —– Ramaswamy Jagannathan To the memory of my parents Hamid Ahmed Khan & Nighat Hamid Khan —– Sameen Ahmed Khan

Contents Preface.......................................................................................................................xi Authors..................................................................................................................... xv Chapter 1

Introduction ..................................................................................... 1

Chapter 2

An Introductory Review of Classical Mechanics ............................ 9 2.1

2.2

Single Particle Dynamics........................................................ 9 2.1.1 Lagrangian Formalism ............................................... 9 2.1.1.1 Basic Theory.............................................. 9 2.1.1.2 Example: Motion of a Charged Particle in an Electromagnetic Field........ 10 2.1.2 Hamiltonian Formalism ........................................... 12 2.1.2.1 Basic Theory............................................ 12 2.1.2.2 Example: Motion of a Charged Particle in an Electromagnetic Field........ 14 2.1.3 Hamiltonian Formalism in Terms of the Poisson Brackets.................................................................... 16 2.1.3.1 Basic Theory............................................ 16 2.1.3.2 Example: Dynamics of a Charged Particle in a Constant Magnetic Field ..... 18 2.1.4 Changing the Independent Variable ......................... 21 2.1.4.1 Basic Theory............................................ 21 2.1.4.2 Example: Dynamics of a Charged Particle in a Constant Magnetic Field ..... 22 2.1.5 Canonical Transformations ...................................... 25 2.1.5.1 Basic Theory............................................ 25 2.1.5.2 Optical Hamiltonian of a Charged Particle Moving Through an Electromagnetic Optical Element with a Straight Axis ......................................... 29 2.1.6 Symplecticity of Canonical Transformations .......... 31 2.1.6.1 Time-Independent Canonical Transformations....................................... 31 2.1.6.2 Time-Dependent Canonical Transformations: Hamiltonian Evolution.............. 32 2.1.6.3 Canonical Invariants: Poisson Brackets... 35 Dynamics of a System of Particles ....................................... 36 vii

viii

Chapter 3

Contents

An Introductory Review of Quantum Mechanics.......................... 41 3.1 3.2

3.3

3.4

Introduction........................................................................... 41 General Formalism of Quantum Mechanics......................... 42 3.2.1 Single Particle Quantum Mechanics: Foundational Principles ........................................... 42 3.2.1.1 Quantum Kinematics ............................... 42 3.2.1.2 Quantum Dynamics ................................. 51 3.2.1.3 Different Pictures of Quantum Dynamics................................................. 64 3.2.1.4 Ehrenfest’s Theorem................................ 68 3.2.1.5 Spin.......................................................... 70 Nonrelativistic Quantum Mechanics .................................... 78 3.3.1 Nonrelativistic Single Particle Quantum Mechanics ................................................................ 78 3.3.1.1 Free Particle............................................. 78 3.3.1.2 Linear Harmonic Oscillator..................... 90 3.3.1.3 Two-Dimensional Isotropic Harmonic Oscillator ............................................... 100 3.3.1.4 Charged Particle in a Constant Magnetic Field....................................... 103 3.3.1.5 Scattering States .................................... 107 3.3.1.6 Approximation Methods, Time-Dependent Systems, and the Interaction Picture ................................. 111 3.3.1.7 Schr¨odinger–Pauli Equation for the Electron.................................................. 117 3.3.2 Quantum Mechanics of a System of Identical Particles.................................................................. 119 3.3.3 Pure and Mixed States: Density Operator.............. 126 Relativistic Quantum Mechanics ........................................ 132 3.4.1 Klein–Gordon Equation ......................................... 132 3.4.1.1 Free-Particle Equation and Difficulties in Interpretation .................. 132 3.4.1.2 Feshbach–Villars Representation .......... 137 3.4.1.3 Charged Klein–Gordon Particle in a Constant Magnetic Field........................ 139 3.4.2 Dirac Equation ....................................................... 141 3.4.2.1 Free-Particle Equation ........................... 141 3.4.2.2 Zitterbewegung ...................................... 149 3.4.2.3 Spin and Helicity of the Dirac Particle ................................................... 150 3.4.2.4 Spin Magnetic Moment of the Electron and the Dirac–Pauli Equation ................ 153 3.4.2.5 Electron in a Constant Magnetic Field ... 154

ix

Contents

3.4.3

3.5

Chapter 4

An Introduction to Classical Charged Particle Beam Optics ...... 173 4.1 4.2 4.3

4.4 4.5

Chapter 5

Foldy–Wouthuysen Transformation....................... 156 3.4.3.1 Foldy–Wouthuysen Representation of the Dirac Equation................................. 156 3.4.3.2 Foldy–Wouthuysen Representation of the Feshbach–Villars form of the Klein–Gordon Equation......................... 168 Appendix: The Magnus Formula for the Exponential Solution of a Linear Differential Equation ......................... 170

Introduction: Relativistic Classical Charged Particle Beam Optics........................................................................ 173 Free Propagation................................................................. 174 Optical Elements with Straight Optic Axis ........................ 178 4.3.1 Axially Symmetric Magnetic Lens: Imaging in Electron Microscopy.............................................. 178 4.3.2 Normal Magnetic Quadrupole ............................... 197 4.3.3 Skew Magnetic Quadrupole................................... 202 4.3.4 Axially Symmetric Electrostatic Lens ................... 204 4.3.5 Electrostatic Quadrupole........................................ 205 Bending Magnet: An Optical Element with a Curved Optic Axis ........................................................... 206 Nonrelativistic Classical Charged Particle Beam Optics........................................................................ 210

Quantum Charged Particle Beam Optics: Scalar Theory for Spin-0 and Spinless Particles....................................................... 213 5.1 5.2

General Formalism of Quantum Charged Particle Beam Optics........................................................................ 213 Relativistic Quantum Charged Particle Beam Optics Based on the Klein–Gordon Equation ................................ 214 5.2.1 General Formalism................................................. 214 5.2.2 Free Propagation: Diffraction ................................ 230 5.2.3 Axially Symmetric Magnetic Lens: Electron Optical Imaging ..................................................... 232 5.2.3.1 Paraxial Approximation: Point-to-Point Imaging.................................................. 232 5.2.3.2 Going Beyond the Paraxial Approximation: Aberrations.................. 250 5.2.3.3 Quantum Corrections to the Classical Results ................................................... 260 5.2.4 Normal Magnetic Quadrupole ............................... 262 5.2.5 Skew Magnetic Quadrupole................................... 267 5.2.6 Axially Symmetric Electrostatic Lens ................... 269

x

Contents

5.3

5.4 5.5 Chapter 6

Quantum Charged Particle Beam Optics: Spinor Theory for Spin- 21 Particles ........................................................................... 287 6.1

6.2 Chapter 7

5.2.7 Electrostatic Quadrupole Lens............................... 270 5.2.8 Bending Magnet..................................................... 271 Effect of Quantum Uncertainties on Aberrations in Electron Microscopy and Nonlinearities in Accelerator Optics.................................................................................. 277 Nonrelativistic Quantum Charged Particle Beam Optics: Spin-0 and Spinless Particles.............................................. 281 Appendix: Propagator for a System with Time-Dependent Quadratic Hamiltonian........................................................ 283

Relativistic Quantum Charged Particle Beam Optics Based on the Dirac–Pauli Equation .................................... 287 6.1.1 General Formalism................................................. 287 6.1.1.1 Free Propagation: Diffraction................ 293 6.1.1.2 Axially Symmetric Magnetic Lens........ 300 6.1.1.3 Bending Magnet .................................... 306 6.1.2 Beam Optics of the Dirac Particle with Anomalous Magnetic Moment .............................. 316 6.1.2.1 General Formalism ................................ 316 6.1.2.2 Lorentz and Stern–Gerlach Forces, and the Thomas–Frenkel–BMT Equation for Spin Dynamics ................. 320 6.1.2.3 Phase Space and Spin Transfer Maps for a Normal Magnetic Quadrupole ...... 323 6.1.2.4 Phase Space and Spin Transfer Maps for a Skew Magnetic Quadrupole .......... 327 Nonrelativistic Quantum Charged Particle Beam Optics: Spin- 12 Particles .................................................................. 330

Concluding Remarks and Outlook on Further Development of Quantum Charged Particle Beam Optics ................................ 335

Bibliography .......................................................................................................... 339 Index ...................................................................................................................... 349

Preface It is surprising that only classical mechanics is required in the design and operation of numerous charged particle beam devices, from electron microscopes to particle accelerators, though the microscopic particles of the beam should be obeying quantum mechanics. First, such a success of the classical charged particle beam optics needs an explanation from the point of view of quantum mechanics. Then, it is certain that though the classical charged particle beam optics is extremely successful in the present-day beam devices, future developments in charged particle beam technology would require quantum theory. With this in mind, we present in this book the formalism of quantum charged particle beam optics. Though the formalism of quantum charged particle beam optics presented in this book needs a lot of further development for it to be a complete theory, we hope that it would be the basis for the theory of any future charged particle beam device that would require quantum mechanics for the design and operation of its optical elements. In a series of papers starting in 1983, Professors N. Mukunda, R. Simon, and E. C. G. Sudarshan had established, using a group theoretical approach, the Fourier optics for the Maxwell Field, which generalizes the paraxial scalar wave optics to paraxial Maxwell wave optics, consistently taking into account polarization. Around 1987, when Professor Sudarshan was the director of The Institute of Mathematical Sciences (IMSc), Chennai, India, he suggested to me, then a junior faculty member of the institute, to study the group theoretical aspects of electron optics based on the Dirac equation, with the aim to extend the traditional nonrelativistic scalar wave theory of electron optics to the Dirac spinor wave theory, taking into account the spin of the electron analogous to the way they had extended the Helmholtz scalar light optics to the Maxwell vector optics, taking into account the polarization of light. Searching the literature on electron optics led to a surprise: electron optics, or in general, charged particle optics, used in beam devices from electron microscopes to particle accelerators, was based essentially on classical mechanics. Of course, in electron microscopy, the image formation was understood on the basis of the nonrelativistic Schr¨odinger equation, ignoring completely the electron spin and the Dirac equation. In accelerator physics, topics like quantum fluctuations of the classical particle trajectories were treated using quantum mechanical concepts, but the design and operation of the accelerator beam elements were based only on classical mechanics. So, we had to start from scratch to understand the quantum mechanics of electron optics based on the Dirac equation. The first result of this study was a paper by Professors Simon, Sudarshan, and Mukunda, and me, in which we derived the focusing action of the round magnetic lens, the central part of any electron microscope, using the Dirac equation ab initio. Professor Sudarshan encouraged me to continue the study further in this direction, and in a subsequent paper, I used the quantum electron beam optics based on the Dirac equation to study other types of electromagnetic lenses. The PhD work of Sameen Ahmed Khan with me, during 1992–1997 at IMSc, xi

xii

Preface

led to the formulation of a more general framework of quantum theory of charged particle beam optics using the Heisenberg picture, Ehrenfest’s theorem, and quantum transfer maps. Sameen and I are fortunate to have had a collaboration in 1996 with Professors M. Conte and M. Pusterla. This collaboration helped us to extend our theory to accelerator optics. Later work has led to a comprehensive formalism of quantum charged particle beam optics applicable to any charged particle beam device. We remember Professor Sudarshan with gratitude for initiating our study of quantum charged particle beam optics, and would like to thank Professors Mukunda, Simon, Conte, and Pusterla for fruitful discussions and collaboration in the early stages of our work. I thank the IMSc for support throughout my academic career. We thank the library of IMSc for its excellent resources, and we also thank Dr Paul Pandian, the librarian, and his colleagues, in particular, Dr P. Usha Devi, for timely help in getting the references required during the writing of this book. I worked on my contribution to the writing of this book during my tenure at the Chennai Mathematical Institute (CMI) as adjunct professor, and I wish to thank CMI for supporting me after retirement from IMSc. I have enjoyed working on the book mostly from the comfort of my home and my daughters’ homes and with the kind support of everyone in my family. It gives me great pleasure to thank Malathy (my wife), Subadhra & Sandeep, and Anisha & Ajay (my elder daughter & sonin-law, and granddaughter & grandson), and Sujatha & Srinivas, and Aditi (my younger daughter & son-in-law, and granddaughter). And, I wish to thank my friend Kuppuswamy, friend from my school days, who has been encouraging me constantly throughout my life and, in a way, was responsible for me joining IMSc in 1971 as a PhD student. Sameen adds: I wish to acknowledge IMSc, where the work on this topic originated, for the support I received as a PhD student (and subsequent visits) and its excellent research environment. During my doctoral study at IMSc, the Internet had just begun to take roots. It was during this time that there was a long-distance collaboration with Professors Conte and Pusterla in Italy, exclusively by e-mail (thanks to the excellent Internet resources at IMSc), leading to the Conte-Jagannathan-KhanPusterla paper without a single face-to-face meeting! This collaboration was followed by my postdoctoral work at Istituto Nazionale di Fisica Nucleare (INFN) with Professor Pusterla when some more work was done on accelerator optics. I would also like to acknowledge Professor K. B. Wolf for a fruitful stay as a Consejo Nacional de Ciencia y Tecnologia (CONACyT) postdoctoral fellow at the Universidad Nacional Autnoma de Mxico (UNAM), Cuernavaca, Mxico. I would like to profusely thank my current workplace, Dhofar University, for constant encouragement to pursue research. My contributions to the book would not have been possible without the support of my family. My wife Noama Khan and daughter Hajira Khan always take a keen interest in my writing endeavors. I wish to acknowledge my elder brothers Mohammed Ahmed Khan and Farooq Ahmed Khan, and remember my parents thankfully, for the academic environment in which I grew up. And, I have to thank my sisters, Ayesha Khan and Bushra Khan, who have been very supportive

Preface

xiii

throughout my life. I also wish to thank my friend Azher Majid Siddiqui from my college days for continuous encouragement. We are grateful to Professor Vasudevan Lakshminarayanan for urging us constantly for a long time to write a book on our work, and the book is here finally! Ramaswamy Jagannathan, with Sameen Ahmed Khan, November 2018.

Authors Professor Ramaswamy Jagannathan retired in 2009 as a senior professor of physics from The Institute of Mathematical Sciences (IMSc), Chennai, India. He is currently an adjunct professor of physics at the Chennai Mathematical Institute (CMI), Chennai. He got his PhD (Theor. Phys.) from the University of Madras, Chennai, India, in 1976, working at IMSc. His PhD work on generalized Clifford algebras was done under the guidance of Professor Alladi Ramakrishnan, the founder director of IMSc known popularly as MATSCIENCE at that time. He has authored/coauthored about 80 research papers in various branches of Physical Mathematics, like Generalized Clifford Algebras and Their Physical Applications, Finite-Dimensional Quantum Mechanics, Applications of Classical Groups, Quantum Groups, Nonlinear Dynamics, Deformed Special Functions, and Quantum Theory of Charged Particle Beam Optics with Applications to Electron Microscopy and Accelerator Optics. In particular, his paper with Professors R. Simon, E. C. G. Sudarshan, and N. Mukunda (1989) on the quantum theory of magnetic electron lenses based on the Dirac equation initiated a systematic study of the Quantum Theory of Charged Particle Beam Optics. This theory was subsequently developed vastly by him and his collaborators (in particular, his PhD student Dr Sameen Ahmed Khan). Dr Sameen Ahmed Khan is an associate professor at the Department of Mathematics and Sciences, College of Arts and Applied Sciences, Dhofar University, Salalah, Sultanate of Oman (http://du.edu.om/). He got his PhD (Theor. Phys.) at the University of Madras, Chennai, India, in 1997. His PhD thesis, done at The Institute of Mathematical Sciences (IMSc), Chennai, under the supervision of Professor Ramaswamy Jagannathan, was on the quantum theory of charged particle beam optics. He did postdoctoral research at INFN, Padova, Italy, and Universidad Nacional Autonoma de Mexico, Cuernavaca, Mexico. He has 16 years of teaching experience in Oman. He has developed a unified treatment of light beam optics and light polarization using quantum methodologies. This formalism describes the beam optics and light polarization from a parent Hamiltonian which is exact and derived from Maxwell’s equations. He has authored three books, fifteen book chapters, and about 75 technical publications in journals and proceedings of repute. He has more than 250 publications on science popularization. Dr Sameen is one of the founding members of the Ibn al Haytham LHiSA Light: History, Science, and Applications (LHiSA) International Society set up during the International Year of Light and Light-based Technologies. He is a signatory to six of the reports on the upcoming International Linear Collider.

xv

1 Introduction This book essentially addresses a question of curiosity: How is Classical Charged Particle Beam Optics, entirely based on classical mechanics, so successful in practice in numerous devices from low-energy electron microscopes to high-energy particle accelerators though the microscopic particles of the beam should be obeying quantum mechanics? To get a detailed answer to this question, this book explores Quantum Charged Particle Beam Optics. A charged particle beam is a collection of charged particles of the same type (electrons, protons, same type of ions, etc.), with the same mass, charge, etc., moving almost in the same direction with almost the same momentum. For any beam device, there is a design trajectory, straight or curved, defining its optic axis along which the beam particle is supposed to move with a specific momentum, say ~p0 = p0~s, where ~s is the unit vector along the tangent to the design trajectory at any point on it. A charged particle moving along this design trajectory with the specified momentum ~p0 , the design momentum, is called the reference particle. If all the particles of a beam propagating through the device along its optic axis have momenta very close to the design momentum, i.e., for any particle of the beam ~p ≈ ~p0 , then all the particles of the beam would move in almost the same direction. Such a beam is called a paraxial beam. Let p0~s be the design momentum of a beam device. If ~p ∦ ~s is the momentum of a particle of a beam propagating through the device, and |~p| = p0 , then its longitudinal and perpendicular components are ~pk = (~s ·~p)~s = pk~s and ~p⊥ = ~p − pk~s, respectively, and p2k = |~p|2 − |~p⊥ |2 = p20 − p2⊥ . Any particle q of a paraxial beam will
satisfy the condition |~p⊥ |  p0 , and it will have pk = p20 − p2⊥ ≈ p0 − p2⊥ /2p0 . At times, for a paraxial beam, we may even take pk ≈ p0 . If the paraxial condition is not satisfied strictly, and we can write only

pk ≈ p0 − p2⊥ /2p0 − p4⊥ /8p30 then the beam can be called a quasiparaxial beam. q If more terms in the Taylor series of p20 − p2⊥ are required to get an approximate expression for pk in terms of p0 and |~p⊥ |, then the beam has to be called nonparaxial. Any beam violating the paraxial condition is nonparaxial, and hence a quasiparaxial beam may also be classified as nonparaxial. A charged particle beam, an ordered flow of charged particles in nearly the same direction, is a complex dynamical system. The dynamics of the particles depends on both the external electric and magnetic fields guiding the beam propagation and the internal fields contributed by the beam particles themselves. There will also be collisions between the particles. Even a low-current beam may contain more than 1010 particles. So, any exact description of the dynamical behavior of the system is impossible. We can have only approximate theories of the collective behavior of the beam. Such theories have to be built by averaging over the behaviors of the large number of individual charged particles of the beam.

1

2

Quantum Mechanics of Charged Particle Beam Optics

It is clear that the first step in the study of charged particle beam physics is the study of the dynamics of a single charged particle of the beam, ignoring all the interparticle and collective, or statistical, interactions. This single particle dynamics is the basis for the design of any charged particle beam device, i.e., designing the required trajectory of the reference particle. Study of the dynamics of a single charged particle under the influence of electric and magnetic fields, and consequent designs of the beam devices, like electron microscopes, electron beam lithography systems, electron beam welding machines, and particle accelerators, is called charged particle beam optics. The basis of design and operation of any device using charged particle beam technology is classical mechanics, and it has been very successful so far without any exception. This is where one becomes curious to know how and why classical mechanics works so well in this context, though the microscopic particles of the beam, like electrons, protons, and ions, should be obeying quantum mechanics. Any electron microscope is designed and operated using electron optics based entirely on classical mechanics. Quantum mechanics, or wave mechanics, is being used for understanding the image formation and resolution in electron microscopes since Glaser initiated the work in this direction (see Glaser [62] and references therein; see the encyclopedic three-volume text book of Hawkes and Kasper ([70, 71, 72]) for a comprehensive account of historical aspects and any technical aspect of geometrical electron optics and electron wave optics). In understanding the image formation in electron microscopy, mostly the nonrelativistic Schr¨odinger equation is used. In high-energy electron microscopy, one starts with the relativistic Klein– Gordon equation and soon approximates it to the nonrelativistic Schr¨odinger equation, often with a relativistic correction essentially based onp replacing the rest mass m of the particle by the so-called relativistic mass γm = m/ 1 − (v2 /c2 ). It is considered that under the conditions obtained in high-energy electron microscopy, use of the Klein–Gordon equation, as approximation of the Dirac equation, is adequate (see, e.g., Ferwerda, Hoenders, and Slump [48, 49], Hawkes and Kasper [72], Groves [68], Lubk [129], and Pozzi [150]). In accelerator physics, there are several quantum effects, like quantum fluctuations in high-energy electron beams due to synchrotron radiation, which are studied using quantum mechanics (see, e.g., Sokolov and Ternov [173], Ternov [181], Bell and Leinass [9], Hand and Skuja [69], Chao, Mess, Tigner, and Zimmermann [20], Chen [22, 23] and Chen and Reil [24] and references therein). The required quantum effects are treated usually as perturbations to the classical trajectories. Accelerator optics, the basic theory for the design and operation of accelerators, itself is based only on classical mechanics (see, e.g., Berz, Makino, and Wan [12], Conte and MacKay [27], Lee [127], Reiser [158], Rosenzweig [159], Seryi [168], Weidemann [187], Wolski [192], Chao, Mess, Tigner, and Zimmermann [20] and references therein). To understand the remarkable effectiveness of classical mechanics in charged particle beam optics, we explore in this book the quantum mechanics of charged particle beam optics at the level of single particle dynamics, using the appropriate basic equations, namely, the Klein–Gordon equation for spin-0 particles

Introduction

3

or high-energy particles without spin, (i.e., when the spin is ignored), the Dirac equation for spin- 12 particles, the nonrelativistic Schr¨odinger equation for lowenergy particles without spin, and the nonrelativistic Schr¨odinger–Pauli equation for low-energy spin- 12 particles. To this end, we closely follow Jagannathan, Simon, Sudarshan, and Mukunda [79], Jagannathan [80], Khan and Jagannathan [90], Jagannathan and Khan [82], Conte, Jagannathan, Khan, and Pusterla [26], Khan [91], Jagannathan [83, 84, 85], Khan [92, 99], and references therein. This chapter, Introduction, gives a brief overview of what to expect in each of the later chapters. Chapter 2, An Introductory Review of Classical Mechanics, presents a summary of the basic concepts of classical mechanics required essentially for classical charged particle beam optics. We recall the Lagrangian and the Hamiltonian formalisms of classical mechanics of a single particle. We discuss the Hamiltonian dynamics in terms of the Poisson brackets, changing the independent variable, canonical transformations, and the symplecticity of canonical transformations. The transfer maps for the observables across the system with respect to the coordinate along the optic axis are described in terms of the Lie transfer operator. The dynamics of a charged particle moving in a constant magnetic field serves as an example for the discussions. When time is the independent variable with respect to which we study the dynamical evolution of a given system, the generator of time evolution, the Hamiltonian, corresponds to the energy of the system. When we change the independent variable from time to, say s, the coordinate along the optic axis of the system, the corresponding Hamiltonian generating evolution of the system along the forward s-direction, the optical Hamiltonian, corresponds to −ps , negative of the momentum canonically conjugate to s. We derive the classical optical Hamiltonian of a relativistic charged particle moving through an electromagnetic optical element with a straight optic axis. Though we are concerned only with single particle dynamics, in this chapter, we also give a brief introduction to the formalism of multiparticle dynamics for the sake of completeness. For more details on classical mechanics, see, e.g., Corben and Stehle [28], Goldstein, Poole, and Safko [63], and Sudarshan and Mukunda [174]. Classical charged particle beam optics is based on classical electrodynamics, besides classical mechanics. We have not given any account of classical electrodynamics. Books on electron optics, charged particle optics, accelerator physics, and beam physics, mentioned earlier contain adequate details of classical electrodynamics. For more details on classical electrodynamics, see, e.g., Griffiths [66] and Jackson [78]. For any mathematical details required, see, e.g., Arfken, Weber, and Harris [3] and Byron and Fuller [18]. The Lie transfer operator method of studying the classical dynamical evolution of the observables of the system introduced in this chapter plays a central role in the modern approach to classical charged particle beam optics and has a straightforward generalization in the formalism of quantum charged particle beam optics. Dragt et al. have developed extensive techniques for applying the Lie transfer operator methods to light optics, classical charged particle beam optics, and accelerator optics (see Dragt [34], Dragt and Forest [35], Dragt et al. [36], Forest, Berz, and Irwin [54], Rangarajan, Dragt, and Neri [156], Forest and Hirata [55], Forest [56], Radli˘cka [154], and references therein; see also Berz [11], Mondrag´on

4

Quantum Mechanics of Charged Particle Beam Optics

and Wolf [135], Wolf [191], Rangarajan and Sachidanand [157], Lakshminarayanan, Sridhar, and Jagannathan [124], and Wolski [192]). Chapter 3, An Introductory Review of Quantum Mechanics, recalls the basic concepts of quantum mechanics required essentially for quantum charged particle beam optics. First, the fundamental principles of quantum mechanics like the wave function, probability interpretation, Hermitian operator representations of observables like position, linear momentum, angular momentum, and energy or Hamiltonian, eigenstates of observables, time-dependent Schr¨odinger equation for the dynamical evolution of a quantum system, measurement of an observable, average (or expectation) values, and the uncertainty principle are recalled. The Schr¨odinger and Heisenberg pictures of quantum dynamics and the Heisenberg equation of motion for observables are introduced. Ehrenfest’s theorem, basic to the formalism of quantum charged particle beam optics, is derived. Time-dependent perturbation theory and the role of interaction picture are explained. The Dyson expression for the time-evolution operator as a time-ordered exponential is given, and its Magnus form as an ordinary exponential, particularly suitable for quantum charged particle beam optics, is presented. As examples of nonrelativistic quantum mechanics, free particle, harmonic oscillator in one and two dimensions, charged particle in a constant magnetic field, Laundau levels, and scattering of a particle by onedimensional potential well and barrier are treated. Concept of spin as a nonclassical property of a particle is introduced using the angular momentum algebra, and the two-component spinor wave function of a spin- 12 particle is introduced. The nonrelativistic Schr¨odinger–Pauli equation for the electron, or any spin- 12 particle, is formulated, and the magnetic moment of electron is discussed. Concepts of pure and mixed states are explained, and the formalism of quantum mechanics in terms of density operator is discussed. The scalar relativistic wave equation, the Klein– Gordon equation, and the problems with its interpretation as a single particle theory are analysed. Examples of free particle and a charged particle in a constant magnetic field are treated. The Feshbach–Villars form of the Klein–Gordon equation and the passage from the Klein–Gordon equation to its nonrelativistic limit, the nonrelativistic Schr¨odinger equation, are discussed. The Dirac equation with four-component spinor wave function is formulated. As examples, free particle and a charged particle in a constant magnetic field are treated. It is shown how a particle obeying the Dirac equation, or a Dirac particle, possesses naturally an intrinsic angular momentum, or spin, with value 12 h¯ . The Dirac–Pauli equation incorporating an anomalous magnetic moment is written down. The Foldy–Wouthuysen transformation of the Dirac equation is presented in detail. This transformation leads to a systematic procedure for approximating the Dirac Hamiltonian as a sum of the nonrelativistic Hamiltonian plus higher order relativistic corrections. In the Foldy–Wouthuysen representation, the four-component wave function has large upper pair of components compared to the vanishingly small lower pair of components when the particle has positive energy, and the Hamiltonian becomes approximately a direct sum of two parts such that it is possible to build an effective two-component theory for the positive energy particle. A Foldy–Wouthuysen-like transformation is applied to

Introduction

5

the Feshbach–Villars form of the Klein–Gordon equation, leading to a systematic procedure for approximating it as the nonrelativistic part plus higher order relativistic corrections. A Foldy–Wouthuysen-like transformation is central to the formalism of quantum charged particle beam optics, in which we study the quantum beam optical Hamiltonian as the sum of a paraxial part plus nonparaxial corrections. For more details on quantum mechanics, see, e.g., Cohen-Tannoudji, Diu, and Lolo¨e [25], Esposito, Marmo, Miele, and Sudarshan [39], Greiner [64], Griffiths and Schroetter [67], Sakurai and Napolitano [166], Shankar [169], Bjorken and Drell [13], Greiner [65], and Parthasarathy [142]. Chapter 4, An Introduction to Classical Charged Particle Beam Optics, gives a brief account of the classical theory of optical behaviors of a few magnetic and electrostatic optical elements which will serve as examples for the formalism of quantum charged particle beam optics. The systems we study are monoenergetic paraxial or quasiparaxial charged particle beams propagating in the forward direction through the optical elements comprising time-independent electromagnetic fields. Starting with the general form of the optical Hamiltonian of a relativistic charged particle moving through an electromagnetic optical element with a straight optic axis, derived in Chapter 2, the relativistic classical charged particle beam optical Hamiltonian, or simply, the classical beam optical Hamiltonian, for any electromagnetic optical element with a straight optic axis is derived. Starting with this classical beam optical Hamiltonian, first we study the propagation of the beam through free space. Then, we study a few optical systems with straight optic axis, namely, axially symmetric magnetic and electrostatic lenses, and magnetic and electrostatic quadrupoles. We derive the well-known results on their optical behaviors by computing the transfer maps for the transverse position coordinates (x, y) and the transverse momentum components (px , py ). These optical systems are studied in the lowest order, the paraxial, approximation. Study of the axially symmetric magnetic lens leads to the understanding of the classical theory of image formation in an electron microscope. Study of the magnetic quadrupole explains its central role in guiding the accelerator beams. Transforming the optical Hamiltonian to a curvilinear coordinate system adapted to the geometry of the design orbit, bending of a charged particle beam by a magnetic dipole element, the simplest optical element with a curved optic axis, is studied in paraxial approximation, and the well-known classical results are obtained. It is found that one can obtain the relativistic results by extending the nonrelativistic results through the replacement of the rest mass m by the so-called relativistic mass γm as is the common practice in classical charged particle beam optics. For more details on classical charged particle beam optics, see, e.g., the references on electron optics, charged particle optics, charged particle beam physics, and accelerator physics, mentioned earlier (Hawkes and Kasper [70, 71], Orloff [141], Groves [68], Lubk [129], Pozzi [150], Berz, Makino, and Wan [12], Conte and MacKay [27], Lee [127], Reiser [158], Rosenzweig [159], Seryi [168], Weidemann [187], Wolski [192], Chao, Mess, Tigner, and Zimmermann [20] and references therein). Chapter 5, Quantum Charged Particle Beam Optics: Scalar Theory for Spin-0 and Spinless Particles, spells out the general framework of the formalism of quantum

6

Quantum Mechanics of Charged Particle Beam Optics

charged particle beam optics and develops the scalar theory applicable to spin-0 and spinless particles. Let a monoenergetic paraxial or quasiparaxial charged particle beam be propagating in the forward direction along the optic axis of an optical system comprising a time-independent electromagnetic field. Let s be coordinate along the optic axis of the system and |ψ(s)i represent the quantum state of the beam in the vertical plane at the point s on the optic axis. Since we are interested in studying the evolution of the beam along the optic axis of the system, the quantum charged particle beam optical evolution equation should have the general form: i¯h

∂ |ψ(s)i co |ψ(s)i, =H ∂s

(1.1)

co is the quantum charged particle beam optical Hamiltonian of the system. where H In quantum charged particle beam optics, Equation (1.1) replaces the Schr¨odinger equation: ∂ |Ψ(t)i c|Ψ(t)i. i¯h =H (1.2) ∂t The quantum systems we are studying are the scattering states corresponding to the beams propagating through optical elements comprising time-independent electromagnetic fields. Once the proper basic quantum mechanical time-evolution equation (nonrelativistic Schr¨odinger equation, Klein–Gordon equation, Dirac equation, . . .) appropriate for the given system is rewritten in the form of (1.1), the quantum charged particle beam optical Schr¨odinger equation, or in short, the quantum beam optical Schr¨odinger equation, all the optical behaviors of system can be deduced using the standard rules of quantum mechanics. This chapter presents the scalar theory of quantum charged particle beam optics, which is applicable to spin-0 and spinless (i.e., spin-ignored) particles. It is based on the relativistic Klein–Gordon equation for relativistic particles and the nonrelativistic Schr¨odinger equation for nonrelativistic particles. In the scalar theory, the wave function has a single component, ψ (x, y, s) = hx, y |ψ(s) i, where x and y are the coordinates in the vertical plane, or the transverse coordinates, at the point s on the optic axis. The optic axis can be straight or curved. Except for the example of the dipole magnet, or the bending magnet, which has a curved optic axis, the systems we study will have straight optic axis. The coordinate s along a straight optic axis will be taken as z. Starting with the chosen basic time-evolution equation, Klein–Gordon, or the nonrelativistic Schr¨odinger, equation in the case of the scalar theory, and the Dirac equation in the case of the spinor theory, we shall first write down the beam optical evolution equation for a general electromagnetic optical system as an z-evolution equation. In case the system to be studied has a curved optic axis, we shall transform the z-evolution equation into an s-evolution equation adapted to the geometry of optic axis of the system. Writing the time-independent Klein–Gordon equation in the Feshbach–Villars-like form and using a Foldy–Wouthuysen-like transformation lead to the scalar quantum beam optical Schr¨odinger equation for a monoenergetic quasiparaxial beam propagating in the forward direction along the optic axis with the quantum beam optical Hamiltonian as the sum of the paraxial part and nonparaxial

Introduction

7

approximations up to any desired level of accuracy. Using this quantum beam optical Schr¨odinger equation, the transfer maps for quantum averages of the transverse posi
tion coordinates (hxi, hyi) and the transverse momentum components h pˆx i , pˆy across the optical elements are obtained in the paraxial approximation. Propagation through free space explains diffraction. Propagation through an axially symmetric magnetic lens explains image formation in an electron microscope. In the classical limit (¯h −→ 0), the results coincide exactly with the classical results. There are quantum corrections which are very small compared to the classical results and vanish in the classical limit. This explains the success of classical mechanics in charged particle beam optics. Image aberrations due to the deviation of the beam from paraxial conditions are also studied, leading to the quantum versions of the aberration coefficients which correspond to the well-known classical expressions plus some tiny quantum corrections. Similarly, the quantum theory for other optical elements considered, namely, the axially symmetric electrostatic lens, the magnetic and electrostatic quadrupoles, and the bending magnet, leads to quantum beam optical Hamiltonians corresponding to the exact classical beam optical Hamiltonians plus correction terms, which vanish in the classical limit. The nonrelativistic quantum charged particle beam optics based on the nonrelativistic Schr¨odinger equation is obtained as the nonrelativistic approximation of the theory based on the Klein–Gordon equation. It is found that the common practice of replacing the rest mass m by γm to get the relativistic results from the corresponding nonrelativistic results is justified in the scalar quantum beam optics. However, as we shall see in the next chapter, this is not true in the spinor theory when the spin of the particle is taken into account. Chapter 6, Quantum Charged Particle Beam Optics: Spinor Theory for Spin- 12 Particles, presents the spinor theory of quantum charged particle beam optics applicable to spin- 12 particles. For the relativistic spin- 12 particles, the Dirac equation and the Dirac–Pauli equation are the fundamental equations on which the quantum charged particle beam optics should be based. For nonrelativistic spin- 21 particles, the quantum charged particle beam optics can be based on the Schr¨odinger–Pauli equation. The Dirac equation for a charged particle in an electromagnetic field being linear in ∂ /∂ z it is straightforward to write down the z-evolution equation in the timeindependent case. Then, using a Foldy–Wouthuysen-like transformation, we build the spinor quantum beam optics for optical elements with straight optic axis. For a monoenergetic quasiparaxial beam propagating in the forward direction along the straight optic axis of an optical element comprising time-independent electromagnetic field, the quantum beam optical spinor, four-component, wave function has the upper pair of components large compared to the vanishingly small lower pair of components. We study the free propagation to understand diffraction. Study of axially symmetric magnetic lens shows that the paraxial part of the spinor quantum beam optical Hamiltonian is same as in the scalar theory, whereas the nonparaxial, aberration parts depend on the spin of the particle. Besides the presence of matrix terms, as is to be expected, the scalar terms in the aberration part of the spinor quantum beam optical Hamiltonian differ from the corresponding terms in the scalar quantum beam optical Hamiltonian. In high-energy electron microscopy, use of the Klein– Gordon equation, as approximation of the Dirac equation, is considered adequate.

8

Quantum Mechanics of Charged Particle Beam Optics

However, we find that a scalar approximation of the quantum beam optics based on the Dirac equation by keeping only the scalar terms in the quantum beam optical Hamiltonian and dropping the matrix terms is not the same as the quantum beam optics based on the Klein–Gordon equation. Propagation of a monoenergetic paraxial charged Dirac particle beam through a bending magnet is studied adopting the formalism of general relativity to transform the z-evolution equation into the appropriate s-evolution equation. Quantum beam optics of propagation of a forward-moving monoenergetic paraxial beam of Dirac particles with anomalous magnetic moment through an optical element with straight optic axis is formulated using the accelerator optics framework. Magnetic quadrupoles are studied as examples. This study shows how the spinor quantum beam optics gives a unified treatment of orbital motion, or the Lorentz force, effect of the Stern–Gerlach force, and the Thomas–Frenkel– Bargmann–Michel–Telegdi spin motion. Finally, the two-component spinor theory of quantum beam optics based on the nonrelativistic Schr¨odinger–Pauli equation is given. We find that while it is possible to approximate the quantum beam optical Dirac Hamiltonian to get the nonrelativistic quantum beam optical Schr¨odinger– Pauli Hamiltonian by taking the limit γ −→ 1, it would be misleading to extend the nonrelativistic results to the corresponding relativistic results when the spin of the particle is also taken into account. Chapter 7, Concluding Remarks and Outlook on Further Development of Quantum Theory of Charged Particle Beam Optics, lists remarks on certain developments in light optics inspired by the quantum charged particle beam optics, the necessity for the use of quantum charged particle beam optics in future charged particle beam devices and particle accelerators, and the main problems to be addressed in the future development of quantum charged particle beam optics. We hope that the formalism of quantum charged particle beam optics presented here would be the basis for the development of the quantum theory of future charged particle beam devices.

Introductory Review of 2 An Classical Mechanics 2.1

SINGLE PARTICLE DYNAMICS

2.1.1 2.1.1.1

LAGRANGIAN FORMALISM Basic Theory

All charged particle beam devices, from low-energy electron microscopes to highenergy particle accelerators, are designed and operated very successfully on the basis of classical mechanics though the beams propagating through them are beams of microscopic particles, like electrons, protons, and ions, which should be obeying quantum mechanics. To understand in depth, we study the quantum mechanics of charged particle beam optics in detail in this book. In this chapter, we give an introductory review of classical mechanics just at the level necessary for understanding the classical charged particle beam optics. For more details on any topic in classical mechanics and advanced perspectives, see, e.g., Goldstein, Poole, and Safko [63], Corben and Stehle [28], and Sudarshan and Mukunda [174]. Classical mechanics of Newton and its relativistic follow  extension by Einstein from a principle of stationary action. Let x(t) = x j (t) | j = 1, 2, 3 be the set of components of the position vector of a particle at time t, in any chosen Cartesian or  curvilinear coordinate system, and x(t) ˙ = x˙ j (t) = dx j (t)/dt | j = 1, 2, 3 be the set of corresponding components of the velocity of the particle at that time. Time t is the independent variable with respect to which we are studying the evolution of the system. An action functional Z tf

dtL (x(t), x(t),t) ˙

S [x(t)] =

(2.1)

ti

is associated with any trajectory of the particle, x(t), from an initial time ti to a final time t f , where L is called the Lagrangian of the particle. There can be several equivalent Lagrangians leading to the same description of the dynamics of the particle. Hamilton’s principle of stationary action says that the actual trajectory of the particle from ti to t f is along a path for which the action is stationary: i.e., the actual path x(t) is such that δ S = S [x(t) + δ x(t)] − S [x(t)] = 0, (2.2) to first order in δ x(t), any arbitrary small deviation in the path between the fixed
initial and final points, x (ti ) and x t f , respectively. Usually, along the actual path, the action takes the least value, and hence Hamilton’s principle is often called the principle of least action. Nature seems to have chosen such variational principles in formulating its basic laws. 9

10

Quantum Mechanics of Charged Particle Beam Optics

Expanding δ S, the variation in action, to first order in δ x, we have (

Z tf

δS =

dt

3

∂L ∂L δxj + δ x˙ j ∂xj ∂ x˙ j



) ∂L ∂L d δxj + (δ x j ) . ∂xj ∂ x˙ j dt

∑

ti

j=1

(

Z tf

=

dt ti

3

)



∑

j=1

(2.3)


Using integration by parts, and the boundary conditions δ x (ti ) = 0 and δ x t f = 0, in the second term, we get (

Z tf

δS =

dt

j=1

(

Z tf

dt ti



∑

ti

=

3

3

∑

j=1



∂L d δxj − ∂xj dt ∂L d − ∂ x j dt





∂L ∂ x˙ j

∂L ∂ x˙ j

)

 δxj

"

3

∂L δxj + ∑ ∂ j=1 x˙ j

#t f ti

)



δxj .

(2.4)

Then, the principle of stationary action (2.2) leads to the set of Euler–Lagrange equations,   d ∂L ∂L , j = 1, 2, 3, (2.5) = dt ∂ x˙ j ∂xj the solutions of which determine the actual paths, given the appropriate initial conditions. The derivation of the Euler–Lagrange equations from the principle of station¯ which also ary action shows that if we have a different, equivalent Lagrangian, say L, leads to the same Euler–Lagrange equations, then we should have dF (x, x,t) ˙ . L¯ = L + dt

(2.6)

We can write (2.5), Lagrange’s equations of motion for the particle, as d dt 2.1.1.2



∂L ∂ x˙

 =

∂L . ∂x

(2.7)

Example: Motion of a Charged Particle in an Electromagnetic Field

As an example of the Lagrangian formalism, let us consider the motion of a particle of mass m and electric charge q in an electromagnetic field. Let us denote x =~r = x~i+y~j +z~k, the position vector of the particle, and x˙ =~r˙ = vx~i+vy~j +vz~k =~v, the velocity of the particle in a right-handed Cartesian coordinate system with (~i, ~j,~k) as the unit vectors along the x-, y-, and z-axes, respectively. Let the electromagnetic

A Review of Classical Mechanics

11

field, electric field ~E(~r,t), and magnetic field ~B(~r,t) be specified by the scalar potential φ (~r,t) and the vector potential ~A(~r,t), such that1 ~ ~E(~r,t) = −~∇φ (~r,t) − ∂ A(~r,t) , ∂t ~B(~r,t) = ~∇ × ~A(~r,t). Now, Lagrange’s equations of motion become   d ∂L ∂L ~ = = ∇L. dt ∂~v ∂~r The relativistic Lagrangian of the particle is p L(~r,~v,t) = −mc2 1 − β 2 − q(φ −~v · ~A),

(2.8)

(2.9)

(2.10)

where c is the speed of light in vacuum and β = |~v| /c = v/c. Then, the x component of (2.9) is !   mvx dAx ∂φ ∂ d p +q =q − + (~v · ~A) . (2.11) dt dt ∂x ∂x 1−β2 Similar equations follow for y and z components. These three equations are the components of the following equation:   d(γm~v) d~A +q = q −~∇φ + ~∇(~v · ~A) , dt dt p

where γ = 1/

(2.12)

1 − β 2 . Substituting the well-known expressions d (· · · ) ∂ (· · · ) = + (~v · ~∇) (· · · ) , dt dt

(2.13)

~∇(~v · ~A) = (~v · ~∇)~A +~v × (~∇ × ~A),

(2.14)

! ~A d(γm~v) ∂ = q −~∇φ − +~v × (~∇ × ~A) , dt ∂t

(2.15)

d(γm~v) = q(~E +~v × ~B), dt

(2.16)

and we get

or

in general, we shall denote the electric field ~E(~r,t), magnetic field ~B(~r,t), scalar potential φ (~r,t), and the vector potential ~A(~r,t), respectively, as ~E, ~B, φ , and ~A suppressing their dependence on~r and t unless it is required to mention it explicitly. 1 Hereafter,

12

Quantum Mechanics of Charged Particle Beam Optics

which is the relativistic equation of motion for the charged particle in an electromagnetic field under the Lorentz force. For details on classical electrodynamics, see, e.g., Griffiths [66], and Jackson [78]. In the nonrelativistic case with β  1, we have γ ≈ 1, and the equation (2.16) becomes, d(m~v) = q(~E +~v × ~B), (2.17) dt Newton’s equation of motion for the charged particle in an electromagnetic field under the Lorentz force. The Lagrangian of the nonrelativistic particle is2 1 L = mv2 − q(φ −~v · ~A), 2

(2.18)

p 1 − β 2 ≈ 1 − β 2 /2 in view of the relation β  1, in obtained by taking approximating the relativistic Lagrangian (2.10), and dropping the constant rest energy term mc2 . 2.1.2 2.1.2.1

HAMILTONIAN FORMALISM Basic Theory

Hamiltonian formalism is an equivalent reformulation of the Lagrangian formalism. Let us now define ∂L pj = , j = 1, 2, 3, (2.19) ∂ x˙ j and

3

H=

∑ x˙ j p j − L,

(2.20)

j=1

where p j s are called the components of the canonical momentum conjugate to the coordinates x j s and H is called the Hamiltonian of the particle. The pair (x j , p j ) is known as a pair of canonically conjugate variables. The total differential of H is given by 3

dH =

3

3

3

∂L

∂L

∂L

∑ x˙ j d p j + ∑ p j d x˙ j − ∑ ∂ x j dx j − ∑ ∂ x˙ j d x˙ j − ∂t dt

j=1

j=1

3

3

j=1

j=1

3



∂L ∂L ∂L = ∑ x˙ j d p j − ∑ dx j − dt + ∑ p j − ∂ x ∂t ∂ x˙ j j j=1 j=1 j=1

 d x˙ j .

(2.21)

In view of the definition (2.19), Lagrange’s equations of motion (2.7) can be written as ∂L p˙ = , (2.22) ∂x 2 In general, for any vector quantity, say ~ V , we shall denote ~V · ~V by ~V 2 , or V 2 , to be understood as ~V · ~V from the context.

13

A Review of Classical Mechanics

and consequently we get 3

dH =

3

∂L

∑ x˙ j d p j − ∑ p˙ j dx j − ∂t dt.

j=1

(2.23)

j=1

Note that the definition of p j s in (2.19) allows us to solve for any x˙ j in terms of all x j s, p j s, and t. Thus, the Hamiltonian H defined in (2.20) becomes a function of only x j s, p j s, and t when x˙ j s are expressed in terms of x j s, p j s, and t. Hence, writing 3


dH x, p,t =

3 ∂H ∂H ∂H dx + j ∑ ∂xj ∑ ∂ p j d p j + ∂t dt, j=1 j=1

(2.24)

and comparing with (2.23), we get x˙ j =

∂H , ∂ pj

p˙ j = −

∂H , ∂xj

j = 1, 2, 3,

(2.25)

and

∂H ∂L =− . (2.26) ∂t ∂t The equations in (2.25) are Hamilton’s equations of motion for the particle, which can be written as ∂H ∂H , p˙ = − . (2.27) x˙ = ∂p ∂x The 6-dimensional space formed by the set of coordinates, {x}, and the set of components of the canonically conjugate momentum, {p}, is called the phase space of the particle. Hamilton’s equations of motion (2.27), or the canonical equations of motion, are completely equivalent to Lagrange’s equations of motion (2.7). However, Hamiltonian formalism has certain advantages over the Lagrangian formalism as we shall see later. Let us now introduce a 6-dimensional column vector ϕ in phase space with components ϕ j = x j , j = 1, 2, 3, ϕ3+ j = p j , j = 1, 2, 3. (2.28) Then, it is obvious that we can write Hamilton’s equations of motion (2.27) as ϕ˙ = J

∂H , ∂ϕ

(2.29)

where ∂ H/∂ ϕ is the 6-dimensional column vector with components ! ! ∂H ∂H ∂H ∂H = , j = 1, 2, 3, = , j = 1, 2, 3, ∂ϕ ∂xj ∂ϕ ∂ pj j

(2.30)

3+ j

and J is the 6 × 6 antisymmetric matrix given by   O I J= , −I O

(2.31)

with I and O as 3 × 3 identity and null matrices, respectively. In (2.29), we have the symplectic form of the canonical equations of motion.

14

2.1.2.2

Quantum Mechanics of Charged Particle Beam Optics

Example: Motion of a Charged Particle in an Electromagnetic Field

Let us now use the Hamiltonian formalism to study the motion of a charged particle in an electromagnetic field. As we have seen earlier, the relativistic Lagrangian of the particle is p (2.32) L(~r,~v,t) = −mc2 1 − β 2 − q(φ −~v · ~A). The canonical momentum vector conjugate to the position vector is given by ~p =

∂L = γm~v + q~A. ∂~v

(2.33)

This shows that ~p − q~A = γm~v

(2.34)

is actually the kinetic, or mechanical, linear momentum of the particle. Hereafter, in general, we shall write ~p − q~A = ~π , (2.35) as a shorthand notation for the kinetic momentum of a particle. Solving for~v in terms of~r and ~p, we have ~π . (2.36) ~v = q 2 m + c12 ~π 2 Then, the relativistic Hamiltonian is H(~r,~p,t) =~v ·~p − L p

1 − β 2 + q(φ −~v · ~A) p =~v · ~π + mc2 1 − β 2 + qφ p = m2 c4 + c2~π 2 + qφ .

=~v ·~p + mc2

(2.37)

In the absence of the electromagnetic field, the particle is free and its Hamiltonian becomes p HF = m2 c4 + c2 p2 . (2.38) Then, note that the Hamiltonian in the presence of the electromagnetic field is obtained from the free particle Hamiltonian by the replacement HF −→ H − qφ and ~p −→ ~π , known as the principle of minimal electromagnetic coupling. In the nonrelativistic case, since ~π 2  m2 c2 , the Hamiltonian becomes ~π 2 + qφ , (2.39) 2m √ 
 taking m2 c4 + c2~π 2 ≈ mc2 1 + ~π 2 /2m2 c2 and dropping the constant rest energy term mc2 . It is clear that the Hamiltonian represents the total energy of the particle. H=

15

A Review of Classical Mechanics

Now, Hamilton’s equations of motion for the particle are ~r˙ =~v = ∂ H , ∂~p ∂ H ~p˙ = − = −~∇H. ∂~r

(2.40)

It is straightforward to check the first part of this equation:  πx ∂ p 2 4 m c + c2~π 2 + qφ = q = vx , ∂ px m2 + c12 ~π 2

(2.41)

and similar equations follow for y and z components. To understand the second part of the equation, we proceed as follows. For the left-hand side, we have d d px = (γmvx + qAx ) dt dt   d(γmvx ) ∂ Ax = +q + (~v · ~∇)Ax , dt ∂t

(2.42)

and similar equations for y and z components. For the right-hand side, we have −

 ∂H ∂ p 2 4 =− m c + c2~π 2 + qφ ∂x ∂x   ∂ ∂φ ~ (~v · A) − =q , ∂x ∂x

(2.43)

and similar equations for y and z components. Equating the left- and right-hand sides, we get d(γm~v) =q dt

"

∂ ~A −~∇φ − ∂t

!

#   + ~∇(~v · ~A) − (~v · ~∇)~A

  = q ~E +~v × (~∇ × ~A)   = q ~E +~v × ~B ,

(2.44)

the relativistic equation of motion for the charged particle in an electromagnetic field under the Lorentz force (2.16).

16

2.1.3

Quantum Mechanics of Charged Particle Beam Optics

HAMILTONIAN FORMALISM IN TERMS OF THE POISSON BRACKETS

2.1.3.1

Basic Theory
Let O x, p,t be an observable, or dynamical variable, of a particle with the Hamil
tonian H x, p,t . The time evolution of O is given by 3



 ∂O ∂O ∂O ∑ ∂ x j x˙ j + ∂ p j p˙ j + ∂t j=1  3  ∂O ∂H ∂O ∂H ∂O , =∑ − + ∂ x ∂ p ∂ p ∂ x ∂t j j j j j=1

dO = dt

(2.45)

because of Hamilton’s equations of motion. Defining the Poisson bracket between

any two functions f x, p and g x, p by 3

{ f , g} =

∑

j=1



∂ f ∂g ∂ f ∂g − ∂xj ∂ pj ∂ pj ∂xj

 ,

(2.46)

we can write (2.45) as dO ∂O = {O, H} + , (2.47) dt ∂t which gives Hamilton’s equation of motion for any observable of the particle. Observe that we can write g ∂ f ∂g { f , g} = J , (2.48) ∂ϕ ∂ϕ where ∂ff /∂ ϕ is the row vector obtained by taking the transpose of the column vector ∂ f /∂ ϕ, and {ϕk , ϕl } = Jkl , k, l = 1, 2, . . . , 6. (2.49) Note that the Poisson bracket is antisymmetric: { f , g} = −{g, f }.

(2.50)

−{g, f } = {−g, f }.

(2.51)

dO ∂O = {−H, O} + . dt ∂t

(2.52)

Also, note that Thus, we can write (2.47) as

If an observable O is not explicitly time-dependent, ∂ O/∂t = 0, and hence
dO x, p = {−H, O}. dt

(2.53)

17

A Review of Classical Mechanics

Thus, the basic Hamilton’s equations of motion (2.27) can be written as x˙ = {−H, x},

p˙ = {−H, p}.

(2.54)

The Poisson brackets, {x j , pk } = δ jk ,

{x j , xk } = 0,

{p j , pk } = 0,

j, k = 1, 2, 3,

(2.55)

are the fundamental Poisson brackets. In calculating the various Poisson brackets, the following basic relations will be often useful: { f , c} = 0, {a f , bg + ch} = ab{ f , g} + ac{ f , h} { f , gh} = { f , g}h + g{ f , h}, { f , {g, h}} + {g, {h, f }} + {h, { f , g}} = 0,

(2.56)

where f , g, and h are functions of x and p, and a, b, and c are constants. The last relation in (2.56) is known as the Jacobi identity. Note that using the antisymmetry of the Poisson bracket (2.50), the Jacobi identity can be rewritten as { f , {g, h}} − {g, { f , h}} = {{ f , g}, h}. (2.57)
If an observable of the particle O x, p is not explicitly time-dependent and has vanishing Poisson bracket with the Hamiltonian, then it will be a constant of motion, i.e., it will be conserved. Thus, for a particle with a Hamiltonian that is not explicitly time-dependent, the energy will be conserved, as one can also verify directly. With ∂ H/∂t = 0, it follows from Hamilton’s equations that
 3  dH x, p ∂H ∂H =∑ x˙ j + p˙ j dt ∂ pj j=1 ∂ x j  3  ∂H ∂H ∂H ∂H =∑ − = 0. (2.58) ∂ pj ∂xj j=1 ∂ x j ∂ p j For a particle with a Hamiltonian not dependent explicitly on
time, the equation of motion (2.53) for any time-independent observable O x, p can be directly integrated. To this end, define the Lie operator 3

: f :=

∑

j=1



∂f ∂ ∂f ∂ − ∂xj ∂ pj ∂ pj ∂xj

 .

(2.59)

Then, one can write : f :0 g = g,

: f : g = { f , g}

: f :3 g = { f , { f , { f , g}}},

: f :2 g = { f , { f , g}},

....

(2.60)

18

Quantum Mechanics of Charged Particle Beam Optics

Thus, Hamilton’s equations of motion are x˙ = : −H : x,

p˙ = : −H : p.

(2.61)

Now, for a time-independent observable O, we get on integrating (2.53) or (2.61) directly, h i O (t) = e{(t−ti ):−H:} O (ti ) # " ∞ (t − ti )n n (2.62) = ∑ : −H : O (ti ) , n! n=0 where ti and t are the initial and final instants of time, respectively. Here, [· · · ] (ti ) means that after the expression [· · · ] has been found, it should be evaluated by substituting the values of the observables in it at the initial time ti . This equation (2.62) defines the Lie transfer operator generated by the Hamiltonian H, relating the initial and final values of any observable of the particle. For the basic phase space variables, in terms of which any observable of the particle can be written, we get the transfer map, for any t ≥ ti , as h i ϕ (t) = e{(t−ti ):−H:} ϕ (ti ) . (2.63) We shall see later how this map can be generalized when the Hamiltonian depends on the independent variable. Often, for the sake of notational convenience, we will write the transfer map (2.63) as ϕ (t) = e{(t−ti ):−H:} ϕ (ti ) . 2.1.3.2

(2.64)

Example: Dynamics of a Charged Particle in a Constant Magnetic Field

As an example of how the Poisson bracket formalism works, let us consider the motion of a particle of mass m and charge q in a constant magnetic field. By constant magnetic field, we mean that the magnetic field does not vary in space and time. Without loss of generality, we shall take the magnetic field to be in the z-direction, ~B = B~k, and the corresponding vector potential to be ~A = 1 (~B ×~r). The relativistic 2 Hamiltonian of the particle is p H(~r,~p) = m2 c4 + c2~π 2 (2.65) ( " #)1/2 2  2 1 1 , = m2 c4 + c2 px + qBy + py − qBx + p2z 2 2 and is time-independent. Thus, the total energy of the particle is a constant of motion. From (2.36), we know that the velocity of the particle is given by ~π ~v = q . m2 + c12 ~π 2

(2.66)

19

A Review of Classical Mechanics

Noting that r m2 +

1 2 H ~π = 2 , c2 c

(2.67)

we can write, in this case, vx =

c2 H

vz =

c2 pz , H



 1 px + qBy , 2   2 1 c py − qBx , vy = H 2 (2.68)

where the components of ~v are to be treated as functions of~r and ~p. Note that : −H :~r =~v.

(2.69)

If any f (~r,~p) is a constant of motion, i.e., : −H : f = 0, then f −1 is also a constant of motion. This follows from the observation
: −H : f f −1 = 0
= f : −H : f −1 + (: −H : f ) f −1
= f : −H : f −1 , (2.70) where the relations in (2.56) have been used. Then, we find c2 pz c2 = : −H : pz = 0, H H   2 1 c px + qBy : −H : vx =: −H : H 2     2  2 1 c qB c : −H : px + qBy = vy , = H 2 H  2  c 1 : −H : vy =: −H : py − qBx H 2     2  2 c 1 c qB = : −H : py − qBx = − vx . H 2 H : −H : vz =: −H :

It follows that

 : −H : (vx + ivy ) = −i

 c2 qB (vx + ivy ) , H

(2.71)

(2.72)

and  n c2 qB : −H : (vx + ivy ) = −i (vx + ivy ) , H n

n = 0, 1, 2, 3, . . . .

(2.73)

20

Quantum Mechanics of Charged Particle Beam Optics

Let us take ti = 0 as the time when the particle enters the magnetic field, and t as the time of observation. From (2.62), (2.69), (2.71), and (2.73), it follows that   [x(t) + iy(t)] = et:−H: (x + iy) (0) " # ∞ n n−1 t = [x(0) + iy(0)] + ∑ (−iωc ) [vx + ivy ] (0) n! n=1  i  −iωc t e − 1 [vx (0) + ivy (0)] . = [x(0) + iy(0)] + ωc   z(t) = et:−H: z (0) = z(0) + vz (0)t, (2.74) where ωc = c2 qB/H(0). Let us now take, without loss of generality, that the particle enters the magnetic field at t = 0 in the yz-plane: vx (0) = 0, vy (0) = v⊥ , vz (0) = vk . Then, the position of the particle at any later time t is given by x(t) = (x(0) + ρL ) − ρL cos (ωct) , y(t) = y(0) + ρL sin (ωct) , z(t) = z(0) + vkt,

(2.75)

where ρL = v⊥ /ωc . From this, we get vx (t) = −v⊥ sin (ωct) ,

vy (t) = v⊥ cos (ωct) ,

vz (t) = vk .

(2.76)

Since the
particle enters the field at t = 0 as a free particle with velocity ~v(0) = 0, v⊥ , vk , we know that its total energy, or Hamiltonian, at the initial time to be H(0) =

q

mc2 = γ(0)mc2 . m2 c4 + c2 p(0)2 = p 1 − β (0)2

(2.77)

where β (0) = v(0)/c. From (2.76), it is clear that v⊥ (t)2 = vx (t)2 + vy (t)2 = vx (0)2 + vy (0)2 = v⊥ (0)2 , vk (t) = vz (t) = vz (0) = vk (0),

(2.78)

and hence v(t)2 = vk (t)2 + v⊥ (t)2 has at any time t the same value as at the initial time, namely v(0)2 . So, γ is a constant of motion for the particle and c2 /H(0) = 1/γm. Thus, we have qB ωc = (2.79) γm known as the cyclotron frequency or the frequency of gyration. Equation (2.75) shows that the particle moves in a helical trajectory of radius ρL = v⊥ /ωc , the radius of gyration called the Larmor radius, and pitch ℘= 2πvk /ωc . The central axis of the helix cuts the xy-plane at the point (x0 , y0 ) = (x(0) + ρL , y(0)), the guiding center, where (x(0), y(0), z(0)) is the point at which the particle enters the magnetic field at the initial time.

21

A Review of Classical Mechanics

2.1.4 2.1.4.1

CHANGING THE INDEPENDENT VARIABLE Basic Theory

Let us rederive Hamilton’s equations of motion by formulating the principle of stationary action in phase space. We can write (2.2) in phase space as " (Z #) 3 tf
dt ∑ p j x˙ j − H x, p,t δS = δ = 0, (2.80) ti

j=1

where the Lagrangian has been substituted in terms of the Hamiltonian. Expanding (2.80), we get "  # Z tf 3 ∂H ∂H δS = dt ∑ p j δ x˙ j + x˙ j δ p j − δxj − δ pj ∂xj ∂ pj ti j=1 "  # Z tf 3 ∂H ∂H δxj − δ pj = dt ∑ − p˙ j δ x j + x˙ j δ p j − ∂xj ∂ pj ti j=1 " #t f 3

+

∑ p jδ x j

j=1

=

t

(

Z tf

dt ti

i    )   ∂H ∂H ∑ − p˙ j + ∂ x j δ x j + x˙ j − ∂ p j δ p j j=1

3

= 0,

(2.81)

where in the second step, integration by parts has been used and the boundary conditions at ti and t f make the last term vanish. Now, for the
action to be stationary for arbitrary variations in the phase space trajectory δ x, δ p , we must have p˙ j = −

∂H , ∂xj

x˙ j =

∂H , ∂ pj

j = 1, 2, 3,

(2.82)

which are Hamilton’s equations of motion. Let us now write the principle of stationary action in phase space (2.80) as (Z " #) 3 tf
δS = δ (2.83) ∑ p j dx j + H x, p,t d(−t) = 0. ti

j=1

It is clear from this that we can regard  (−t, H) as a canonically conjugate pair, and all the canonically conjugate pairs (−t, H), (x j , p j ) | j = 1, 2, 3 have equal status. This shows that instead of time, we can choose any other coordinate as the independent variable, and the corresponding Hamilton’s equations will give the evolution of the state of the particle in terms of that variable. If we choose, say, xk as the independent variable, then we can write the variational principle (2.80) as (Z " #)  3 xk (t) dx j d(−t) +E −H = 0, (2.84) δS = δ dxk ∑ pj dxk dxk xk (ti ) j=1,6=k

22

Quantum Mechanics of Charged Particle Beam Optics

with the new Hamiltonian H = −pk (x, p1 , p2 , . . . , pk−1 , pk+1 , . . . , pn , E,t) ,

(2.85)

where pk has been obtained by solving the equation H(x, p,t) = E, and the old Hamiltonian H has been denoted by E, representing the total energy of the particle. Equation (2.84) has the same form as (2.80), except for the difference in the independent variable, and hence this will lead to Hamilton’s equations of motion dx j ∂H ∂ pk = =− , dxk ∂ pj ∂ pj ∂H ∂ pk dt = =− , − dxk ∂E ∂E

dpj ∂H ∂ pk =− = , dxk ∂xj ∂xj dE ∂H ∂ pk = =− . dxk ∂t ∂t

j(6= k) = 1, 2, 3, (2.86)

Introducing the notations x0 = −t and p0 = E, we can define a Poisson bracket   3 ∂ f ∂g ∂ f ∂g { f , g}xk = ∑ − , (2.87) ∂ pj ∂xj j=0,6=k ∂ x j ∂ p j and write the equations of motion (2.86) as dx j = {−H , x j }xk = {pk , x j }xk , dxk dpj = {−H , p j }xk = {pk , p j }xk , dxk j(6= k) = 0, 1, 2, 3.

(2.88)

It is straightforward to write down the equations analogous to (2.52) and (2.64) in this case. 2.1.4.2

Example: Dynamics of a Charged Particle in a Constant Magnetic Field

As an example of changing the independent variable, we shall consider the case of a charged particle moving in a constant magnetic field. We shall write x =~r = (~r⊥ , z) and p = ~p = (~p⊥ , pz ). We can write the principle of stationary action as Z t  f δS = δ dt (px x˙ + py y˙ + pz z˙ − H(~r,~p,t)) = 0. (2.89) ti

If we now choose z as the independent variable, instead of t, we can write (2.89) as
follows: with zi = z (ti ), z f = z t f , Z z   f dx dy dt δS = δ dz px + py − E − H = 0, (2.90) dz dz dz zi where H = −pz (~r⊥ ,~p⊥ , E,t, z) ,

(2.91)

23

A Review of Classical Mechanics

with pz obtained by solving the equation H(~r,~p,t) = E and E denoting the total energy of the particle. The new Hamiltonian is H , and the corresponding Hamilton’s equations of motion are ∂ pz dx ∂ H = =− , dz ∂ px ∂ px d px ∂H ∂ pz =− = , dz ∂x ∂x dt ∂H ∂ pz − = =− , dz ∂E ∂E

dy ∂ H ∂ pz = =− , dz ∂ py ∂ py d py ∂H ∂ pz =− = , dz ∂y ∂y dE ∂H ∂ pz = =− . dz ∂t ∂t

(2.92)

Note that the evolution of a system from a time ti to a time t f through Hamilton’s equations of motion with a Hamiltonian H corresponds to t f > ti , i.e., in the forward time direction. Similarly, the z-evolution of a system with −pz as the Hamiltonian would correspond to the propagation of the system in the forward z, or +z, direction. For a charged particle moving in a constant magnetic field ~B = B~k, the Hamiltonian is time-independent and is given by ( H(~r,~p) =

m2 c4 + c2

"

#)1/2 2  2 1 1 px + qBy + py − qBx + p2z , 2 2

(2.93)

representing the total energy E of the particle. Solving for pz , we can write the Hamiltonian for z-evolution as H = −pz ( " 2  2 #)1/2 1 1 1 2 2 4 2 . =− E −m c −c px + qBy + py − qBx c 2 2

(2.94)

The corresponding canonical equations of motion (2.92) for z-evolution become   dx ∂ pz 1 1 vx =− = px + qBy = , dz ∂ px pz 2 vz   vy dy ∂ pz 1 1 =− = py − qBx = , dz ∂ py pz 2 vz   qBvy d px ∂ pz qB 1 = = py − qBx = , dz ∂x 2pz 2 2vz   d py ∂ pz qB 1 qBvx = =− px + qBy = − , dz ∂y 2pz 2 2vz dt ∂ pz E 1 − =− =− 2 =− , dz ∂E c pz vz dE ∂ pz =− = 0. (2.95) dz ∂t

24

Quantum Mechanics of Charged Particle Beam Optics

These equations can be easily verified with the help of the earlier discussions on the dynamics of a charged particle in an electromagnetic field and a constant magnetic field (see (2.42), (2.44, (2.68)). The last equation expresses the constancy of the total energy of the particle along the trajectory. For any observable of the charged particle moving in the constant magnetic field, O (~r⊥ ,~p⊥ ,t, E, z), we can write ∂O ∂O dO = {−H , O}z + = {pz , O}z + , dz ∂z ∂z

(2.96)

where 

   ∂ f ∂g ∂ f ∂g ∂ f ∂g ∂ f ∂g { f , g}z = + − − ∂ x ∂ px ∂ px ∂ x ∂ y ∂ py ∂ py ∂ y   ∂ f ∂g ∂ f ∂g − − . ∂t ∂ E ∂ E ∂t

(2.97)

For an observable O not explicitly dependent on z, dO = {−H , O}z = {pz , O}z . dz

(2.98)

{ f , g}z =: f :z g,

(2.99)

Introducing the notation we would have, on integrating (2.98), h i O(z) = e(z−zi ):−H :z O (zi ) h i = e(z−zi ):pz :z O (zi ) # " ∞ (z − zi )n n : pz :z O (zi ) , = ∑ n! n=0

(2.100)

where zi is the initial point of the trajectory, and z is any point of observation on the trajectory. Applying the relation (2.100) to O = x + iy, we have, with zi = 0, and with the help of (2.95), [x(z) + iy(z)] = [ez:pz :z (x + iy)] (0) = [x(0) + iy(0)] "

# n z + ∑ (−iωc )n−1 [vx + ivy ] (0) n!vnz n=1 ∞

= [x(0) + iy(0)] i i h (−iωc z/vz ) + e − 1 [vx (0) + ivy (0)] . ωc

(2.101)

25

A Review of Classical Mechanics

Similarly, we get E(z) = [ez:pz :z E] (0) = E(0), showing that the total energy of the particle is a constant of motion, and   E z t(z) = [ez:pz :z t] (0) = t(0) + z 2 (0) = , c pz vz (0)

(2.102)

(2.103)

where t(z) refers to the time when the particle reaches the position at z if it starts at z = 0 at t = 0. From (2.101) and (2.103), we have   ωc z x(z) = (x(0) + ρL ) − ρL cos , vk   ωc z y(z) = y(0) + ρL sin , vk z t(z) = , (2.104) vk where we have taken vx (0) = 0, vy (0) = v⊥ , vz (0) = vk as earlier. The trajectory is seen to be a helix of radius ρL = v⊥ /ωc and pitch ℘= 2πvk /ωc such that x(z +℘) = x(z), y(z +℘) = y(z). 2.1.5 2.1.5.1

CANONICAL TRANSFORMATIONS Basic Theory

Once we have a Hamiltonian of a particle with a set of phase space variables and an independent variable, it is sometimes desirable to change the set of phase space variables to suit a particular description of the evolution of the state of the particle preserving the form of Hamilton’s equations of motion. Such a transformation of the phase space variables is called a canonical transformation. We shall denote the independent variable as t generally; it can be z or any other independent variable of the particle.
For a particle with the Hamiltonian H x, p,t , and the equations of motion x˙ =

∂H , ∂p

p˙ = −

∂H , ∂x

let us consider a change of phase space coordinates given by

x j −→ X j x, p,t , p j −→ Pj x, p,t , j = 1, 2, 3.

(2.105)

(2.106)

If there exists a corresponding Hamiltonian, say K (X, P,t), such that the resulting Hamilton’s equations of motion are in the canonical form, i.e., ∂K X˙ = , ∂P

∂K P˙ = − , ∂X

(2.107)

26

Quantum Mechanics of Charged Particle Beam Optics

then the transformation (2.106) is said to be a canonical transformation. As a simple example, let us consider the following scale transformation: x j −→ X j =

xj , µx

p j −→ Pj =

pj , µp

for all j = 1, 2, 3.

(2.108)

If we take the corresponding Hamiltonian to be given by

K (X, P,t) =

1 H (µx X, µ p P,t) , µx µ p

(2.109)

then the equations in (2.107) are obviously satisfied in view of (2.105). This implies that we must have (Z

tf

δ ti

"

#)

3

dt ∑ Pj X˙ j − K (X, P,t)

= 0,

(2.110)

j=1



where δ X (ti ) = 0, δ P (ti ) = 0, δ X t f = 0, δ P t f = 0, since δ x (ti ) = 0, δ p (ti ) = 0,

δ x t f = 0, δ p t f = 0. This is seen to happen because "

#

3

1 ∑ Pj X˙ j − K (X, P,t) = µx µ p j=1

"

3

∑ p j x˙ j − H

x, p,t

#


.

(2.111)

j=1

Now, let us consider a nonsimple example. Let a function of the old coordinates and new coordinates, and possibly time, F1 (x, X,t), be such that

pj =

∂ F1 , ∂xj

Pj = −

∂ F1 , ∂ Xj

j = 1, 2, 3.

(2.112)

Define K(X, P,t) = H(x, p,t) +

∂ F1 , ∂t

(2.113)

where x and p have been substituted in terms of X and P on the right-hand side. Note that from the first part of the equation (2.112), it is possible to solve for all X j s in terms of all x j s and p j s, and from the second part of the equation (2.112), it is possible to solve for all Pj s. Now, observe that

27

A Review of Classical Mechanics 3

dF1 ∑ Pj X˙ j − K + dt = j=1

3

  ∂ F1 ˙ ∑ Pj X j − H + ∂t j=1 3

3 ∂ F1 ∂ F1 ∂ F1 ∑ ∂ x j x˙ j + ∑ ∂ X j X˙ j + ∂t j=1 j=1

+ 3

=

!

∂ F1

∑ ∂ x j x˙ j − H

j=1

" # 3  ∂ F1 ∂ F1 ∂ F 1 − + + ∑ Pj X˙ j + X˙ j ∂t ∂t ∂ Xj j=1 3

=

∑ p j x˙ j − H.

(2.114)

j=1

Thus, we have (Z " tf

dt

δ ti

#)

3

∑ p j x˙ j − H

(Z

tf

=δ ti

j=1

(Z

tf

=δ

"

3

dF1 dt ∑ Pj X˙ j − K + dt j=1 " #) 3

ti

(Z

+ δ F1 |tti

∑ Pj X˙ j − K

dt

#)

j=1 tf

=δ ti

"

#)

3

dt ∑ Pj X˙ j − K

= 0,

(2.115)

j=1

leading to the canonical equations of motion (2.107) in terms of the new phase space t coordinates X and P. Here, δ F1 |tif vanishes because δ x (ti ) = 0, and δ X (ti ) = 0, since we also have δ p (ti ) = 0. The function F1 (x, X,t) is called a generating function of a canonical transformation. Let us see another example of a generating function of a canonical transformation. Let F2 (x, P,t) be such that pj =

∂ F2 , ∂xj

and define

Xj =

∂ F2 , ∂ Pj

j = 1, 2, 3,


∂ F2 K (X, P,t) = H x, p,t + . ∂t

(2.116)

(2.117)

Now, observe 3

dF ∑ Pj X˙ j − K + dt = j=1 where

3

∑ p j x˙ j − H,

(2.118)

j=1 3

F = F2 (x, P,t) − ∑ X j Pj . j=1

(2.119)

28

Quantum Mechanics of Charged Particle Beam Optics

Thus, (Z

"

tf

∑ p j x˙ j − H

dt

δ

#)

3

ti

(Z

tf

=δ ti

j=1

(Z

tf

=δ

"

dF dt ∑ Pj X˙ j − K + dt j=1 #) "

#)

3

∑ Pj X˙ j − K

dt ti

(Z

3

t

+ δ F|tif

j=1 tf

=δ ti

"

#)

3

dt ∑ Pj X˙ j − K

= 0,

(2.120)

j=1

leading to the canonical equations of motion (2.107). In this case, F2 is the generating function. Note that if we take the generating function to be 3

F2 (x, P,t) =

(2.121)

∑ x j Pj

j=1

then it leads to a canonical transformation pj =

∂ F2 = Pj , ∂xj

Xj =

∂ F2 = x j, ∂ Pj

j = 1, 2, 3,

(2.122)

i.e., X = x and P = p, which is the identity transformation. There are other choices for the generating functions. One can choose 3
F = F3 p, X,t + ∑ x j p j ,

(2.123)

j=1

with xj = −

∂ F3 , ∂ pj

Pj = −

∂ F3 , ∂ Xj

j = 1, 2, 3,

(2.124)

and K =H+

∂ F3 . ∂t

(2.125)

Another choice is 3 3
F = F4 p, P,t + ∑ x j p j − ∑ X j Pj , j=1

(2.126)

j=1

with xj = −

∂ F4 , ∂ pj

Xj = −

∂ F4 , ∂ Pj

j = 1, 2, 3

(2.127)

and K =H+

∂ F4 . ∂t

(2.128)

29

A Review of Classical Mechanics

The aforementioned four types do not exhaust all the possibilities. One may also choose a mixture of the four types. For example, for a particle moving in a plane, one can choose F = F 0 (x1 , X2 , P1 , p2 ,t) − X1 P1 + x2 p2 . Then, to have dF = p1 x˙1 + p2 x˙2 − H, P1 X˙1 + P2 X˙2 − K + dt

(2.129)

we must have ∂ F0 ∂ F0 , x2 = − , ∂ x1 ∂ p2 ∂ F0 ∂ F0 , P2 = − , X1 = ∂ P1 ∂ X2 ∂ F0 K =H+ . ∂t p1 =

2.1.5.2

(2.130)

Optical Hamiltonian of a Charged Particle Moving Through an Electromagnetic Optical Element with a Straight Axis

As we have seen earlier, the Hamiltonian of a charged particle moving in an electromagnetic field is given by p (2.131) H = m2 c4 + c2~π 2 + qφ . Let us consider an optical element of a charged particle beam optical system comprising a time-independent electromagnetic field and having its optic axis along the z-direction. If we now change the independent variable from t to z and replace H by E, as discussed earlier, the z-evolution Hamiltonian for a particle of a charged particle beam propagating through the optical element in the forward z-direction will be H = −pz = −

1 c

q

(E − qφ )2 − m2 c4 − c2~π⊥2 − qAz .

(2.132)

This H is the basic Hamiltonian, the classical charged particle beam optical Hamiltonian, for studying the optical behavior of any electromagnetic optical element with a straight optic axis along the z-direction. Often one makes further transformations for convenience. If we make a scale transformation as ~p¯ =

~p , p0

~a =

q~A , p0

(2.133)

where p0 is the magnitude of the design momentum, the corresponding z-evolution Hamiltonian, or the beam optical Hamiltonian, will become s i (E − qφ )2 − m2 c4 h 2 2 H¯ = − − ( p ¯ − a ) + ( p ¯ − a ) − az . (2.134) x x y y c2 p20

30

Quantum Mechanics of Charged Particle Beam Optics

Note that E = pt and −t are canonically conjugate variables. A further change of phase space variables is made usually in accelerator beam dynamics: x −→ x, p¯x −→ p¯x ,

y −→ y,

−t −→ ζ = z − v0t E − E0 , p¯y −→ p¯y , p¯t −→ pζ = v0 p0

(2.135)

where E0 is the energy of the reference particle with momentum p0 and velocity v0 . This transformation is effected through the generating function  
E0 F2 x, y, −t, p¯x , p¯y , pζ , z = x p¯x + y p¯y + (z − v0t) pζ + , v0 p0

(2.136)

such that ∂ F2 = p¯x , ∂x ∂ F2 = x, ∂ p¯x

∂ F2 = p¯y , ∂y ∂ F2 = y, ∂ p¯y

∂ F2 E = p¯t = , ∂ (−t) p0 ∂ F2 = ζ. ∂ pζ

(2.137)

The resulting z-evolution Hamiltonian becomes
f x, y, ζ , p¯x , p¯y , pζ , z = H¯ + ∂ F2 H ∂z s i (E − qφ )2 − m2 c4 h 2 2 − ( p ¯ − a ) + ( p ¯ − a ) , = pζ − az − x x y y c2 p20

(2.138)

after dropping the constant additional term E0 /v0 p0 . The longitudinal position coordinate ζ = z − v0t is a measure of delay in the arrival time of the tracked particle relative to the reference particle at the position z of the optics axis, namely, the z-axis. The longitudinal momentum coordinate pζ is a measure of deviation in the energy of the tracked particle relative to the reference particle, i.e., v0 p0 pζ = ∆E = E − E0 . In what follows, we shall consider only the propagation of monoenergetic charged particle beams through optical elements comprising constant electromagnetic fields. In such systems, the energy will be conserved, and the beam optics is concerned only with the transfer maps across the optical elements for the 4-dimensional phase space comprising the transverse position coordinates and momentum components. For studying the beam dynamics of systems in which the energy of the particle changes during the propagation of the beam, like in an accelerating cavity, we will have to consider the transfer maps for the 6-dimensional phase space, including the longitudinal position coordinate ζ and the longitudinal momentum coordinate pζ .

31

A Review of Classical Mechanics

2.1.6 2.1.6.1

SYMPLECTICITY OF CANONICAL TRANSFORMATIONS Time-Independent Canonical Transformations

Let us now analyse the conditions which a phase space coordinate transformation has to satisfy in order to be canonical. First, we shall consider a transformation  



or, ϕ −→ Φ ϕ (2.139) x, p −→ X x, p , P x, p which does not involve time, or the independent variable, which we will generally denote by t. The transformation is independent of the particle Hamiltonian. If the dynamics of a particle is described in terms of the phase space variables ϕ and the Hamiltonian H(ϕ,t), then its canonical equations of motion are given by ϕ˙ = J

∂H . ∂ϕ

(2.140)

Let the Hamiltonian of the particle become K (Φ,t) under the given transformation (2.139). For this transformation to be canonical, we should have ∂K Φ˙ = J . ∂Φ

(2.141)

From the transformation relations (2.139), and the equations of motion (2.140) for ϕ, we can write 6

Φ˙ j =

6 6 ∂Φj ∂Φj ∂H Jkl , ϕ˙ k = ∑ ∑ ∂ ϕl k=1 ∂ ϕk k=1 l=1 ∂ ϕk

∑

(2.142)

or ∂H Φ˙ = MJ , ∂ϕ

(2.143)

where the elements of the 6 × 6 matrix M, the Jacobian matrix of the transformation, are given by ∂Φj , j, k = 1, 2, . . . , 6. (2.144) M jk = ∂ ϕk Observing that 6 6 ∂H ∂ H ∂ Φk ∂H =∑ = ∑ Mk j , ∂ ϕ j k=1 ∂ Φk ∂ ϕ j k=1 ∂ Φk

(2.145)

∂H e ∂H , =M ∂ϕ ∂Φ

(2.146)

e ∂H . Φ˙ = MJ M ∂Φ

(2.147)

or

we get

32

Quantum Mechanics of Charged Particle Beam Optics

From the discussion on canonical transformations in terms of generating functions, we know already that for any time-independent transformation, the new Hamiltonian is the same as the old Hamiltonian expressed in terms of  the new  variables, using the inverse of the transformation (2.139), i.e. K (Φ) = H ϕ (Φ) . Thus, the equation (2.147) becomes e ∂K . Φ˙ = MJ M (2.148) ∂Φ Comparison of this equation with (2.141) shows that for any time-independent transformation of the phase space coordinates to be canonical, the corresponding Jacobian matrix M should satisfy the condition e = J, MJ M

(2.149)

known as the symplectic condition. Any matrix M satisfying the condition (2.149) is called a symplectic matrix. Note that in view of the relations  −1 e −1 J −1 M −1 = J −1 , e =M MJ M

J −1 = −J,

(2.150)

the symplectic condition (2.149) can also be written as e MJM = J.

(2.151)

As we shall see later, this is the  condition  also for time-dependent transformations. We have taken K (Φ) = H ϕ (Φ) in the above discussion. In case the transfor  mation involves a scale transformation, we should take K (Φ) = λ H ϕ (Φ) where λ is a constant (see (2.109), where λ = 1/µx µ p ). Then, the condition for canone = λ J, or MJM e ical transformation (2.149) becomes, obviously, MJ M = λ J. If a canonical transformation corresponds to λ 6= 1, it is called an extended canonical transformation. 2.1.6.2

Time-Dependent Canonical Transformations: Hamiltonian Evolution

Let us now consider a phase space coordinate transformation  



ϕ = x, p −→ Φ ϕ,t = X x, p,t , P x, p,t

(2.152)

which depends continuously on t, representing time or the chosen independent variable. In the transformation, Φ(t) −→ Φ(t + δt) during an infinitesimal time interval δt the change in Φ, namely δ Φ, should result from a transformation close to identity. So, we shall take the generating function of this transformation to be 3

F2 =

∑ x j Pj + δtG(x, P,t),

j=1

(2.153)

33

A Review of Classical Mechanics

differing only infinitesimally from the identity transformation. Then, the relation between the old and the new coordinates is given by ∂G ∂ F2 = x j + δt , j = 1, 2, 3, ∂ Pj ∂ Pj ∂ F2 ∂G ∂G p j (t) = = Pj + δt = p j (t + δt) + δt , ∂xj ∂xj ∂xj X j = x j (t + δt) =

j = 1, 2, 3.

(2.154)

In other words, ∂G , ∂ Pj ∂G , δ p j = p j (t + δt) − p j = −δt ∂xj δ x j = x j (t + δt) − x j = δt

j = 1, 2, 3, j = 1, 2, 3.

(2.155)


Since Pj differs from p j only infinitesimally, we can take G (x, P,t) = G x, p,t , and
∂ G/∂ Pj = ∂ G/∂ p j for all j = 1, 2, 3. The function G x, p,t is called the generating function of the infinitesimal canonical transformation. Then, we can write (2.155) as δ ϕ = δtJ

∂G . ∂ϕ

It is clear that

(2.156)

∂G . ∂ϕ

(2.157)

∂δϕ ∂Φ ∂ 2G =I+ = I + δtJ , ∂ϕ ∂ϕ ∂ ϕ∂ ϕ

(2.158)

Φ = ϕ(t + δt) = ϕ + δ ϕ = ϕ + δtJ Thus, the Jacobian matrix of the transformation is M=

where I is the 6 × 6 identity matrix. Since J is an antisymmetric matrix and ∂ 2 G/∂ ϕ∂ ϕ, with elements ∂ 2 G/∂ ϕ j ∂ ϕk , is a symmetric matrix, we have 2 e = I − δt ∂ G J. M ∂ ϕ∂ ϕ

(2.159)

As a result we have, upto first order in δt, ! ! 2G 2G ∂ ∂ e = I + δtJ MJ M J I − δt J ∂ ϕ∂ ϕ ∂ ϕ∂ ϕ ≈ J + δtJ = J,

∂ 2G ∂ 2G J − δtJ J ∂ ϕ∂ ϕ ∂ ϕ∂ ϕ (2.160)

34

Quantum Mechanics of Charged Particle Beam Optics

which is the symplectic condition for the transformation to be canonical. Thus, a canonical transformation during an infinitesimal time interval δt satisfies the syme = J. Let M1 and M2 be two symplectic matrices, i.e., plectic condition, i.e., MJ M ff f1 = J and M2 J M f2 = J. Note that M2 M1 J M ^ M1 J M 2 M1 = M2 M1 J M1 M2 = J, and simi^ larly M1 M2 J M1 M2 = J, i.e., any product of symplectic matrices is also a symplectic matrix. From this, it follows that since a succession of infinitesimal transformations leads to a finite transformation, with the Jacobian matrix of the finite transformation being equal to the product of the symplectic Jacobian matrices of the individual   transformations, any time-dependent canonical transformation ϕ (ti ) −→ Φ ϕ,ti + t will satisfy the symplectic condition.
Let us now take the Hamiltonian of the particle H x, p,t to be the generating function of an infinitesimal canonical transformation depending on time continuously. Then, for a time interval δt, we have, as seen from (2.156), δ ϕ = δtJ

∂H , ∂ϕ

(2.161)

which implies δϕ δt

=J

∂H . ∂ϕ

(2.162)

On taking the limit δt −→ 0, these equations become just Hamilton’s equations of motion: ∂H ϕ˙ = J . (2.163) ∂ϕ This shows that the Hamiltonian evolution of the state of a particle, i.e., time evolution of the trajectory of the particle following Hamilton’s equations of motion, is a continuous canonical transformation generated by the Hamiltonian. Thus, if
the phase space variables of a particle have values x (ti ) , p (ti ) at time ti , and
values x (ti + t) , p (ti + t) at time ti + t, then the time evolution in phase space,

x (ti ) , p (ti ) −→ x (ti + t) , p (ti + t) , is a canonical transformation obeying the symplectic condition. This makes the transfer map (2.64), a symplectic map. While deriving the optical Hamiltonian of a charged particle moving through an electromagnetic optical element with a straight optic axis, we have made two transformations of the corresponding phase space. Let us check how these transformations satisfy the conditions for being canonical. The first is a scale transformation:   px py pz . (2.164) ϕ = (x, y, z, px , py , pz ) −→ Φ = x, y, z, , , p0 p0 p0 The Jacobian matrix of this transformation has the elements ( δ jk j = 1, 2, 3 k = 1, 2, . . . , 6, M jk = δ jk j = 4, 5, 6 k = 1, 2, . . . , 6. p0

(2.165)

35

A Review of Classical Mechanics

e = λ J with λ = 1/p0 , i.e., it is an extended canonical satisfying the condition MJ M transformation. The second is the transformation:     px py E px py E − E0 ϕ = x, y, −t, , , −→ Φ = x, y, z − v0t, , , . (2.166) p0 p0 p0 p0 p0 v0 p0 The corresponding Jacobian matrix has the elements   j = 1, 2, 4, 5 k = 1, 2, . . . , 6,  δ jk v δ j = 3 k = 1, 2, . . . , 6, 0 jk M jk =   δ jk j = 6 k = 1, 2, . . . , 6, v0

(2.167)

and satisfies the symplectic condition for a canonical transformation, namely, e = J. MJ M 2.1.6.3

Canonical Invariants: Poisson Brackets

As we have seen earlier, the Poisson bracket of two functions of the phase space variables of a particle, say f and g, representing two observables of the particle, is given by g ∂ f ∂g J , (2.168) { f , g}ϕ = ∂ϕ ∂ϕ where the subscript indicates that the Poisson bracket is evaluated with respect to  the  set of phase space variables ϕ. If we make a canonical transformation ϕ −→ Φ ϕ , then the Poisson bracket of f and g with respect to Φ is given by { f , g}Φ =

g ∂ f ∂g J . ∂Φ ∂Φ

(2.169)

If M is the Jacobian matrix of the transformation, i.e., M jk = ∂ Φ j /∂ ϕk , then g g ∂f ∂f = M, ∂ϕ ∂Φ

∂f e∂f , =M ∂ϕ ∂Φ

(2.170)

e = J since the transformation is canoniand M obeys the symplectic condition MJ M cal. In view of these relations, Equation (2.168) becomes { f , g}ϕ =

g g ∂f e ∂ g = ∂ f J ∂ g = { f , g}Φ , MJ M ∂Φ ∂Φ ∂Φ ∂Φ

(2.171)

showing that the Poisson bracket of any two functions of the phase space variables is an invariant under canonical transformations, i.e., it has the same value irrespective of with respect to which set of phase space variables it is calculated if the sets of variables are related to each other by canonical transformations.

36

Quantum Mechanics of Charged Particle Beam Optics

For example, let us calculate the fundamental Poisson brackets between the new phase space variables Φ with respect to the old phase space variables ϕ. We have {Φk , Φl }ϕ =

  6 6 ∂g Φk ∂ Φl e J = ∑ ∑ Mkr Jrs Mls = MJ M ∂ϕ ∂ϕ kl r=1 s=1

= Jkl = {Φk , Φl }Φ ,

k, l = 1, 2, . . . , 6,

(2.172)

showing the invariance of the fundamental Poisson brackets under canonical transformations. This fact can also be used to test whether a given transformation of the phase space coordinates is canonical. For example, for the canonical transformation (2.166), it is easy to verify that {Φk , Φl }ϕ = {Φk , Φl }Φ = Jkl ,

2.2

k, l = 1, 2, . . . , 6.

(2.173)

DYNAMICS OF A SYSTEM OF PARTICLES

Though we shall be dealing only with single particle dynamics, for the sake of completeness, let us see how to extend the Lagrangian and the Hamiltonian formalisms of single particle dynamics to a system of particles like a charged particle beam, which is a system of identical particles. In classical mechanics, identical particles can be distinguished, labeled, and tracked individually. Let us consider a system of N particles. The Lagrangian of the system will be o a n (i) function of 3N configuration space coordinates x = x j | i = 1, 2, . . . , N; j = 1, 2, 3 n o (i) and their time derivatives x˙ = x˙ j where the subscript j corresponds to the position coordinates of a particle and the superscript (i) corresponds to the labeling of the particles. The trajectories of all the particles of the system will follow the same principle of stationary action where the action has the same definition Z tf

S [x(t)] =

dtL (x(t), x(t),t) ˙ ,

(2.174)

ti

as in the single particle case, except that now x and x˙ have 3N components. Consequently, the Euler–Lagrange equations of motion for the system become   d  ∂L  ∂L = (i) , i = 1, 2, . . . , N; j = 1, 2, 3, (2.175) dt ∂ x˙(i) ∂x j

j

or

d dt



∂L ∂ x˙

 =

∂L . ∂x

(2.176)

Defining the canonically conjugate momentum coordinates by (i)

pj =

∂L (i)

∂ x˙ j

,

i = 1, 2, . . . , N; j = 1, 2, 3,

(2.177)

37

A Review of Classical Mechanics

the Hamiltonian of the system is given by N 3
(i) (i) ˙ , H x, p = ∑ ∑ x˙ j p j − L (x, x)

(2.178)

i=1 j=1

n o (i) where p = p j | i = 1, 2, . . . , N; j = 1, 2, 3 . As a consequence of Hamilton’s principle of stationary action in phase space, the canonical equations of motion for the system become (i)

x˙ j =

∂H (i) ∂ pj

,

∂H

(i)

p˙ j = −

(i)

,

i = 1, 2, . . . , N; j = 1, 2, 3,

(2.179)

∂xj

or x˙ =

∂H , ∂p

p˙ = −

∂H . ∂x

(2.180)

Now, the phase space of the system is 6N-dimensional. Let us represent a point in the phase space by
ϕ = x, p , (2.181) where (i)

x = {x1 , x2 , . . . , x3N } ,

x3(i−1)+ j = x j ,

i =1, 2, . . . , N; j = 1, 2, 3, (i)

p = {p1 , p1 , . . . , p3N } ,

p3(i−1)+ j = p j ,

i =1, 2, . . . , N; j = 1, 2, 3.

(2.182)

Then, ϕ j = x j,

j = 1, 2, . . . , 3N,

ϕ3N+ j = p j ,

j = 1, 2, . . . , 3N.

(2.183)

Hamilton’s equations of motion can be written as ϕ˙ = J

∂H , ∂ϕ

(2.184)

where J is the antisymmetric matrix  J=

O I −I O

 ,

with O as the 3N × 3N null matrix and I as the 3N × 3N identity matrix. A coordinate transformation in the phase space,  



ϕ = x, p −→ Φ ϕ,t = X x, p,t , P x, p,t ,

(2.185)

(2.186)

38

Quantum Mechanics of Charged Particle Beam Optics

will be a canonical transformation, i.e., will preserve the form of Hamilton’s equations of motion (2.184), if the symplectic condition is satisfied: e = J, MJ M

(2.187)

where M is the 6N-dimensional Jacobian matrix of the transformation with elements Mkl =

∂ Φk , ∂ ϕl

k, l = 1, 2, . . . , 6N.

(2.188)



For any two functions f x, p and g x, p , the Poisson bracket is defined by  3N  g ∂ f ∂g ∂ f ∂g ∂ f ∂g { f , g}ϕ = J =∑ − ∂ϕ ∂ϕ ∂ pj ∂xj j=1 ∂ x j ∂ p j   N 3  ∂ f ∂g  ∂ f ∂g − . =∑∑  ∂ x(i) ∂ p(i) ∂ p(i) ∂ x(i)  i=1 j=1 j

j

j

(2.189)

j

The Poisson bracket is a canonical invariant, i.e., its value is the same whether it is calculated with respect to a set of phase space variables ϕ or with respect to a set of phase space variables Φ related to ϕ by a canonical transformation. Time evolution of the system under Hamilton’s equations of motion is a continuous canonical transformation. The transfer map, i h ϕ (t) = e(t−ti ):−H: ϕ (ti ) # " ∞ (t − ti )n n (2.190) = ∑ : −H : ϕ (ti ) , n! n=0 with : −H : f = {−H, f }, is a symplectic map. Besides the Poisson brackets, there are several other canonical invariants. Let us look at one more such canonical invariant, namely, the magnitude of a volume element in the phase space of thesystem When we make a canonical trans of particles.



formation, ϕ = x, p −→ Φ ϕ,t = X x, p,t , P x, p,t the volume element (dϕ) = dx1 dx2 . . . dx3N d p1 d p2 . . . d p3N

(2.191)

(dΦ) = dX1 dX2 . . . dX3N dP1 dP2 . . . dP3N .

(2.192)

transforms into As is well known from the rule for change of variables in multivariate calculus, these volume elements are related: (dΦ) = |det(M)|(dϕ),

(2.193)

where M is the Jacobian matrix of the transformation, det(M) is the determinant of M, and |det(M)| is the absolute value of det(M). From the relation (2.187), we get     e = det(M)det(J)det M e = det(M)2 det(J) = det(J), det MJ M (2.194)

A Review of Classical Mechanics

39

and hence we find that |det(M)| = 1. Thus, (dΦ) = (dϕ), or dx1 dx2 . . . dx3N d p1 d p2 . . . d p3N = dX1 dX2 . . . dX3N dP1 dP2 . . . dP3N .

(2.195)

This proves the canonical invariance of the volume element in phase space for an N-particle system evolving in time according to Hamilton’s equations. It follows R R that the volume of any arbitrary region in phase space · · · (dϕ) occupied by an N-particle system evolving in time according to Hamilton’s equations is a canonical invariant.

Introductory Review of 3 An Quantum Mechanics 3.1

INTRODUCTION

All physical phenomena are quantum mechanical at the fundamental level. Classical mechanics we observe in the macroscopic world is an approximation. That is why it raises a curiosity when we find classical mechanics to be very successful at a microscopic level in the design and operation of beam devices like an electron microscope or a particle accelerator. In this chapter, we give an introductory review of quantum mechanics just at the level necessary for presenting the formalism of quantum charged particle beam optics. For more details on any topic in quantum mechanics at introductory and advanced levels, see, e.g., Cohen-Tannoudji, Diu, and Lolo¨e [25], Esposito, Marmo, Miele, and Sudarshan [39], Greiner [64], Griffiths and Schroetter [67], Sakurai and Napolitano [166], Shankar [169], Bjorken and Drell [13], Greiner [65], and Parthasarathy [142]. In classical mechanics, we specify the state of a particle by its position and velocity vectors in configuration space (Lagrangian description), or by its position and canonical momentum vectors in phase space (Hamiltonian description). This classical description, based on experience in the macroscopic domain, breaks down in the microscopic—molecular, atomic, nuclear, and subnuclear—domains. Experimental realization of the failure of classical mechanics in the atomic domain led to the discovery of quantum mechanics, which shows that it is not possible to know, or determine precisely, both the position and momentum of a particle like a single electron or even a single atom in a molecule. Of course, quantum mechanics approaches classical mechanics, as should be, in the macroscopic domain. With the impossibility of knowing precisely both the position ~r and canonical momentum ~p of a particle, at any time t, it has been only possible to specify the state of the particle by a function, Ψ(~r,t), which is in general a complex function of~r and t. We shall be using the righthanded Cartesian coordinate system unless stated otherwise. This function Ψ(~r,t) is called the wave function of the particle. It does not represent any time-dependent physical wave in space but represents the amplitude of a probability wave. It has been found that Ψ(~r,t) is the probability amplitude for the particle to have its position at~r, at time t, if its position is determined, i.e., |Ψ(~r,t)|2 = Ψ∗ (~r,t)Ψ(~r,t) gives the probability of finding the particle at~r if its position is determined at time t. Since the particle has to be found somewhere in the entire space the wave function Ψ(~r,t) of a particle has to satisfy, at any time t, the normalization condition Z ∞Z ∞Z ∞

dxdydz |Ψ(~r,t)|2 = 1.

(3.1)

−∞ −∞ −∞

41

42

Quantum Mechanics of Charged Particle Beam Optics

Since |Ψ(~r,t)|2 dxdydz can be interpreted as the probability of finding the particle within an infinitesimal volume element dxdydz around the point at ~r, the expression |Ψ(~r,t)|2 is called the position probability density. If there are N particles with the same wave function Ψ(~r,t), then we can say that the number of particles found at time t in an infinitesimal volume dxdydz around the point ~r will be N|Ψ(~r,t)|2 dxdydz. Thus, when dealing with a beam of particles, we can identify |ψ(~r,t)|2 with the intensity of the beam at the position~r at time t. Hereafter, we shall write Z Z Z Z ∞

∞

∞

dxdydz {

}=

d3r {

},

(3.2)

−∞ −∞ −∞

with d 3 r = dxdydz, without specifying the limits whenever the integral is over the entire space. We shall now see how the quantum mechanics of a particle is described in terms of the wave function Ψ(~r,t). Quantum theory is essentially based on a set of postulates, supported by experimental verifications of their consequences, without any indication of any violation so far. First, we shall summarize the principles of quantum kinematics and then consider the quantum dynamics.

3.2

GENERAL FORMALISM OF QUANTUM MECHANICS

3.2.1 3.2.1.1

SINGLE PARTICLE QUANTUM MECHANICS: FOUNDATIONAL PRINCIPLES Quantum Kinematics

I Any observable of a particle, taking real values on measurement, is to be repreb Time t is a parameter as in classical physics. sented by a Hermitian operator, say O. Notes: b denoted by O b† , is defined The Hermitian conjugate, or adjoint, of an operator O, by the relation Z Z  ∗ b r) = d 3 r O b† f (~r) g(~r), d 3 r f ∗ (~r)Og(~ (3.3) for any arbitrary f (~r) and g(~r). Here ∗ denotes the complex conjugate. It is understood that the integral is over the entire space, as in (3.2), if the limits are not specified explicitly. Taking complex conjugate of both sides of this equation (3.3), we have Z

Z  ∗ b b† f (~r) d r Og(~r) f (~r) = d 3 r g(~r)∗ O   ∗ Z † b† g(~r) f (~r), = d3r O 3

(3.4)

 † b† = O. b A Hermitian operator H b is defined by the condition implying that O b † = H, b H

(3.5)

43

A Review of Quantum Mechanics

or Z

b r) = d 3 r f ∗ (~r)Hg(~

Z

∗ b f (~r) d 3 r g∗ (~r)H

(3.6)

b for any arbitrary f (~r) and g(~r). This implies that for a Hermitian operator H,

R 3 ∗ b f (~r) is real. Such Hermitian operators have only real eigenvalues, i.e., d r f (~r)H

the equation b n (~r) = λn ϕn (~r) Hϕ

(3.7)

admits only real eigenvalues λn . Further, the eigenfunctions ϕn (~r) and ϕn0 (~r) corresponding to two distinct eigenvalues λn and λn0 , respectively, are orthogonal in the sense that Z d 3 r ϕn∗ (~r)ϕn0 (~r) = 0. (3.8) A nondegenerate eigenvalue λn will have only one eigenfunction ϕn (~r). A degenerate eigenvalue has more than one eigenfunctions, and all those degenerate eigenfunctions can be chosen to be orthogonal to each other. Thus, all the eigenfunctions of a Hermitian operator can be chosen to be orthonormal: Z

d 3 r ϕn∗ (~r)ϕn0 (~r) = δnn0 .

(3.9)

Also, the set of all orthonormal eigenfunctions of a Hermitian operator form a complete set, such that
(3.10) ∑ ϕn (~r)ϕn∗ (~r0 ) = δ ~r −~r0 , n

where δ

(~r −~r0 )

is the three-dimensional Dirac delta function given by the definition Z

ψ(~r) =


d 3 r0 δ ~r −~r0 ψ(~r0 ),

(3.11)

for any ψ(~r). As a result, any function ψ(~r) can be expanded in terms of the complete set of orthonormal eigenfunctions of a Hermitian operator, {ϕn (~r)}, as ψ(~r) = ∑ an ϕn (~r),

(3.12)

n

with

Z

an =

d 3 r ϕn (~r)∗ ψ(~r).

(3.13)

This is seen to follow from (3.10) and (3.11). If ψ(~r) is a zero function, then all an s are zero, showing the linear independence of the basis functions {ϕn (~r)}. For more details on Hermitian operators, see e.g., Arfken, Weber, and Harris [3], and Byron and Fuller [18]. b corresponding The set of all real eigenvalues {λn } of the Hermitian operator O to an observable O is called the spectrum of the observable O. The spectrum of an observable can be completely discrete or continuous or mixed. If the wave function

44

Quantum Mechanics of Charged Particle Beam Optics

Ψ(~r,t) of a particle is expanded in terms of {ϕn (~r)}, the complete set of orthonormal eigenfunctions of an observable, we have Ψ(~r,t) = ∑ an (t)ϕn (~r),

(3.14)

n

with Z

an (t) =

d 3 r ϕn (~r)∗ Ψ(~r,t).

(3.15)

Now, the normalization condition (3.1) becomes, in view of (3.9), Z

d 3 r Ψ∗ (~r,t)Ψ(~r,t) = ∑ ∑ a∗n (t)an0 (t)

Z

d 3 rϕn∗ (~r)ϕn0 (~r)

n n0

2 Z = ∑ |an (t)|2 = ∑ d 3 r ϕn∗ (~r)Ψ(~r,t) = 1. n

(3.16)

n

I The operators corresponding to the Cartesian position coordinates of a particle (x, y, z) are just multiplication by (x, y, z), respectively, i.e., xb = x×,

yb = y×,

b z = z×,

(3.17)

or xbψ(~r) = xψ(~r),

ybψ(~r) = yψ(~r),

b zψ(~r) = zψ(~r),

(3.18)

for any arbitrary ψ(~r). The operators corresponding to the Cartesian components of linear momentum of the particle, canonically conjugate to the respective Cartesian position coordinates, are pbx = −i¯h

∂ , ∂x

pby = −i¯h

∂ , ∂y

pbz = −i¯h

∂ , ∂z

(3.19)

or pbx ψ(~r) = −i¯h

∂ ψ(~r) , ∂x

pby ψ(~r) = −i¯h

∂ ψ(~r) , ∂y

pbz ψ(~r) = −i¯h

∂ ψ(~r) , ∂z

(3.20)

where h¯ = h/2π, and h = 6.62607004 × 10−34 J· s is Planck’s constant. We can write ~b rψ(~r) =~rψ(~r) and ~pbψ(~r) = −i¯h~∇ψ(~r). Notes: Hermiticity of position operators is obvious. For xb, we have Z

Z

dx f ∗ (x)b xg(x) = dx f ∗ (x)xg(x) Z ∗ Z ∗ ∗ ∗ dx g (x)b x f (x) , = dx g (x)x f (x) =

(3.21)

45

A Review of Quantum Mechanics

and we get similar results for yb and b z. Hermiticity of pbx is seen as follows:   Z Z ∂ g(x) dx f ∗ (x) pbx g(x) = dx f ∗ (x) −i¯h ∂x     Z Z ∂ ∂ f ∗ (x) = dx −i¯h ( f ∗ (x)g(x)) + dx i¯h g(x) ∂x ∂x   Z ∂ f ∗ (x) = f ∗ (x)g(x)|∞ + dx i¯ h g(x) −∞ ∂x Z   ∂ f (x) ∗ = dx g∗ (x) −i¯h ∂x ∗ Z dx g∗ (x) pbx f (x) , (3.22) = where we have assumed that f (x) −→ 0 and g(x) −→ 0 when x −→ ±∞ as required of functions normalizable in the sense of (3.1). Similar results for pby and pbz prove their Hermiticity. Representing~b r by~r and ~pb by −i¯h~∇ is known as the position representation. It is seen that the operators xb and pbx do not commute with each other, i.e., xb pbx ψ(~r) 6= pbx xbψ(~r) for any arbitrary ψ(~r). Similarly, yb and b z do not commute with pby and pbz , respectively. The precise commutation relation between xb and pbx is seen to be (b x pbx − pbx xb) ψ(~r) = xb pbx ψ(~r) − pbx xbψ(~r)   ∂ ψ(~r) ∂ = −i¯h x − (xψ(~r)) = i¯hψ(~r). ∂x ∂x

(3.23)

Since this relation is valid for any arbitrary ψ(~r), we write xb pbx − pbx xb = [b x, pbx ] = i¯h,

(3.24)

which is known as the Heisenberg canonical commutation relation. The expression h i b1 , O b2 = O b1 O b2 − O b2 O b1 O (3.25) b1 and O b2 . Similarly, we have is known as the commutator of O [b y, pby ] = i¯h,

[b z, pbz ] = i¯h.

(3.26)

The commutators [b x, pby ], [b y, pbx ], etc., vanish. If we write (b x, yb,b z) = (b x1 , xb2 , xb3 ) and ( pbx , pby , pbz ) = ( pb1 , pb2 , pb3 ), then we have [b xi , xbj ] = 0, [ pbi , pbj ] = 0, [b xi , pbj ] = i¯hδi j , i, j = 1, 2, 3. (3.27) h i   b1 , O b2 , one has to find O b1 O b2 − O b2 O b1 ψ (~r) for an arbitrary ψ (~r). If To evaluate O we replace the minus (−) sign in the commutator bracket by the plus (+) sign, we get the anticommutator bracket: n o b1 , O b2 = O b1 O b2 + O b2 O b1 . O (3.28)

46

Quantum Mechanics of Charged Particle Beam Optics

†  b2 = O b† O b† b b1 and O b2 are two operators O b1 O If O 2 1 . So, if two Hermitian operators O1 b2 is a Hermitian operator: b2 commute with eath other, the product operator O b1 O and O †  b b b b b2 = O b† O b† b b b1 O O 2 1 = O2 O1 = O1 O2 . If the two Hermitian operators O1 and O2 do b2 is not Hermitian operator. b1 O not commute with each other, the product operator O This has to be taken into account while forming the quantum mechanical operators b is not Hermitian, it can corresponding to classical observables. If an operator, say O, be written as the sum of a Hermitian and an anti-Hermitian operator:     b=O bH + O bA , bH = 1 O b+O b† , bA = 1 O b−O b† , O O O 2 2 b† = O bH , b† = −O bA . O O (3.29) H A bH and O bA as the Hermitian and anti-Hermitian parts of O, b respectively. We shall call O For example, corresponding to the classical observable xpx , the quantum operator cannot be xb pbx , since xb and pbx do not commute and hence xb pbx is not Hermitian. We can take the corresponding quantum operator to be the symmetrical combina † 1 1 tion 2 xb pbx + (b x pbx ) = 2 (b x pbx + pbx xb) = 12 {b x , pbx }. Note that the antisymmetrical   † x pbx − pbx xb) = 21 [b x , pbx ] = 12 i¯h is anti-Hermitian. x pbx ) = 12 (b combination, 12 xb pbx − (b In general, the ‘Classical −→ Quantum’ correspondence rule used for getting the Hermitian quantum operator corresponding to any classical observable O(~r,~p,t) is: Replace ~r and ~p by ~b r and ~pb, respectively, and make the resulting expression Hermitian, if it is not already Hermitian. To make an operator Hermitian, one may   1 b † b is Hermitian, or some ordering rule like the Weyl use the result that 2 O + O ordering rule:   1 m m m n xbm− j pbnx xbj . (3.30) classical x px −→ quantum m ∑ j 2 j=0 The algebra of commutators is exactly similar to the algebra of the Poisson brackets (2.56). We have b B] b b = −[B, b A], [A, b BbC] b = [A, b B] b C], b b Cb + B[ b A, [A, b bBb + cC] b = ab[A, b B] b C], b b + ac[A, [aA, b [B, b + [B, b A]] b + [C, b [A, b B]] b C]] b [C, b = 0, [A,

(3.31)

where a, b, and c are constants, and the last relation is the Jacobi identity for commutators. Note that, using the antisymmetry of the commutator, we can rewrite the Jacobi identity as b [B, b − [B, b C]] b = [[A, b B], b b C]] b [A, b C]. [A, (3.32) It may be noted that while the commutator of any two Hermitian operators is antiHermitian, the anticommutator of any two Hermitian operators is Hermitian.

47

A Review of Quantum Mechanics

Besides the position representation, we can have other equivalent representations e p) as any too. For example, we have the momentum representation in which, with ψ(~ momentum space function, we have e p) = px ψ(~ e p), pbx ψ(~

e p) = py ψ(~ e p), pby ψ(~

e p) = pz ψ(~ e p), pbz ψ(~

(3.33)

e p) ∂ ψ(~ . ∂ pz

(3.34)

and e p) = i¯h xbψ(~

e p) ∂ ψ(~ , ∂ px

e p) = i¯h ybψ(~

e p) ∂ ψ(~ , ∂ py

e p) = i¯h b zψ(~

Thus, in momentum representation ~b r = i¯h~∇~p ,

~pb = ~p×,

(3.35)

acting on functions in ~p-space. Note that the Heisenberg commutation relations are valid in the same form (3.27) in this representation also. One can work in any representation of the operators which preserves the Heisenberg canonical commutation relations. In the momentum representation, the wave function would be a function of 2 e p,t), such that Ψ(~ e p,t) gives the probability of momentum of the particle, say Ψ(~ finding the particle to have the momentum ~p, at time t, if its momentum is measured. It would be useful to note down the following commutation relations: ∂ f (~r) ∂ f (~r) ∂ f (~r) , [ pby , f (~r)] = −i¯h , [ pbz , f (~r)] = −i¯h , ∂x ∂y ∂z          h  i h  i h  i ∂ f ~pb ∂ f ~pb ∂ f ~pb xb, f ~pb = i¯h , yb, f ~pb = i¯h , b z, f ~pb = i¯h . ∂ pbx ∂ pby ∂ pbz (3.36) [ pbx , f (~r)] = −i¯h

The first set of relations follow by finding ( pbx f (~r) − f (~r) pbx ) ψ (~r), etc., for an arbitrary ψ (~r). The second set of relations follow from the observation that in the momentum representation f ~pb = f (~p) and xb = i¯h(∂ /∂ px ), etc. Thus, the second set of relations are got by calculating them first in the momentum representation and then converting the results to the position representation. Another way to obtain the x, pbnx ] =  second  set of relations is to use, repeatedly, the relation [b n−1 n−1 i¯h pbx + pbx xb, pbx , got using the second of the relations in (3.31), and note that the final result can be written as [b x, pbnx ] = i¯h (∂ pbnx /∂ pbx ).1 I If a particle has the wave function Ψ(~r,t), at time t, and an observable O of the particle is measured, then O will be found to have only one of the eigenvalues, b Let us assume that the eigenvalues {λn }, of the corresponding Hermitian operator O. {λn } are all distinct and nondegenerate, i.e., to each eigenvalue λn , there is only one 1 Since x b, yb, and b z are simply multiplications by x, y, and z, respectively, we shall hereafter write them simply as x, y, and z, and write~b r as~r, unless it is necessary to use the operator (b) notation.

48

Quantum Mechanics of Charged Particle Beam Optics

b n (~r) = λn ϕn (~r). The probability that O is found to eigenfunction ϕn (~r) such that Oϕ have a value λn is given by 2 Z 3 ∗ (3.37) P (O = λn ) = d r ϕn (~r)Ψ(~r,t) . If  the eigenvalue λn is dn -fold degenerate, there would be dn eigenfunctions, say b n j (~r) = λn ϕn j (~r), for all j = 1, 2, . . . , dn . In ϕn j (~r) | j = 1, 2, . . . , dn , such that Oϕ that case, the probability that O is found to have a value λn is given by Z 2 3 ∗ P (O = λn ) = ∑ d r ϕn j (~r)Ψ(~r,t) .

(3.38)

∑ P (O = λn ) = 1,

(3.39)

dn

j=1

Note that n

as should be, in view of the normalization of the wave function (3.16). Notes: For the position operator xb, the eigenvalue equation xbξ (x0 , x) = x0 ξ (x0 , x) has the solution ξ (x0 , x) = δ (x − x0 ), i.e., xδ (x − x0 ) = x0 δ (x − x0 ), −∞ < x0 < ∞, where δ (x − x0 ) is the one-dimensional Dirac delta function defined by Z ∞ −∞


dx0 δ (x − x0 )ψ x0 = ψ(x).

(3.40)

Taking ψ (x0 ) = 1, we see that Z ∞ −∞

dx0 δ (x − x0 ) = 1.

(3.41)

Effectively,
δ x − x0 =



−→ ∞ at x = x0 , 0 for x 6= x0 .

(3.42)

The Dirac delta function is a generalization of the Kronecker delta symbol δnn0 and the defining relation (3.11), or (3.40), is analogous to ∑n0 δnn0 ψn0 = ψn . Note that δ ∗ (x − x0 ) = δ (x − x0 ). Also, δ (x − x0 ) = δ (x0 − x), and hence we can write (3.40) also as Z ∞ dx0 δ (x0 − x)ψ(x0 ) = ψ(x). (3.43) −∞

The Dirac delta function can arise in several ways. For example, 0 1 ∞ dk eik(x−x ) , δ x−x = 2π −∞ (x−x0 )2
1 − δ x − x0 = lim √ e 2σ 2 . σ →0 2π σ

0




Z

(3.44)

49

A Review of Quantum Mechanics

Further, the orthonormality relation for these position eigenfunctions takes the form Z ∞ −∞

dx δ ∗ (x − x0 )δ (x − x00 ) = δ (x0 − x00 ).

(3.45)

The eigenfunctions of yb and b z will be similarly given by {δ (y − y0 ) |−∞ < y0 < ∞ } 0 0 and {δ (z − z ) |−∞ < z < ∞ }, respectively, with similar orthonormality properties. With δ (~r −~r0 ) = δ (x − x0 )δ (y − y0 )δ (z − z0 ), (3.46) we have the orthonormality relation Z

d 3 r δ ∗ (~r −~r0 )δ (~r −~r00 ) = δ (~r0 −~r00 ).

(3.47)

Note that the position eigenfunction δ (~r −~r0 ) cannot be normalized in the sense of (3.1) since the position eigenvalues {~r0 } span the entire continuous space and R 3 d r |δ (~r −~r0 )|2 −→ ∞ for any ~r0 . This is because if a particle has the wave function δ (~r −~r0 ), at any time t, it is localized at the position ~r0 at that instant, and the probability of finding it at that position becomes ∞ (!), and the probability of finding it elsewhere is zero. This happens whenever we have continuous eigenvalues for any observable, and the normalization of corresponding eigenfunctions has to be done in the sense of (3.47). It is to be observed that the three position coordinates of any particle are still assumed to take all continuous values from −∞ to ∞, like in classical physics, without any quantization. May be the space itself is quantized at a deeper level (see, e.g., Singh and Carroll [172]). b1 and O b2 be two Hermitian operators with a complete set of Let O b1 ϕ (1) (2) (~r) = common orthonormal eigenfunctions ϕ (1) (2) (~r) such that O (1)

λn ϕ

(1) (2) λn ,λm

(1) (2)

λn λm ϕ

(1)

(2)

λn ,λm

(2) λm .

λn ,λm λn ,λm (2) b1 O b2 ϕ (1) (2) (~r) = r). Then, O (1) (2) (~ λn ,λm λn ,λm (1) (2) (1) b b O2 O1 ϕ (1) (2) (~r) = λm λn ϕ (1) (2) (~r) for any λn and λn ,λm λn ,λm

b2 ϕ (~r) and O (~r) and

Or, for any ϕ

(1)

(2)

λn ,λm

(1) (2) λn ,λm

(~r) = λm ϕ

b1 O b2 ϕ (~r), O

(1)

(~r). Consequently, b1 O b2 ψ(~r) = O b2 O b1 ψ(~r). r), we have O (2) (~

(2)

λn ,λm

b2 O b1 ϕ (~r) = O

(1)

(2)

λn ,λm

for any arbitrary ψ(~r) = ∑n,m C (1) (2) ϕ (1) λn ,λm λn ,λm b1 and O b2 must commute with each other. So, only commuting In other words, O Hermitian operators can have a complete set of simultaneous orthonormal eigenfunctions. The three components of the position operator ~b r commute with each other and have {δ (~r −~r0 )} as their simultaneous eigenfunctions. We can write the three eigenvalue equations xbδ (~r −~r0 ) = x0 δ (~r −~r0 ),

ybδ (~r −~r0 ) = y0 δ (~r −~r0 ),

b zδ (~r −~r0 ) = z0 δ (~r −~r0 ), (3.48)

~b rδ (~r −~r0 ) =~r0 δ (~r −~r0 ).

(3.49)

as

50

Quantum Mechanics of Charged Particle Beam Optics

b =~b r and λn =~r0 in (3.37), we see that the probability of finding the particle Taking O at the position~r0 is given by 2 2

Z 3 ∗ 0 0 (3.50) P ~r =~r = d r δ ~r −~r Ψ(~r,t) = Ψ(~r0 ,t) , as stated in the beginning. The three components of the momentum operator pb commute with each other. Their common eigenfunction is given by φ~p (~r) =

i 1 e h¯ ~p·~r , (2π h¯ )3/2

(3.51)

such that pbx φ~p (~r) = px φ~p (~r),

pby φ~p (~r) = py φ~p (~r),

pbz φ~p (~r) = pz φ~p (~r),

(3.52)

where the eigenvalues of the momentum components, (px , py , pz ), take all real values from −∞ to ∞. We can write ~pbφ~p (~r) = ~pφ~p (~r). (3.53)  These momentum eigenfunctions φ~p (~r) corresponding to a continuous spectrum of momentum eigenvalues, like the position eigenfunctions, cannot be normalized in the sense of (3.1) and are to be normalized only in the sense of (3.47). They are orthonormal in the sense Z

d 3 r φ~p∗ (~r)φ~p0 (~r) = δ (~p −~p0 ).

(3.54)  They form a complete set such that any ψ(~r) can be expanded in terms of φ~p (~r) as Z

ψ(~r) =

e p)φ~p (~r), d 3 p ψ(~

(3.55)

where d 3 p = d px d py d pz , the integral is over the entire momentum space, and Z

e p) = ψ(~

3

d r

φ~p∗ (~r)ψ(~r) =

1 (2π h¯ )3/2

Z

i

d 3 r e− h¯ ~p·~r ψ(~r).

(3.56)

This shows that for a particle with the wave function Ψ(~r,t), which can be expanded in terms of the momentum eigenfunctions as Z

Ψ(~r,t) =

e p,t)φ~p (~r), d 3 p Ψ(~

the probability of finding it with momentum ~p0 is given by Z 2 e 0 2 P(~p = ~p0 ) = Ψ(~ p ,t) = d 3 r φ~p∗0 (~r)Ψ(~r,t) 2 Z 1 3 − h¯i ~p0 ·~r d re Ψ(~r,t) . = 3/2 (2π h¯ )

(3.57)

(3.58)

e p,t) is the Fourier transform of the wave function Ψ(~r,t) and represents Note that Ψ(~ the state in the momentum representation.

51

A Review of Quantum Mechanics

3.2.1.2

Quantum Dynamics

I The wave function Ψ(~r,t) evolves in time according to the Schr¨odinger equation i¯h

∂ b r,~pb,t)Ψ(~r,t), Ψ(~r,t) = H(~ ∂t

(3.59)

b r,~pb,t) is the Hamiltonian operator corresponding to the energy of the syswhere H(~ tem, when the system is left undisturbed by any action on the system like a measurement of an observable of the system. When an observable O of the system is measured, the wave function collapses, immediately after the measurement, to an eigenfunction of the observable corresponding to the observed eigenvalue. From that moment onwards, the wave function evolves deterministically according to the Schr¨odinger equation (3.59) until a further measurement. Notes: b r,~pb.t) is derived from the classical HamilThe quantum Hamiltonian operator H(~ tonian function H(~r,~p,t) by the Classical −→ Quantum transition, or correspondence, rule mentioned earlier, for getting the Hermitian quantum operator corresponding to any classical observable. This procedure works well for nonrelativistic quantum mechanics. But, in the case of relativistic quantum mechanics, the quantum Hamiltonian is to be obtained by a different approach as we shall see later. Also, the quantum operators are to be expressed first in Cartesian coordinates and then transformed to curvilinear coordinates if needed. If a particle has the wave function Ψ(~r,t) and its position is determined to be ~r =~r0 , at time t, then immediately after the observation, its wave function collapses to δ (~r −~r0 ). If at that instant the momentum of the particle is measured, the probability for it to be ~p0 is given by 2 Z 1 3 − h¯i ~p0 ·~r 0 d re δ (~r −~r ) P(~p = ~p ) = 3/2 (2π h¯ ) 0

=

2 1 1 − h¯i ~p0 ·~r0 e = , 3/2 (2π h¯ )3 (2π h¯ )

(3.60)

showing that the probability of finding it to be any value ~p0 is the same (ignoring the value of probability given by this equation since the position eigenfunction is not normalized in the sense of (3.1)). In other words, if we try to locate the particle precisely at a particular position, its momentum will be spread over its infinite spectrum of values with equal probability. Similarly, if a particle is found to have a definite momentum, say ~p0 , after a momentum measurement, it will have its wave function i 0 collapsed to (2π h1)3/2 e h¯ ~p ·~r immediately after the measurement. If at that instant the ¯

position of the particle is measured, the probability of finding it to be~r0 is given by

52

Quantum Mechanics of Charged Particle Beam Optics

2 Z 1 3 0 h¯i ~p0 ·~r P(~r =~r ) = d r δ (~r −~r )e 3/2 (2π h¯ ) 2 i 0 0 1 1 h¯ ~p ·~r = = , e (2π h¯ )3 (2π h¯ )3/2 0

(3.61)

showing that the probability of finding it at any position~r0 is the same (again, ignoring the value of probability given by this equation since the momentum eigenfunction is not normalized in the sense of (3.1)). This means that a particle with a well-defined value of momentum is likely to be found at any position with equal probability. In general, it is impossible to say that a particle has well-defined values for position and momentum both, or for any two observables associated with noncommuting Hermitian operators that cannot have a complete set of simultaneous eigenfunctions. When the value of one of them is determined, the state of the particle collapses to the eigenfunction corresponding to the eigenvalue obtained for that observable. Since this eigenfunction is not an eigenfunction of the other observable for any of its eigenvalues, it cannot take a definite value on a measurement of the other observable. This is the essence of Heisenberg’s uncertainty principle. To proceed further, we have to recall some basic concepts of finite-dimensional vector spaces. An N-dimensional vector N

|vi =

∑ v j | ji,

(3.62)

j=1

where {| ji| j = 1, 2, . . . , N} are N linearly independent vectors forming a basis, can be represented as   v1  v2    (3.63) |vi =  .  .  ..  vN Following Dirac, |vi or its representative column vector |vi is called a ket vector, and its adjoint, or the transpose conjugate, row vector |vi† = (v∗1 v∗2 · · · v∗N ) ,

(3.64)

is called a bra vector and represents N

hv| =

∑ v∗j h j|.

(3.65)

j=1

The inner product of two vectors, say |ui and |vi, is given by N

hu|vi =

∑ u∗j v j ,

j=1

(3.66)

53

A Review of Quantum Mechanics

and the norm of any vector is N

k |vi k2 = hv|vi =

2

.

∑ v j

(3.67)

j=1

It follows that hv|vi = 0 =⇒ |vi = 0,

(3.68)

where |vi = 0 means v j = 0, for j = 1, 2, . . . , N. Note that hv|ui = hu|vi∗ .

(3.69)

The basis vectors {| ji| j = 1, 2, . . . , N} are orthonormal: h j|ki = δ jk ,

j, k = 1, 2, . . . , N.

(3.70)

Such a basis is called a unitary basis. The basis bra vectors {h j|} are said to be dual to the basis ket vectors {| ji} in the sense of (3.70). From this, it follows that for any vector |vi, v j = h j|vi, j = 1, 2, . . . , N, (3.71) and hence N

|vi =

!

N

∑ | jih j|vi = ∑ | jih j|

j=1

|vi.

(3.72)

j=1

Thus, the completeness of the basis vectors {| ji| j = 1, 2, . . . , N} is expressed as N

∑ | jih j| = I,

(3.73)

j=1

where I is the identity operator, or N × N identity matrix in this case. This relation (3.73) is known as the resolution of identity. Note that | jih j| is the projection operator that projects out of |vi its j-th component v j | ji. We can change the unitary basis from the given set of complete orthonormal {| ji| j = 1, 2, . . . , N} to another set of complete orthonormal vectors, say, vectors  |ϕ j i| j = 1, 2, . . . , N . We can write N

 ϕ j =

ϕ j =

k=1

k=1

N

N

∑ ϕk∗j hk| = k=1 N

hϕm | ϕn i =

N

 k| ϕj ,

j = 1, 2, . . . ,

 ϕ j |k hk|,

j = 1, 2, . . . ,

∑ ϕk j |ki = ∑ |ki ∑ k=1

N

∑ hϕm | ji h j | ϕn i =

j=1

∑ ϕ ∗jm ϕ jn = δmn ,

j=1

m, n = 1, 2, . . . , N.

(3.74)

54

Quantum Mechanics of Charged Particle Beam Optics

The completeness of the new basis vectors means, we have N

I=

∑ |ϕ j ihϕ j |.

(3.75)

j=1

To see this, let us observe N

∑ ϕ ∗jm ϕ jn = δmn

N

=⇒

j=1

∑ ϕ ∗jn ϕ jm = δmn

j=1

N

=⇒

∑ ϕn j ϕm∗ j = δmn =⇒

j=1

N

∑



 n | ϕ j ϕ j | m = δmn

j=1

N

=⇒

∑ |ϕ j ihϕ j | = I.

(3.76)

j=1

Then, for any vector |vi, we have N

|vi =

N

∑ |ϕ j ihϕ j |vi,

hv| =

j=1

∑ hv|ϕ j ihϕ j |,

(3.77)

j=1

and the inner product of |ui and |vi becomes N

hu|vi =

!

N

∑ hu|ϕ j ihϕ j |vi = hu| ∑ |ϕ j ihϕ j |

j=1

|vi.

(3.78)

j=1

Note that if we form an N × N matrix U with the N columns, i.e.,   U = |ϕ1 i, |ϕ2 i, . . . , |ϕN i ,  ϕ11 ϕ12 . . .  ϕ21 ϕ22 . . .  . . . . =  .  . . . . . ϕN1 ϕN2 . . .

orthonormal basis vectors as its

ϕ1N ϕ2N . . ϕNN

   ,  

(3.79)

then, it is a unitary matrix: N

∑ ϕ ∗jm ϕ jn = δmn

=⇒ U †U = I,

j=1 N

∑ ϕn j ϕm∗ j = δmn

=⇒ UU † = I.

(3.80)

j=1

 If M is an N × N matrix with elements M jk | j, k = 1, 2, . . . , N , we can write N

M=

N

∑ ∑ M jk | jihk|,

j=1 k=1

(3.81)

55

A Review of Quantum Mechanics

and M jk = h j|M|ki,

j, k = 1, 2, . . . , N.

(3.82)

For any vector |vi, we have N

M|vi =

N

N

!

N

∑ ∑ | jiM jk hk|vi = ∑ ∑ M jk vk

j=1 k=1

j=1

| ji,

(3.83)

k=1

as  expected. If the basis is changed from the old basis to any other unitary basis, |ϕ j i| j = 1, 2, . . . , N , then for any vector |vi represented in the new basis as N

|vi =

∑ |ϕk ihϕk |vi

(3.84)

k=1

the action of M becomes N

M|vi =

N

∑ ∑ |ϕ j ihϕ j |M|ϕk ihϕk |vi.

(3.85)

j=1 k=1

Thus, the matrix elements of the linear operator M in the new basis become N

hϕ j |M|ϕk i =

N

∑ ∑ hϕ j |lihl|M|mihm|ϕk i,

j, k = 1, 2, . . . , N.,

(3.86)

l=1 m=1

where {hl|M|mi = Mlm |l, m = 1, 2, . . . , N} are the matrix elements of M in the old basis. If we denote the matrix in the old basis as M, and in the new basis as Mϕ , then it follows from (3.86) that Mϕ = U † MU. (3.87) For a Hermitian matrix H, one should have in any basis H † = H, or H †jk = Hk∗j = H jk .

(3.88)

If H −→ Hϕ under the change to the new unitary basis, then its hermiticity is seen to be preserved:
† Hϕ† = U † HU = U † HU = Hϕ . (3.89) An arbitrary vector |vi can be written as N

|vi =

N



 ϕ j |k hk|vi.

∑ ∑ ϕ j

(3.90)

j=1 k=1

This equation becomes, in terms of the components of |vi in the old and new bases, N

 ϕ j |v =

∑ k=1

N

 ϕ j |k hk|vi =

∑ ϕk∗j hk|vi, k=1

j = 1, 2, . . . , N.,

(3.91)

56

Quantum Mechanics of Charged Particle Beam Optics

or

E vϕ = U † |vi,

(3.92)

where U is the unitary matrix defined in (3.79). The inverse of this relation E |vi = U vϕ ,

(3.93)

gives the components of |vi in the old basis in terms of its components in the new basis. The two equations (3.92) and (3.93) imply that if all the vectors of the vector space are transformed by premultiplication by a unitary matrix U (in the above case U = U † ), the resulting vectors represent the same vectors in a new basis consisting of the columns of the unitary matrix U† (in the above case U† = U). Correspondingly, any linear transformation matrix M acting on the vector space gets transformed into UMU† in the new representation. In fact, one can generalize this result to nonunitary transformations. Let us make a linear transformation of an arbitrary vector |vi by a nonsingular matrix, say T , as 0 v = T |vi, det(T ) 6= 0, (3.94) or N

v0j =

∑ Tjk v j ,

j = 1, 2, . . . , N.

(3.95)

k=1

The inverse relation,  |vi = T −1 v0 ,

(3.96)

or N

vj =

∑

T −1




v0 , jk k

j = 1, 2, . . . , N,

(3.97)

k=1

implies that the components of the new vector |v0 i are the components of the original vector |vi in a new representation in which the columns of T −1 are the basis vectors. Since T −1 is not unitary, its columns will not be orthonormal, and hence the linear transformation effected by T changes the original basis to a nonorthogonal basis. If M is a linear transformation acting on the vector space, it will become T MT −1 in the new representation, since   (Mv)0 = T (M|vi) = T MT −1 v0 . (3.98)   For a nonorthonormal basis, with a set of basis vectors ξ j , it is possible to con struct a dual, or reciprocal, basis with the set of dual basis vectors ξ˜ j , such that D E ξ˜ j | ξk =

∑ ξ˜ j`∗ ξ`k

= δ jk ,

`

 D ∑ ξ j ξ˜ j = I. j

(3.99)

57

A Review of Quantum Mechanics

Hence, any vector |vi can the expanded in the nonorthonormal basis  |vi = ∑ v j ξ j ,

(3.100)

j

in which

D E v j = ξ˜ j | v .

(3.101)

Let us now extend the above discussion to the space of functions of ~r, which can be regarded as an infinite-dimensional vector space, i.e., N −→ ∞ which can be discrete or continuous. In the finite-dimensional vector space, the canonical basis vector | ji has its components as hk| ji = δ jk . In the function space we can take the canonical basis vectors as {|~ri} with h~r0 |~ri = δ (~r0 −~r). Then, identifying f (~r0 ) with h~r0 | f i, the relation Z f (~r0 ) =

d 3 r δ (~r0 −~r) f (~r)

(3.102)

translates to the representation |fi =

Z

d 3 r |~rih~r| f i,

(3.103)

with the completeness relation reading Z

d 3 r |~rih~r| = I.

(3.104)

Corresponding to the ket | f i, we can write the dual bra vector as hf| =

Z

d 3 rh f |~rih~r|

(3.105)

where h f |~ri = h~r| f i∗ . Now, the inner product of two functions f (~r) and g(~r) can be written as Z d 3 r f ∗ (~r)g(~r) = h f |gi. (3.106) Note that h f |gi = hg| f i∗ . The norm of a function f (~r), k f k, is given by k f k2 = h f | f i, and h f | f i = 0 implies that f (~r) = 0. The normalization condition for the wave function of a particle (3.1) now reads hΨ(t)|Ψ(t)i = 1,

(3.107)

where Ψ∗ (~r,t) = hΨ(t)|~ri,

Ψ(~r,t) = h~r|Ψ(t)i,

(3.108)

and |ψ(t)i is called the state vector of the particle. Sometimes, we will be using the terms state vector and wave function interchangeably. the basis to another complete set of orthonormal functions,  If we now change ϕ j (~r) = h~r|ϕ j i , which can be the eigenfunctions of a Hermitian operator representing some observable of a particle, we can write any arbitrary |ψi, with h~r|ψi = ψ(~r), as |ψi = ∑ |ϕ j ihϕ j |ψi, (3.109) j

58

Quantum Mechanics of Charged Particle Beam Optics

with the completeness relation, or the resolution of identity, for the new basis, say ϕ-basis, reading (3.110) ∑ |ϕ j ihϕ j | = I. j

In the position representation, the resolution of identity reads





∗

∑ ~r | ϕ j ϕ j |~r0 = ∑ ϕ j (~r) ϕ j ~r0 = ~r |~r0 = δ ~r −~r0 . j

(3.111)

j

Here, ∑ j runs over all the infinite values of j, discrete, continuous, or mixed; it will become an integral over ranges where j is continuous. A basis with a complete set of orthonormal functions is called a unitary basis for the same reason as in the finiteb is a linear operator, we shall write dimensional case. If O b f i = h~r|O| b f i, h~r|O

(3.112)

or, independent of representation, b f i = O| b f i. |O

(3.113)

b f i = ∑hϕ j |O|ϕ b k ihϕk | f i, hϕ j |O

(3.114)

In the ϕ-basis, we can write this as k

n o b k i are the matrix elements of O b in the ϕ-basis. For any Hermitian where hϕ j |O|ϕ b we should have operator H, b k i = hϕk |H|ϕ b j i∗ , hϕ j |H|ϕ

(3.115)

for all j and k, such that for any arbitrary | f i = ∑ j |ϕ j ihϕ j | f i and |gi = ∑k |ϕk ihϕk |gi we will have b = hg|H| b f i∗ , h f |H|gi (3.116) b h f |H| b f i is as demanded in (3.6). This implies that for any Hermitian operator H, real.  When we change the basis from one complete orthonormal set ϕ (~ r) to another j o n complete orthonormal set ϕ 0j (~r) , we go from the representation | f i = ∑ |ϕ j ihϕ j | f i,

(3.117)

| f i = ∑ |ϕ 0j ihϕ 0j | f i.

(3.118)

j

to the representation j

The components of | f i in the two representations are related by hϕ 0j | f i = ∑hϕ 0j |ϕk ihϕk | f i. k

(3.119)

59

A Review of Quantum Mechanics

Observe that the connection coefficients between the two representations, i.e., n o 0 hϕ j |ϕk i satisfy the relation

∑hϕ 0j |ϕk ihϕk |ϕm0 i = δ jm ,

(3.120)

k

since both the bases are complete orthonormal bases. Writing hϕ 0j |ϕk i = U jk , the jk-th element of a matrix U, the above relation becomes
∗ = UU † jm = δ jm , (3.121) ∑ U jkUmk k

showing that U is a unitary matrix. In other words, the change of basis effected is a b unitary transformation. The matrix elements of a Hermitian operator n o nO, correspondo b k and hϕ 0 |O|ϕ b 0 are ing to an observable O, given in the two bases by hϕ j |O|ϕ j k related by b k0 i = ∑ ∑hϕ 0j |ϕl ihϕl |O|ϕ b m ihϕm |ϕk0 i. hϕ 0j |O|ϕ (3.122) l

m

b in ϕ-basis and ϕ 0 -basis as [O] This shows that if we call the matrices representing O 0 and [O ], respectively, then the two matrices are related by a unitary transformation  0 O = U [O]U † . (3.123) It is seen that under such unitary transformations the commutator relations between the operators remain invariant, i.e., if [[O1 ] , [O2 ]] = [O3 ] in ϕ-basis, then [[O01 ] , [O02 ]] = [O03 ] in ϕ 0 -basis. Thus on change of basis, the commutation relations like [b x, pbx ] = i¯h remain invariant. In other words, classical canonical transformations, under which the Poisson bracket relations are invariant, become in quantum mechanics unitary transformations under which the corresponding commutator bracket relations remain invariant. In the position representation, or the canonical basis, we can write the matrix b r,~p,t) as elements of any operator O(~ b ~b b ~b h~r|O( r,~pb,t)|~r0 i = O( r,~pb,t)δ (~r −~r0 ).

(3.124)

In the ϕ-basis, we get b ~b hϕ j |O( r,~pb,t)|ϕk i =

Z Z

b ~b d 3 rd 3 r0 hϕ j |~rih~r|O( r,~pb,t)|~r0 ih~r0 |ϕk i

Z Z

b ~b d 3 rd 3 r0 ϕ ∗j (~r)O( r,~pb,t)δ (~r −~r0 )ϕk (~r0 )

= Z

=

b ~b d 3 r ϕ ∗j (~r)O( r,~pb,t)ϕk (~r).

(3.125)

Let us make a unitary transformation of the function space such that any vector |ψi in it gets transformed as 0 b ψ = U|ψi. (3.126)

60

Quantum Mechanics of Charged Particle Beam Optics

  In a representation with φ j as the complete set of orthonormal basis kets, we have   

0  ψ = ∑ φ j φ j ψ 0 . (3.127) |ψi = ∑ φ j φ j |ψ , j

j

Inverting the relation (3.126), we get  b † ψ 0 , |ψi = U

(3.128)

  b † φ j φ j ψ 0 , |ψi = ∑ U

(3.129)

which can be written as j

showing that |ψ 0 i in the old representation represents the vector |ψi in annew repre o b † φ j . sentation for which the complete set of orthonormal basis kets is given by U   b in the old representation, with φ j as the If a linear operator is represented by O n o b † φ j basis kets, in the new representation with U as the basis kets, it will be † bO b . This is seen as follows: bU represented by U      0   b O|ψi bO b † ψ 0 = O Oψ b b bU b0 ψ 0 . = U =U

(3.130)

Analogous to what we found in the finite-dimensional case, we can also effect a linear transformation of the function space using any invertible operator. Let such a transformation in the function space lead to the transformation of any vector in it as 0 ψ = Tb|ψi, (3.131) where Tb has an inverse Tb−1 such that Tb−1 Tb = I. Then, we can write  |ψi = Tb−1 ψ 0 ,   or, in any orthonormal basis φ j ,   0  |ψi = ∑ Tb−1 φ j φ j ψ ,

(3.132)

(3.133)

j

0 showing that |ψi in a new representation with a nonorthonormal basis n |ψ i represents o −1 b φ j . In this case, the relation between the expressions for any linear given by T b and the new representation (O b0 ) becomes operator in the old representation (O),

b0 = TbO bTb−1 . O This is seen as follows:      0   Oψ b b bTb−1 ψ 0 = O b0 ψ 0 . b O|ψi = T = TbO

(3.134)

(3.135)

61

A Review of Quantum Mechanics

As in the finite-dimensional case, it is possible to construct a dual basis for a  nonorthonormal basis. If a set ofnfunctions ξ (x) form a nonorthonormal basis, j o ˜ then a set of dual basis functions ξ j (x) can be defined such that D E Z ξ˜ j | ξk = dx ξ˜ j (x)∗ ξk (x) = δ jk ,

∑ ξ j (x)ξ˜ j

x0


∗


= δ x − x0 .

(3.136)

j

Then, any function φ (x) can be expanded as Z

φ (x) = ∑ f j ξ j (x),

fj =

dx ξ˜ j (x)∗ φ (x).

(3.137)

j

In connection with unitary and nonunitary transformations, it is useful to recall b and B, b here an operator identity. For any pair of operators A h i h h ii h h h iii b −A b Bb + 1 A, b A, b Bb + 1 A, b A, b A, b Bb + · · · . b b = Bb + A, eA Be (3.138) 2! 3! This can be proved as follows. Let b −λ A b λA b

fb(λ ) = e Be

b d f = fb(0) + λ dλ

λ =0

λ 2 d 2 fb + 2! dλ 2

λ =0

λ 3 d 3 fb + 3! dλ 3

+··· ,

λ =0

(3.139) where the infinite series represents the Taylor series expansion of fb(λ ). Observe that   d fb b −λ A b b −λ A b b b = Aeλ A Be − eλ A Be A dλ λ =0 λ =0 h i h i b , fb(λ ) b , Bb , = A = A λ =0 h h ii d 2 fb b, A b , Bb , and so on. = A (3.140) dλ 2 λ =0

Substituting these results in the Taylor series for fb(λ ) and setting λ = 1, we get the identity (3.138). Introducing the notation h i b : Bb = A, b , Bb , :A (3.141) analogous to the case of the Poisson bracket, we can write the identity (3.138) as   1 b 2 1 b 3 b −A b b A b b b e Be = I+ : A : + : A : + : A : + · · · Bb = e:A: B. (3.142) 2! 3! Also note that for any constant a e:aA: = ea:A: . b

b

(3.143)

62

Quantum Mechanics of Charged Particle Beam Optics

Using the Dirac hbra|c|keti notation, we can write the Schr¨odinger quantum dynamical time-evolution equation (3.59) in a representation-independent way2 as i¯h

∂ |Ψ(t)i b = H|Ψ(t)i. ∂t

(3.144)

Taking the adjoint of the above equation on both sides, we get the time-evolution equation for hΨ(t)| as i¯h

∂ hΨ(t)| b = −hΨ(t)|H. ∂t

(3.145)

If we choose to work in a ϕ-representation, with an orthonormal basis, then multiplying both sides of (3.144) from left by I = ∑ j |ϕ j ihϕ j | and equating the coefficients of the linearly independent basis vectors |ϕ j i on both sides, we get i¯h

∂ b b k ihϕk |Ψ(t)i. hϕ j |Ψ(t)i = hϕ j |H|Ψ(t)i = ∑hϕ j |H|ϕ ∂t k

(3.146)

b k i, we have Or, with Ψ j (t) = hϕ j |Ψ(t)i and H jk = hϕ j |H|ϕ 

.. .



   Ψ j−1 (t)   ∂  Ψ j (t)  i¯h   ∂t   Ψ j+1 (t)    .. .  · · · ·  · · · ·   · · · ·   · · · H j−1 j−1  =  · · · H j j−1  · · · H j+1 j−1   · · · ·   · · · · · · · ·

· · ·

· · ·

H j−1 j Hj j H j+1 j · · ·

H j−1 j+1 H j j+1 H j+1 j+1 · · ·

· · · · · · · · ·

· · · · · · · · ·

· · · · · · · · ·

    ..  .     Ψ j−1 (t)      Ψ j (t)  .     Ψ j+1 (t)     ..  .  (3.147)

2 Hereafter we will write, in general, H( b ~b b b ~b r,~pb,t) and other observables, such as O( r,~pb,t), simply as H b respectively, without mentioning their dependence on~b and O, r,~pb and t whenever such dependence is clear from the context. Similarly, we shall write Ψ(~r,t) and ψ(~r) simply as Ψ and ψ, respectively, unless for the sake of better clarity, we have to indicate their dependence on space, and time, explicitly.

63

A Review of Quantum Mechanics

Note that in the position representation, with hϕ j |Ψ(t)i −→ h~r|Ψ(t)i = Ψ(~r,t) b k i −→ h~r|H|~ b r0 i = Hδ b (~r −~r0 ) H jk = hϕ j |H|ϕ b k ihϕk |Ψ(t)i −→ ∑hϕ j |H|ϕ k

Z

=

Z

b ~b r,~pb,t)|~r0 ih~r0 |Ψ(t)i d 3 r0 h~r|H( b ~b r,~pb,t)δ (~r −~r0 )Ψ(~r0 ,t) d 3 r0 H(

b ~b = H( r,~pb,t)Ψ(~r,t),

(3.148)

the equation (3.147) becomes the differential equation of Schr¨odinger (3.59). For a particle in a state |Ψ(t)i, the average (or expectation, or mean) value for any observable O, at time t, can be defined as b hOiΨ(t) = hΨ(t)|O|Ψ(t)i.

(3.149)

b Ψ . The definition (3.149) can be Sometimes, we shall write hOiΨ also as hOi b corresponding to the understood as follows. Let the Hermitian quantum operator O {λn }, where λn is dn -fold degenerate. Let the observable O have real eigenvalues  b n j i = λn |ϕn j i corresponding eigenvectors be |ϕn j i| j = 1, 2, . . . , dn such that O|ϕ for all j = 1, 2, . . . , dn These eigenvectors would form a complete orthonormal set, such that hϕn j |ϕmk i = δnm δ jk,

j = 1, 2, . . . , dn , k = 1, 2, . . . , dm ,

(3.150)

with the resolution of identity dn

∑ ∑ |ϕn j ihϕn j | = I.

(3.151)

n j=1

b as From this we get the spectral decomposition of O dn

dn

b = OI b =O b ∑ ∑ |ϕn j ihϕn j | = ∑ ∑ λn |ϕn j ihϕn j |. O n j=1

(3.152)

n j=1

b in the right-hand side of (3.149), we get Substituting this expression for O dn

hOiΨ(t) = hΨ(t)| ∑ ∑ λn |ϕn j ihϕn j |Ψ(t)i n j=1

( = ∑ λn n

as claimed.

dn

∑ |hϕn j |Ψ(t)i|2

j=1

) = ∑ λn P (O = λn ) , n

(3.153)

64

Quantum Mechanics of Charged Particle Beam Optics

3.2.1.3

Different Pictures of Quantum Dynamics

We can formally integrate the Schr¨odinger equation (3.144) as b (t,ti ) |Ψ (ti )i, |Ψ(t)i = U

t ≥ ti ,

(3.154)

where ti is the initial time at which the system starts evolving with the initial state b (t,ti ) is the time-evolution operator depending on the vector given as |Ψ (ti )i, and U b of the system. The Hamiltonian may or may not depend explicitly on Hamiltonian H time. The adjoint of (3.154) is seen to be b † (t,ti ) . hΨ(t)| = hΨ (ti ) |U

(3.155)

b (ti ,ti ) = I, U

(3.156)

Obviously, we should have b (ti ,ti ). From the normalization condition for the state as the initial condition for U vector, namely hΨ(t)|Ψ(t)i = 1 at any time t, or the conservation of probability, we get b † (t,ti ) U b (t,ti ) |Ψ (ti )i = hΨ (ti ) |Ψ (ti )i = 1. hΨ(t)|Ψ(t)i = hΨ (ti ) |U

(3.157)

b (t,ti ) must be a unitary operator such that This shows that U b † (t,ti ) U b (t,ti ) = I. U

(3.158)

b † (t,ti ) |Ψ(t)i = |Ψ (ti )i, U b (t,ti ) U b † (t,ti ) |Ψ(t)i = U b (t,ti ) |Ψ (ti )i = |Ψ(t)i, U

(3.159)

Then, note that the relations

for any |Ψ(t)i, imply that b (t,ti ) U b † (t,ti ) = I. U

(3.160)

Substituting the solution (3.154) for |Ψ(t)i in the Schr¨odinger equation (3.144), we have    ∂ ∂ b b (t,ti ) |Ψ (ti )i U (t,ti ) |Ψ (ti )i = i¯h U i¯h ∂t ∂t b b = HU (t,ti ) |Ψ (ti )i. (3.161) b (t,ti ) must satisfy the differential equation This implies that U i¯h

∂ b bU b (t,ti ) , U (t,ti ) = H ∂t

(3.162)

with the initial condition b (ti ,ti ) = I. U

(3.163)

65

A Review of Quantum Mechanics

Taking the adjoint of both sides of (3.162) implies that i¯h

b † (t,ti ) ∂U b † (t,ti ) H. b = −U ∂t

(3.164)

Multiplying both sides of this equation from left by hΨ (ti )|, we get the Schr¨odinger equation for hΨ(t)| as given in (3.145). Another important property of the timeevolution operator is to be noted. Let t > t 0 > ti . From



 b t 0 ,ti |Ψ (ti )i , b t,t 0 Ψ t 0 , Ψ t 0 = U |Ψ(t)i = U b (t,ti ) |Ψ (ti )i , |Ψ(t)i = U (3.165) it is clear that

b (t,ti ) = U b t,t 0 U b t 0 ,ti . U

(3.166)

In general, for t > tn > tn−1 > · · · > t2 > t1 > ti , we have b (t,ti ) = U b (t,tn ) U b (tn ,tn−1 ) . . . U b (t2 ,t1 ) U b (t1 ,ti ) . U

(3.167)

This property of the time-evolution operator is referred to as semigroup property. b we can integrate (3.162) For a system with a time-independent Hamiltonian H, directly to get b (t,ti ) = e− h¯i (t−ti )Hb , (3.168) U which obviously satisfies the initial condition (3.163). The Hamiltonian can be written, using the spectral decomposition, as b = ∑ En |nihn|, H

(3.169)

n

in terms of its eigenvalues {En } and the corresponding eigenkets {|ni} and bras {hn|}. Note that the projection operators {Pn = |nihn|} satisfy the relation Pn Pn0 = δnn0 Pn .

(3.170)

b and the relation (3.170), we get Using the expression (3.169) for H, b (t,ti ) = e− h¯i (t−ti ) ∑n En |nihn| = ∑ e− h¯i (t−ti )En |nihn|, U

(3.171)

n

b (t,ti ), in terms of the eigenvalues and an expression for the time-evolution operator, U b (t,ti ) eigenvectors of the time-independent Hamiltonian. We shall see later how U can be calculated when the Hamiltonian depends explicitly on time. In position representation, we can write (3.154) as Z E D b h~r|Ψ(t)i = d 3 ri ~r U (t,ti ) ~ri h~ri | Ψ (ti )i , (3.172) or

Z

Ψ(~r,t) =

d 3 ri K (~r,t;~ri ,ti ) Ψ (~ri ,ti ) ,

(3.173)

66

Quantum Mechanics of Charged Particle Beam Optics

where the kernel of propagation b (t,ti ) |~ri i K (~r,t;~ri ,ti ) = h~r|U

(3.174)

is called the propagator. From (3.173), it is clear that b (t,ti ) δ (~r −~ri ), K (~r,t;~ri ,ti ) = U

t ≥ ti ,

(3.175)

which means that K (~r,t;~ri ,ti ) would be the wave function of a particle with the b at time t if at an earlier time ti it was localized at ~ri with the wave Hamiltonian H function δ (~r −~ri ). This means that the propagator K (~r,t;~ri ,ti ) can be interpreted as the probability amplitude for finding the particle at~r at time t if it was found at~ri at time ti . In position representation, the equation (3.162) becomes   ∂ b i¯h − H K (~r,t;~ri ,ti ) = 0. (3.176) ∂t D E   b b ~b as can be seen by taking ~r H r,~pb,t δ (~r −~ri ). In other words, ~ri = H b (ti ,ti ) = I K (~r,t;~ri ,ti ) satisfies the Schr¨odinger equation. The initial condition U implies that K (~r,ti ;~ri ,ti ) = δ (~r −~ri ) . (3.177) When the Hamiltonian is time-independent, we have explicitly, using (3.171), i

K (~r,t;~ri ,ti ) = ∑ e− h¯ (t−ti )En h~r|nihn|~ri i n

i

= ∑ e− h¯ (t−ti )En ψn (~r)ψn∗ (~ri ),

(3.178)

n

b corresponding to the eigenvalue En . The where ψn (~r) is the n-th eigenfunction of H validity of equations (3.176) and (3.177) can be directly checked in this case. One can absorb the condition t ≥ ti in the definition of the propagator and write G (~r,t;~ri ,ti ) = θ (t − ti ) K (~r,t;~ri ,ti ) , where θ (t − ti ) is the Heaviside step function defined by  0 for t < ti , θ (t − ti ) = 1 for t ≥ ti .

(3.179)

(3.180)

The step function satisfies the differential equation dθ (t − ti ) = δ (t − ti ) . dt

(3.181)

Then,   ∂ b i¯h − H G (~r,t;~ri ,ti ) = i¯hδ (t − ti ) K (~r,t;~ri ,ti ) ∂t   ∂ b + θ (t − ti ) i¯h − H K (~r,t;~ri ,ti ) ∂t = i¯hδ (t − ti ) δ (~r −~ri ) ,

(3.182)

67

A Review of Quantum Mechanics

where, in the last step, we have used equations (3.176) and (3.181), and the fact that δ (t − ti ) K (~r,t;~ri ,ti ) vanishes for t 6= ti and becomes δ (t − ti ) δ (~r −~ri ) as t −→ ti . The function G (~r,t;~ri ,ti ) is known as Green’s function for the operator b [(i¯h∂ /∂t) − H]. Substituting the formal solution (3.154) of the Schr¨odinger equation in the expression (3.149) for the average of an observable, we get bU b † (t,ti ) O b (t,ti ) |Ψ (ti )i. hOiΨ(t) = hΨ (ti ) |U

(3.183)

This expression (3.183) for the average of O can be reinterpreted as the average of the time-dependent operator bH (t) = U bU b † (t,ti ) O b (t,ti ) , O

(3.184)

in the fixed state of the system at the initial time ti , namely |Ψ (ti )i. Note that ( !  b bH (t) ∂ b† ∂ O dO bU b (t,ti ) + U b † (t,ti ) b (t,ti ) = i¯h U (t,ti ) O U i¯h dt ∂t ∂t   b ∂U b (t,ti ) b † (t,ti ) O +U ∂t n o bU bH b † (t,ti ) H bO b (t,ti ) + U b † (t,ti ) O bU b (t,ti ) = −U b † (t,ti ) + i¯hU

b ∂O b (t,ti ) U ∂t

(3.185)

It is easy to see that we can rewrite this equation as bH (t) dO i h b bi = H, O + dt h¯ H

b ∂O ∂t

! ,

(3.186)

H

or bH (t) i hb b i dO = HH , OH + dt h¯

b ∂O ∂t

! ,

(3.187)

H

b need not commute with since (O1 O2 )H = (O1 )H (O2 )H . Note that a time-dependent H b (t,ti ) and hence H bH 6= H, b unless H b is time-independent. For any Hermitian operaU   † † † b b b b b b b b tor O, the operator OH defined by (3.184) is Hermitian U OU = U OU and is called the Heisenberg operator corresponding to the observable O. The equation (3.187) is the Heisenberg equation of motion for the quantum observables. This is the Heisenberg picture of quantum dynamics. From this picture, it follows that an observable without an explicit time dependence will be a conserved quantity if its bH (t) = U bU b † (t,ti ) O b (t,ti ) = operator commutes with the Hamiltonian. In that case, O b independent of time. O In the Schr¨odinger picture, the state vector |Ψ(t)i evolves in time according to the Schr¨odinger equation (3.144), and the quantum observables do not change with

68

Quantum Mechanics of Charged Particle Beam Optics

time unless they have an explicit time dependence. In the Heisenberg picture the state vector is fixed at the initial time, say ti , as |Ψ (ti )i, and the observables evolve in time according to the Heisenberg equation of motion (3.187). Rewriting the Heisenberg equation of motion (3.187) as bH (t) 1h b bi dO −HH , OH + = dt i¯h

b ∂O ∂t

! ,

(3.188)

H

we see its striking resemblance to the classical Hamilton’s equation of motion in the Poisson bracket formalism (2.52), ∂O dO = {−H, O} + . dt ∂t

(3.189)

Dirac’s rule of correspondence between classical and quantum mechanics is quantum

1 hb b i O1 , O2 −→ classical {O1 , O2 } . i¯h

(3.190)

It is seen that the passage from the Heisenberg equation of motion for the observables (3.188) to the classical Hamilton’s equation of motion for the observables in the Poisson bracket formalism (3.189) is in accordance with Dirac’s rule of correspondence between classical and quantum mechanics (3.190). Writing the Heisenberg equation of motion (3.188) for observables in matrix form, with the matrix elements of the operators calculated in some orthonormal basis, takes us to Heisenberg’s matrix mechanics discovered earlier to Schr¨odinger’s wave mechanics based on differential equations. Anyway, it was soon realized that the two forms of quantum mechanics were the same. Heisenberg’s matrix mechanics was the closest to classical mechanics in form because of the correspondence (3.190). Besides the Schr¨odinger and the Heisenberg pictures, there is an important third picture, namely, the Dirac picture or the interaction picture. In this picture, intermediate between the Schr¨odinger and the Heisenberg pictures, both the state and the observables change with time. We shall consider this picture later.

3.2.1.4

Ehrenfest’s Theorem

Let us now find the equation of motion for the quantum average of an observable. To this end, let us start with the Heisenberg equation of motion written as bH (t) dO i  b b = [H, O] + dt h¯ H

b ∂O ∂t

! . H

(3.191)

69

A Review of Quantum Mechanics

Taking averages on both sides of this equation in the state |Ψ (ti )i, we get   E d d D b b Ψ (ti ) O Ψ (ti ) OH Ψ (ti ) = H Ψ (ti ) dt dt  E  iD b b = O] Ψ (ti ) Ψ (ti ) [H, h¯ * + ! H ∂O b + Ψ (ti ) Ψ (ti ) , ∂t

(3.192)

H

where d/dt has been pulled out of the term in the left-hand side since hΨ (ti ) | and |Ψ (ti )i are time-independent vectors. Since D E D E bH |Ψ (ti ) = Ψ (ti ) |U bU b † (t,ti ) O b (t,ti ) |Ψ (ti ) Ψ (ti ) |O b = hΨ(t)|O|Ψ(t)i = hOiΨ(t) ,

(3.193)

the average of the observable O in the Schr¨odinger picture, we can write (3.192) as   d i b b ∂O . (3.194) hOiΨ(t) = h[H, O]iΨ(t) + dt ∂t Ψ(t) h¯ This is the equation of motion for the average of a quantum observable, representing the general form of the Ehrenfest theorem. To see what the equation (3.194) implies, let us take O = x and 2 b = pbx +V (x), H (3.195) 2m the quantum Hamiltonian of a particle of mass m moving nonrelativistically in one dimension (x-direction) in the field of a potential V (x). Then, from (3.194), we find  2  dhxiΨ(t) i pbx = +V (x) , x dt h¯ 2m Ψ(t)  2  i pbx 1 = ,x = hpx iΨ(t) , (3.196) h¯ 2m m Ψ(t)

where we have used the fact that x is time-independent, and the commutation relations  2  pbx pbx , x = −i¯h , [V (x) , x] = 0, (3.197) 2m m derived with the help of (3.31). For the same system, if we take O = px , we find  2  d hpx iΨ(t) i pbx = +V (x) , pbx dt h¯ 2m Ψ(t)   i ∂V (x) = h[V (x) , pbx ]iΨ(t) = − h¯ ∂x Ψ(t) = hF(x)iΨ(t) ,

(3.198)

70

Quantum Mechanics of Charged Particle Beam Optics

where F(x) is the force experienced by the particle. Equations (3.196) and (3.198) are the classical Newton’s equations if we replace the quantum averages hxiΨ(t) , hpx iΨ(t) , and hF(x)iΨ(t) , for position of the particle, momentum of the particle, and the force acting on the particle, respectively, by the corresponding classical observables. Thus, the Ehrenfest theorem suggests that since classical mechanics is only an approximation of quantum mechanics, classical observables are the averages of the corresponding quantum operators.

3.2.1.5

Spin

In classical mechanics, the orbital angular momentum of a particle is given by ~L =~r ×~p.

(3.199)

Explicitly writing, the components of ~L are Lx = ypz − zpy ,

Ly = zpx − xpz ,

Lz = xpy − ypx ,

(3.200)

{Lz , Lx } = Ly .

(3.201)

with the Poisson bracket relations,  Lx , Ly = Lz ,

 Ly , Lz = Lx ,

In quantum mechanics, the components of the orbital angular momentum are represented by Hermitian operators   bx = y pbz − z pby = −i¯h y ∂ − z ∂ , L ∂z ∂y   by = z pbx − x pbz = −i¯h z ∂ − x ∂ , L ∂x ∂z   bz = x pby − y pbx = −i¯h x ∂ − y ∂ , L ∂y ∂x

(3.202)

as obtained by the classical −→ quantum transition rule. We can write ~L b =~r ×~pb = −i¯h~r × ~∇.

(3.203)

The commutation relations between the components of the angular momentum operator become h i bx , L by = i h¯ L bz , L

h i by , L bz = i h¯ L bx , L

h i bz , L bx = i h¯ L by . L

(3.204)

71

A Review of Quantum Mechanics

Let us collect here, for later use, the commutation relations between the components of the angular momentum operator and the components of the position and momentum operators: h i h i h i bx , x = 0, bx , y = i¯h z, bx , z = −i¯h y, L L L h i h i h i by , x = −i¯h z, by , y = 0, by , z = i¯h x, L L L h i h i h i bz , x = i¯h y, bz , y = −i¯h x, bz , z = 0, L L L h i h i h i bx , pbx = 0, bx , pby = i¯h pbz , bx , pbz = −i¯h pby , L L L h i h i h i by , pbx = −i¯h pbz , by , pby = 0, by , pbz = i¯h pbx , L L L h i h i h i bz , pbx = i¯h pby , bz , pby = −i¯h pbx , bz , pbz = 0, L L L h i h i ~b 2 ~b b2 L , r = 0, L , p = 0. (3.205) In deriving these relations, we have to use the commutator algebra (3.31). bx , L by , and L bz do not commute with each other, they are incompatible Since L observables, i.e., it is not possible to measure more than one of them simultaneously and they cannot have simultaneous eigenstates. There exists an operator that commutes with all of them: the square of the total angular momentum b2 = L bx2 + L by2 + L bz2 . L Using the commutation relations straightforward to verify that h i b2 , L bx = 0, L

(3.206)

(3.204) and the commutator algebra (3.31), it is h i b2 , L by = 0, L

h i b2 , L bz = 0. L

(3.207)

b2 Thus, it is possible to have simultaneous  eigenstates for L and any one of the angular bx , L by , L bz . The traditional choice is to use the simultanemomentum components L b2 and L bz . Let us denote a simultaneous eigenket of L b2 and L bz by ous eigenstates of L |λ mλ i which satisfies the eigenvalue equations b2 |λ mλ i = λ h¯ 2 |λ mλ i, L

bz |λ mλ i = mλ h¯ |λ mλ i. L

(3.208)

bz and L b2 in units of h¯ and h¯ 2 , respectively, since We have taken the eigenvalues of L the angular momentum has the same dimensions as h¯ , as seen from (3.202). Now, define b± = L bx ± iL by . L (3.209)  † b± = L b∓ . Observe that Note that L h i b± , bz , L b± = ±¯hL L

h i b2 , L b± = 0. L

(3.210)

72

Quantum Mechanics of Charged Particle Beam Optics

As a consequence of these relations, we get     b2 L b± |λ mλ i = L b± L b2 |λ mλ i = λ h¯ 2 L b± |λ mλ i , L      bz L b± |λ mλ i = L b± L bz ± h¯ |λ mλ i L   b± |λ mλ i . = (mλ ± 1) h¯ L

(3.211)

b+ |λ mλ i and L b− |λ mλ i are simultaneous eigenkets of L b2 and L bz This shows that L 2 2 b and eigenvalues (mλ + 1)¯h and corresponding to the same eigenvalue λ h¯ for L bz . For this reason L b+ and L b− are, respectively, called the (mλ − 1)¯h, respectively, for L bz being a component of ~L, b its value cannot become raising and lowering operators. L greater than the total value of angular momentum. Hence, for some maximum value of mλ , say `, we should have b2 |λ `i = λ h¯ 2 |λ `i, L

bz |λ `i = `¯h|λ `i, L

b+ |λ `i = 0. L

(3.212)

Now, from the identity   b2 = L b− L b+ + L bz L bz + h¯ , L

(3.213)

b2 |λ `i = `(` + 1)¯h2 |λ `i. L

(3.214)

λ = `(` + 1).

(3.215)

it follows that Hence, b− acts repeatedly on any eigenket |λ mλ i, the eigenWhen the lowering operator L value mλ h¯ is reduced by h¯ at each step. This process also cannot go on and should stop for some minimum value of mλ , say `, such that b2 |λ `i = λ h¯ 2 |λ `i, L

bz |λ `i = `¯h|λ `i, L

b− |λ `i = 0. L

(3.216)

Now, from the identity   b2 = L b+ L b− + L bz L bz − h¯ , L

(3.217)

b2 |λ `i = ` (` − 1) h¯ 2 |λ `i. L

(3.218)

λ = ` (` − 1) .

(3.219)

it follows that Hence, For this result to be consistent with (3.215), it must be that either ` = ` + 1 or ` = −`. The first choice is obviously absurd, and so we have ` = −`. Then, we get `(` + b2 in terms of the maximum value of mλ , namely, `. The 1)¯h2 as the eigenvalue of L minimum value of mλ is −`. Thus, mλ takes all values from −` to ` in integer steps.

73

A Review of Quantum Mechanics

If the number of integer steps from −` to ` is N, then 2` = N. Thus, relabeling the eigenket |λ mλ i as |`m` i, we have b2 |`m` i = `(` + 1)¯h2 |`mi, bz |`m` i = m` h¯ |`m` i, L L ` = 0, 1/2, 1, 3/2, . . . , m` = − `, −(` − 1), . . . , (` − 1), `.

(3.220)

Being eigenkets of Hermitian operators, {|`m` i | ` = 0, 1/2, 1, . . . , m` = −`, . . . , `} are orthonormal: h`0 m`0 |`m` i = δ``0 δm` m`0 . (3.221) Thus, we have solved completely the eigenvalue problem of quantum angular momentum algebraically. From the identity (3.213), we get   b− L b+ |`m` i = h`m` |L b2 − L bz L bz + h¯ |`m` i h`m` |L = h¯ 2 (`(` + 1) − m` (m` + 1)).

(3.222)

This shows that we can take p b+ |`m` i = h¯ `(` + 1) − m` (m` + 1)|`(m` + 1)i, L p b− |`m` i = h¯ `(` + 1) − m` (m` − 1)|`(m` − 1)i. L

(3.223)

b2 and L b± between the various From this, we can write down the matrix elements of L eigenkets as b2 |`m` i = `(` + 1)¯h2 δ``0 δm m 0 , h`0 m`0 |L ` ` p 0 b h` m`0 |L+ |`m` i = h¯ `(` + 1) − m` (m` + 1)δ``0 δm`0 (m` +1) , p b− |`m` i = h¯ `(` + 1) − m` (m` − 1)δ``0 δm (m −1) , h`0 m`0 |L ` `0 bz |`m` i = m¯hδ``0 δm m 0 . h`0 m`0 |L ` `

(3.224)

b2 and L bz matrices are Hermitian and L b+ and L b− matrices Note that, for any `, L  † b+ = L b− . It is clear from these matrix elements that the action are such that L b± and L bz on any linear combination of the 2` + 1 degenerate eigenkets of of L 2 b L corresponding to the eigenvalue `(` + 1)¯h2 , namely {|`mi}, takes it to another linear combination of the same 2` + 1 eigenkets. Thus, the (2` + 1)-dimensional vector space spanned by the kets {|`m` i} is invariant under the action of angular momentum operators. Hence, the vector space spanned by {|`m` i} carriesa b± , L bz , (2` + 1)-dimensional representation of the angular momentum operators L n     o bx = L b+ − L b− /2, L by = i L b− − L b+ /2, L bz . Let us spell out these angular or L n o  2 (`) (`) (`) and L(`) explicitly for ` = 0, 1/2, 1. : momentum matrices Lx , Ly , Lz

74

Quantum Mechanics of Charged Particle Beam Optics

` = 0 : In this case, there is only one basis vector |0 0i. In the corresponding onedimensional representation, (0)

Lx = 0,

(0)

Ly = 0,

(0)

Lz = 0,

(3.225)

and  2 L(0) = 0.

(3.226)

Note that, though this representation satisfies the algebra (3.204) trivially, it is not a faithful representation. ` = 12 : In this case, there are two basis vectors | 12 21 i and | 12 − 21 i, which can be     1 0 represented by and , respectively. The corresponding 2-dimensional 0 1 representation can be worked out from (3.224) to be       h¯ h¯ h¯ 0 1 0 −i 1 0 (1) (1) (1) Lx 2 = , Ly 2 = , Lz 2 = , i 0 2 1 0 2 2 0 −1 (3.227) and   1 2 3  1 0 L( 2 ) = h¯ 2 . (3.228) 0 1 4 Note that all the above 2 × 2 matrices are Hermitian. This representation satisfies the algebra (3.204) exactly and is a faithful representation. ` = 1 : In this case, there are  three  basis  namely  1i, |1 0i, and |1 − 1i,  vectors,  |1 1 0 0 which can be represented by  0 ,  1 , and  0 , respectively. Now the 0 0 1 corresponding 3-dimensional representation becomes     0 1 0 0 −1 0 h i¯ h ¯ (1) (1) 0 −1  , Ly = √  1 Lx = √  1 0 1  , 2 2 0 1 0 0 1 0     1 0 0 1 0 0  2 (1) 0 , Lz = h¯  0 0 L(1) = 2¯h2  0 1 0  . (3.229) 0 0 −1 0 0 1 Note that all the above 3 × 3 matrices are Hermitian. This representation is a faithful representation of the algebra (3.204). Let us now look at the angular momentum eigenvalue equations (3.220) as partial differential equations in position representation. To this end, let us transform the angular momentum operators in Cartesian coordinates (3.202) to spherical polar coordinates defined by x = r sin θ cos φ ,

y = r sin θ sin φ ,

z = r cos θ .

(3.230)

75

A Review of Quantum Mechanics

where 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π. The result is   ∂ ∂ b Lx = −i¯h − sin φ − cos φ cot θ , ∂θ ∂φ   bx = −i¯h cos φ ∂ − sin φ cot θ ∂ , L ∂θ ∂φ bz = −i¯h ∂ , L ∂φ and b2 = −¯h2 L



1 ∂ sin θ ∂ θ

   ∂ 1 ∂2 sin θ + 2 . ∂θ sin θ ∂ φ 2

(3.231)

(3.232)

Note that the angular momentum operators in spherical polar coordinates depend b2 and L bz can now be only on θ and φ and not on r. The eigenvalue equations for L written as     1 ∂ ∂ 1 ∂2 − h¯ 2 sin θ + 2 hθ φ |`m` i sin θ ∂ θ ∂θ sin θ ∂ φ 2 = `(` + 1)¯h2 hθ φ |`m` i, − i¯h

∂ hθ φ |`m` i = m` h¯ hθ φ |`m` i. ∂φ

(3.233)

The solution for hθ φ |`m` i is given by m

hθ φ |`m` i = Y` ` (θ , φ ) s (2` + 1)(` − |m` |)! m` =ε P` (cos θ )eim` φ 4π(` + |m` |)!

(3.234)

where m P` ` (x) =

P` (x) =

1−x 1 2` `!


2 |m` |/2



d dx

`



d dx

|m` | P` (x),


` x2 − 1 , m

(3.235)

ε is equal to (−1)m` for m` ≥ 0, and 1 for m` ≤ 0. Y` ` (θ , φ ) is called a spherical harm monic, P` ` (x) is the associated Legendre function, and P` (x) is the Legendre polynomial (for the details on these special functions and polynomials, see, e.g., Arfken, Weber, and Harris [3], Byron and Fuller [18], and the book of Lakshminarayanan and Varadharajan [126], which is an excellent source for all important special functions from a practical point of view with computational codes in Python). Now, it is to be seen that when φ increases by 2π we return to the same point in space so that we have to require hθ , φ + 2π|`m` i = hθ φ |`m` i. (3.236)

76

Quantum Mechanics of Charged Particle Beam Optics

This implies that we must have eim` (φ +2π) = eim` φ

(3.237)

m` = 0, ±1, ±2, . . . .

(3.238)

In other words, e2πim` = 1, i.e.,

This means that `, called the orbital angular momentum quantum number, can take only nonnegative integer values. Also, only for nonnegative integer values of `, the definition of P` (x) in the solution p (3.234) makes sense. The orbital angular momenb2 . tum of the particle is given by `(` + 1)¯h, the square root of the eigenvalue of L If the orbital angular momentum quantum number can take only nonnegative inte~b ger values, is there any physical significance for the matrix representations of L for ~b half integer values of `? The answer is that there is an intrinsic L-like dynamical variable associated with particles, called the spin, which can take any one of the allowed values of ` = 0, 21 , 1, 23 , . . . . While the orbital angular momentum quantum number of a particle ` can take all nonnegative integer values, the spin quantum number of a particle takes only a fixed value. Let us relabel ` as s to refer to the spin quantum number and relabel the corresponding matrices (Lx , Ly , Lz ) as (Sx , Sy , Sz ). Then, for any particle the spin quantum number s has a fixed value 0, or 12 , or 1, or 32 , or . . .. For a particle with spin s = 0 there is only one spin state, and so we can consider it simply as a spectator variable. In other words, we can continue to use the singlecomponent state vector |Ψ(t)i in single particle dynamical problems, ignoring the presence of spin. However, in dealing with the physics of multiple identical spin-0 particles, one has to consider them as bosons obeying the Bose–Einstein statistics. Any number of identical bosons can occupy a single quantum state. All integer spin particles, i.e., particles with s = 0, 1, 2, . . ., are bosons. For a spin- 21 particle spin is an observable vector quantity ~S with three components (Sx , Sy , Sz ). Since spin has no classical analog, we have to only postulate the corresponding Hermitian operators. Since the z-component of the spin of electron is found, experimentally, to have two values, namely ± 12 , and to get coupled to the orbital angular momentum in atomic systems, the components of ~S are taken to obey the same algebra as the components of ~L. Thus, for a spin- 12 particle, we take       h¯ h¯ h¯ 0 1 0 −i 1 0 Sx = , Sy = , Sz = , (3.239) i 0 2 1 0 2 2 0 −1 following the two-dimensional matrix representation of the angular momentum algebra corresponding to ` = 1/2 as given in (3.227). In terms of the Pauli matrices,       0 1 0 −i 1 0 σx = , σy = , σz = , (3.240) 1 0 i 0 0 −1 we write

h¯ Sx = σx , 2

h¯ Sy = σy , 2

h¯ Sz = σz . 2

(3.241)

77

A Review of Quantum Mechanics

  Hence for a spin- 12 particle, there are two spin states: 12 12 and 12 − 12 . Since s 1 it in the specification of states and write simply has 1 1a fixed 1 value 12 , we1 can drop  1 2 2 = 2 and 2 − 2 = − 2 . Physically, when a magnetic field is switched on in the z-direction, a charged spin- 12 particle aligns its spin either in the z-direction, i.e., parallel to the field, or in the −z-direction, i.e., antiparallel to the field. When  its spin is aligned parallel to the field, its spin state is 12 and Sz = h2¯ , and when it 1 − and Sz = − h¯ . So the states is aligned antiparallel to the field, its spin state is 2 2 1 1 and − are also referred to as up and down states and written as | ↑i and | ↓i, 2 2     1 0 respectively. We can also write | ↑i = and | ↓i = . Then, we have 0 1 h¯ h¯ Sz | ↑i = | ↑i, Sz | ↓i = − | ↓i, 2 2 3 3 S2 | ↑i = h¯ 2 | ↑i, S2 | ↓i = h¯ 2 | ↓i. 4 4

(3.242)

The state vector of a spin- 12 particle can be represented as  |Ψ(t)i = |Ψ1 (t)i| ↑i + |Ψ2 (t)i| ↓i =

|Ψ1 (t)i |Ψ2 (t)i

 .

(3.243)

Note that the quantum operator of any classical observable O commutes with all the components of spin, ~S, and so any one component of ~S can be measured precisely along with O. If position and spin of the particle are measured simultaneously in the above state |Ψ(t)i, the probability for the result {position =~r, spin = ↑} is |h~r|Ψ1 (t)i|2 and the probability for the result {position =~r, spin = ↓} is |h~r|Ψ2 (t)i|2 . Hence, the normalization condition now becomes Z

d 3 r |h~r|Ψ1 (t)i|2 + |h~r|Ψ2 (t)i|2 Z

=




d 3 r (hΨ1 (t)|~rih~r|Ψ1 (t)i + hΨ2 (t)|~rih~r|Ψ2 (t)i)

= hΨ1 (t)|Ψ1 (t)i + hΨ2 (t)|Ψ2 (t)i   |Ψ1 (t)i = (hΨ1 (t)|hΨ2 (t)|) |Ψ2 (t)i = hΨ(t)|Ψ(t)i = 1.

(3.244)

Note that |h~r|Ψ1 (t)i|2 + |h~r|Ψ2 (t)i|2 is the probability for finding the particle at ~r with its spin up or down. The probability for finding the particle with its spin up, independent of its position, is hΨ1 (t)|Ψ1 (t)i and the probability for finding the particle with its spin down, independent of its position, is hΨ2 (t)|Ψ2 (t)i. Particles with half odd integer spin, i.e., s = 21 , 32 , . . ., are fermions that obey the Fermi–Dirac statistics. They obey the Pauli exclusion principle according to which two identical fermions cannot be in the same quantum state.

78

3.3

Quantum Mechanics of Charged Particle Beam Optics

NONRELATIVISTIC QUANTUM MECHANICS

All physical systems are basically quantum mechanical, and the classical behavior of macroscopic bodies is an approximation. Similarly, basic physics is relativistic, and nonrelativistic phenomena are approximations. The classical Hamiltonian of a particle of mass m and charge q moving in an electromagnetic field and the field of a potential V is given by p (3.245) H = m2 c4 + c2~π 2 + qφ +V, where φ and ~A are the scalar and vector potentials of the electromagnetic field, respectively, and the potential V gives rise to the force −~∇V (~r,t) acting on the particle. The Hamiltonian H represents the total energy of the particle. When the motion of the particle is nonrelativistic (|~v|  c), such that its kinetic energy  √ m2 c4 + c2~π 2 − mc2 is very small compared to its rest energy mc2 , we can approximate the classical Hamiltonian as H ≈ mc2 +

~π 2 + qφ +V. 2m

(3.246)

Since the rest energy term mc2 is a constant, we can set it as the zero of the energy scale and take H − mc2 as the nonrelativistic Hamiltonian of the particle. Hence, we take the classical nonrelativistic Hamiltonian of the particle to be ~π 2 + qφ +V. (3.247) 2m To study the quantum mechanics of the system, we have to get the quantum Hamilb from the classical Hamiltonian H(~r,~p,t) by replacing in it ~p by tonian operator H ~pb and making the resulting expression Hermitian by suitable symmetrization procedure. Thus, we get the quantum Hamiltonian of a particle of mass m and charge q moving nonrelativistically in the field of a potential V and an electromagnetic field with φ and ~A as the scalar and vector potentials to be H=

~b2 b = π + qφ +V. H 2m

(3.248)

2 Note that in expanding ~πb = (~pb − q~A) · (~pb − q~A), one has to be careful, since the components of ~r and ~pb do not commute. We shall now look at how nonrelativistic quantum mechanics works.

3.3.1 3.3.1.1

NONRELATIVISTIC SINGLE PARTICLE QUANTUM MECHANICS Free Particle

Single free particle is the simplest physical system. For a particle moving in free space, with ~A = 0, φ = 0, and V = 0, the Hamiltonian (3.248) becomes 2 b = pb . H 2m

(3.249)

79

A Review of Quantum Mechanics

The dynamics is given by the Schr¨odinger equation i¯h

∂ |Ψ(t)i b = H|Ψ(t)i. ∂t

(3.250)

Since the Hamiltonian is time-independent, we can integrate this equation immediately to get b (t,ti ) |Ψ (ti )i = e− h¯i (t−ti )Hb |Ψ (ti )i. |Ψ(t)i = U (3.251) Now, we have to choose a representation. Let us choose the energy representation in which the eigenvectors of the Hamiltonian form the complete orthonormal basis. To this end, let us look at the eigenvalue equation for the Hamiltonian pb2 b H|ψi = |ψi = E|ψi. 2m

(3.252)

We have already solved the eigenvalue equation for the momentum operator. We know that   ~pb φ~p = ~p φ~p , (3.253) with


φ~p |φ~p0 = δ ~p −~p0 . It is obvious that

pb2  p2  φ~p = φ~p , 2m 2m

(3.254)

with p2 = |~p|2 .

(3.255)

b has a continuous eigenvalue Hence, we find that the free-particle Hamiltonian H 2 spectrum E(p) = p /2m | 0 ≤ p ≤ ∞ . In position representation

 ~r | φ~p = φ~p (~r) =

i 1 e h¯ ~p·~r . (2π h¯ )3/2

(3.256)

The free-particle wave function, corresponding to a particle of momentum ~p and energy p2 /2m, is   i 1 h¯ e Ψ(~r,t) = (2π h¯ )3/2

2

p ~p·~r− 2m t

.

(3.257)

In one dimension, a particle of momentum p moving along the +x-direction has the wave function   p2 i 1 h¯ px− 2m t Ψ(x,t) = e , (3.258) (2π h¯ )1/2 and the wave function − h¯i 1 Ψ(x,t) = e (2π h¯ )1/2



2

p px+ 2m t



represents a particle of momentum p moving along the −x-direction.

(3.259)

80

Quantum Mechanics of Charged Particle Beam Optics

 Note  that each eigenvalue E(p) is infinitely degenerate with the set of eigenvectors φ~p | |~p| = p belonging to the same eigenvalue E(p). This means that a ket vector defined by Z  |ψ p i = d 3 p C(~p) φ~p , (3.260) |~p|=p

with arbitrary coefficients {C(~p)}, subject to normalization of |ψ p i, will be an eigenb corresponding to the eigenvalue E(p) = p2 /2m. If we specify the eigenvector of H ~ b values  of both H and pb, then the degeneracy is lifted, and we get a unique eigenvector φ~p . Now, let us take ti = 0 in (3.251) and consider |Ψ(0)i =

Z

 d 3 p C(~p, 0) φ~p ,

(3.261)

with the coefficients {C(~p, 0) = hφ~p |Ψ(0)i} satisfying the relation Z

d 3 p |C(~p, 0)|2 = 1,

(3.262)

as required by the normalization condition hΨ(0)|Ψ(0)i =

Z Z

d 3 p0 d 3 p C∗ (~p0 , 0)C(~p, 0)hφ~p0 |φ~p i

Z Z

d 3 p0 d 3 p C∗ (~p0 , 0)C(~p, 0)δ ~p −~p0

= Z

=

d 3 p |C(~p, 0)|2 = 1.


(3.263)

Then, we have |Ψ(t)i =

Z

  Z ip2 t d 3 p e− 2m¯h C(~p, 0) φ~p = d 3 p C(~p,t) φ~p .

(3.264)

If we represent the state vectors |Ψ(t)i and |Ψ(0)i by the column vectors of the coefficients, then the above equation becomes      .. .. .. .. .. .. .. . . .  . .    . .   C(~p,t)  =  · · e−ip2 t/2m¯h · ·   C(~p, 0)  , (3.265)      .. .. .. .. .. .. .. . . . . . . . b 0) = showing that the matrix representing the unitary time-evolution operator U(t, b h −it H/¯ e is diagonal in the energy representation. In other words, the Hamiltonian is diagonal in the energy representation. In the position representation, corresponding to (3.261), we have h~r|Ψ(0)i =

Z

 d 3 p C(~p, 0) ~r φ~p ,

(3.266)

81

A Review of Quantum Mechanics

or Z

Ψ(~r, 0) = =

d 3 p C(~p, 0)φ~p (~r)

1 (2π h¯ )3/2

Z

i

d 3 p C(~p, 0)e h¯ ~p·~r .

(3.267)

For the particle with this initial wave function Ψ(~r, 0) at time 0, the wavefunction at a later time t will be, following (3.264), Ψ(~r,t) =

1 (2π h¯ )3/2

Z

i

d 3 p C(~p, 0)e h¯ (~p·~r−E(p)t) ,

(3.268)

where E(p) = p2 /2m. This represents a wave packet solution of the time-dependent Schr¨odinger equation (3.250) in the position representation. This wave packet is not an eigenstate of the Hamiltonian, since it is a linear superposition of several eigenstates with different energy eigenvalues. It has an average energy b hEiΨ(t) = hΨ(t)|H|Ψ(t)i Z 1 = d 3 r Ψ∗ (~r,t) pb2 Ψ(~r,t) 2m Z 1 1 2 = d 3 p |C(~p, 0)|2 p2 = hp iΨ(0) . 2m 2m

(3.269)

Note that the average energy is a constant of motion, in accordance with (3.194), b comsince the quantum operator for energy, the time-independent Hamiltonian, H, mutes with itself. Similarly, the averages of the momentum components will also be constants of motion for this state, since the momentum operator commutes with the Hamiltonian, i.e., h~piΨ(t) = h~piΨ(0) . For the average position of the particle, represented by the above wave packet, the equation of motion (3.194) shows that dh~riΨ(t) 1 = h~piΨ(t) , dt m

(3.270)

in accordance with the classical relation between the velocity and momentum, d~r/dt = ~p/m, (´a la Ehrenfest’s theorem).  Inthe Heisenberg picture, we can write the time-evolution, or transfer, map of ~b r,~pb as ~b r(t) ~pb(t)

!



 it pb2 −it pb2 2m¯h~ 2m¯h b e r e =  it pb2 −it pb2  e 2m¯h ~pbe 2m¯h !  ~b r(0) + mt ~pb(0) 1 = = ~pb(0) 0

t m

1



~b r(0) ~pb(0)

! .

(3.271)

82

Quantum Mechanics of Charged Particle Beam Optics

2 2 2 2 b b −A b Note that eit pb /2m¯h~b re−it pb /2m¯h and eit pb /2m¯h~pbe−it pb /2m¯h are of the form eA Be , which can be calculated using the identity (3.142)   1 b 2 1 b 3 b −A b b A :A: b b b b (3.272) e Be = e B = I+ : A : + : A : + : A : + · · · B, 2! 3!

b and b b If the series terminates because by substituting it pb2 /2m¯h for A, r and pb for B. of the vanishing of a commutator at any step, we get an exact expression as in the present case. Let us now have a closer look at the Heisenberg uncertainty principle in the case of the conjugate pair of dynamical variables x and px . For any observable O, the uncertainty in a state |Ψi is defined by s  2  b (∆O)Ψ = O − hOiΨ ,

(3.273)

Ψ

which is a measure of the spread, or dispersion, of its values about the average, or the mean value, and (∆O)2Ψ is known as variance, second moment about the mean, or the second-order central moment. Note that D E b − hOiΨ O = 0. (3.274) Ψ

Since  2  D E b b2 + hOi2Ψ − 2OhOi b Ψ O − hOiΨ = O Ψ Ψ D E 2 2 2 b + hOiΨ − 2hOiΨ = O Ψ

2 = O Ψ − hOi2Ψ ,

(3.275)

we can also write (∆O)Ψ =

q hO2 iΨ − hOi2Ψ .

(3.276)

b (∆O) = 0. When |Ψi is an eigenstate of O, Ψ b1 O b2 6= O b2 O b1 ) with uncerLet O1 and O2 be two incompatible observables (i.e., O tainties (∆O1 )Ψ and (∆O2 )Ψ in a state |Ψi. We shall drop the subscript Ψ to be understood from the context. Let b1 − hO1 i , δc O1 = O

b2 − hO2 i , δc O2 = O

(3.277)

which are Hermitian operators. Now, we can write E E  D  D (∆O1 )2 = Ψ δc O1 δc O1 Ψ , (∆O2 )2 = Ψ δc O2 δc O2 Ψ . (3.278)

83

A Review of Quantum Mechanics

According to the Cauchy–Schwarz inequality, or the Schwarz inequality (see, e.g., Arfken, Weber, and Harris [3], and Byron and Fuller [18]), for any two functions f and g, h f | f ihg|gi ≥ |h f |gi|2 . (3.279) Taking | f i = δc O1 |Ψi and |gi = δc O2 |Ψi, we get (∆O1 )2 (∆O2 )2 ≥ |hΨ|δc O1 δc O2 |Ψi|2 .

(3.280)

Note that hΨ|δc O1 δc O2 |Ψi is a complex number and recall that h f |gi∗ = hg| f i. For any complex number z = x + iy, |z|2 = x2 + y2 ≥ y2 = [(z − z∗ ) /2i]2 . Thus, we have  hD E D Ei2 1 (∆O1 )2 (∆O2 )2 ≥ Ψ δc O1 δc O2 Ψ − Ψ δc O2 δc O1 Ψ 2i  hD h i Ei2 1 c c Ψ δ O1 , δ O2 Ψ = . (3.281) 2i Since

h i h i h i b1 − hO1 i , O b2 − hO2 i = O b1 , O b2 , δc O1 , δc O2 = O

(3.282)

we get  1 hb b i 2 O1 , O2 , (3.283) 2i or  h i 1 b1 , O b2 . (∆O1 ) (∆O2 ) ≥ O (3.284) 2i h i  b1 and O b2 , O b1 , O b2 /2i is a Hermitian Note that for any two Hermitian operators O operator. If we now take O1 = x and O2 = px in (3.284), we get the Heisenberg uncertainty relation h¯ (∆x)(∆px ) ≥ . (3.285) 2 The free-particle eigenstate with specific momentum and energy eigenvalues, ~p and E(~p) = |~p|2 /2m, given in position representation by, (∆O1 )2 (∆O2 )2 ≥

Ψ(~r,t) =



i 1 e h¯ (~p·~r−E(~p)t) , 3/2 (2π h¯ )

(3.286)

has ∆px = ∆py = ∆pz = 0 and ∆x = ∆y = ∆z −→ ∞. Note that this free-particle wave function represents a plane wave Ψ(~r,t) =

1 ~ ~ ei(k·~r−ω(k)t) , 3/2 (2π h¯ )

with h¯~k = ~p,

h¯ ω(~k) = E(~p) =

h¯ 2 k2 . 2m

(3.287)

(3.288)

84

Quantum Mechanics of Charged Particle Beam Optics

This takes us to the beginning of wave mechanics, or the early quantum mechanics, when de Broglie associated this plane wave with a free particle. The wavelength of this wave λ = 2π/|~k| = 2π h¯ /|~p| = h/p is called the de Broglie wavelength of the particle, and the frequency of the wave is ν = ω/2π in accordance with the relation E = hν postulated by Planck for a harmonic oscillator in the context of blackbody radiation, and adopted by Einstein and Bohr in their theories of light and atomic spectra, respectively. History of quantum mechanics starts with Planck’s relation E = hν. In case the motion of the particle is not nonrelativistic, the associated de Broglie wave p will have the same plane waveform (3.287) with h¯ ω(~k) = m2 c4 + c2 p2 and will obey a relativistic generalization of the nonrelativistic Schr¨odinger equation known as the Klein–Gordon equation, which we shall study later. Association of a physical wave with a particle was abandoned later when Born’s interpretation of |Ψ(~r,t)|2 as the position probability density was established. As an example of the Heisenberg uncertainty principle, let us consider the dynamics of a free-particle wave packet in one dimension (x-direction). Let the particle be, at t = 0, in the state Z ∞

|Ψ(0)i =

−∞

d px C (px , 0) |φ px i,

(3.289)

where  1/4 r 2 2 2 (∆x)0 − (∆x)0 ( p2x −p0 ) h¯ C (px , 0) = , e h¯ π

(3.290)

a Gaussian function such that Z ∞ −∞

d px |C (px , 0) |2 = 1.

(3.291)

To verify (3.291), use the well-known result on the Fourier transform of the Gaussian function, (see e.g., Arfken, Weber, and Harris [3], and Byron and Fuller [18]) Z ∞

−aχ 2 +iχη

dχ e

r =

−∞

π − η2 e 4a , a

(3.292)

√ with the assumption that a has a positive real part and its square root a is also chosen to have a positive real part. In position representation, the wave function of the particle is, with p¯ = px − p0 ,  1/4 r Z (∆x)20 ( px −p0 )2 1 2 (∆x)0 ∞ − + h¯i px x h¯ 2 Ψ(x, 0) = √ d px e h¯ −∞ 2π h¯ π s  1/4 Z ∞ (∆x)20 p¯2 i p x (∆x)0 2 − + h¯i px ¯ 0 h ¯ = e d p¯ e h¯ 2 2 π −∞ 2π h¯ 1

2 − x 2 + h¯i p0 x 4(∆x)

= √
1/2 e 2π(∆x)0

0

,

(3.293)

85

A Review of Quantum Mechanics

where we have used again (3.292). Using (3.268), the wave function of the wave packet at t > 0 is found to be given by   2  1/4 r Z ∞ (∆x)20 ( px −p0 )2 i p x− px t x 2 (∆x)0 1 − 2m h ¯ h¯ 2 √ Ψ(x,t) = e d px e π h¯ 2π h¯ −∞    1/4 r p20 i 2 (∆x)0 1 h¯ p0 x− 2m t √ = e h¯ π 2π h¯

×

2 (∆x)2 0 p¯ − h¯ 2

Z ∞

i h¯

  p p¯2 − 2m t+ p¯(x− m0 t )

d p¯ e e s    1/4 p2 (∆x)0 h¯i p0 x− 2m0 t 2 = e π 2π h¯ 2 −∞

×

p¯2 (∆x)2 0 h¯ 2

−

Z ∞

d p¯ e

 1+



i¯ht 2m(∆x)2 0

p0

p¯

ei h¯ (x− m t )

−∞ −

1

= √
1/2 e 2π(∆x)0 (1 + iδ ) with δ=

x¯2 + h¯i 4(∆x)2 0 (1+iδ )

h¯ t , 2m(∆x)20

x¯ = x −



p2

p0 x− 2m0 t



,

p0 t. m

(3.294)

(3.295)

The last step can be verified again using (3.292). Note that as t −→ 0, Ψ(x,t) −→ Ψ(x, 0). For the initial wave packet Ψ(x, 0), the average value of x is given by hxi(0) =

Z ∞

Z ∞

dx Ψ(x, 0)∗ xΨ(x, 0) =

−∞

dx x|Ψ(x, 0)|2

−∞

1 =√ 2π(∆x)0

2 − x 2 2(∆x)

Z ∞

dx xe

0

= 0,

(3.296)

−∞

where the integral vanishes since the integrand is an odd function of x. This result shows that initially, at t = 0, the wave packet is centered at x = 0. The average of x2 for Ψ(x, 0) is given by 2

hx i(0) =

Z ∞

Z ∞

∗ 2

dx Ψ(x, 0) x Ψ(x, 0) = −∞

dx x2 |Ψ(x, 0)|2

−∞

1 =√ 2π(∆x)0

Z ∞

2 − x 2 2(∆x)

dx x2 e

0

−∞

= (∆x)20 ,

(3.297)

where we have used the integral Z ∞ −∞

2 −aχ 2

dχ χ e

r =

π , 4a3

(3.298)

86

Quantum Mechanics of Charged Particle Beam Optics

which follows by differentiating the integral in (3.292), with η = 0, with respect to a. Thus, the uncertainty in x for the wave packet Ψ(x, 0) is q (3.299) (∆x)(0) = hx2 i(0) − hxi(0)2 = (∆x)0 . The average value of px for Ψ(x, 0) is Z ∞

dx Ψ∗ (x, 0) pbx Ψ(x, 0)   ∂ ∗ = dx Ψ (x, 0) −i¯h Ψ(x, 0) ∂x −∞

hpx i (0) =

−∞ Z ∞

2

∞ − x 2 − h¯i p0 x 1 dx e 4(∆x)0 =√ 2π(∆x)0 −∞ " !# x2 + i p x − 0 ∂ 2 h ¯ × −i¯h e 4(∆x)0 ∂x   Z ∞ − i¯hx 1 + p dx =√ 0 e 2 2(∆x)0 2π(∆x)0 −∞

Z

x2 2(∆x)20

= p0 .

(3.300)

Calculating the average of p2x for Ψ(x, 0), we get Z

2 px (0) =

∞

dx Ψ∗ (x, 0) pb2x Ψ(x, 0)   ∂2 = dx Ψ∗ (x, 0) −¯h2 2 Ψ(x, 0) ∂x −∞ −∞ Z ∞

1 =√ 2π(∆x)0 " ∂2 × −¯h2 2 ∂x

2 − x 2 − h¯i p0 x 4(∆x)

Z ∞

1 =√ 2π(∆x)0 ×e

dx e

0

−∞

e

2 − x 2 + h¯i p0 x 4(∆x) 0



Z ∞

dx −∞

!#

h¯ 2 h¯ 2 x2 i¯h p0 x − + + p20 2 4 2(∆x)0 4(∆x)0 (∆x)20



2 − x 2 2(∆x)

= p20 +

0

h¯ 2 . 4(∆x)20

Thus the uncertainty in px for the wave packet Ψ(x, 0) is q (∆px ) (0) = hp2x i (0) − hpx i (0)2 =

(3.301)

h¯ . 2(∆x)0

(3.302)

87

A Review of Quantum Mechanics

This shows that for the initial wave packet Ψ(x, 0), the uncertainty product is h¯ h¯ = , 2(∆x)0 2

(∆x∆px ) (0) = (∆x)0

(3.303)

i.e., Ψ(x, 0) is a minimum uncertainty wave packet. Calculating the average value of x for the wave packet Ψ(x,t), we find hxi(t) =

Z ∞

dx Ψ(x,t)∗ xΨ(x,t) =

−∞

Z ∞

dx x|Ψ(x,t)|2

−∞

1 =√ 2π(∆x)t p0 = t, m

 x¯2 p0  − 2(∆x) 2 t d x¯ x¯ + t e m −∞

Z ∞

(3.304)

where (∆x)t = (∆x)0

q

(1 + δ 2 ).

(3.305)

The result (3.304) shows that the center of the wave packet is moving with the velocity p0 /m as if it corresponds to the position of a classical free particle of mass m moving with momentum p0 . This is Ehrenfest’s theorem. The average of x2 for Ψ(x,t) is hx2 i(t) =

Z ∞

dx Ψ(x,t)∗ x2 Ψ(x,t) =

−∞

Z ∞

dx x2 |Ψ(x,t)|2

−∞

2  p0 2 − x¯ 2 d x¯ x¯ + t e 2(∆x)t m −∞  Z ∞  p 2 2p x¯  − x¯2 1 2 0 0 d x¯ x¯2 + t + t e 2(∆x)t =√ m m 2π(∆x)t −∞  p 2 0 = (∆x)t2 + t . m

1 =√ 2π(∆x)t

Z ∞

(3.306)

Thus, the uncertainty in x for Ψ(x,t) is q

hx2 i (t) − hxi(t)2 = (∆x)t "  2 #1/2 h¯ t = (∆x)(0) 1 + , 2m(∆x)(0)2

(∆x)(t) =

(3.307)

showing that the wave packet is spreading or dissipating. This spreading of the freeparticle wave packet implies that we cannot think of the de Broglie waves as real physical waves as we don’t observe any such dissipation of a free particle.

88

Quantum Mechanics of Charged Particle Beam Optics

The average of px for Ψ(x,t) is given by Z ∞

dx Ψ∗ (x,t) pbx Ψ(x,t)   ∂ ∗ = dx Ψ (x,t) −i¯h Ψ(x,t) ∂x −∞

hpx i (t) =

−∞ Z ∞



2

p2



x¯ ∞ − − h¯i p0 x− 2m0 t 1 2 dx e 4(∆x)0 (1−iδ ) =√ 2π(∆x)t −∞      2 i p x− p0 t x¯2   + − 0 ∂ 2 2m h¯  × −i¯h e 4(∆x)0 (1+iδ )   ∂x

Z

1 =√ 2π(∆x)t = p0 ,

 2 − x¯ 2 i¯hx¯ 2(∆x)t d x¯ p0 + e 2(∆x)20 (1 + iδ ) −∞

Z ∞



(3.308)

as should be, since the momentum is conserved for a free particle. This is also in accordance with Ehrenfest’s theorem. Similarly, we can find the average of p2x for Ψ(x,t): Z

2 px (t) =

∞

dx Ψ∗ (x,t) pb2x Ψ(x,t)   2 ∗ 2 ∂ Ψ(x,t) = dx Ψ (x,t) −¯h ∂ x2 −∞ −∞ Z ∞

2



p2



x¯ ∞ − − h¯i p0 x− 2m0 t 1 2 =√ dx e 4(∆x)0 (1−iδ ) 2π(∆x)t −∞      2 x¯2 i p x− p0 t   2 − + 0 ∂ 2 2m h¯  × −¯h2 2 e 4(∆x)0 (1+iδ )   ∂x   Z ∞ 1 1 h¯ 2 =√ d x¯ p20 + + i¯h p0 x¯ (∆x)20 (1 + iδ ) 2 2π(∆x)t −∞  2 − x¯ 2 h¯ 2 x¯2 2(∆x)t e − 4(∆x)20 (1 + iδ )

Z

= p20 +

h¯ 2 , 4(∆x)20

(3.309)

same as for the initial wave packet Ψ(x, 0). This is again due to conservation of momentum for the free particle. Thus, for Ψ(x,t) the uncertainty in px is (∆px ) (t) =

q hp2x i (t) − hpx i (t)2 =

h¯ , 2(∆x)0

(3.310)

89

A Review of Quantum Mechanics

same as for the initial wave packet Ψ(x, 0). The uncertainty product for Ψ(x,t) is "



h¯ t (∆x∆px ) (t) = (∆x)(0) 1 + 2m(∆x)(0)2

2 #1/2

h¯ 2(∆x)(0)

" 2 #1/2  h¯ h¯ h¯ t > , = 1+ 2 2m(∆x)(0)2 2

for t > 0,

(3.311)

showing that as the minimum uncertainty free-particle wave packet evolves in time, it does not remain a minimum uncertainty wave packet. The uncertainty principle implies that we cannot assign definite values for both the position and the momentum of particle at any time. Thus, the concept of trajectory loses its meaning since we can only specify the probability of finding the particle at any position at any time. Let us now look at the propagator for a free particle. From (3.178), we have i

Kfree (~r,t;~ri ,ti ) = ∑ e− h¯ (t−ti )En h~r|nihn|~ri i n

Z

=

d 3 p e−

1 = (2π h¯ )3

ip2 (t−ti ) 2m¯h

Z

3

d pe

φ~p (~r)φ~p∗ (~ri ) i h¯

  p2 (t−t ) ~p·(~r−~ri )− 2m i

.

(3.312)

The integral in the last step can be performed exactly using the integral in (3.292). The result is  3/2 im|~r−~ri |2 m Kfree (~r,t;~ri ,ti ) = e 2¯h(t−ti ) . (3.313) 2πi¯h (t − ti ) As an example, let us find Ψ(x,t) when Ψ(x, 0) is the free-particle minimum uncertainty wave packet (3.293): Z ∞

Ψ(x,t) = −∞

 =  =



dx0 Kfree x,t; x0 , 0 Ψ x0 , 0

m (2π)3/2 i¯ht(∆x)0

1/2 Z

m 3/2 (2π) i¯ht(∆x)0

1/2

∞

dx0 e

2 im(x−x0 ) 2¯ht

02 − x 2 + h¯i p0 x0 4(∆x)

e

0

−∞

e

im 2 2¯ht x

Z ∞

 −

dx0 e

1 − im 4(∆x)20 2¯ht

   p0 t 0 x0 2 − im h¯ t x− m x

−∞ 2

−

1 = √  1/2 e i¯ht 2π(∆x)0 1 + 2m(∆x) 2 0

(x− pm0 t )

i¯ht 4(∆x)2 0 1+ 2m(∆x)2 0

!+ i h¯



p2

p0 x− 2m0 t



, (3.314)

90

Quantum Mechanics of Charged Particle Beam Optics

where, in the last step, we have to use the integral (3.292) and some algebraic manipulations. Thus, using the propagator, we have arrived at the same result for Ψ(x,t) as in (3.294) starting with Ψ(x, 0). 3.3.1.2

Linear Harmonic Oscillator

Before considering the quantum mechanics of a nonrelativistic linear harmonic oscillator, let us recall its classical mechanics. Consider a particle of mass m moving along the x-axis under the influence of a restoring force F(x) = −κx, where κ is the spring constant. The classical Hamiltonian of the particle is H=

p2 1 2 + κx , 2m 2

(3.315)

where p2 /2m is the kinetic energy and V (x) = κx2 /2 is the potential energy of the particle at the position x. Since the particle is moving only along the x-direction, we have denoted px simply by p. Hamilton’s equations of motion are   2   dx p 1 2 =: −H : x = − + κx ,x dt 2m 2   p p2 ,x = , = − 2m m   2   dp p 1 2 =: −H : p = − + κx , p dt 2m 2   1 = − κx2 , p = −κx. (3.316) 2 Defining κ = mω 2 , we can rewrite these equations as      d x 0 1 x =ω . p p −1 0 dt mω mω

(3.317)

This equation can be readily integrated to give the classical phase space transfer map 

x(t) p(t) mω

ωt

 =e

 =

0 −1 cos ωt − sin ωt

1 0

!



x(0)



p(0) mω

sin ωt cos ωt



x(0) p(0) mω

 ,

(3.318)

where (x(0), p(0)) and (x(t), p(t)) are the values of x and p at times 0 and t, respectively. Since the Hamiltonian is time-independent, the energy of the oscillator is a constant of motion, say E. When the particle is at the extreme position of oscillation x = A, the amplitude of motion, its energy is completely potential energy, and the

91

A Review of Quantum Mechanics

kinetic energy is zero since p = 0. Thus, E = mω 2 A2 /2. When the particle is at any other position x, between 0 and A, its energy will be distributed between the kinetic and potential energies such that 1 p2 1 + mω 2 x2 = mω 2 A2 . 2m 2 2

(3.319)

Dividing both sides of this equation by mω 2 /2, we get  p 2 + x 2 = A2 , mω

(3.320)

a relation to be satisfied at any position of the particle. Thus, at any position, we can write x = A cos ϕ, and p/mω = ±A sin ϕ depending on whether the particle is moving in the +x-direction or −x-direction. Let us now choose the initial conditions at t = 0 as x(0) = A cos ϕ, with 0 ≤ ϕ ≤ π/2, and p(0)/mω = A sin ϕ, without loss of generality. Then, we get x(t) = A cos(ωt − ϕ),

p(t) = −mωA sin(ωt − ϕ),

(3.321)

as the general solution for the position and momentum of the oscillating particle. Note that p(t) = mx(t). ˙ The constant ϕ is the initial phase of the oscillator, fixed by the initial conditions at t = 0, and the circular frequency ω gives the frequency with which the phase changes, i.e., ωT = 2π, where T is the period of the oscillator such that x(t + T ) = x(t), p(t + T ) = p(t). The energy of the oscillator is, at any time, H(t) =

p(t)2 1 1 + mω 2 x(t)2 = mω 2 A2 , 2m 2 2

(3.322)

constant, equal to the potential energy of the particle at its extreme positions, or the turning points, where the kinetic energy vanishes. Let us observe that we can write H(t) in a factorized form as    mωx(t) + ip(t) mωx(t) − ip(t) √ √ H(t) = = a(t)a∗ (t). (3.323) 2m 2m Substituting the solutions for x(t) and p(t), we have r m ωAe−i(ωt−ϕ) = a(0)e−iωt , a(t) = 2 r m ∗ a (t) = ωAei(ωt−ϕ) = a∗ (0)eiωt , 2

(3.324)

such that the relation (3.323) is consistent with (3.322). Since the amplitude of oscillation A can take continuous values, we see that the energy of the classical harmonic oscillator can vary continuously. Any oscillatory motion is approximately a linear harmonic motion as long as the amplitude is small and that is what makes the linear

92

Quantum Mechanics of Charged Particle Beam Optics

harmonic oscillator a basic paradigm for understanding a wide variety of physical phenomena. Let us now look at the quantum mechanics of the linear harmonic oscillator. The quantum Hamiltonian of the nonrelativistic linear harmonic oscillator of mass m and circular frequency ω along the x-axis is given by 2 b = pb + 1 mω 2 x2 . H 2m 2

(3.325)

Let us work in the position representation. Then, the Schr¨odinger equation for Ψ(x,t) reads   h¯ 2 ∂ 2 ∂Ψ 1 2 2 i¯h = − + mω x Ψ. (3.326) ∂t 2m ∂ x2 2 This partial differential equation can be solved by the method of separation of variables. To this end, let Ψ(x,t) = T (t)ψ(x). This turns (3.326) into   h¯ 2 d 2 1 1 i¯h dT 2 2 = − + mω x ψ. (3.327) T dt ψ 2m dx2 2 Since the left-hand side of this equation is a function of only t and the right-hand side is a function of only x, both sides should be equal to some constant, say E. Thus, we get dT i¯h = ET, dt   h¯ 2 d 2 1 − + mω 2 x2 ψ = Eψ. 2m dx2 2

(3.328) (3.329)

The first equation is solved immediately: T (t) = e−iEt/¯h . The second equation is the eigenvalue equation for the Hamiltonian of the oscillator. Hence, we identify the constant E with the energy eigenvalue. The equation (3.329) is called the timeindependent Schr¨odinger equation for the harmonic oscillator. In general, the eigenvalue equation for the Hamiltonian is known as the time-independent Schr¨odinger equation of the system. Let {En } be the set of all eigenvalues and let {ψn (x)} be the respective eigenfunctions allowed by (3.329). In this case, we will find that the eigenvalues are discrete, nondegenerate, and can be labeled as n = 0, 1, 2, . . . ,. Since the Hamiltonian operator is Hermitian, these eigenvalues will be real, and the corresponding eigenfunctions will form a complete orthonormal set. Note that the Hamiltonian of any nonrelativistic system, of the type pb2 /2m +V (~b r,t), will be Hermitian. Thus, the particular solutions of (3.326) are given by i

Ψn (x,t) = e− h¯ En t ψn (x),

n = 0, 1, 2, . . . .

(3.330)

These wave functions correspond to stationary states in which the position probability density, |Ψn (x,t)|2 , is independent of time since the time dependence cancels out. Similarly, the mean value of any time-independent observable hOiΨn (t) is also seen to

93

A Review of Quantum Mechanics

be time independent. The general solution of the time-dependent Schr¨odinger equation (3.326), Ψ(x,t), is any linear combination of the particular solutions, subject to the only condition of normalization hΨ(t)|Ψ(t)i = 1. Thus, the general solution is Ψ(x,t) = ∑ Cn Ψn (x,t),

with

n

∑ |Cn |2 = 1.

(3.331)

n

Let us now look at the solutions ψn (x) of the time-independent Schr¨odinger equation (3.329), the eigenvalue equation for the time-independent Hamiltonian of the system. To this end, we have to solve the differential equation   h¯ 2 d 2 1 2 2 − + mω x ψ(x) = Eψ(x). (3.332) 2m dx2 2 p Now, introducing the dimensionless variable ξ = mω/¯h x, this equation becomes   d2 2 − 2 + ξ ψ(ξ ) = Kψ(ξ ), dξ

(3.333)

where K = 2E/¯hω is the energy in units of h¯ ω/2. Normalizable solutions for ψ(x) exist only when K = 2n + 1, with n = 0, 1, 2, . . . ,. Thus, the quantized energy spectrum of the oscillator consists of eigenvalues   1 h¯ ω, n = 0, 1, 2, . . . . (3.334) En = n + 2 All these energy eigenvalues are nondegenerate. Note that except for the addition of the extra energy hν/2 to each energy level, and the zero-point energy E0 = hν/2, the formula for the energy spectrum of the harmonic oscillator (3.334) coincides exactly with the postulate of Planck that the energy of oscillators in a black body is quantized as E = nhν, with n = 1, 2, 3, . . .. The origin of quantum mechanics in 1900 lies in this postulate of Planck. When the oscillator makes a transition, or jumps down, from the (n + 1)-th energy level to the n-th energy level, it emits a quantum of energy hν (photon) and by absorbing a quantum of energy hν it can move, or jump up, from a lower energy level to the next higher energy level. The normalized eigenfunctions, corresponding to the energy eigenvalues En , orthogonal to each other, are ψn (x) =

 mω 1/4 π h¯

1 2 1 √ Hn (ξ )e− 2 ξ , n 2 n!

n = 0, 1, 2, . . . ,

(3.335)

where Hn (ξ ) are called the Hermite polynomials given by the Rodrigues formula (for details, see, e.g., Arfken, Weber, and Harris [3], and Byron and Fuller [18], and Lakshminarayanan and Varadharajan [126]):  n 2 d n ξ2 Hn (ξ ) = (−1) e e−ξ . (3.336) dξ

94

Quantum Mechanics of Charged Particle Beam Optics

The first few Hermite polynomials are H0 (ξ ) = 1,

H2 (ξ ) = 4ξ 2 − 2,

H1 (ξ ) = 2ξ ,

H3 (ξ ) = 8ξ 3 − 12ξ .

(3.337)

The time-dependent energy eigenstates, the stationary states, are given by 1

i

Ψn (x,t) = e− h¯ En t ψn (x) = e−i(n+ 2 )ωt ψn (x).

(3.338)

The general solution (3.331) for the time-dependent Schr¨odinger equation (3.326) can now be written explicitly as 1

Ψ(x,t) = ∑ Cn e−i(n+ 2 )ωt ψn (x) n

i

i

= e− h¯ t H ∑ Cn ψn (x) = e− h¯ t H Ψ(x, 0), b

b

(3.339)

n

where ∑n |Cn |2 = 1. The lowest eigenstate, the ground state (n = 0), corresponding to energy E0 = h¯ ω/2, has the wave function Ψ0 (x,t) =

 mω 1/4 π h¯

1

e− 2 (

mω x2 +iωt h¯

).

(3.340)

The nonclassical nature of the quantum oscillator, besides the quantization of energy, is evident now. If we think classically, for the oscillatorp in its ground state p with energy E0 = h¯ ω/2, the amplitude of oscillation would be A = 2E0 /mω 2 = h¯ /mω. Classically, for the oscillating particle, x = ±A are the turning points beyond which the particle cannot go. But, for the quantum oscillator in the ground state, we have |Ψ0 (x,t)|2 =

 mω 1/2 π h¯

e−

mω x2 h¯

,

(3.341)

a Gaussian function, showing that there is a definite probability for the particle to be found beyond the classical turning points, in principle up to x = ±∞. Higher (n > 0), excited, states of the oscillator also have this property. Further, it is seen that
the wave function of n-th excited level, with energy En = n + 12 h¯ ω, vanishes at n points, called nodes, where the probability of finding the particle is zero! We have found that all the wave functions of the linear harmonic oscillator corresponding to R discrete energy eigenvalues are normalizable in the sense dx |Ψ(x,t)|2 = 1 and Ψ(x,t) −→ 0 as x −→ ±∞. Such states are known as bound states. Note that freeparticle eigenstates are not bound states. There is an alternative way of finding the energy spectrum and the corresponding eigenstates of the harmonic oscillator, the algebraic method, similar to the case of angular momentum we have already seen. To this end, let us define the quantum operators corresponding to the complex classical dynamical variables a and a∗ in (3.323) as follows: mωx + i pb ab = √ , 2m¯hω

mωx − i pb ab† = √ . 2m¯hω

(3.342)

95

A Review of Quantum Mechanics

Since ab and ab† correspond to the classical complex a and a∗ , they are not Hermitian operators. Note that ab and ab† are dimensionless, ab† ab is Hermitian, and in terms of b becomes them the Hamiltonian H   b = ab† ab + 1 h¯ ω, (3.343) H 2 as can be verified directly. Further, we have the following commutation relations:   ab , ab† = 1, (3.344) and  †  ab ab , ab† = ab† ,

 †  ab ab , ab = −b a.

(3.345)

An operator for which the expectation value in any state is nonnegative is called a positive operator. Now, ab† ab is seen to be a positive operator because hϕ|b a† ab|ϕi = hb aϕ|b aϕi ≥ 0,

(3.346)

and hϕ|b a† ab|ϕi = 0

implies ab|ϕi = 0.

(3.347)

b be a positive operator and |ϕλ i be its normalized eigenvector corresponding Let P b λ i = λ ≥ 0. Thus, all the eigenvalues of to an eigenvalue λ . Then, we have hϕλ |P|ϕ any positive operator are ≥ 0. Now, from (3.343), it follows that the eigenvalues of b should be ≥ h¯ ω/2 since the lowest eigenvalue of the positive operator ab† ab should H be ≥ 0. From (3.343) and (3.345), it follows that b ab† = ab† (H b + h¯ ω), H

b ab = ab(H b − h¯ ω). H

(3.348)

b with the eigenvalue En , i.e., These relations imply that if |ni is an eigenstate of H b H|ni = En |ni, then

b ab† |ni = (En + h¯ ω) ab† |ni , b (b H H a|ni) = (En − h¯ ω) (b a|ni) . (3.349) For this reason ab† and ab, called the raising and lowering operators, respectively, are b cannot become negative, known as the ladder operators. Since the eigenvalue of H the action of the lowering operator must stop at some stage. Let |0i be the ground state corresponding to this lowest eigenvalue E0 , such that ab|0i = 0.

(3.350)

If ab|0i were nonzero, then it has to be an eigenstate with the eigenvalue E0 − h¯ ω contradicting the assumption that E0 is the lowest eigenvalue. Then, it is obvious that E0 = h¯ ω/2 since   h¯ 1 † b |0i = ω|0i. (3.351) H|0i = h¯ ω ab ab + 2 2

96

Quantum Mechanics of Charged Particle Beam Optics

Now, applying the raising operator repeatedly on the ground state, we get the comb Thus, we have plete spectrum and the eigenstates of H.   b = n + 1 h¯ ω|ni, H|ni 2

|ni = Nn ab†


n

|0i,

n = 0, 1, 2, . . . ,

(3.352)

where Nn is the normalization constant that can be found by assuming that the ground state is normalized, i.e., h0|0i = 1. The normalization constant Nn can be found as follows. First, let us observe that b H 1 − h¯ ω 2

†

ab ab|ni =

! |ni = n|ni,

n = 0, 1, 2, . . . .

(3.353)

b is Thus, the spectrum of ab† ab consists of all integers ≥ 0. Hence, ab† ab, denoted by N, † b called the number operator. Let |ni be the normalized eigenvector of H, or ab ab. The normalized eigenvector |n + 1i can be written as |n + 1i = Nn+1 ab†


n+1

|0i = Nn+1 ab† ab†


n

|0i =

Nn+1 † ab |ni. Nn

(3.354)

Now, with hn|b a† ab|ni = n, the normalization condition for |n + 1i becomes 

Nn+1 Nn

2



Nn+1 Nn

2



Nn+1 Nn

2

hn + 1|n + 1i = = =

hn|b a ab† |ni
hn| 1 + ab† ab |ni (n + 1) = 1,

(3.355)

†a b, following from the commutation where we have used the relation abab† = 1 + ab√ relation (3.344). This implies that Nn+1 = Nn / n + 1. By definition N0 = 1. Hence, we get

1 Nn = √ , n!

n = 0, 1, 2, . . . .

(3.356)

b can be written as Thus, the normalized eigenstates of H
n 1 |ni = √ ab† |0i, n!

with ab|0i = 0.

(3.357)

97

A Review of Quantum Mechanics

b is Hermitian, its eigenvectors {|ni} are orthogonal. In the orthonormal basis Since H {|ni} we find, from (3.357),
n+1 1 |0i ab† |ni = √ ab† n! " # √
1 † n+1 = n+1 p ab |0i (n + 1)! √ = n + 1 |n + 1i.

(3.358)

Similarly, 


1 † n ab|ni = ab √ ab |0i n! " #
abab† 1 n−1 p = √ ab† |0i n (n − 1)!
√ 1 + ab† ab √ = |n − 1i = n |n − 1i, n

(3.359)

where we have used the relation ab† ab|n − 1i = (n − 1)|n − 1i. The eigenfunctions b namely ψn (x) = hx|ni, can be obtained from the above formalism as follows. of H, In the position representation, the defining equation for the ground state, ab|0i = 0, becomes the simple equation   d ψ0 (x) = 0, (3.360) mωx + h¯ dx with the solution

mω 2

ψ0 (x) = N0 e− 2¯h x ,

(3.361)

where N0 is the normalization constant such that Z

2

dx |ψ0 (x)|

= N02

Z

2 − mω h¯ x

dx e

r = N02

π h¯ = 1, mω

(3.362)

using the result on the Gaussian integral we have already seen (3.292). Thus, N0 = b is (mω/π h¯ )1/4 and the ground state eigenfunction of H ψ0 (x) =

 mω 1/4 π h¯

mω 2

e− 2¯h x .

(3.363)

The excited state eigenfunctions can be obtained from (3.357), which becomes in position representation !n r r 1 mω h¯ d ψn (x) = √ x− ψ0 (x). (3.364) 2¯h 2mω dx n!

98

Quantum Mechanics of Charged Particle Beam Optics

p Using the dimensionless variable ξ = ( mω/¯h)x, we have   mω 1/4 1  d n −1ξ2 √ ψn (x) = ξ− e 2 . π h¯ dξ 2n n!

(3.365)

Using the identity   n  2 d d n −1ξ2 n 21 ξ 2 2 = (−1) e e e−ξ , ξ− dξ dξ

(3.366)

easily proved by induction, we can rewrite ψn (x) =

 mω 1/4 π h¯

1 2 1 √ e− 2 ξ Hn (ξ ), n 2 n!

where Hn (ξ ) = (−1)n eξ

2



d dξ

n

2

e−ξ .

(3.367)

(3.368)

Thus, we have reconstructed the earlier result (3.335). The existence of a nonzero ground state energy, so-called zero-point energy, can be understood intuitively in terms of the uncertainty principle. As seen in (3.363), the ground state wave function is a Gaussian with a finite width. It is not localized means that the position of the oscillating particle is fluctuating around the equilibrium even in the ground state unlike the classical oscillator which will be at rest in the equilibrium position in the ground state. As a consequence of this uncertainty in position, say ∆x, there will be a finite uncertainty in the momentum of the particle, ∼ h¯ /∆x, as dictated by the Heisenberg uncertainty principle. This intrinsic ground state motion implies the existence of a minimum energy. Let us make a precise calculation of the uncertainty product ∆x ∆p for the harmonic oscillator eigenstates. From the definitions of ab and ab† in (3.342), we have r r

m¯hω † h¯ † ab + ab , pb = i ab − ab . (3.369) x= 2mω 2 Calculating the averages of x and p in the n-th eigenstate, we get r
 h¯ † n ab + ab n = 0, hxin = hn|x|ni = 2mω r
 m¯hω † hpin = hn| pb|ni = i n ab − ab n = 0, 2

(3.370)

where we have used the relations (3.358) and (3.359) and the orthogonality of the eigenstates {|ni}. Averages of x2 and p2 can be calculated as follows: hx2 in = hn|x2 |ni
2 E h¯ D † = n ab + ab n 2mω

99

A Review of Quantum Mechanics

i E h¯ D h †
2 n ab + ab2 + ab† ab + abab† n 2mω
 h¯ † = n ab ab + abab† n 2mω  h¯ 1 = n+ , mω 2 =

(3.371)

hp2 in = hn| pb2 |ni
2 i E m¯hω D h = n − ab† − ab n 2 i E
2 m¯hω D h n − ab† − ab2 + ab† ab + abab† n = 2
 m¯hω † = n ab ab + abab† n 2   1 = m¯hω n + , 2

(3.372)

where we have used the relations (3.358) and (3.359) and the orthonormality of the eigenstates {|ni}. From (3.276), we find that the uncertainties in x and p in the n-th eigenstate are given by s   q q 1 h¯ 2 2 2 n+ (∆x)n = hx in − hxin = hx in = mω 2 s   q q 1 (∆p)n = hp2 in − hpi2n = hp2 in = m¯hω n + . (3.373) 2 Now, for the n-th eigenstate |ni, the uncertainty product ∆x∆p becomes   1 h¯ . (∆x)n (∆p)n = n + 2

(3.374)

For the ground state |0i h¯ (3.375) (∆x)0 (∆p)0 = . 2 Thus, the ground state is a minimum uncertainty state. It should be noted that, unlike in the case of a minimum uncertainty free-particle wave packet, the ground state of the harmonic oscillator is a minimum uncertainty state at any time since it is a stationary state and hence ∆x and ∆p do not depend on time. The propagator for the linear harmonic oscillator (lho), for t ≥ ti , is given by
Klho x,t; x0 ,ti =

∞

n=0

r =

i

∑ e− h¯ En (t−ti ) ψn (x)ψn∗ (x0 ) mω π h¯

∞

1 1 −1 ∑ 2n n! e−i(n+ 2 )ω(t−ti ) Hn (ξ )Hn (ξ 0 ) e 2 n=0

  ξ 2 +ξ 0 2

,

(3.376)

100

Quantum Mechanics of Charged Particle Beam Optics

p p where ξ = mω/¯h x and ξ 0 = mω/¯h x0 . Now, it is possible to sum the series in the above expression exactly, using the Mehler formula (for a derivation of this formula, see, e.g., Lakshminarayanan and Varadharajan [126]), ∞

1

∑ 2n n! Hn (z)Hn (z0 )ζ n e

− 21

  z2 +z0 2

n=0

−

1

=p

1−ζ2

  2 z2 +z0 −4ζ zz0

(1+ζ 2 )

e

(

2 1−ζ 2

)

,

(3.377)

taking z = ξ , z0 = ξ 0 , and ζ = e−iω(t−ti ) . Thus, we get 0




Klho x,t; x ,ti =



mω 2πi¯h sin ω (t − ti )

1/2 e

imω 2¯h sin ω (t−ti )

h  i x2 +x0 2 cos ω(t−ti )−2xx0

.

(3.378) When ω −→ 0, the linear harmonic oscillator becomes a free particle in one dimension. Note that when ω −→ 0, Klho (x,t; x0 ,ti ) −→ Kfree (x,t; x0 ,ti ) as should be. 3.3.1.3

Two-Dimensional Isotropic Harmonic Oscillator

A classical two-dimensional isotropic harmonic oscillator in the xy-plane has the Hamiltonian H (x, y, px , py ) =


1
1 p2x + p2y + mω 2 x2 + y2 . 2m 2

Hamilton’s equations are given by    x 0 px   −1 d   mω  = ω   0 dt  y  py 0 mω

 x 1 0 0  px 0 0 0    mω 0 0 1  y py 0 −1 0 mω

(3.379)

  , 

(3.380)

showing the independence of motions in the x- and y-directions. From the treatment of the linear harmonic oscillator, we can write down the solution of (3.380) as x(t) = Ax cos (ωt − ϕx ) , y(t) = Ay cos (ωt − ϕy ) ,

px (t) = −Ax mω sin (ωt − ϕx ) , py (t) = −Ay mω sin (ωt − ϕy ) ,

(3.381)

where Ax and Ay are the amplitudes of motion in the x- and y-directions, respectively, and ϕx and ϕy are the corresponding initial phases. If there is no relative phase between the x and y motions initially, i.e., ϕx = ϕy , then at all times y(t) = (Ay /Ax ) x(t) and so the motion of the particle will be a straight line. If the initial phase difference ϕx − ϕy is ±π/2, then at all times (x(t)/Ax )2 + (y(t)/Ay )2 = 1, and hence the trajectory of the particle will be an ellipse with its axes parallel to the x and y axes; if Ax = Ay , the motion will be circular. In general, the particle will have an elliptical orbit with the orientation of its axes depending on the initial phase difference ϕx − ϕy .

101

A Review of Quantum Mechanics

The two-dimensional isotropic quantum harmonic oscillator in the xy-plane has the Hamiltonian

b = 1 pb2x + pb2y + 1 mω 2 x2 + y2 H 2m 2   2
h¯ 2 ∂2 1 ∂ =− + 2 + mω 2 x2 + y2 . 2 2m ∂ x ∂y 2

(3.382)

We can write this as b=H bx + H by , H

(3.383)

where 2 bx = pbx + 1 mω 2 x2 , H 2m 2

by = H

pb2y 1 + mω 2 y2 . 2m 2

(3.384)

bx and H by , Hamiltonians of linear oscillators along x and y axes, respecNote that H tively, commute with eath other. Hence, the complete set of orthonormal solutions of b the eigenvalue equation for H, b Hψ(x, y) = Eψ(x, y),

(3.385)

the time-independent Schr¨odinger equation of the system, are given by ψnx ny (x, y) = ψnx (x)ψny (y),

Enx ny = (nx + ny + 1) h¯ ω,

(3.386)

  1 b Hy ψny (y) = ny + h¯ ω. 2

(3.387)

where   1 b h¯ ω, Hx ψnx (x) = nx + 2 Explicitly, 1/2 mω 2 2 mω e− 2¯h (x +y ) ψnx ny (x, y) = π h¯ 2nx +ny nx !ny ! r  r  mω mω × Hnx x Hny y . h¯ h¯ 

(3.388)

The energy eigenvalue Enx ny is seen to be (nx + ny + 1)-fold degenerate since nx + ny = n can be partitioned into an ordered pair of nonnegative integers in (n + 1) ways. One can define the lowering and raising operators mωx + i pbx abx = √ , 2m¯hω mωy + i pby aby = √ , 2m¯hω

mωx − i pbx ab†x = √ 2m¯hω mωy − i pby ab†y = √ 2m¯hω

(3.389)

102

Quantum Mechanics of Charged Particle Beam Optics

which obey the commutation relations h i abj , ab†k = δ jk , [b a j , abk ] = 0,

h i ab†j , ab†k = 0,

j, k = x, y.

(3.390)

Then, the Hamiltonian can be written as
b = ab†x abx + ab†y aby + 1 h¯ ω, H

(3.391)

so that the energy eigenvalues are given by (nx + ny + 1) h¯ ω. The corresponding eigenstates can be written as
nx †
ny 1 aby |0, 0i, ab†x nx !ny !

|nx , ny i = p

(3.392)

where the ground state is defined by abx |0, 0i = 0,

aby |0, 0i = 0.

(3.393)

b is seen to have rotational symmetry, i.e., if the coordinates x The Hamiltonian H and y are replaced by x cos φ + y sin φ and −x sin φ + y cos φ , respectively, it remains invariant. This suggests using plane polar coordinates for solving the differential b Changing from (x, y) to the equation to get the eigenvalues and eigenfunctions of H. plane polar coordinates (r, φ ), defined by x = r cos φ and y = r sin φ , we have    2  1 ∂ ∂ 1 ∂2 1 b = − h¯ H r + 2 + mω 2 r2 (3.394) 2 2m r ∂ r ∂r r ∂φ 2 and the time-independent Schr¨odinger equation (3.385) becomes       h¯ 2 1 ∂ ∂ 1 ∂2 1 2 2 − r + 2 + mω r ψ(r, φ ) = Eψ(r, φ ). 2m r ∂ r ∂r r ∂φ2 2

(3.395)

The complete set of orthonormal eigenfunctions of this equation are given by   mωr2 |m | mωr2 imφ φ e , ψnr ,mφ (r, φ ) ∼ r|mφ | e− 2¯h Lnr φ h¯ nr = 0, 1, 2, . . . , mφ = 0, ±1, ±2, . . . , (3.396) apart from normalization factors, where Lnα (x) =

x−α ex d n −x n+α
e x , n! dxn

(3.397)

is called an associated Laguerre polynomial (see, e.g., Arfken, Weber, and Harris [3], Byron and Fuller [18], and Lakshminarayanan and Varadharajan [126]). The respective energy eigenvalues are
Enr ,mφ = 2nr + |mφ | + 1 h¯ ω, (3.398)

A Review of Quantum Mechanics

103

where nr is the radial quantum number and mφ is the two-dimensional angular bz = x pby − y pbx = momentum quantum number, with mφ h¯ being the eigenvalue of L imφ φ −i¯h∂ /∂ φ corresponding to the eigenfunction e . We can write Enr ,mφ = (n+1)¯hω
with n = 2nr + |mφ | . It is seen that for a fixed value of n, there are

n+1 choices for the pair nr , mφ : e.g., n = 0 corresponds to n = 0, m = 0 , n
= 1 r φ
corresponds to
nr = 0, mφ = ±1 , n = 2 corresponds to
nr = 0, mφ = ±2 and
nr = 1, mφ = 0 , n = 3 corresponds to
nr = 0, mφ = ±3
and nr = 1, mφ = ±1
, n = 4 corresponds to nr = 0, mφ = ±4 , nr = 1, mφ = ±2 , and nr = 2, mφ = 0 , etc. Thus, the energy eigenvalues of the two-dimensional isotropic oscillator are given by En = (n + 1)¯hω, with n = 0, 1, 2, . . ., and the eigenvalue En is (n + 1)-fold degenerate, as we already found earlier in the treatment using the Cartesian coor 2 dinates. Note that the position probability distribution, ψnr ,mφ (r, φ ) , has rotational symmetry in the xy-plane with only r-dependence and no φ -dependence for any state |nr mφ i. This is to be compared with the classical mechanics where the particle has, in general, elliptical orbits. 3.3.1.4

Charged Particle in a Constant Magnetic Field

We have found that, according to classical mechanics, a charged particle moving in a constant magnetic field has, in general, a helical trajectory with the direction of the magnetic field as its axis and its conserved energy can take any nonzero value. Such a helical trajectory is a superposition of free-particle motion along the axis of the magnetic field and a circular motion in the plane perpendicular to the axis. Now, we shall see how a charged particle moving in a constant magnetic field behaves when it obeys quantum mechanics. For a particle of mass m and charge q moving in an electromagnetic field with φ and ~A as the scalar and vector potentials, the nonrelativistic Schr¨odinger equation is   1 ~2 ∂Ψ = πb + qφ Ψ (3.399) i¯h ∂t 2m related to the free-particle equation through the principle of minimal electromagnetic coupling, i.e., i¯h∂ /∂t −→ (i¯h∂ /∂t) − qφ , ~pb −→ ~πb = ~pb − q~A. For a charged particle moving in a constant magnetic field ~B, there is no scalar potential, and the timeindependent vector potential can be taken in the symmetric gauge as ~A(~r) = 21 ~B × ~r, which is such that ~∇ · ~A = 0. The corresponding time-independent Schr¨odinger equation is  2 b r) = 1 ~pb − q~A ψ(~r) Hψ(~ 2m  i 1 h 2 = pb + q2 A2 − q ~A ·~pb +~pb · ~A ψ(~r) 2m  2  pb q2 2 q ~ ~ = + A − A · pb ψ(~r) 2m 2m m

104

Quantum Mechanics of Charged Particle Beam Optics

 =

 q ~ ~b pb2 q2 h 2 2 ~ 2 i B r − (B ·~r) − + B · L ψ(~r) 2m 8m 2m

= Eψ(~r),

(3.400)

b =~r ×~pb is the angular momentum operator, and we have used the identity where L   (~A × ~B)2 = A2 B2 − (~A · ~B)2 , and the relation ~A · ~∇ + ~∇ · ~A ψ(~r) = 2~A · ~∇ψ(~r) since ~∇ · ~A = 0. This shows that the Hamiltonian of the particle is  b= H

 pb2 q2 h 2 2 ~ 2 i q ~ ~b + B r − (B ·~r) − B·L , 2m 8m 2m

(3.401)

in which the third term represents the potential energy due to the interaction of the orbital magnetic dipole moment, or simply called the orbital magnetic moment, of the charged particle, q ~b ~µ b= L, (3.402) 2m b is the quantum operator corresponding to the with the magnetic field. Note that ~µ classical magnetic moment of a charged particle ~µ = (q/2m)~L, where ~L is the angular momentum, and recall that a magnetic moment ~µ placed in a magnetic field ~B has the interaction energy −~µ · ~B. Let us now take the magnetic field to be in the z-direction: ~B = B~k. The corresponding vector potential in the symmetric gauge is given by ~A = (−By/2, Bx/2, 0). Then, the Hamiltonian is " # 2  2 1 1 1 b= H pbx + qBy + pby − qBx + pb2z 2m 2 2  2
1
qB 1 2 1 qB bz , = pb + pb2 + pb2y + m L (3.403) x2 + y2 − 2m z 2m x 2 2m 2m which, apart from the last magnetic interaction term, represents a free particle moving along the z-direction, the direction of the magnetic field, and a two-dimensional isotropic harmonic oscillator in the perpendicular xy-plane with a frequency qB/2m. Let us write b=H bz + H bxy , H (3.404) with bz = 1 pb2z , H 2m

bxy = 1 pb2x + pb2y + 1 mωL2 x2 + y2 − ωL (x pby − y pbx ) , H 2m 2

(3.405)

where ωL = qB/2m, called the Larmor frequency, is half of the nonrelativisbz and H bxy commute with each other, the tic cyclotron frequency ωc . Since H b can be chosen to be simultaneous eigenfunctions of H bz and eigenfunctions of H

105

A Review of Quantum Mechanics

bxy , and the eigenvalues of H b can be expressed as the sum of the eigenvalH b b bz form a continuous spectrum given by ues of Hz and Hxy . The eigenvalues of H  2 pz /2m| − ∞ < pz < ∞ corresponding to the free-particle eigenfunctions in the √  bxy can be found by z-direction eipz z/¯h / 2π h¯ | − ∞ < pz < ∞ . The spectrum of H applying the algebraic method used earlier for the linear harmonic oscillator and the two-dimensional isotropic oscillator. To this end, let us define mωL x + i pbx abx = √ , 2m¯hωL mωL y + i pby , aby = √ 2m¯hωL which obey the commutation relations h i abj , ab†k = δ jk , [b a j , abk ] = 0,

mωL x − i pbx ab†x = √ 2m¯hωL by mω Ly − i p ab†y = √ 2m¯hωL

i h ab†j , ab†k = 0,

j, k = x, y.

In terms of these operators, we have 
 bxy = h¯ ωL ab†x abx + ab†y aby + 1 + i ab†x aby − ab†y abx . H

(3.406)

(3.407)

(3.408)

Let us introduce the operators b = √1 (b ax + ib ay ) , A 2


b† = √1 ab†x − ib a†y , A 2

which obey the commutation relation h i b, A b† = 1. A

(3.409)

(3.410)

Now, it is found that     b† A b + 1 = h¯ ωc A b† A b+ 1 , bxy = 2¯hωL A H 2 2

(3.411)

bxy are with ωc as the nonrelativistic cyclotron frequency. Then, the eigenvalues of H b are seen to be {(n + (1/2))¯hωc | n = 0, 1, 2, . . .}. Thus, the eigenvalues of H   p2 1 h¯ ωc + z , En,pz = n + 2 2m

n = 0, 1, 2, . . . , −∞ < pz < ∞.

(3.412)

b is not over. Let us solve the equation The story of the spectrum of H b Aψ(x, y) ∼ [mωL (x + iy) + (i pbx − pby )] ψ(x, y)    mωc ∂ ∂ = (x + iy) + +i ψ(x, y) = 0. 2¯h ∂x ∂y

(3.413)

106

Quantum Mechanics of Charged Particle Beam Optics

If we transform this equation to plane polar coordinates (r, φ ), such that x = r cos φ and y = r sin φ , we get    ∂ i ∂ iφ mωc r+ + ψ(r, φ ) = 0. (3.414) e 2¯h ∂r r ∂φ Writing ψ(r, φ ) = R(r)Φ(φ ) leads to separate equations for R(r) and Φ(φ ). Solving these equations, we find that ψ0,mφ (r, φ ) = reiφ


mφ

e−

mωc 2 4¯h r

,

mφ = 0, 1, 2, . . . .

(3.415)

The value of mφ is restricted to nonnegative integers since solution is to be single valued for any φ and φ + 2π and should be regular at the origin r = 0. Note that mφ h¯ bz = (x pby − y pbx ) = −i¯h(∂ /∂ φ ). is the eigenvalue of the angular momentum operator L In terms of Cartesian coordinates these ground state wave functions are ψ0,mφ (x, y) = N (x + iy)mφ e−

mωc 4¯h

(x2 +y2 ) ,

mφ = 0, 1, 2, . . . ,

(3.416)

where N is the normalization constant. It is seen that we have an infinity of degenbxy all with the same energy E0,m = h¯ ωc /2 independent of erate ground states for H φ mφ , which labels the angular momentum state (for two-dimensional motion, there is bxy are obtained by only one angular momentum (x pby − y pbx )). The excited states of H † b on each of the ground states. We get the repeated action of A 1  b† n ψn,mφ (x, y) = √ A ψ0,mφ (x, y), n!

n = 0, 1, 2, . . . , mφ = 0, 1, 2, . . . . (3.417)

Thus, for a particle of charge q and mass m moving in a constant magnetic field ~B = B~k, the complete set of orthonormal eigenstates are given by  n i 1 b† ψ0,m (x, y), e h¯ pz z A ψn,mφ ,pz (x, y, z) = √ φ 2π h¯ n! mωc 2 2 ψ (x, y) = N (x + iy)mφ e− 4¯h (x +y ) , 0,mφ

n = 0, 1, 2, . . . , mφ = 0, 1, 2, . . . ,

−∞ < pz < ∞, (3.418)

corresponding to the energy spectrum, called the Landau levels,   p2 1 h¯ ωc + z , En,mφ ,pz = n + 2 2m

(3.419)

where N is the normalization constant and ωc = qB/m is the nonrelativistic cyclotron frequency. Each eigenvalue is infinitely degenerate, and each degenerate eigenstate has distinct angular momentum eigenvalue mφ h¯ on which the energy eigenvalue does not depend.

107

A Review of Quantum Mechanics

3.3.1.5

Scattering States

Let us now consider a particle of mass m moving along the x-axis in which it experiences a potential well   0 for −∞ < x < −L region I −V for −L ≤ x ≤ L region II V (x) = (3.420)  0 for L 0, its kinetic energy is positive

110

Quantum Mechanics of Charged Particle Beam Optics

throughout. The above result implies that this is not so for a particle obeying quantum mechanics. It can get reflected from the boundaries where the potential energy changes abruptly. This example presents a quantum system for which there are both discrete and continuous energy eigenvalues. The discrete energy levels correspond to bound states with normalizable wave functions that vanish at ±∞. The eigenstates belonging to the continuous energy spectrum correspond to the particle incident on the potential well from, say, −∞ and getting scattered (reflected/transmitted) to ∓∞. The eigenfunctions of these states are not normalizable and do not vanish at ±∞. Such states are called scattering states. Let us now consider a particle of mass m moving along the x-axis in which it encounters a potential barrier  for −∞ < x < −L region I  0 V > 0 for −L ≤ x ≤ L region II (3.434) V (x) =  0 for L V will go through the potential barrier region with reduced momentum, but it will not be reflected from anywhere since it has positive kinetic energy everywhere. The particle obeying quantum mechanics is seen to be reflected from the boundaries of the potential barrier where the potential changes abruptly. 3.3.1.6

Approximation Methods, Time-Dependent Systems, and the Interaction Picture

There are very few quantum systems, like free particle, harmonic oscillator, and hydrogen-like atoms, for which the Schr¨odinger equation can be solved exactly. Therefore, approximation methods become necessary for many practical applications. For example, to study anharmonic oscillator with the
the one-dimensional
b = pb2 /2m + kx2 /2 + εx4 , where ε > 0 is a small parameter, Hamiltonian H time-independent perturbation theory is used treating εx4 as a small perturbation to the harmonic oscillator Hamiltonian. Time-independent perturbation theory is a systematic procedure of getting approximate solutions to the perturbed problem in terms of the known exact solutions to the unperturbed problem. Variational method can be used to obtain approximately the bound-state energies and wave functions of a time-independent Hamiltonian. The Jeffreys–Wentzel–Kramers–Brillouin (JWKB) approximation technique is useful in analyzing any one-dimensional timeindependent Schr¨odinger equation with a slowly varying potential. We will not be concerned here with these time-independent perturbation theory techniques. We shall now consider time-dependent systems. The time evolution of a system b is given by the Schr¨odinger equation with the time-dependent Hamiltonian H(t) i¯h

∂ |Ψ(t)i b = H(t)|Ψ(t)i. ∂t

(3.439)

Equivalently, we can write, as seen already (3.154–3.163), b (t,ti ) |Ψ (ti )i , |Ψ(t)i = U

t ≥ ti ,

(3.440)

b (t,ti ) satisfying the relawith ti as the initial time and the time-evolution operator U tions i¯h

∂ b b U b (t,ti ) , U (t,ti ) = H(t) ∂t

(3.441)

112

Quantum Mechanics of Charged Particle Beam Optics

b (t,ti )† U b (t,ti ) = I, U

b (ti ,ti ) = I. U

(3.442)

If the Hamiltonian is time-independent, we can integrate (3.441) exactly to write b i )/¯h . But, when H(t) b (t,ti ) = e−iH(t−t b is time-dependent, this cannot be done. This U can be seen as follows. Let t − ti = N∆t, where N is a large number and ∆t is an infinitesimally small time interval. We can assume that the Hamiltonian does not change within the time interval ∆t. Then, we can write n i i i b b b e− h¯ ∆t H(ti +N∆t) e− h¯ ∆t H(ti +(N−1)∆t) e− h¯ ∆t H(ti +(N−2)∆t) · · · N→∞ ∆t→0 o i i b b · · · e− h¯ ∆t H(ti +2∆t) e− h¯ ∆t H(ti +∆t) |Ψ (ti )i. (3.443)

|Ψ(t)i = lim lim

b b (ti + N∆t) = H b (ti + (N − 1)∆t) = If H(t) is time-independent, then we have H b b b b H (ti + (N − 2)∆t) = · · · = H (ti + 2∆t) = H (ti + ∆t) = H and hence the above equation can be written as i

|Ψ(t)i = lim e− h¯ ∑ ∆t H |Ψ (ti )i = e ∆t→0 b − i H(t−t i)

=e

h¯

b

R b − h¯i tt dt H i

|Ψ (ti )i

b (t,ti ) |Ψ (ti )i. |Ψ (ti )i = U

(3.444)

b When the Hamiltonian is time-dependent, H(t)s at different times do not commute with each other and the exponents cannot be added, as done earlier, in taking the b b b b bBb 6= BbA. b The Baker–Campbell– product of the exponentials since eA eB 6= eA+B if A b Bb A b b is b Hausdorff (BCH) formula for the product e e , when AB 6= BbA, b b 1 b b

1

eA eB = eA+B+ 2 [A , B]+ 12 ([A , [A , B]]+[[A , B] , B])+··· . b b

b b b

b b b

(3.445)

For a proof of the BCH formula, and for more operator techniques useful in physics, see, e.g., Bellman and Vasudevan [10], and Wilcox [188]. When the Hamiltonian is time-dependent, we can proceed as follows to find b (t,ti ). From (3.441), we have U Z t

 dt

ti

 Z ∂ b i t b 1 )U b (t1 ,ti ) , dt1 H(t U (t,ti ) = − ∂t h¯ ti

(3.446)

or Z t i t b b b 1 )U b (t1 ,ti ) , U (t,ti ) = U (t,ti ) − I = − dt1 H(t h¯ ti ti

(3.447)

leading to the formal solution b (t,ti ) = I − i U h¯

Z t ti

b 1 )U b (t1 ,ti ) . dt1 H(t

(3.448)

113

A Review of Quantum Mechanics

Iterating this formal solution, we get  Z  Z Z t2 i t i 2 t b b b (t2 ) H b (t1 ) U (t,ti ) = I − dt2 dt1 H (t1 ) + − dt1 H h¯ ti h¯ ti ti  Z  Z t3 Z t2 i 3 t b (t3 ) H b (t2 ) H b (t1 ) dt3 dt2 dt1 H + − h¯ ti ti ti + ··· ,  Z Z ∞  i n ··· dtn dtn−1 · · · dt2 dt1 =I+ ∑ − h¯ n=1 t>tn >tn−1 >···>t2 >t1 >ti

b (tn ) H b (tn−1 ) · · · H b (t1 ) . ×H

(3.449)

This series expression can be written in a compact form, symbolically, by introducing the time ordering operator:  A (t1 ) B (t2 ) , if t1 > t2 T [A (t1 ) B (t2 )] = (3.450) B (t2 ) A (t1 ) , if t2 > t1 Now, note that "Z T ti

t

Z t  Z t  2 # =T dt2 A (t2 ) dt1 A (t1 ) dt A(t) ti

Z t

= ti

ti

Z t

dt2

ti

dt1 T [A (t2 ) A (t1 )]

ZZ

=

dt2 dt1 A (t2 ) A (t1 )

t>t2 >t1 >ti

ZZ

+

dt2 dt1 A (t1 ) A (t2 ) .

(3.451)

t>t1 >t2 >ti

The t1 ←→ t2 symmetry between the two integrals on the right-hand side shows that the two integrals must be equal. Hence, we have "Z 2 # ZZ t 1 dt2 dt1 A (t2 ) A (t1 ) = T dt A(t) . (3.452) 2 ti t>t2 >t1 >ti

Extending this argument leads easily to the result that Z

···

Z

dtn dtn−1 · · · dt2 dt1 A (tn ) A (tn−1 ) · · · A (t1 )

t>tn >tn−1 >···>t2 >t1 >ti

1 = T n!

Z ti

t

n  dt A(t) .

(3.453)

114

Quantum Mechanics of Charged Particle Beam Optics

Thus, we can write (3.449) as   Z t n  ∞ i n 1 b b − U (t,ti ) = I + ∑ T dt H(t) h¯ ti n=1 n! "  Z t n #  ∞ i n 1 b − dt H(t) =T ∑ h¯ ti n=0 n!  i Rt b  − dt H(t) = T e h¯ ti ,

(3.454)

where the expression on the right-hand side is known as the Dyson time-ordered exponential. It should be noted that the time-ordered exponential is not a true exponential and is only a notation for the series that has to be computed by truncating it up to any desired order. b (t,ti ) is the Magnus An alternate expression for the time-evolution operator U formula (Magnus [130]): b (t,ti ) = e− h¯i Tb(t,ti ) , (3.455) U b where T (t,ti ) is an infinite series with the first few terms given by  Z t Z t Z t2 h i b (t1 ) + 1 − i b (t2 ) , H b (t1 ) Tb (t,ti ) = dt1 H dt2 dt1 H 2 h¯ ti ti ti  2 Z t Z t3 Z t2 nhh i i i 1 b (t3 ) , H b (t2 ) , H b (t1 ) − dt3 + dt2 dt1 H 6 h¯ ti t t hh i i ioi b b b + H (t1 ) , H (t2 ) , H (t3 ) + · · · . (3.456) b (t,ti ) is unitary. In the Magnus formula Note that Tb (t,ti ) is Hermitian such that U b (3.455 and 3.456), U (t,ti ) is a true exponential (see the Appendix at the end of this chapter for derivation). The Magnus formula is completely equivalent to the Dyson time-ordered exponential formula. Let us now consider a system for which the Hamiltonian is time-dependent and can be written as b =H b0 + H b 0 (t), H(t) (3.457) 0 b b where H0 is the unperturbed Hamiltonian and H (t) is a time-dependent perturbation b0 is chosen to be time-independent. With Hamiltonian. Usually H i¯h

∂ |Ψ(t)i b = H(t)|Ψ(t)i, ∂t

(3.458)

define  Ψ (t) = U b † (t,ti ) |Ψ(t)i, i b0 H

(3.459)

b b (t,ti ) is the unitary time-evolution operator corresponding to the unperwhere U H0 b0 such that turbed Hamiltonian H i¯h

b b (t,ti ) ∂U H0 ∂t

b0U b b (t,ti ) , =H H0

i¯h

b † (t,ti ) ∂U b H0

∂t

b † (t,ti ) H b0 . = −U b H0

(3.460)

115

A Review of Quantum Mechanics

b corresponding to an observable O, define For the Hermitian operator O b (t) = U bU b † (t,ti ) O b b (t,ti ) . O i b0 H0 H

(3.461)

From (3.459–3.461) we have  b † (t,ti ) ∂U ∂ Ψi (t) b0 H b b † (t,ti ) H(t)|Ψ(t)i i¯h = i¯h |Ψ(t)i + U b0 H ∂t ∂t b0 |Ψ(t)i + U b † (t,ti ) H(t)|Ψ(t)i b b † (t,ti ) H = −U b b0 H H   0 b † (t,ti ) H(t) b −H b0 |Ψ(t)i = U b † (t,ti ) H b 0 (t)|Ψ(t)i =U b b H0

H0

b † (t,ti ) H b 0 (t)U b b (t,ti ) U b † (t,ti ) |Ψ(t)i =U b0 b0 H0 H H  0 b = Hi (t) Ψi (t) .

(3.462)

From (3.460) and (3.461) we have   ! b † (t,ti ) ∂U b b dO b ∂ O H † 0 i bU b b (t,ti ) + U b (t,ti ) i¯h b b (t,ti ) O i¯h = i¯h U b0 H0 H0 H dt ∂t ∂t ! b b (t,ti ) ∂ U H 0 b i¯h b † (t,ti ) O +U b0 H ∂t ! b ∂ O † † bU b0 O b b (t,ti ) + U b (t,ti ) H b (t,ti ) i¯h b b (t,ti ) = −U U b0 b0 H0 H0 H H ∂t bH b † (t,ti ) O b0U b b (t,ti ) +U b0 H0 H !    b ∂O bU b † (t,ti ) O b b (t,ti ) U b † (t,ti ) H b0U b b (t,ti ) + U = i¯h b0 b0 H0 H0 H H ∂t i    bU b † (t,ti ) H b0U b b (t,ti ) U b † (t,ti ) O b b (t,ti ) − U b0 b0 H0 H0 H H ! h  i b ∂O b , H b0 = O + i¯h i ∂t i i! i h   b b + i¯h ∂ O , b0 , O = − H (3.463) i ∂t i i or h   i b dO i= 1 − H b + b0 , O i dt i¯h i

b ∂O ∂t

!

. (3.464) i Equations (3.462) and (3.464) constitute the interaction picture, introduced by Dirac. It is seen to be intermediate between the Schr¨odinger and the Heisenberg pictures. b b0 −→ 0 and H b 0 (t) −→ H(t). It becomes the Schr¨odinger picture when H It becomes

116

Quantum Mechanics of Charged Particle Beam Optics

b0 −→ H(t) b and H b 0 (t) −→ 0. Note that the equation of the Heisenberg picture when H motion for the interaction picture observables (3.464) is exactly like the equation of motion for the Heisenberg picture observables (3.188). The interaction picture is designed for use in time-dependent perturbation theory, where the goal is to study the time evolution of a system with a small  time-dependent perturbation Hamiltonian. The time-evolution equation for Ψi (t) , (3.462), can be formally integrated to give   Ψ (t) = U b b 0 (t,ti ) Ψ (ti ) , (3.465) i i H i where   i Rt b0 b b 0 (t,ti ) = T e− h¯ ti dt Hi (t) . U H i  From the definition of Ψi (t) , (3.459), we get   b b (t,ti ) U b b 0 (t,ti ) Ψ (ti ) b b (t,ti ) Ψ (t) = U |Ψ(t)i = U i i H0 H H0 i b b (t,ti ) U b b 0 (t,ti ) |Ψ (ti )i =U H0 H i b (t,ti ) |Ψ (ti )i , =U

(3.466)

(3.467)

relating the state of the system at t to its initial state at ti . In the second step above, we have used the relation Ψi (ti ) = |Ψ (ti )i following directly from the definition b0 is chosen to be time-independent (3.459). When the unperturbed Hamiltonian H b0 (t−ti ) − h¯i H b UHb0 (t,ti ) = e and, when the time-dependent perturbation Hamiltonian 0 b b b 0 (t,ti ) only the first few terms of the infinite series H (t) is small, to compute U H i in the Dyson, or the Magnus, formula need to be taken. We can directly verb b (t,ti ) U b b 0 (t,ti ) = U b (t,ti ), the time-evolution operator corresponding to ify that U H0 H i b =H b0 + H b 0 (t): H(t)   ∂ b b i¯h UHb0 (t,ti ) UHb 0 (t,ti ) ∂t i      ∂ b ∂ b b b UHb0 (t,ti ) UHb 0 (t,ti ) + UHb0 (t,ti ) i¯h UHb 0 (t,ti ) = i¯h ∂t ∂t i i b 0 (t)U b b 0 (t,ti ) b b (t,ti ) H b0U b b (t,ti ) U b b 0 (t,ti ) + U =H H H0 H H0 i i i b b 0 (t,ti ) + U b b (t,ti ) U b † (t,ti ) H b 0 (t)U b b (t,ti ) U b b 0 (t,ti ) b0U b b (t,ti ) U =H b0 H0 H H0 H0 H H i i    b0 + H b 0 (t) U b b (t,ti ) U b b 0 (t,ti ) = H H H0   i b b b (t,ti ) U b b 0 (t,ti ) . = H(t) U (3.468) H0 H i

117

A Review of Quantum Mechanics

3.3.1.7

Schr¨odinger–Pauli Equation for the Electron

Electron has the nonclassical property, spin, the intrinsic angular momentum, with two values, ±¯h/2, for its z-component. In the presence of a magnetic field, it aligns itself either parallel or antiparallel to the field. As already discussed, the spin- 21 operators, components of the vector ~S, are to be represented in terms of the Pauli matrices as h¯ h¯ h¯ Sx = σx , Sy = σy , Sz = σz . (3.469) 2 2 2 The spin magnetic dipole moment, or simply called the spin magnetic moment, of the electron, with charge −e, is found to be ~µe = −µB~σ = −

e~S e¯h ~σ = − , 2me me

(3.470)

where µB = e¯h/2me is the Bohr magneton, the unit of magnetic moment, and me is the mass of the electron. It is seen that ~µe is twice the value of the magnetic moment it would have if the spin ~S is like a classical orbital angular momentum (see (3.402)). This factor of 2, known as the Land´e g-factor, will be seen to be a direct consequence of Dirac’s relativistic equation for the electron to be discussed later. Actually, the electron is observed to have an anomalous magnetic moment corresponding to a value of g slightly higher than 2, and the successful explanation of this g − 2 ≈ 0.00232 is provided by quantum electrodynamics. The nonrelativistic Hamiltonian of an electron in an electromagnetic field associated with the vector and scalar potentials ~A and φ , respectively, is, including spin,   ~πb2 b= − eφ  I + µB~σ · ~B, with ~πb = ~pb + e~A, (3.471) H 2me where the last term, the Pauli term, is the interaction energy of the electron spin magnetic moment in the magnetic field and I is the 2 × 2 unit matrix inserted to be consistent with the last 2 × 2 matrix term. The Schr¨odinger equation now becomes    2   ~ πb ∂ |Ψ(t)i =  − eφ  I + µB~σ · ~B |Ψ(t)i, (3.472) i¯h  2me  ∂t the Schr¨odinger–Pauli equation, or the Pauli equation, for the electron, where the state vector is a two-component column vector,   |Ψ1 (t)i |Ψ(t)i = , (3.473) |Ψ2 (t)i since the 2 × 2 matrix Hamiltonian has to act on a two-component column vector. The nonrelativistic electron state vector |Ψ(t)i is called a two-component spinor. Note that we can write the Pauli equation (3.472) also as   ∂ |Ψ(t)i 1  ~ 2 ~σ · πb − eφ I |Ψ(t)i i¯h = (3.474) ∂t 2me

118

Quantum Mechanics of Charged Particle Beam Optics

To derive (3.472) from (3.474), one has to proceed as follows. First, use the algebra of the Pauli matrices σx2 = I,

σy2 = I,

σx σy = −σy σx ,

σz2 = I,

σy σz = −σz σy ,

σz σx = −σx σz ,

to observe that  2 2 ~σ ·~πb = ~πb + σx σy [πbx , πby ] + σy σz [πby , πbz ] + σz σx [πbz , πbx ] .

(3.475)

(3.476)

Then, use the relations between the Pauli matrices σx σy = iσz ,

σy σz = iσx ,

σz σx = iσy ,

(3.477)

[πbz , πbx ] = −ie¯hBy .

(3.478)

and the commutation relations [πbx , πby ] = −ie¯hBz ,

[πby , πbz ] = −ie¯hBx ,

We can now write  |Ψ(t)i = |Ψ1 (t)i

1 0



 + |Ψ2 (t)i

= |Ψ1 (t)i | ↑i + |Ψ2 (t)i | ↓i.

0 1

 (3.479)

As already discussed, |h~r|Ψ1 (t)i|2 gives the probability for the electron to be found at~r with spin up and |h~r|Ψ2 (t)i|2 gives the probability for the electron to be found at ~r with spin down. For an electron in a constant magnetic field ~B = B~k, the Schr¨odinger–Pauli equation is # ) ( " 2  2 ∂ |Ψ(t)i 1 1 1 i¯h = pbx − eBy + pby + eBx + pb2z I + µB Bσz |Ψ(t)i. ∂t 2me 2 2 (3.480) For the stationary states, we can write |Ψ(t)i = e−iEt/¯h |ψi, where |ψi satisfies the time-independent Pauli equation ( " # ) 2  2 1 1 1 2 pbx − eBy + pby + eBx + pbz I + µB Bσz |ψi = E|ψi. (3.481) 2me 2 2 From the earlier discussion on the stationary states of a particle of mass m and charge q moving in a constant magnetic field, without considering spin, we know that the Hamiltonian " # 2  2 1 1 1 2 bNRL = H pbx + qBy + pby − qBx + pbz (3.482) 2m 2 2

119

A Review of Quantum Mechanics

has the energy spectrum En,pz = [n + (1/2)]¯hωc , where ωc = |q|B/m, n n = 0, 1, 2, . .o. ,

and −∞ < pz < ∞. The corresponding eigenfunctions are given by ψn,mφ ,pz (~r) , as in (3.418), where mφ (= 0, 1, 2, . . . ,) represents an angular momentum quantum number with respect to which each energy level is infinitely degenerate. The subscript NRL in this Hamiltonian is to indicate that its energy spectrum corresponds to nonrelativistic Landau levels. Now, it is clear that equation (3.481) can be written as      bNRL + µB B H 0 ψ1 (~r) ψ1 (~r) = E . (3.483) bNRL I − µB B ψ2 (~r) ψ2 (~r) 0 H The general solution of this equation is     D E c1 ψn−1,mφ ,pz (~r) ψ1 (~r) = , ~r ψ = ψ(~r) = ψ2 (~r) c2 ψn,mφ ,pz (~r)

(3.484)

with |c1 |2 + |c2 |2 = 1 and n = 1, 2, . . .. The energy eigenvalue is, with ωc = eB/me ,   p2 h¯ eB 1 En,pz = z + (n − 1) + h¯ ωc + 2me 2 2me   p2z 1 h¯ eB + n+ h¯ ωc − = 2me 2 2me =

p2z + n¯hωc , 2me

(3.485)

with infinite degeneracy due to the angular momentum quantum number mφ = 0, 1, 2, . . .. Note that in the energy spectrum (3.485) all levels are shifted up by h¯ ωc /2 compared to the Landau level spectrum (3.419) obtained ignoring spin. The probability for the particle to be found at ~r with spin up is given by |c1 ψn−1,mφ ,pz (~r)|2 , and the probability for the particle to be found at ~r with spin down is given by |c2 ψn,mφ ,pz (~r)|2 . Note that Z

  d 3 r |c1 ψn−1,mφ ,pz (~r)|2 + |c2 ψn,mφ ,pz (~r)|2 Z

= 3.3.2

d 3 r ψ † (~r)ψ(~r) = |c1 |2 + |c2 |2 = 1.

(3.486)

QUANTUM MECHANICS OF A SYSTEM OF IDENTICAL PARTICLES

When we have a quantum system of several, say N, particles moving in a common external field of force and interacting with each other, the Hamiltonian of the system and the state vectors will be functions of the dynamical variables of all the particles. We take the quantum operators of the corresponding observables belonging to two different particles to be commuting with each other. For example, the position operator of j-th particle (~b r j ) will commute with the momentum operator of the k-th particle ( j) (~pbk = −i¯h~∇~rk ), and any component of the spin operator of the j-th particle (say Sx )

120

Quantum Mechanics of Charged Particle Beam Optics (k)

will commute with any component of the spin operator of the k-th particle (say Sy ). Just to fix the notations, let us consider the Hamiltonian of two electrons moving in the electromagnetic field of vector potential ~A and scalar potential φ given by    1     2 (1) (2) ~pb + e~A(~r1 ,t) − eφ (~r1 ,t) ~ ~ ~ ~ b ,t = , ~r2 , pb2 , S H ~r1 , pb1 , S 1 2me  2 1 ~ ~ pb + eA(~r2 ,t) − eφ (~r2 ,t) I + 2me 2 + µB~σ (1) · ~B(~r1 ,t) + µB~σ (2) · ~B(~r2 ,t), (3.487) where we have ignored completely the electron–electron interaction. The corresponding Schr¨odinger equation will be i¯h

     ∂ |Ψ(t)i b ~r1 ,~pb1 , ~S(1) , ~r2 ,~pb2 , ~S(2) ,t |Ψ(t)i. =H ∂t

(3.488)

The components of spin matrix operators ~σ (1) and ~σ (2) commute with each other. Hence, their matrix representations have to be (1)

σy = σy ⊗ I,

(2)

σx = I ⊗ σy ,

σx = σx ⊗ I, σx = I ⊗ σx ,

(1)

(2)

(1)

σz

= σz ⊗ I,

(2)

σx = I ⊗ σz ,

(3.489)

where I is the 2 × 2 identity matrix and ⊗ denotes the direct product. The direct product of two 2 × 2 matrices, say A and B, is the 4 × 4 matrix defined by     a11 a12 b11 b12 A⊗B = ⊗ b21 b22 a21 a22  = 

a11 B a12 B a21 B a22 B

a11 b11  a11 b21 =  a21 b11 a21 b21



a11 b12 a11 b22 a21 b12 a21 b22

a12 b11 a12 b21 a22 b11 a22 b21

 a12 b12 a12 b22  . a22 b12  a22 b22

(3.490)

In general, the direct product of two n × n matrices, A and B, is the n2 × n2 matrix with the matrix elements given by (A ⊗ B) jk,lm = A jk Blm ,

j, k, l, m = 1, 2, . . . , n.

(3.491)

121

A Review of Quantum Mechanics

The direct product of matrices has the property (A ⊗ B)(C ⊗ D) = AC ⊗ BD.

(3.492)

Hence, if AC = CA and BD = DB, then (A ⊗ B)(C ⊗ D) = (C ⊗ D)(A ⊗ B). From this, it follows that (1) (2) (2) (1) σ j σk = σk σ j , j, k = x, y, z, (3.493) as required. The direct product of two 2-dimensional vectors is the 4-dimensional vector defined by  (1) (2)  V1 V1 !  (2)   (1) (2)  (1) E E V 1 V1 V V (1)  = ⊗ V (2) = ⊗ (3.494) V  1(1) 2(2)  . (1)  V2 V1  (2) V2 V2 (1) (2) V2 V2 Similarly, the direct product of two n-dimensional vectors will be an n2 -dimensional vector with the components  E E (1) (1) (2) ⊗ V (2) = V j Vk , j, k = 1, 2, . . . , n. (3.495) V jk

Now, if A and B are two n × n matrices and V (1) and V (2) are two n-dimensional vectors, then  E E E E (A ⊗ B) V (1) ⊗ V (2) = A V (1) ⊗ B V (2) . (3.496) For more details on the direct product, or the Kronecker product, see, e.g., Arfken, Weber, and Harris [3], and Byron and Fuller [18]. Let us now consider the direct products of two spin- 21 vectors:   1      0  1 1  | ↑i ⊗ | ↑i = ⊗ =  0 , 0 0 0   0      1  1 0  | ↑i ⊗ | ↓i = ⊗ =  0 , 0 1 0   0      0  0 1  | ↓i ⊗ | ↑i = ⊗ =  1 , 1 0 0 

 | ↓i ⊗ | ↓i =

0 1



 ⊗

0 1



 0  0   =  0 . 1

(3.497)

122

Quantum Mechanics of Charged Particle Beam Optics

Then, it follows from (3.496) that h¯ (1) (1) Sz (| ↑i ⊗ | ↑i) = σz (| ↑i ⊗ | ↑i) = 2 h¯ = (| ↑i ⊗ | ↑i) , 2 h¯ (2) (2) Sz (| ↑i ⊗ | ↑i) = σz (| ↑i ⊗ | ↑i) = 2 h¯ = (| ↑i ⊗ | ↑i) , 2

h¯ (σz ⊗ I) (| ↑i ⊗ | ↑i) 2

h¯ (I ⊗ σz ) (| ↑i ⊗ | ↑i) 2 (3.498)

Similarly, one can verify that h¯ (| ↑i ⊗ | ↓i) , 2 h¯ (1) Sz (| ↓i ⊗ | ↑i) = − (| ↓i ⊗ | ↑i) , 2 h¯ (1) Sz (| ↓i ⊗ | ↓i) = − (| ↓i ⊗ | ↓i) , 2 (1)

Sz (| ↑i ⊗ | ↓i) =

h¯ (2) Sz (| ↑i ⊗ | ↓i) = − (| ↑i ⊗ | ↓i) , 2 h¯ (2) Sz (| ↓i ⊗ | ↑i) = (| ↓i ⊗ | ↑i) , 2 h¯ (2) Sz (| ↓i ⊗ | ↓i) = − (| ↓i ⊗ | ↓i) . 2 (3.499)

This means that the spin state of a system of two spin- 21 particles will be represented by |↑i ⊗ |↑i if both the particles have their spins up. If the first particle has its spin up and the second particle has its spin down, then the state is |↑i ⊗ |↓i. If the first particle has its spin down and the second particle has its spin up, then the state is |↓i ⊗ |↑i. If bothEthe particles state is |↓i ⊗ spins E have their E down, E then the E E |↓i. We (1) (2) (1) (2) (1) (2) shall write V ⊗ V simply as V , as if V and V are one V dimensional, to be understood as a direct product from the context. It is now clear that the Hamiltonian in (3.487) is a 4-dimensional matrix with operators as elements. Hence, the state vector |Ψ(t)i in the Schr¨odinger equation (3.488) should be a 4-dimensional vector. Let   |Ψ11 (t)i  |Ψ12 (t)i   |Ψ(t)i =  (3.500)  |Ψ21 (t)i  . |Ψ22 (t)i From (3.497), we have |Ψ(t)i = |Ψ11 (t)i | ↑i| ↑i + |Ψ12 (t)i | ↑i| ↓i + |Ψ21 (t)i | ↓i| ↑i + |Ψ22 (t)i | ↓i| ↓i.

(3.501)

Thus, |h~r1 ,~r2 |Ψ11 (t) i|2 = |Ψ11 (~r1 ,~r2 ,t)|2 gives the probability for the two particles to be found at positions~r1 and~r2 with both their spins up. Similarly, the other components of |Ψ(t)i can be interpreted. It is straightforward to extend this formalism to a system of several particles. When the spin of the particle is ignored, |Ψ(t)i is treated only as a single-component state vector.

123

A Review of Quantum Mechanics

When we have a system of identical particles, the Hamiltonian of the system should not change under the interchange of position coordinates, and other   variables b like spin, of any pair of particles. Let ξ j denote the set of operators ~b r j ,~pbj , ~S( j) belonging to the j-th particle. Then, the invariance of the Hamiltonian under the interchange of any pair of particles means     b ξb1 , ξb2 , . . . , ξbj , . . . , ξbk , . . . , ξbN ,t = H b ξb1 , ξb2 , . . . , ξbk , . . . , ξbj , . . . , ξbN ,t Pjk H   b ξb1 , ξb2 , . . . , ξbj , . . . , ξbk , . . . , ξbN ,t , =H (3.502) where Pjk is the exchange operator that interchanges the particles j and k. Let |1, 2, . . . , Ni be an eigenstate of a time-independent Hamiltonian   of a system  b ξb1 , ξb2 , . . . , ξbN . Since the Hamiltonian H b ξb1 , ξb2 , . . . , ξbN of N particles, say H is invariant under the interchange of any pair of particles, it is found that Pjk |1, 2, . . . , j, . . . , k, . . . , Ni = |1, 2, . . . , k, . . . , j, . . . , Ni differs only by a multiplicative factor ±1 from |1, 2, . . . , j, . . . , k, . . . , Ni. This multiplicative factor is ±1 since 2 = I, the identity operator. Since the particles are indistinguishable, the multiplicaPjk tive factor, +1 or −1, cannot depend on which pair of particles are interchanged. In other words, the eigenstate |1, 2, . . . , Ni must be either totally symmetric or totally antisymmetric under the interchange of any pair of particles. This should be true for all eigenstates of any time-independent Hamiltonian of the system. Since any state vector |Ψ(t)i of the system evolving under a time-dependent or time-independent Hamiltonian can be expressed as a linear combination of eigenstates of a timeindependent Hamiltonian of the system, the state vector of the system has to be either totally symmetric or totally antisymmetric. It is found that for integer spin particles, bosons, the state vector is totally symmetric and for half integer spin particles, fermions, the state vector is totally antisymmetric. Let us look at an example. Consider a system of N identical particles with the time-independent Hamiltonian   b ξb1 , ξb2 , . . . , ξbN = H

N

  b H ∑ ξbj ,

(3.503)

j=1

n   o b ξbj j = 1, 2, . . . , N are all identical, where the single-particle Hamiltonians H except for the labeling of the variables. The N-particle Hamiltonian H is obviously invariant under any permutation of the particles. The form of H shows that the system is such that each of the N particles may be in a common external force field and the interparticle interactions have been ignored. For example, take the electronic Hamiltonian of an Z-electron atom,   ~pb2 Z Z 2 e2 b A = ∑  j − Ze  + ∑ (3.504) H 2 , 4πε0 |~r j | j=1 2me j zr ) ≈ 0. We are concerned with a monoenergetic paraxial beam with design momentum p0 such that |~p| = p0 ≈ pz for all its constituent particles. For a paraxial beam propagating through the round magnetic lens, along the optic axis, the vector potential ~A can be taken as   ~A ≈ ~A0 = − 1 B(z)y, 1 B(z)x, 0 , (4.22) 2 2

179

Classical Charged Particle Beam Optics

corresponding to the field   ~B ≈ ~B0 = − 1 B0 (z)x, − 1 B0 (z)y, B(z) . 2 2

(4.23)

In the paraxial approximation, only the lowest order terms in ~r⊥ are considered to contribute to the effective field felt by the particles moving close to the optic axis. Further, it entails approximating the z-evolution, or optical, Hamiltonian by dropping from it the terms of order higher than second in |~p⊥ | /p0 in view of the condition |~p⊥ | /p0  1. Thus, from (4.2) and (4.22), we can write down the classical beam optical Hamiltonian of the system as q Ho = − p20 − π⊥2 − qAz π⊥2 2p0 i 1 h = −p0 + (px − qAx )2 + (py − qAy )2 2p0 " 2  2 # 1 1 1 = −p0 + px + qB(z)y + py − qB(z)x 2p0 2 2   1 1 2 2 2 2 = −p0 + p⊥ + q B(z) r⊥ − qB(z) (xpy − ypx ) 2p0 4   1 1 2 2 2 2 = −p0 + p⊥ + q B(z) r⊥ − qB(z)Lz , 2p0 4

≈ −p0 +

(4.24)

where  B(z)

6= 0 =0

in the lens region (z` ≤ z ≤ zr ) outside the lens region (z < z` , z > zr ) ,

(4.25)

and Lz is the z-component of the orbital angular momentum of the particle. With the notation qB(z) , (4.26) α(z) = 2p0 we shall write Ho = −p0 +

1 2 1 2 p + p0 α 2 (z)r⊥ − α(z)Lz . 2p0 ⊥ 2

(4.27)

Hamilton’s equations of motion are 

  x(z) 0 α(z) 1 0   y(z) d  −α(z) 0 0 1  p (z)    x = 2 (z) −α 0 0 α(z) dz  p0  py (z) 0 −α 2 (z) −α(z) 0 p0

  x(z)  y(z)     .    pxp(z)   0 py (z) p0

(4.28)

180

Quantum Mechanics of Charged Particle Beam Optics

It is seen that in the regions outside the lens, where α(z) = 0, this equation of motion becomes (4.6) with p0 ≈ pz for a paraxial beam. Let us write (4.28) as 

   x(z) x(z)  y(z)   d   y(z)     px (z)  = [µ(z) ⊗ I + I ⊗ ρ(z)]  px (z)  , dz  p0   p0  py (z) p0

(4.29)

py (z) p0

where  µ(z) =

0 1 −α 2 (z) 0

 ,

(4.30)

.

(4.31)

and  ρ(z) = α(z)

0 −1

1 0



We shall now integrate (4.29). Note that µ(z) ⊗ I and I ⊗ ρ(z) commute with each other. Let     x(z) x (zi )  y(z)   y (zi )   p (z)    (4.32)  x  = M (z, zi ) R (z, zi )  px (zi )  , for any z ≥ zi ,  p0   p0  py (z) p0

py (zi ) p0

where dM (z, zi ) = (µ(z) ⊗ I) M (z, zi ) , dz dR (z, zi ) = (I × ρ(z)) R (z, zi ) , dz

M (zi , zi ) = I, R (zi , zi ) = I.

(4.33) (4.34)

It is clear that we can write M (z, zi ) = M (z, zi ) ⊗ I,

dM (z, zi ) dM (z, zi ) = ⊗ I, dz dz

(4.35)

with dM (z, zi ) = µ(z)M (z, zi ) , dz

(4.36)

and R (z, zi ) = I ⊗ R (z, zi ) ,

dR (z, zi ) dR (z, zi ) =I⊗ , dz dz

(4.37)

with dR (z, zi ) = ρ(z)R (z, zi ) . dz

(4.38)

181

Classical Charged Particle Beam Optics

Note that M (z, zi ) and R (z, zi ) commute with each other, dM (z, zi ) /dz commutes with R (z, zi ), and dR (z, zi ) /dz commutes with M (z, zi ). Then,     x(z) x (zi )    i)  dM (z, zi ) dR (z, zi )  d   y(z)   yp (z  R (z, zi ) + M (z, zi )  px (z)  =  x (zi )  dz  p0  dz dz  p0  py (z) p0

py (zi ) p0



x (zi )  y (zi )  = [µ(z) ⊗ I + I ⊗ ρ(z)]M (z, zi ) R (z, zi )  px (zi )  p0 py (zi ) p0

    



 x(z)  y(z)    = [µ(z) ⊗ I + I ⊗ ρ(z)]  px (z)  ,  p0 

(4.39)

py (z) p0

as required by (4.29). The equation for R (z, zi ), (4.34), can be readily integrated by ordinary exponentiation since [ρ (z0 ) , ρ (z00 )] = 0 for any z0 and z00 . Thus, we have R (z, zi ) = e

Rz

zi

dz ρ(z)

    0 1 = exp I ⊗ θ (z, zi ) −1 0   cos θ (z, zi ) sin θ (z, zi ) =I⊗ − sin θ (z, zi ) cos θ (z, zi ) = I ⊗ R (z, zi ) , with

(4.40)

Z z

θ (z, zi ) =

dz α(z) = zi

q 2p0

Z z

dz B(z).

(4.41)

zi

Let us now introduce an XY z-coordinate system with its X and Y axes rotating along the z-axis at the rate dθ (z)/dz = α(z) = qB(z)/2p0 , such that we can write     x(z) X(z) = R (z, zi ) . (4.42) y(z) Y (z) or 

X(z) Y (z)



−1



x(z) y(z)



= R (z, zi )    cos θ (z, zi ) − sin θ (z, zi ) x(z) = . sin θ (z, zi ) cos θ (z, zi ) y(z)

(4.43)

182

Quantum Mechanics of Charged Particle Beam Optics

Then, we have 

px (z) py (z)



 = R (z, zi )



PX (z) PY (z)

,

(4.44)

or 

PX (z) PY (z)





cos θ (z, zi ) − sin θ (z, zi )

=





 sin θ (z, zi )

cos θ (z, zi )



px (z) py (z)

,

(4.45)

where PX and PY are the components of momentum in the rotating coordinate system. Note that in the vertical plane at z = zi , where the particle enters the system, XY z coordinate system coincides with the xyz coordinate system. Thus, we can write (4.32) as, for any z ≥ zi ,   X (zi ) X(z)  Y (zi )  Y (z)     R (z, zi )  PX (z)  = M (z, zi ) R (z, zi )  PX (zi )  p0  p0 





PY (z) p0

   

PY (zi ) p0



 X (zi )  Y (zi )    = R (z, zi ) M (z, zi )  PX (zi )  .  p0 

(4.46)

PY (zi ) p0

In other words, the position and momentum of the particle with reference to the rotating coordinate system evolve along the optic axis according to 

  X(z) X (zi )  Y (z)   Y (zi )     PX (z)  = M (z, zi )  PX (zi )  p0   p0 PY (z) p0

PY (zi ) p0

   , 

for any z ≥ zi .

(4.47)

From (4.30) and (4.33), the corresponding equations of motion follow: ! !  !  ~ ~R⊥ (z) ~R⊥ (z) R⊥ (z) d 0 1 = µ(z) = , ~P⊥ (z) ~P⊥ (z) ~P⊥ (z) −α 2 (z) 0 dz p p p 0

0

(4.48)

0

with ~R⊥ (z) =



X(z) Y (z)

 ,

~P⊥ (z) =



PX (z) PY (z)

From this, it follows that !   ~R⊥ (z) d2 −α 2 (z) 0 = ~ P⊥ (z) −2α(z)α 0 (z) −α 2 (z) dz2 p 0

 .

~R⊥ (z) ~P⊥ (z) p0

(4.49)

! ,

(4.50)

183

Classical Charged Particle Beam Optics

or ~R00⊥ (z) + α 2 (z)~R⊥ = 0, 1 1 ~ 00 P⊥ (z) + 2α(z)α 0 (z)~R⊥ (z) + α 2 (z)~P⊥ (z) = 0. p0 p0

(4.51) (4.52)

The first of the above equations (4.51) is the paraxial equation of motion in the rotating coordinate system. The second equation is not independent of the first equation since it is just a consequence of the relation ~P⊥ (z)/p0 = d ~R⊥ (z)/dz. Equation (4.48) cannot be integrated by ordinary exponentiation since µ (z0 ) and µ (z00 ) do not commute when z0 6= z00 if α 2 (z0 ) 6= α 2 (z00 ). So, to integrate (4.48) we have to follow the same method used to integrate the time-dependent Schr¨odinger b (t,t0 ) given by the series expression in (3.449). equation (3.439) as in (3.440) with U b Thus, replacing t by z, and −iH(t)/¯ h by µ(z), we get ~R⊥ (z)

!

~R⊥ (zi )



 g (z, zi ) h (z, zi ) = g0 (z, zi ) h0 (z, zi ) ! ~R⊥ (zi ) = M (z, zi ) , ~P⊥ (zi )

~P⊥ (z) p0

!

~P⊥ (zi ) p0

(4.53)

p0

where Z z

M (z, zi ) = I + Z

zi z

+ zi

Z z

dz1 µ (z1 ) + Z z3

dz3

zi

zi

Z z2

dz2

zi

Z z2

dz2

zi

dz1 µ (z2 ) µ (z1 )

dz1 µ (z3 ) µ (z2 ) µ (z1 )

+ ··· .

(4.54)

Note that g (z, zi ) and h (z, zi ) are two linearly independent solutions of the paraxial equation of motion (4.51) corresponding to the initial conditions ~R⊥ (zi ) ~P⊥ (zi ) p0

!

 =

1 0

~R⊥ (zi )

 ,

!

~P⊥ (zi ) p0

 =

0 1

 .

(4.55)

It can be directly verified that dM (z, zi ) = µ(z)M (z, zi ) , dz

(4.56)

and hence d dz

~R⊥ (z) ~P⊥ (z) p0

!

dM (z, zi ) = dz

~R⊥ (zi ) ~P⊥ (zi ) p0

! = µ(z)

~R⊥ (zi ) ~P⊥ (zi ) p0

! ,

(4.57)

184

Quantum Mechanics of Charged Particle Beam Optics

as required in (4.48). Explicit expressions for the elements of M can be written down from (4.54) as follows: g (z, zi ) = 1 −

Z z zi

Z z2

dz2

Z z

Z z4

+

dz4

zi

zi

h (z, zi ) = (z − zi ) − Z z

dz4

zi

Z z zi

h0 (z, zi ) = 1 −

zi

Z z zi

Note that

dz3 α 2 (z3 )

zi

zi

Z z2

dz2

zi

Z z3 zi

Z z2

dz2

zi

dz1 α 2 (z1 ) − · · · ,

dz1 α 2 (z1 ) (z1 − zi )

dz3 α 2 (z3 )

dz1 α 2 (z1 ) +

Z z

+

dz1 α 2 (z1 )

Z z

Z z4

+ g0 (z, zi ) = −

zi

Z z zi

Z z3 zi

Z z2

dz2

dz3 α 2 (z3 )

zi

dz1 α 2 (z1 ) (z1 − zi ) − · · · ,

Z z3 zi

Z z2

dz2

zi

dz1 α 2 (z1 ) − · · · ,

dz1 α 2 (z1 ) (z1 − zi )

dz3 α 2 (z3 )

Z z3 zi

Z z2

dz2

dg (z, zi ) = g0 (z, zi ) , dz

zi

dz1 α 2 (z1 ) (z1 − zi ) − · · · .

(4.58)

dh (z, zi ) = h0 (z, zi ) , dz

(4.59)

consistent with the relation d ~R⊥ (z)/dz = ~P⊥ (z)/p0 . From the differential equation, (4.56) for M we know that we can write n M (z, zi ) = lim lim e∆zµ(zi +N∆z) e∆zµ(zi +(N−1)∆z) e∆zµ(zi +(N−2)∆z) · · · N→∞ ∆z→0 o (4.60) · · · e∆zµ(zi +2∆z) e∆zµ(zi +∆z) , with N∆z = (z − zi ). Since the trace of µ(z) is zero, each of the factors in the above product expression, of the type e∆zµ(zi + j∆z) , has unit determinant. Thus, the matrix M has an unit determinant, i.e., g (z, zi ) h0 (z, zi ) − h (z, zi ) g0 (z, zi ) = 1,

(4.61)

for any (z, zi ). In a simplified picture of an electron microscope, let us consider the monoenergetic paraxial beam comprising of electrons scattered elastically from the specimen (the object to be imaged) being illuminated. The beam transmitted by the specimen carries the information about the structure of the specimen. This beam going through the magnetic lens gets magnified and is recorded at the image plane. Let us take the

185

Classical Charged Particle Beam Optics

positions of the object plane and the image plane in the z-axis, the optic axis, as zob j and zimg , respectively. As already mentioned, we shall take the practical boundaries of the lens to be at z` and zr . The magnetic field of the lens is practically confined to the region between z` and zr and zob j < z` < zr < zimg . The region between zob j and z` and the region between zr and zimg are free spaces. Thus, the electron beam transmitted by the specimen (object) travels first through the free space between zob j and z` , then the lens between z` and zr , and finally the free space between zr and zimg where it is recorded (image). Hence, the z-evolution matrix, or the transfer matrix,
M zimg , zob j can be written as

M zimg , zob j = MD (zimg , zr ) ML (zr , z` ) MD z` , zob j , (4.62) where subscript D indicates the drift region and L indicates the lens region. Since α 2 (z) = 0 in the free space regions, or drift regions, we get from (4.58)
   
1 (zimg − zr ) 1 z` − zob j MD (zimg , zr ) = , MD z` , zob j = . 0 1 0 1 (4.63) For the lens region, we have from (4.58)     g (zr , z` ) h (zr , z` ) gL hL ML (zr , z` ) = = . (4.64) g0 (zr , z` ) h0 (zr , z` ) g0L h0L Thus, we get




M zimg , zob j =



1 (zimg − zr ) 0 1



gL g0L

hL h0L



1 0

z` − zob j 1





gL z` − zob j + hL
+g0L (zimg − zr ) z` − zob j + h0L (zimg − zr )]

 gL + g0 (zimg − zr ) L  =  
g0L g0L z` − zob j + h0L

  g zimg , zob j
h zimg , zob j
= . g0 zimg , zob j h0 zimg , zob j

      (4.65)


Note that MD z` , zob j and MD (zimg , zr ) have unit determinant. From the general theory, we know that ML (zr , z` ) should also have unit determinant, i.e., gL h0L − hL g0L = 1.

(4.66)
Since zimg is the position of the image plane h zimg , zob j , the 12-element of the
matrix M zimg , zob j , should vanish such that
~R⊥ (zimg ) ∝ ~R⊥ zob j .

(4.67)

186

Quantum Mechanics of Charged Particle Beam Optics

This means that we should have

gL z` − zob j + hL + g0L (zimg − zr ) z` − zob j + h0L (zimg − zr ) = 0.

(4.68)

Now, let
h0 − 1 u = z` − zob j + L 0 , gL

v = (zimg−zr ) +

gL − 1 , g0L

(4.69)

and f =−

1 . g0L

(4.70)

We are to interpret u as the object distance, v as the image distance, and f as the focal length of the lens. Using straightforward algebra, and the relation (4.66), one can verify that the familiar lens equation, 1 1 1 + = , u v f

(4.71)

implies (4.68). From (4.69) it is seen that the principal planes from which u and v are to be measured in the case of a thick lens are situated at
h0L − 1 = z` + f 1 − h0L , g0L gL − 1 = zr − = zr − f (1 − gL ) , g0L

zPob j = z` + zPimg

(4.72)

such that u = zPob j − zob j ,

v = zimg − zPimg .

(4.73)

The explicit expression for the focal length f follows from (4.70) and (4.58): 1 = f =

Z zr

dz α 2 (z) −

z` q2

Z zr

4p20

z`

Z zr z`

dz2 α 2 (z2 )

q4 dz B (z) − 16p40 2

Z zr z`

Z z2 z`

Z z1

dz1 2

dz α 2 (z) + · · ·

z`

dz2 B (z2 )

Z z2 z`

Z z1

dz1

dz B2 (z) + · · · . (4.74)

z`

To understand the behavior of the lens, let us consider the idealized model in which B(z) = B is a constant in the lens region and zero outside. Then, we get   1 qB qBw = sin , (4.75) f 2p0 2p0 where w = (zr − z` ) is the width, or thickness, of the lens. This shows that the focal length is always positive to start with and is then periodic with respect to the variation of the field strength. Thus, the round magnetic lens is convergent up to a certain strength of the field. Round magnetic lenses commonly used in electron microscopy

187

Classical Charged Particle Beam Optics

are always convergent. Note that zz`r dz α 2 (z) has the dimension of reciprocal length. The round magnetic lens is considered weak when R

Z zr

dz α 2 (z) 

z`

1 . w

(4.76)

For such a weak lens, the expression for the focal length (4.74) can be approximated as 1 ≈ f

Z zr

dz α 2 (z) =

z`

q2 4p20

Z zr

dz B2 (z) =

z`

q2 4p20

Z ∞

dz B2 (z).

(4.77)

−∞

A weak lens is said to be thin since in this case 1/ f  1/w, or f  w. The formula (4.77), known as the Busch formula, for the focal length of a thin axially symmetric magnetic lens was derived in 1927 by Busch [17]. The paraxial z-evolution matrix from the object plane to the image plane (4.65) becomes, taking into account (4.68), zimg −zr f − 1f

gL −

M(zimg , zob j ) =  =

 =

h0L −

gL − v+ f (gf L −1) − 1f 1 − vf − 1f

=

M − 1f

!

0

0 1 − uf  0 , 1

z` −zob j f

0



 u+ f (h0L −1) h0L − f !   − uv 0 = − 1f − uv (4.78)

M

where M denotes the magnification. In our notation, both u and v are positive, and hence M is negative, showing the inverted nature of the image, as should be in the case of imaging by a convergent lens. For a thin lens with f  w, we can take w ≈ 0, and the two principal planes collapse into a single plane at the center of the lens, i.e., we can take zP = (z` + zr ) /2. Then, from (4.58), it is clear that for the thin lens we can make the approximation  MT L (zr , z` ) ≈

1 − 1f

0 1

 ,

(4.79)

188

Quantum Mechanics of Charged Particle Beam Optics

where the subscript T L stands for thin lens. Thus, corresponding to (4.65), we have for the thin lens
   
1 0 1 (zimg − zP ) 1 zP − zob j M zimg , zob j ≈ − 1f 1 0 1 0 1 

  1 − 1 (zimg − zP ) f  =   − 1f


zP − zob j
− 1f (zimg − zP ) zP − zob j + (zimg − zP )] 1 − 1f zP − zob j

   .  


(4.80)




Since zP − zob j = u and (zimg − zP ) = v, and 1/u + 1/v = 1/ f , we get !
1 − vf u − uvf + v M zimg , zob j = − 1f 1 − uf  v    −u 0 M 0 = = , 1 − 1f − uv − 1f M

(4.81)

where M = −v/u is the magnification. So far, we have been describing the behavior of the lens in the rotated XY zcoordinate system. Now, if we return to the original xyz-coordinate system, we would have, as seen from (4.32), (4.35), and (4.37), !
!

~r⊥ zob j ~r⊥ (zimg )
. (4.82) = M zimg , zob j ⊗ R zimg , zob j ~p⊥ ~p⊥ p0 (zimg ) p0 zob j Since the rotation of the image is the effect of only the lens magnetic field  
cos θL sin θL R zimg , zob j = R (z` , zr ) = , − sin θL cos θL where

(4.83)

zr q dz B(z). (4.84) 2p0 z` From (4.80) and (4.82), it is clear that apart from rotation and drifts through fieldfree regions in the front and back of the lens,the effect of a thin convergent lens

Z

θL = θ (z` , zr ) =

is essentially described by the transfer matrix

1 −1/ f

0 1

. This can also be seen

simply as follows. If a particle of a paraxial beam enters a thin lens in the transverse xy-plane with the coordinates (x, 0) and momenta (0, 0), i.e., in the xz-plane and parallel to the optic (z) axis, then we have       1 0 0 1 f x = . (4.85) − xf − 1f 1 0 1 0

Classical Charged Particle Beam Optics

189

This means that all the particles of the paraxial beam hitting the lens parallel to the optic axis in the xz-plane meet the optic axis at a distance f from the lens independent of the x coordinate at which it enters the lens. This implies that the lens is a convergent lens with the focal length f . If the lens is a thin divergent lens of focal length − f , then correspondingly we would have       1 0 0 x 1 −f , (4.86) = x 1 1 0 0 1 f f implying that the particles appear virtually to meet behind the lens at a distance f . Thus, the effectof a thin divergent lens is essentially described by the transfer matrix  A is C

1 1/ f

0 1

. Note that, in general, if the transfer matrix of a lens system  B , then it will be focusing or defocusing depending on whether C is

D

negative or positive, respectively. This is so because if C is negative, then px /p0 decreases proportional to the x coordinate of the beam particle driving it towards the optic axis, and if C is positive, then px /p0 increases proportional to the x coordinate of the beam particle driving it away from the optic axis. In this discussion, we have considered the beam to be paraxial and approximated the optical Hamiltonian by keeping only terms up to second order in~r⊥ and ~p⊥ . This paraxial, or Gaussian, approximation has resulted in perfect imaging in which there is a one-to-one relationship between the object points and the image points. When the beam deviates from the ideal paraxial condition, as is usually the case in practice, we have to go beyond the paraxial approximation and retain in the Hamiltonian terms of order higher than second in ~r⊥ and ~p⊥ . The resulting theory accounts for the image aberrations. We shall not deal with the classical theory of aberrations here. We shall treat aberrations in the quantum theory of electron optical imaging, and the classical theory of aberrations will emerge in the classical limit of the quantum theory. The discussion so far is based on Hamilton’s equations of motion. Let us now understand the results based on the Lie transfer operator method. Extensive techniques for applying the Lie transfer operator methods to light optics, charged particle beam optics, and accelerator optics have been developed by Dragt et al. (see Dragt [34], Dragt and Forest [35], Dragt et al. [36], Forest, Berz, and Irwin [54], Rangarajan, Dragt, and Neri [156], Forest and Hirata [55], Forest [56], Radli˘cka [154], and references therein; see also Berz [11], Mondragon and Wolf [135], Wolf [191], Rangarajan and Sachidanand [157], Lakshminarayanan, Sridhar, and Jagannathan [124], and Wolski [192]). In general, from Hamilton’s equations, we have ! ! ~r⊥ (z) ~r⊥ (z) d =: −Ho (z) :z . (4.87) ~p⊥ (z) ~p⊥ (z) dz p0 p0 We shall now follow the treatment of the Schr¨odinger equation with time-dependent Hamiltonian in quantum mechanics. Let us write the solution of the above equation as

190

Quantum Mechanics of Charged Particle Beam Optics

~r⊥ (z) ~p⊥ (z) p0

! = T (z, zi )

~r⊥ (zi ) ~p⊥ (zi ) p0

! ,

z ≥ zi ,

~r⊥ (z)

!!

(4.88)

where it is understood that T (z, zi )

~r⊥ (zi )

! T (z, zi )

.

(4.89)

T (zi , zi ) = 1.

(4.90)

 Zz ∂ T (z, zi ) = dz dz1 : −Ho (z1 ) :z T (z1 , zi ) , ∂z zi

(4.91)

=

~p⊥ (zi ) p0

~p⊥ (z) p0

z=zi

Then, we should have ∂ T (z, zi ) =: −Ho (z) :z T (z, zi ) , ∂z Writing Z z zi



and integrating, we get T (z, zi )|zzi = T (z, zi ) − 1 =

Z z zi

dz1 : −Ho (z1 ) :z T (z1 , zi ) .

(4.92)

This leads to the formal solution T (z, zi ) = 1 −

Z z zi

dz1 : −Ho (z1 ) :z T (z1 , zi ) .

(4.93)

Iterating this formal solution, we get T (z, zi ) = 1 +

Z z zi

dz1 : −Ho (z1 ) :z

Z z

+ z

Z z2

dz2

Z iz

+ zi

z

dz1 : −Ho (z2 ) :z : −Ho (z1 ) :z

Z iz3

dz3

+··· .

zi

Z z2

dz2

zi

dz1 : −Ho (z3 ) :z : −Ho (z2 ) :z : −Ho (z1 ) :z (4.94)

Introducing an z-ordering operator, analogous to the time-ordering operator (3.450) in quantum mechanics, we can write Z z n  ∞ 1 dz : −Ho (z) :z T (z, zi ) = 1 + ∑ P zi n=1 n! "  n # Z z ∞ 1 =P ∑ dz : −Ho (z) :z zi n=0 n!  Rz  dz :−Ho (z):z = P e zi , (4.95)

191

Classical Charged Particle Beam Optics

which may be called the z-ordered, or path ordered, exponential of : −Ho (z) :z . In exact analogy with (3.455 and 3.456) T (z, zi ) can also be written in the Magnus form as T (z, zi ) = eT (z,zi ) ,

(4.96)

where T (z, zi ) is an infinite series with the first few terms given by Z z

T (z, zi ) =

zi

dz : −Ho :z (z)

z2 1 z dz2 dz1 [: −Ho :z (z2 ) , : −Ho :z (z1 )] 2 zi zi Z Z z3 Z z2 1 z dz3 dz2 + dz1 6 zi zi zi {[[: −Ho :z (z3 ) , : −Ho :z (z2 )] , : −Ho :z (z1 )] + [[: −Ho :z (z1 ) , : −Ho :z (z2 )] , : −Ho :z (z3 )]} +··· ,

(4.97)

[: −Ho :z (z1 ) , : −Ho :z (z2 )] = : −Ho :z (z1 ) : −Ho :z (z2 ) − : −Ho :z (z2 ) : −Ho :z (z1 ) = : {−Ho (z1 ) , −Ho (z2 )}z :,

(4.98)

Z

Z

+

with

as follows from (2.57). When Ho is z-independent, the commutators in (4.97) vanish making T (z, zi ) an ordinary exponential: Rz

T (z, zi ) = e

zi

dz :−Ho (z):z

.

(4.99)

Note that we can write  Rz  dz :−Ho (z):z T (z, zi ) = P e zi = eT (z,zi ) .

(4.100)

In general, for any observable O, we would have O(z) = eT (z,zi ) O (zi ) .

(4.101)

Analogous to the semigroup property of the time-evolution operator in quantum mechanics (3.167), the Lie transfer operator T (z, zi ) has the semigroup property,

T (z, zi ) = T z, z0 T z0 , zi ,

(4.102)

192

Quantum Mechanics of Charged Particle Beam Optics

which follows from the observation ! ~r⊥ (z) ~p⊥ (z) p0

~r⊥ (z)

!
= T z, z0

~p⊥ (z) p0

~r⊥ (z0 ) ~p⊥ (z0 )

= T (z, zi )

~r⊥ (zi )

! ,

~p⊥ (zi ) p0

~r⊥ (z0 ) ~p⊥ (z0 )

! ,

p0

! 0

= T z , zi




p0

~r⊥ (zi )

!

~p⊥ (zi ) p0

,

(4.103)

In general, we have, for z > zn > zn−1 > · · · > z2 > z1 > zi , T (z, zi ) = T (z, zn ) T (zn , zn−1 ) . . . T (z2 , z1 ) T (z1 , zi ) ,

(4.104)

For the round magnetic lens, we have Ho = Ho,L + Ho,R ,

(4.105)

1 2 1 2 p + p0 α 2 (z)r⊥ , 2p0 ⊥ 2 Ho,R = −α(z)Lz .

(4.106)

where Ho,L = −p0 +

Let us now note the following:  2 r⊥ , Lz z = 0, where for any two time-independent f and g     ∂ f ∂g ∂ f ∂g ∂ f ∂g ∂ f ∂g + . − − { f , g}z = ∂ x ∂ px ∂ px ∂ x ∂ y ∂ py ∂ py ∂ y

(4.107)

(4.108)

Proof of (4.107) is as follows:  2  r⊥ , Lz z = x2 + y2 , xpy − ypx z     = x2 , xpy z − x2 , ypx z + y2 , xpy z − y2 , ypx z   = −y x2 , px z + x y2 , py z = −2yx + 2xy = 0.

(4.109)

Similarly we have  as follows: 

p2⊥ , Lz

z

p2⊥ , Lz

z

= 0,

(4.110)

 = p2x + p2y , xpy − ypx z     = p2x , xpy z − p2x , ypx z + p2y , xpy z − p2y , ypx z   = p2x , x z py − p2y , y z px = −2px py + 2py px = 0.

(4.111)

193

Classical Charged Particle Beam Optics

In view of (4.107) and (4.110), we have {Ho,L , Ho,R }z = 0.

(4.112)

Then, the commutator terms between Ho,L and Ho,R in (4.97) vanish, leading to the result that for the round magnetic lens T (z, zi ) = M (z, zi ) R (z, zi )

(4.113)

with  Rz  dz :−Ho,L (z):z M (z, zi ) = P e zi ,

 Rz  dz :−Ho,R (z):z R (z, zi ) = P e zi .

(4.114)

It is easy to see that : −Ho,R (z0 ) :z commutes with : −Ho,R (z00 ) :z for any two z0 and z00 , and hence we can write Rz

R (z, zi ) = e

zi

dz :−Ho,R (z):z

=e

q 2p0

R

z zi

 dzB(z) :Lz :z

= eθ (z,zi ):Lz :z .

(4.115)

Note that 

Ho,L z0 , Ho,L z00 z    
2
2 1 2 1 1 2 1 = p⊥ + p0 α 2 z0 r⊥ p⊥ + p0 α 2 z00 r⊥ , 2p0 2 2p0 2 z


2 0 2 00 = α z − α z (~r⊥ ·~p⊥ ) , (4.116) showing that : −Ho,L (z0 ) :z does not commute with : −Ho,L (z00 ) :z if α 2 (z0 ) 6= α 2 (z00 ). Thus, the expression for M (z, zi ) cannot be simplified further. Let us first consider the effect of R (z, zi ) on~r⊥ and ~pp⊥0 . To this end, we proceed as follows:   θ2 θ3 : Lz :2z + : Lz :3z + · · · x eθ :Lz :z x = 1 + θ : Lz :z + 2! 3! 2 o θ  θ3 n  = x + θ {Lz , x}z + Lz , {Lz , x}z z + Lz , Lz , {Lz , x}z z · · · 2! 3! z     4 3 5 2 θ θ θ θ + −··· +y θ − + −··· = x 1− 2! 4! 3! 5! = cos θ x + sin θ y.

(4.117)

Similarly, it is seen that eθ :Lz :z y = − sin θ x + cos θ y, py px px eθ :Lz :z = cos θ + sin θ , p0 p0 p0 py px θ :Lz :z py e = − sin θ + cos θ . p0 p0 p0

(4.118)

194

Quantum Mechanics of Charged Particle Beam Optics

This shows that we can write   x (zi )  y (zi )    R (z, zi )  px (zi )   p0  py (zi ) p0

  x (z )  i cos θ (z, zi ) sin θ (z, zi ) 0 0 y (z )   − sin θ (z, zi ) cos θ (z, zi )  i 0 0   px (zi )  =   0 0 cos θ (z, zi ) sin θ (z, zi )   p0  py (zi ) 0 0 − sin θ (z, zi ) cos θ (z, zi ) p0   x (zi )  y (zi )    = [I ⊗ R (z, zi )]  px (zi )  . (4.119)  p0  

py (zi ) p0

We can derive this result in the following way also. Observe that       {Lz , x}z x x  {Lz , y}z  o   y   y   n px  = L  px  , px  =  : Lz :z  L ,  z p0  p   p    n oz  p0y p0y py Lz , p0 p0 p0

(4.120)

z

where



0 1 0  −1 0 0 L=  0 0 0 0 0 −1 It can be verified directly that      x x x      y y 2 3 y   : L :2z   ppx  = L  ppx  , : L :z  ppx p0y p0

p0y p0

p0y p0

 0 0  . 1  0 

(4.121)



 x   y   = L3  px  , . . . .   p  p0y p0

Hence we can write 

  x (zi ) x (zi )  y (zi )   y (zi )    R (z, zi )  px (zi )  = eθ (z,zi ):Lz :z  px (zi )  p0   p0 py (zi ) p0

py (zi ) p0

     

 x (zi )  y (zi )  ∞ 1   = ∑ θ (z, zi )n : Lz :nz  px (zi )   p0  n=0 n! py (zi ) p0

(4.122)

195

Classical Charged Particle Beam Optics



 x (zi )  y (zi )  ∞ 1   = ∑ θ (z, zi )n Ln  px (zi )  n!   p 0 n=0 py (zi ) p0



 x (zi )  y (zi )    = [I ⊗ R (z, zi )]  px (zi )  ,  p0 

(4.123)

py (zi ) p0

as found already in (4.119). To find the effect of M (z, zi ) on (x, y, px /p0 , py /p0 ), let us proceed as follows. Observe that : −Ho,L (z) :z x = {−Ho,L (z) , x}z     1 2 1 2 = − p⊥ + p0 α 2 (z)r⊥ ,x 2p0 2 z   px p2x ,x = , = − 2p0 p0 z

(4.124)

and   px px : −Ho,L (z) :z = −Ho,L (z) , p0 p0 z     1 2 1 px 2 2 = − p + p0 α (z)r⊥ , 2p0 ⊥ 2 p0 z   p 1 x = −α 2 (z)x. = − p0 α 2 (z)x2 , 2 p0 z

(4.125)

Similar equations follow for the Poisson brackets of −Ho,L (z) with y and py /p0 . Thus, we get      x x 0 0 1 0  y    y  0 0 0 1    px    : −Ho,L (z) :z   ppx  =  −α 2 (z)  0 0 0  p0  p0y py 2 0 −α (z) 0 0 p0 p0   x     y  0 1  = ⊗I   ppx  −α 2 (z 0 0 py p0



 x  y   = (µ(z) ⊗ I)   ppx  . p0y

p0

(4.126)

196

Quantum Mechanics of Charged Particle Beam Optics

It can be verified directly that  x  y   : −Ho,L (z2 ) :z : −Ho,L (z1 ) :z   ppx  

p0y p0



 x  y   = (µ (z2 ) ⊗ I) (µ (z1 ) ⊗ I)   ppx  p0y p0

 x  y   = (µ (z2 ) µ (z1 ) ⊗ I)   ppx  . 

(4.127)

p0y p0

From this, it follows that we can write    x (zi ) x (zi )  y (zi )   Rz   y (zi )   dz:−Ho,L (z):z  M (z, zi )  px (zi )  = P e zi  px (zi )  p0   p0 py (zi ) p0

py (zi ) p0

    



 x (zi )  Rz   y (zi )   dz µ(z)⊗I  = P e zi  px (zi )   p0  py (zi ) p0



 x (zi )  i  y (zi )  h  Rz   dz µ(z) ⊗ I  px (zi )  = P e zi  p0  py (zi ) p0

 x (zi )  y (zi )    = (M (z, zi ) ⊗ I)  px (zi )  ,  p0  

(4.128)

py (zi ) p0

where M (z, zI ) is given by (4.54). transfer map as    x(z) x (zi )  y(z)   y (zi )   p (z)   x  = T (z, zi )  px (zi )  p0   p0 py (z) p0

py (zi ) p0

Now, from (4.113), we get the full phase space 



 x (zi )   y (zi )      = (M (z, zi ) ⊗ R (z, zi ))  px (zi )  .   p0  py (zi ) p0

(4.129)

197

Classical Charged Particle Beam Optics

Thus, we have analyzed the performance of an axially symmetric magnetic lens using direct integration of Hamilton’s equations and also using the Lie transfer operator method. The paraxial approximation is seen to lead to a linear relationship between the initial transverse coordinates and momenta to the final coordinates and momenta for a particle of a paraxial beam traveling through the system close to the optic axis. If the beam deviates from the ideal paraxial condition, the transfer map from the initial to the final transverse coordinates and momenta will become nonlinear leading to aberrations in the imaging systems. Then, one has to go beyond the paraxial approximation and use perturbation techniques to understand the performance of the system. As already mentioned, we shall not deal with the classical theory of aberrations here, and later we shall see that the classical theory of aberrations can be understood as an approximation of the quantum theory. 4.3.2

NORMAL MAGNETIC QUADRUPOLE

Let us now consider a paraxial beam propagating through a normal magnetic quadrupole lens. Let the optic axis be along the z-direction and the practical boundaries of the quadrupole be at z` and zr with z` < zr . The field of the normal magnetic quadrupole is ~B(~r) = (−Qn y, −Qn x, 0) , (4.130) where

 Qn =

constant in the lens region (z` ≤ z ≤ zr ) 0 outside the lens region (z < z` , z > zr )

(4.131)

There is no electric field and hence φ (~r) = 0. The vector potential of the magnetic field can be taken as  
~A(~r) = 0, 0, 1 Qn x2 − y2 . (4.132) 2 Now, from (4.2), the classical beam optical Hamiltonian is seen to be, with Kn = qQn /p0 , q p2 Ho = − p20 − π⊥2 − qAz ≈ −p0 + ⊥ − qAz 2p0
p2⊥ 1 (4.133) = −p0 + − p0 Kn x2 − y2 . 2p0 2 Let us consider the propagation of the beam from the transverse xy-plane at z = zi < z` to the transverse plane at z > zr . The beam propagates in free space from zi to z` , passes through the lens from z` to zr , and propagates through free space again from zr to z. Let us specify the transfer map for the transverse coordinates and momenta of a beam particle by     x(z) x (zi )  px (z)   px (zi )   p0    (4.134)   = T (z, zi )  p0  .  y(z)   y (zi )  py (z) p0

py (zi ) p0

198

Quantum Mechanics of Charged Particle Beam Optics

In view of the semigroup property of the transfer operator (4.104), we can write T (z, zi ) = T (z, zr ) T (zr , z` ) T (z` , zi ) = TD (z, zr ) TL (zr , z` ) TD (z` , zi )   2 R zr p⊥ 2 1 p K x2 −y2 p2 − :z Rzz` dz :− p⊥ :z dz :− ⊥: ) ( n 0 z dz :− 2p 2 ` zr 0 2p0 z 2p0 i e e

Rz

=e

 2  p⊥ 1 p2 p2 2 2 (z−zr ):− 2p⊥ :z (zr −z` ):− 2p0 − 2 p0 Kn (x −y ) :z (z` −zi ):− 2p⊥ :z

=e

0

e

e

0

,

(4.135)

where the subscripts D and L denote drift and propagation through the lens. Note that TD and TL are expressible as ordinary exponentials since Ho is z-independent in the free and lens regions. Note that we have dropped the additive constant term −p0 from Ho in the transfer operators, since its Poisson brackets with the coordinates and momenta vanish. As we already know, we have      x x 0 1 0 0  ppx   0 0 0 0   ppx  p2 0 =  0 , : − ⊥ :z  (4.136) 2p0  y   0 0 0 1   y  py py 0 0 0 0 p p 0

0

and hence x (zr )



 px (zr )  px (z)    p0   = TD (z, zr )  p0   y (zr )  y(z) 

   



x(z)





py (z) p0

py (zr ) p0



1 (z − zr )  0 1 =  0 0 0 0

  x (z ) r 0 0 px (zr )  0 0    p0 1 (z − zr )   y (zr ) py (zr ) 0 1

   . 

p0

(4.137) Similarly, we have    x (zi ) x (z` )  px (z` )   px (zi )  p0    = TD (z` , zi )  p0   y (zi )  y (z` )  py (z` ) p0

py (zi ) p0



1 (z` − zi )  0 1 =  0 0 0 0

    

  x (z ) i 0 0 px (zi )  0 0   p0 1 (z` − zi )   y (zi ) py (zi ) 0 1 p0

   . 

(4.138)

199

Classical Charged Particle Beam Optics

To get TL (zr , z` ), let us observe that    x 0  2  px  p   Kn
p⊥ 1 2 2 0   − p0 Kn x − y :− :z   y = 0 2p0 2 py 0 p 0

Then, we get  x (zr )  px (zr )  p0   y (zr ) py (zr ) p0





x (z` )



  px (z` )    = TL (zr , z` )  p0   y (z` )

   

py (z` ) p0

 

0   Kn  = exp  w  0 0

1 0 0 0 0 0 0 −Kn √
cosh w Kn √
√ Kn sinh w Kn

=

√
cos w Kn √ √
− Kn sin w Kn

1 0 0 0 0 0 0 −Kn

  x 0 px   0    p0  .   y  1 py 0 p0 (4.139)

  x (z )  ` 0 px (z` )   0      p0  1   y (z` )  py (z` ) 0 p0 √
! √1 sinh w Kn Kn √
⊕ cosh w Kn  x (z` ) !!
√  px (z` ) √1 sin w Kn  p0 Kn √
  y (z` ) cos w Kn py (z` ) p0

with w = (zr − z` ) .

   ,  (4.140)

Here,  (A ⊕ B) =

A O

O B

 (4.141)

is the direct sum of the matrices A and B with O as the 2 × 2 null matrix with all its entries as zero. Thus, we have     x(z) x (zi )  px (z)   px (zi )   p0   p0  = T (z, z ) T (z , z ) T (z , z )   , D r L r ` D ` i   y(z)   y (zi )  py (z) p0

py (zi ) p0



 0 0  0 0 , 1 (z − zr )  0 1 √
√
! √1 sinh w Kn cosh w Kn K n √ √
√
⊕ Kn sinh w Kn cosh w Kn

1 (z − zr )  0 1 TD (z, zr ) =   0 0 0 0 TL (zr , z` ) =

200

Quantum Mechanics of Charged Particle Beam Optics

√
cos w Kn √
√ − Kn sin w Kn 

1 (z` − zi )  0 1 TD (z` , zi ) =   0 0 0 0

√
!! sin w Kn √
cos w Kn 

√1 Kn

0 0  0 0 . 1 (z` − zi )  0 1

(4.142)

It is readily seen from this map that when Kn > 0 the lens is divergent in the xz-plane and convergent in the yz-plane. In other words, the normal magnetic quadrupole lens produces a line focus. When Kn < 0 the lens is convergent in the xz-plane and divergent in the yz-plane. Note that Kn has the dimensions of length−2 . In the weak field p case, when w |Kn |  1, the aforementioned transfer map becomes     x(z) 1 (z − zr ) 0 0  px (z)    1 0 0  p0   0 ×  ≈ y(z) 0 0 1 (z − z )   r  py (z) 0 0 0 1 p0   1 0 0 0  wKn 1 0 0  ×   0 0 1 0  0 0 −wKn 1   x (z )   i 1 (z` − zi ) 0 0 px (zi )    0 1 0 0    p0   (4.143) ,  0 0 1 (z` − zi )   y (zi )  p (z ) y i 0 0 0 1 p0

showing that the lens can be considered a thin lens (w ≈ 0) with the focal lengths 1 = −wKn , f (x)

1 = wKn . f (y)

(4.144)

When the beam is not ideally paraxial, more terms will have to be included in the optical Hamiltonian, and this will lead to nonlinearities in the transfer map. Let us consider two thin lenses of focal lengths f1 and f2 kept separated by a distance d. The transfer matrix from the entrance of the lens with focal length f1 to the exit of the lens with focal length f2 will be !     1 − fd1 d 1 0 1 0 1 d   = , − f12 1 − f11 1 0 1 1 − fd2 − f11 + f12 − f1df2 (4.145) showing that the combination has the equivalent focal length given by 1 1 1 d = + − . f f1 f2 f1 f2

(4.146)

201

Classical Charged Particle Beam Optics

This shows that a lens of focal length ± f can be considered effectively to be two lenses of focal length ±2 f , respectively, joined together, i.e., f1 = f2 = ±2 f , and d = 0 in the formula (4.146). This also shows that a doublet of focusing and defocusing lenses of focal lengths f and − f with a distance d between them would have the effective focal length f 2 /d > 0, and hence would behave as a focusing lens. Let us consider a triplet of identical quadrupoles, with focal lengths f and − f in the two transverse planes, arranged as follows. The first and the third quadrupoles, situated symmetrically about the middle quadrupole at a distance d from them, are focusing in the xz-plane and defocusing in the yz-plane. The middle quadrupole is rotated by 90◦ with respect to first (and the third) so that it is defocusing in the xzplane and focusing in the yz-plane. The transfer matrix from the center of the first lens to the center of the third lens is given by

1  − 21f   0 0 

0 1 0 0

0 0 1 1 2f

 0 1 d  0 1 0   0  0 0 0 0 1 

1 d  0 1  × 0 0 0 0 

 1 0 0  1f 0 0   1 d  0 0 1 0

 1 0 0 1   − 0 0  2f 1 d  0 0 1 0

0 1 0 0

0 0 1 0 0 1 0 − 1f 0 0 1 1 2f

 0 0  × 0  1

 0 0   0  1

  1 − 2df 2 2d 1 + 2df 0 0     1 − 2df 2 0 0  − 2df 2 1 − 2df   =  d 2d 1 − 2df 0 0 1− 2f2     0 0 − 2df 2 1 + 2df 1 − 2df 2

     , (4.147)   

corresponding to net focusing effect in both the transverse planes when d < 2 f . In electron optical technology, for electron energies in the range of tens or hundreds of kilovolts to a few megavolts, magnetic quadrupole lenses are used, if at all, as components in aberration-correcting units for round lenses and in devices required to produce a line focus. It is mainly at higher energies, where round lenses are too weak, quadrupole lenses are utilized to provide the principal focusing field. As seen above, a triplet of quadrupoles can be arranged to have net focusing effect in both the transverse planes. Series of such quadrupole triplets are the main design elements of long beam transport lines or circular accelerators to provide a periodic focusing structure called a FODO-channel. FODO stands for F(ocusing)O(nonfocusing)D(efocusing)O(nonfocusing), where O can be a drift space or a bending magnet.

202

4.3.3

Quantum Mechanics of Charged Particle Beam Optics

SKEW MAGNETIC QUADRUPOLE

A magnetic quadrupole associated with the magnetic field ~B(~r) = (−Qs x, Qs y, 0) ,

(4.148)

corresponding to the vector potential ~A(~r) = (0, 0, −Qs xy) ,

(4.149)

is known as a skew magnetic quadrupole. If z` and zr > z` are the boundaries of the quadrupole, then Qs is a nonzero constant for z` ≤ z ≤ zr and Qs = 0 outside the quadrupole (z < z` , z > zr ). For a paraxial beam propagating through this skew magnetic quadrupole along its optic axis, z-axis, the classical beam optical Hamiltonian will be   −p + p2⊥ , for z < z` , z > zr , 0 2p0 Ho = (4.150) 2 p  −p0 + ⊥ + p0 Ks xy, for z ≤ z ≤ zr , ` 2p0 where Ks = qQs /p0 . Now, if we make the transformation   1 0 1 0 x  0  px  1 1 0 1     y  = √2  −1 0 1 0 0 −1 0 1 py 

  x0   p0x    0  = R  y   p0y 

 x0 p0x  , y0  p0y

the Hamiltonian is seen to become  0 2  −p0 + p⊥ , for z < z` , z > zr , 2p 0   Ho0 = 2  −p + p0⊥ − 1 p K x0 2 − y0 2 , for z ≤ z ≤ z , r 0 ` 2p0 2 0 s

(4.151)

(4.152)

same as for the propagation of the paraxial beam through a normal magnetic quadrupole (4.133) in the new (x0 , y0 ) coordinate system, with Kn replaced by Ks . The above transformation corresponds to a clockwise rotation of the (x, y) coordinate axes by π4 about the optic axis. Hence, the transfer map for the skew magnetic quadrupole can be obtained from the transfer map for the normal magnetic quadrupole as Tsq = RTnq R−1 ,

(4.153)

where the subscripts sq and nq stand for the skew magnetic quadrupole and the normal magnetic quadrupole, respectively. The beam optical Hamiltonian for propagation through free space is axially symmetric. Thus, the free space transfer map through any distance, say z0 to z00 , TD (z00 , z0 ), is seen to satisfy the relation

R(θ )TD z00 , z0 R(θ )−1 = TD z00 , z0 ,

for any θ ,

(4.154)

203

Classical Charged Particle Beam Optics

where 

cos θ  0 R(θ ) =   − sin θ 0

0 cos θ 0 − sin θ

1 z00 − z0
 0 1 z00 , z0 =   0 0 0 0 

TD

sin θ 0 cos θ 0

 0 sin θ  , 0  cos θ

 0 0  0 0 . 1 z00 − z0  0 1

Note that R(θ )−1 = R(−θ ) and R in (4.151) is R (4.153) and (4.154), we get

π 4

(4.155)


. Then, from the relations

Tsq (z, zi ) = TD (z, zr ) Tsq,L (zr , z` ) TD (z` , zi ) = TD (z, zr ) RTnq,L (zr , z` ) R−1 TD (z` , zi ) ,

(4.156)

where R is as in (4.151), and Tnq,L (zr , z` ) is the same as TL (zr , z` ) in (4.142) with Kn replaced by Ks . One can get Tsq,L (zr , z` ) directly as follows. Observe that    x   ppx   p2⊥ 0   :− + p0 Ks xy :z   y = 2p0 

py p0

0 0 0 −Ks

1 0 0 −Ks 0 0 0 0

 x 0  ppx 0   0 1  y py 0 p

   . (4.157) 

0

Then, one gets 

x (zr )

 px (zr )  p0   y (zr ) py (zr ) p0





x (z` )



  px (z` )    = Tsq,L (zr , z` )  p0   y (z` )

   

py (z` ) p0

 

0   0   = exp w  0 −Ks  C+ √  1  − Ks S− =  C− 2 √ − Ks S+

1 0 0 −Ks 0 0 0 0 √1 S+ Ks +

C

√1 S− Ks −

C

with w = (zr − z` ) .

  x (z ) ` 0  px (z` )   0   p0  1   y (z` ) py (z` ) 0

    

p0

−

C √ − Ks S+ C+ √ − Ks S−

√1 S− Ks −

C

√1 S+ Ks +

C



x (z` )

  px (z` )   p0    y (z` ) py (z` ) p0

   ,  (4.158)

204

Quantum Mechanics of Charged Particle Beam Optics

where √ √ C± = cos(w Ks ) ± cosh(w Ks ), √ √ S± = sin(w Ks ) ± sinh(w Ks ).

(4.159)

It can be verified that Tsq,L = RTnq,L (zr , z` ) R−1 . 4.3.4

AXIALLY SYMMETRIC ELECTROSTATIC LENS

An axially symmetric, round, electrostatic lens with the axis along the z-direction consists of the electric field corresponding to the potential ∞

φ (~r⊥ , z) =

(−1)n

∑ (n!)2 4n φ (2n) (z)r⊥2n

n=0

1 1 2 4 = φ (z) − φ 00 (z)r⊥ + φ 0000 (z)r⊥ −··· , 4 64

(4.160)

inside the lens region (z` < z < zr ). Outside the lens, i.e., (z < z` , z > zr ), φ (z) = 0. And, there is no magnetic field, i.e., ~A (~r) = (0, 0, 0). Starting with (4.2), the classical paraxial beam optical Hamiltonian of a general electrostatic lens, in the lens region, is obtained as follows: r 1 Ho = − p20 − π⊥2 − 2 2qEφ − qAz c r 1 = − p20 − p2⊥ − 2 2qEφ c s  2  p⊥ 2qEφ = −p0 1 − + p20 c2 p20 (    2 ) 1 p2⊥ 2qEφ 1 p2⊥ 2qEφ + 2 2 − + 2 2 ≈ −p0 1 − 2 p20 8 p20 c p0 c p0 ≈ −p0 +

p2⊥ qEφ qEφ p2⊥ q2 E 2 φ 2 + 2 + + . 2p0 c p0 2c2 p30 2c4 p30

(4.161)

For the round electrostatic lens, we get the classical beam optical Hamiltonian in the lens region, in the paraxial approximation, by using (4.160) in (4.161) as follows:     p2⊥ qE p2 qE 1 1 2 2 + 2 φ (z) − φ 00 (z)r⊥ + 2 ⊥3 φ (z) − φ 00 (z)r⊥ 2p0 c p0 4 4 2c p0   2 2 q E 1 2 + 4 3 φ (z)2 − φ (z)φ 00 (z)r⊥ 2 2c p0

Ho ≈ −p0 +

Classical Charged Particle Beam Optics

    2 p⊥ qE q2 E 2 2 qE ≈ − 1 − 2 2 φ (z) − 4 4 φ (z) p0 + 1 + 2 2 φ (z) 2p c p0 2c p0 c p0 0   qE qE 2 − 2 1 + 2 2 φ (z) φ 00 (z)r⊥ . 4c p0 c p0

205

(4.162)

In the field-free regions outside the lens, the optical Hamiltonian will be −p0 +
p2⊥ /2p0 . The aforementioned optical Hamiltonian of the round electrostatic lens is seen to be similar to the optical Hamiltonian of the round magnetic lens (4.24), except for the absence of the rotation term and the presence of an z-dependent coefficient for the drift term p2⊥ /2p0 . The first constant term, though z-dependent, can be ignored as before, since it has vanishing Poisson bracket with the transverse coordinates and momenta. The paraxial Hamiltonian in (4.162), being quadratic in r⊥ and p⊥ , leads to a linear transfer map for the transverse coordinates and momenta, and it is straightforward to calculate the map following the same procedure used in the case of the round magnetic lens. When the beam is not paraxial, there will be aberrations. We shall not pursue the performance of this lens further. Some uses of electrostatic lenses are in the extraction, preparation, and initial acceleration of electron and ion beams in a variety of applications. 4.3.5

ELECTROSTATIC QUADRUPOLE

The electric field of an ideal electrostatic quadrupole lens with the optic axis along the z-direction corresponds to the potential  φ (~r⊥ , z) =

1 2 Qe

x2 − y2 0




in the lens region (z` < z < zr ) , outside the lens (z < z` , z > zr ) ,

(4.163)

where Qe is a constant and zr − z` = w is the width of the lens. Substituting this expression for φ and ~A = (0, 0, 0), since there is no magnetic field, in the general form of the optical Hamiltonian of an electrostatic lens (4.161), we get Ho ≈ −p0 +


p2⊥ qEQe 2 + x − y2 , 2p0 2c2 p0

(4.164)

as the classical beam optical Hamiltonian of the electrostatic quadrupole lens in the paraxial approximation. Simply by comparing the Hamiltonian of the normal magnetic quadrupole lens (4.133) with the Hamiltonian of the electrostatic quadrupole lens (4.164), it is readily seen that the electrostatic quadrupole lens is convergent in the xz-plane and divergent in the yz-plane when qEQe Ke = 2 2 > 0. (4.165) c p0

206

Quantum Mechanics of Charged Particle Beam Optics

When Ke < 0, this lens is divergent in the xz-plane and convergent in the yz-plane. Note that Ke has the dimension of length−2 . In the weak field case, when w2  1/ |Ke |, the lens can be considered as thin and has focal lengths 1 f (x)

= wKe ,

1 f (y)

= −wKe .

(4.166)

Deviations from the ideal behavior result from nonparaxial conditions of the beam. We shall not pursue the performance of this lens further here.

4.4

BENDING MAGNET: AN OPTICAL ELEMENT WITH A CURVED OPTIC AXIS

Circular accelerators and storage rings require dipole magnets for bending the beams to guide them along the desired curved paths. By applying a constant magnetic field in the vertical direction using a dipole magnet, the beam is bent along a circular arc in the horizontal plane. The circular arc of radius of curvature ρ, the design trajectory of the particle, is the optic axis of the bending magnet. It is natural to use the arclength, say S, measured along the optic axis from some reference point as the independent coordinate, instead of z. Let the reference particle moving along the design trajectory carry an orthonormal XY -coordinate frame with it. The X-axis is taken to be perpendicular to the tangent to the design orbit and in the same horizontal plane as the trajectory, and the Y -axis is taken to be in the vertical direction perpendicular to both the X-axis and the trajectory. The curved S-axis is along the design trajectory and perpendicular to both the X and Y axes at any point on the design trajectory. The instantaneous position of the reference particle in the design trajectory at an arclength S from the reference point corresponds to X = 0 and Y = 0. Let any particle of the beam have coordinates (x, y, z) with respect to a fixed right-handed Cartesian coordinate frame, with its origin at the reference point on the design trajectory from which the arclength S is measured. Then, the two sets of coordinates of any particle of the beam, (X,Y, S) and (x, y, z), will be related as follows:     S S − ρ, z = (ρ + X) sin , y=Y (4.167) x = (ρ + X) cos ρ ρ To study the motion of the beam particles through the dipole magnet with a circular arc as the optic axis, we have to transform the optical Hamiltonian to the (X,Y, S)coordinate system. This can be done through a canonical transformation with the generating function chosen as     S F3 (px , py , pz , X,Y, S) = − (ρ + X) cos ( − ρ px ρ    S − (ρ + X) sin pz −Y py , (4.168) ρ such that xj = −

∂ F3 , ∂ pj

Pj = −

∂ F3 . ∂ Xj

(4.169)

207

Classical Charged Particle Beam Optics

The first part of the above equation just reproduces (4.167) giving the relation between (x, y, z) and (X,Y, S). The second part gives the new momentum components conjugate to (X,Y, S): with 1/ρ = κ, the curvature, and ζ = 1 + κX, 

    PX cos(κS) sin(κS) 0 px  1 PS  =  − sin(κS) cos(κS) 0   pz  ζ 0 0 1 py PY

(4.170)

Since X and S axes are rotated clockwise through the angle S/ρ relative to the x and z axes, the components of the vector potential in the two frames are related as      AX cos(κS) sin(κS) 0 Ax  AS  =  − sin(κS) cos(κS) 0   Az  . (4.171) AY 0 0 1 Ay For more details, see, e.g., Wolski [192]. The general Hamiltonian of the particle in an electromagnetic field (2.37) is q (4.172) H (x, y, z, px , py , pz ,t) = m2 c4 + c2 (~p − q~A)2 + qφ , in the Cartesian coordinate system. If we change it to (X,Y, S) coordinate system using the relations (4.170) and (4.171), we get H (X,Y, S, PX , PY , PS ,t) n h = m2 c4 + c2 (PX − qAX )2 + (PY − qAY )2 1/2 1 2 + 2 (PS − qζ AS ) + qφ , ζ

(4.173)

in which (x, y, z) are expressed in terms of (X,Y, S) using (4.167). Now, we have to change the independent variable in the above Hamiltonian from t to S, and the desired classical beam optical Hamiltonian in the chosen curvilinear coordinate system is given as the solution for −PS : fo = −PS H 1 =− ζ c

r (E − qφ )2 − m2 c4 − c2

!  2 ~P⊥ − q~A⊥ − ζ qAS .

(4.174)

From (4.167), we find that ds, the distance between two infinitesimally close points, is given by ds2 = dx2 + dy2 + dz2 = dX 2 + dY 2 + ζ 2 dS2 . (4.175) In general, in an orthogonal curvilinear (u, v, w)-coordinate system with the line element ds given by ds2 = h2u du2 + h2v dv2 + h2w dw2 , (4.176)

208

Quantum Mechanics of Charged Particle Beam Optics

the curl of any vector ~A has the following expression (see, e.g., Arfken, Weber, and Harris [3]):   ∂ (hw Aw ) ∂ (hv Av ) 1 − , hv hw ∂v ∂w   1 ∂ (hu Au ) ∂ (hw Aw ) − , (~∇ × ~A)v = hw hu ∂w ∂u   1 ∂ (hv Av ) ∂ (hu Au ) (~∇ × ~A)w = − . hu hv ∂u ∂v (~∇ × ~A)u =

(4.177)

Thus, in the (X,Y, S)-coordinate system, we have   1 ∂ (ζ AS ) ∂ AY ~ ~ − , BX = (∇ × A)X = ζ ∂Y ∂S   1 ∂ AX ∂ (ζ AS ) BY = (~∇ × ~A)Y = − , ζ ∂S ∂X ∂ AY ∂ AX − . BS = (~∇ × ~A)S = ∂X ∂Y

(4.178)

The dipole magnetic field is a uniform field perpendicular to the design trajectory such that the beam particle is bent along the design trajectory. So, we should have BX = 0,

BY = B0 ,

BS = 0.

(4.179)

Using (4.178), we find that the corresponding vector potential in the curved coordinate system can be taken as AX = 0,

AY = 0,

  κX 2 AS = −B0 X − . 2ζ

(4.180)

Let us now find the optical Hamiltonian obtained by substituting in (4.174) φ = 0 and choosing the vector potential as in (4.180). The resulting classical beam optical Hamiltonian for the dipole magnet, in the paraxial approximation, is   q 1 κX 2 2 2 2 4 2 f Ho = − ζ E − m c − c P⊥ + ζ qB0 X − c 2ζ   q 1 = −ζ p20 − P⊥2 + qB0 Xζ − κX 2 2     2 P⊥ 1 2 ≈ ζ −p0 + + qB0 X + κX 2p0 2 ≈ −p0 +

P⊥2 1 + (qB0 − κ p0 ) X + qB0 κX 2 . 2p0 2

(4.181)

209

Classical Charged Particle Beam Optics

Let us now find what would be the effect of the term ∝ X in the above Hamiltonian: with ∆S = S − Si ,  RS

e

Si

X (Si )



PX (Si )   p  Y (S0 )  i PY (Si )  X (Si ) PX (Si )  p = e:−∆S(qB0 −κ p0 )X:S   Y (S0 )

ds :−(qB0 −κ p0 )X:S  

i

   

PY (Si )  ∞

=∑ 0

  = 

X (Si )



 PX (Si )  1  p : −∆S (qB0 − κ p0 ) X :nS   Y (S0 )  n! i PY (Si )  X (Si ) 1  p0 (PX (Si ) + ∆S (qB0 − κ p0 ))  ,  Y (Si ) PY (Si )

(4.182)

where Si and S > Si are two points on the curved S-axis or the design trajectory. This shows that the presence of the term ∝ X in the Hamiltonian causes a change in PX and hence a deflection of the particle in the horizontal plane perpendicular to the design trajectory. If the curvature of the design trajectory is properly matched to the dipole magnetic field, i.e., qB0 = κ p0

or, B0 ρ =

p0 , q

(4.183)

then the term (qB0 − κ p0 ) X in the Hamiltonian vanishes, and if a particle is initially on the design trajectory, it will remain on the design trajectory through the dipole. The condition (4.183) follows simply from the fact that if a constant magnetic field of magnitude B bends the path of a particle of charge q moving with a speed v perpendicular to it in a circular arc of radius ρ, then we must have γmv2 = qBv, ρ

or,

γmv p = = Bρ. q q

(4.184)

The quantity Bρ is called magnetic rigidity. Thus, if a beam of design momentum p0 is to be bent by a circular arc of radius ρ by a dipole magnet, the magnetic field should have the magnitude B0 = p0 /(qρ). With the condition (4.183) satisfied, the classical beam optical Hamiltonian of the dipole magnet becomes

210

Quantum Mechanics of Charged Particle Beam Optics 2 fo,d = −p0 + P⊥ + 1 p0 κ 2 X 2 H 2p0 2  2  PX 1 P2 2 2 = −p0 + + p0 κ X + Y . 2p0 2 2p0

(4.185)

which is independent of S. Then, the transfer map for the dipole is seen to be     X(S) X (Si )  PX (S)   PX (Si )  RS fo,d (s):S   p0   ds :−H   = e Si  p0   Y (S)   Y (Si )  PY (S) p0

PY (Si ) p0





0   −κ 2  = exp  ∆S  0 0  cos(κ∆S)  −κ sin(κ∆S)  = 0 0

  X (S )  i 1 0 0  PX (Si )    0 0 0   p0    0 0 1   Y (Si )  P (S ) Y i 0 0 0 p0   X (S ) 1 i sin(κ∆S) 0 0 κ  PX (Si )  cos(κ∆S) 0 0   p0  0 1 ∆S   Y (Si ) PY (Si ) 0 0 1

   . 

p0

(4.186) This shows that a particle entering the dipole magnet along the design trajectory (X = 0,Y = 0, PX = 0, PY = 0) will follow exactly the curved design trajectory. Such a steering of the beam is the purpose of the dipole magnet. Any other particle of the paraxial beam making a small angle with the design trajectory will have a small oscillation in the X-direction about the design trajectory and will have free propagation along the Y -direction.

4.5

NONRELATIVISTIC CLASSICAL CHARGED PARTICLE BEAM OPTICS

We shall now see how the nonrelativistic classical charged particle beam optics becomes an approximation of the relativistic classical charged particle beam optics. We have been so far using the exact relativistic expression for the energy of a charged particle in an electromagnetic field, p E = m2 c4 + c2~π 2 + qφ , (4.187) in deriving the classical beam optical Hamiltonians for the various charged particle optical elements. The expression (4.187) for energy is exact and valid for all values of particle momentum irrespective of its velocity. In high-energy particle accelerators, the velocities of the particles being accelerated are very high, and so one has to use

211

Classical Charged Particle Beam Optics

the relativistic expression (4.187). However, in applications like the common electron microscopy, or some ion accelerators, where the velocities of the beam particles are not so high it would be appropriate to use the nonrelativistic approximation. So, let us try to understand the nonrelativistic approximation of the above formalism. In nonrelativistic situations when the velocities involved are  c, the expression for energy can be approximated as r ~π 2 2 E = mc 1 + 2 2 + qφ m c   ~π 2 2 ≈ mc 1 + 2 2 + qφ (∵ |~π |  mc) 2m c ~π 2 = mc2 + + qφ , (4.188) 2m which is the sum of rest energy, kinetic energy, and potential energy of the particle. Since the rest energy is not involved in any nonrelativistic interactions, one can consider the sum of kinetic energy and potential energy as the total energy of the particle. Thus, we take ~π 2 E= + qφ (4.189) 2m as the total nonrelativistic energy of a particle. Correspondingly, the nonrelativistic Hamiltonian of the particle becomes HNR =

~π 2 + qφ . 2m

(4.190)

Now, if we use the procedure of changing the independent variable from time t to z, the variable along the straight optic axis of a system, starting with this Hamiltonian, we obtain q Ho,NR = −pz = − 2m(E − qφ ) − ~π⊥2 − qAz q = − p20 − ~π⊥2 − 2mqφ − qAz , (4.191) p2

taking p0 as the design momentum such that E = 2m0 is the total conserved energy of the beam particle in the system. Taking the nonrelativistic expression for the momentum p0 = mv0 , we can write q Ho,NR = − m2 v20 − ~π⊥2 − 2mqφ − qAz . (4.192) We have found in (4.2) the relativistic classical beam optical Hamiltonian of a general electromagnetic optical element to be r 2Eqφ Ho = − p20 − π⊥2 − 2 − qAz , (4.193) c

212

Quantum Mechanics of Charged Particle Beam Optics

derived under the condition E  qφ as is usually the case in optical elements and which is anyway true for purely magnetic optical elements q with φ = 0. In (4.193), E is the total conserved energy of the particle given by m2 c4 + c2 p20 = γmc2 , with p0 = γmv0 . Thus, we can write q Ho = − γ 2 m2 v20 − π⊥2 − 2γmqφ − qAz . (4.194) Now, comparing (4.192) and (4.194), it is clear that the relativistic optical Hamiltonian can be obtained from the nonrelativistic optical Hamiltonian by the replacement m −→ γm. (4.195) Thus, we find that one can extend the results based on the nonrelativistic optical Hamiltonian to get the corresponding relativistic results by replacing the rest mass m of the particle by γm (the so-called relativistic mass) under the condition E  qφ . This is a practice often used in classical charged particle optics. Similarly, one can make nonrelativistic approximations of the results derived from the relativistic optical Hamiltonian by using the limit γ −→ 1 carefully.

Charged Particle 5 Quantum Beam Optics Scalar Theory for Spin-0 and Spinless Particles 5.1

GENERAL FORMALISM OF QUANTUM CHARGED PARTICLE BEAM OPTICS

In the classical theory of charged particle beam optics, our interest is to study the evolution of the state (position and momentum) of the beam particle along the optic axis of an optical system through which the beam is propagating. Such a study leads to an understanding of the performance of the concerned optical system based on classical mechanics. In the quantum theory of charged particle beam optics, our interest is to study the evolution of the state (wave function, or density operator) of the beam particle along the optic axis of an optical system through which the beam is propagating. This study should lead to an understanding of the performance of the concerned optical system based on quantum mechanics, and the understanding based on classical mechanics should emerge as an approximation. As we have seen already, the fundamental equations of quantum mechanics prescribe the time evolution of the state of a particle through a Hamiltonian operator governing the system. Exactly like the independent variable was changed from time to the coordinate along the optic axis of the concerned system in the classical Hamiltonian theory of charged particle beam optics, we have to rewrite the basic quantum mechanical equations as equations for the evolution of the quantum state of the particle along the coordinate, say z, of the optic axis of the concerned system, i.e., we have to cast the concerned evolution equations in the form ∂ ψ (~r⊥ , z) co ψ (~r⊥ , z) . i¯h =H (5.1) ∂z Note that i¯h(∂ /∂ z) = − pbz corresponds to the quantum charged particle beam optical co . Hamiltonian operator H In this chapter, we shall consider the scalar quantum theory of charged particle beam optics based on the relativistic Klein–Gordon equation and the nonrelativistic Schr¨odinger equation. The Klein–Gordon equation can be used for spin-0 particles, and all spinless relativistic particles, (i.e., particles for which we can ignore the spin). The nonrelativistic Schr¨odinger equation can be used for any nonrelativistic particle for which we can ignore the spin. Since the nonrelativistic Schr¨odinger equation is an approximation of the Klein–Gordon equation, we shall consider first the 213

214

Quantum Mechanics of Charged Particle Beam Optics

Klein–Gordon equation. This chapter is essentially based on the work of Jagannathan and Khan [82], Khan and Jagannathan [90], and Khan [91], on the nonrelativistic scalar theory of quantum charged particle beam optics based on the nonrelativistic Schr¨odinger equation, and the relativistic scalar theory of quantum charged particle beam optics based on the Klein–Gordon equation, following the earlier work by Jagannathan, Simon, Sudarshan, and Mukunda [79] and Jagannathan [80] on the spinor theory of quantum charged particle beam optics based on the Dirac equation. Spinor theory of quantum charged particle beam optics based on the Dirac–Pauli equation was developed later by Conte, Jagannathan, Khan, and Pusterla [26]. In the formulation of the relativistic quantum charged particle beam optics, we shall follow the philosophy of Hawkes and Kasper [72]: There is no need for a Lorentz-covariant formulation, and in practically all situations, the description in the laboratory frame is perfectly adequate.

5.2 5.2.1

RELATIVISTIC QUANTUM CHARGED PARTICLE BEAM OPTICS BASED ON THE KLEIN–GORDON EQUATION GENERAL FORMALISM

We are concerned with a beam of identical particles of charge q and rest mass m propagating through a stationary electromagnetic field, constituting a charged particle optical system, such as a round magnetic lens in an electron microscope or a magnetic quadrupole in a particle accelerator. Let us ignore the spin of the particle and hence consider the Klein–Gordon equation as the appropriate equation for treating the quantum dynamics of the particle when its motion is relativistic. We consider the optical system to be, in practice, localized in a definite region of space so that the regions outside the system can be considered field-free or free space. We shall assume, throughout, the beam to be monoenergetic, paraxial or quasiparaxial, and forward moving along the optic axis of theq system. All the beam particles are

assumed to have the same positive energy E = m2 c4 + c2 p20 , where p0 is magnitude of the design momentum. We assume that all the particles of the beam enter the optical system from the free space outside the system with this constant energy. Let φ (~r) and ~A(~r) be the electric scalar potential and the magnetic vector potential of the time-independent electromagnetic field of the optical system. In passing through the system or in the process of scattering of the particle by the system, the total energy of the particle E will be conserved, and hence it is time-independent. After passing through the system, the particle will emerge outside in the free space with the same energy E it had when entering the system. In general, we shall consider the system to have a straight optic axis along the z-direction, except in the case of the bending magnet. The scalar Klein–Gordon equation governing the quantum mechanics of the particle in an electromagnetic field is 2    2  ∂ 2 4 2 ~ ~ Ψ(~r,t), i¯h − qφ (~r,t) Ψ(~r,t) = m c + c pb − qA(~r,t) ∂t

(5.2)

215

Scalar Theory: Spin-0 and Spinless Particles

as we have already seen (3.586). Since we are dealing with a time-independent system, we can take the wave function of the beam particle to be Ψ(~r,t) = e−iEt/¯h ψ(~r).

(5.3)

Then, the equation (5.2) becomes
e−iEt/¯h (E − qφ )2 ψ(~r) = e−iEt/¯h m2 c4 + c2 πb2 ψ(~r),

(5.4)

  (E − qφ )2 − m2 c4 − c2 πb⊥2 − c2 πbz2 ψ(~r) = 0.

(5.5)

or We can rewrite this equation as   c2 πbz2 ψ(~r) = E 2 − m2 c4 − qφ (2E − qφ ) − c2 πb⊥2 ψ(~r).

(5.6)

Dividing throughout by c2 , noting that E 2 − m2 c4 = c2 p20 , and introducing the notation qφ (2E − qφ ) = c2 p˜2 , we get
πbz2 ψ(~r) = p20 − p˜2 − πb⊥2 ψ(~r). (5.7) Getting the clue from the Feshbach–Villars formalism of the Klein–Gordon equation, let us introduce a two-component wave function !   ψ ψ1 = . (5.8) πbz ψ2 p0 ψ Then, we can write (5.7) equivalently as !    0 1 πbz ψ1 ψ1
. = 1 p20 − p˜2 − πb⊥2 0 ψ2 ψ2 p0 p20

(5.9)

Now, let 

ψ+ ψ−



 =M =

1 2

ψ1 ψ2



ψ+ ψ−

= πbz p0 ψ πbz p0 ψ

   1 1 1 ψ1 ψ2 2 1 −1 ! .

(5.10)

Then,      1 ∂ πbz ψ+ ψ+ = −i¯h − qAz ψ− ψ− p0 p0 ∂z !   0 1 ψ+ −1
=M 1 p20 − p˜2 − πb⊥2 0 M ψ− p20

!  1 − 2p1 2 πb⊥2 + p˜2 − 2p1 2 πb⊥2 + p˜2 ψ+ 0 0

= . (5.11) 1 ψ− πb⊥2 + p˜2 −1 + 2p1 2 πb⊥2 + p˜2 2p2 0

0

216

Quantum Mechanics of Charged Particle Beam Optics

Rearranging this equation, we get     ∂ ψ+ ψ+ b =H i¯h , ψ− ∂ z ψ− b b H = −p0 σz + Eb + O,
1 Eb = −qAz I + πb⊥2 + p˜2 σz , 2p0
i 2 πb⊥ + p˜2 σy . Ob = 2p0

(5.12)

Note that b H has been partitioned, apart from the leading term −p0 σz , into an even b term E that does not couple ψ+ and ψ− and an odd term Ob that couples ψ+ and ψ− . To understand (5.12) better, let us see what it implies in the case of propagation of the beam in free space. In free space, with φ = 0 and ~A = (0, 0, 0), we have     ∂ ψ+ ψ+ b i¯h =H ψ− ∂ z ψ− b H = −p0 σz + Eb + Ob i 2 1 2 pb σz + pb σy = −p0 σz + 2p0 ⊥ 2p0 ⊥   pb2 pb2 −p0 + 2p⊥0 + 2p⊥0 . = pb2 pb2 − 2p⊥0 p0 − 2p⊥0 If we take ψ(~r) = φ~p0 (~r) =

i 1 e h¯ ~p0 ·~r , 3/2 (2π h¯ )

a plane wave of momentum ~p0 = (p0x , p0y , p0z ), we have       i¯h ∂ φ~p0 1+ − φ ψ+  ~p0 p0 ∂ z  1  =   = 1  2  2 ∂ φ~p ψ− 1− φ~p0 + pi¯h ∂ z0 0

 It can be easily checked that this

ψ+ ψ−

(5.13)

(5.14)

p0z p0



p0z p0



 φ~p0  . 

(5.15)

φ~p0

 satisfies (5.13). For a particle of a quasi-

paraxial beam moving along and close to the +z-direction, associated with the plane wave φ~p0 (~r) with p0z > 0 and p0z ≈ p0 , it is clear from the above equation that ψ+  ψ− . By extending this observation, it is easy to see that for any wave packet of the form Z

ψ(~r) =

d 3 p0 C (~p0 ) φ~p0 (~r),

Z

with

d 3 p0 |C (~p0 )|2 = 1,

|~p0 | = p0 , p0z ≈ p0 , p0z > 0,

(5.16)

217

Scalar Theory: Spin-0 and Spinless Particles

representing a monochromatic quasiparaxial beam moving forward in the z-direction,   R     3 p C (p ) 1 + p0z φ (~r) d 0 0 1 ψ+ (~r) p0  ~p0 ,  =  R p0z 3 ψ− (~r) 2 φ~p (~r) d p0 C (p0 ) 1 − p0

0

with |~p0 | = p0 , p0z ≈ p0 , p0z > 0,

(5.17)

is such that ψ+ (~r)  ψ− (~r). Thus, in general, in the representation of (5.12), ψ+ is large compared to ψ− for any monochromatic quasiparaxial beam passing along and close to the +z-direction through any system supporting beam propagation. The purpose of casting (5.6) in the form of (5.12) is obvious when we compare the latter with the form of the Dirac equation in (3.711). Let us recall briefly. The Dirac equation for a spin- 21 particle of charge q and mass m in an electromagnetic field is     ∂ Ψu (~r,t) Ψu (~r,t) b i¯h =H , (5.18) Ψ` (~r,t) Ψ` (~r,t) ∂t where  Ψu =

Ψ1 Ψ2



 ,

Ψ` =

b b = mc2 β + Eb + O, H   I 0 Eb = qφ , 0 I

Ψ3 Ψ4



Ob = c

,

~σ ·~πb 0 ~ ~σ · πb I

! .

(5.19)

For any positive energy Dirac Ψ in the nonrelativistic situation (|c~π |  mc2 ), the upper components (Ψu ) are larger compared to the lower components (Ψ` ). The even operator (Eb) does not couple the large and small components and the odd operator b does couple them. Using mainly the algebraic property β Ob = −Oβ b , the Foldy– (O) Wouthuysen technique expands the Dirac Hamiltonian in a series with 1/mc2 as the expansion parameter. This leads to a good understanding of the nonrelativistic limit of the Dirac equation by showing how the Dirac Hamiltonian can be seen as consisting of a nonrelativistic part and a systematic series of relativistic correction terms. We have also seen that the Foldy–Wouthuysen transformation technique can be used to expand the Feshbach–Villars Hamiltonian of the Klein–Gordon equation in a series of nonrelativistic and relativistic correction terms. The analogy between (5.12) and the Dirac equation is clear now. There is a correspondence as follows: Forward propagation of the beam close to the z-direction

−→

Positive energy Dirac particle

Paraxial beam (|~π⊥ |  p0 )

−→

Nonrelativistic motion (|~π |  mc)

Deviation from paraxial condition (aberrating system)

−→

Deviation from nonrelativistic situation (relativistic motion)

218

Quantum Mechanics of Charged Particle Beam Optics

Also, it should be noted that in (5.12) σz plays the role of β in the Dirac Hamilb z analogous to the relatonian and Ob in (5.12) and σz obey the relation σz Ob = −Oσ b obtained in the case of the Dirac Hamiltonian. Hence, by applying tion β Ob = −Oβ a Foldy–Wouthuysen-like technique to (5.12), it should be possible to analyse the beam propagation through an electromagnetic optical system in a systematic way up to any desired order of approximation starting with the paraxial approximation. In the Foldy–Wouthuysen theory, a series of transformations are used to make the odd terms in the Hamiltonian as small as desired. We shall apply this technique to (5.12) to arrive at a representation in which the odd term will be as small as we desire. In this case, 1/p0 can be taken as the expansion parameter. The order of smallness of an odd term can be labeled by the lowest power of 1/p0 in it and higher order smallness will correspond to higher power. Thus, the odd term in (5.12) is of first order in 1/p0 . Following the Foldy–Wouthuysen theory, let us define !   (1) i ψ+ ψ+ iSb1 b σz O. (5.20) =e , with Sb1 = (1) ψ 2p − ψ− 0 The equation (5.12) is now transformed, as follows from (3.689), into !  ! (1) (1)   ∂ ψ+ ψ+ −iSb1 iSb1 b −iSb1 iSb1 ∂ = e He − i¯he e i¯h (1) (1) ∂z ∂z ψ− ψ− ! (1) ψ+ =b H(1) , (1) ψ−

(5.21)

b replaced by b where b H(1) is to be calculated using (3.706), with H H, Sb replaced by Sb1 , and t replaced by z. The result is b H(1) = −p0 σz + Eb(1) + Ob(1) ,  
2
1 1 2 2 2 2 (1) b E = −qAz I + πb + p˜ + 3 πb⊥ + p˜ σz 2p0 ⊥ 8p0
  1  2 − πb⊥ + p˜2 , πb⊥2 , qAz 4 32p0 !# ~A⊥ ∂ ~A⊥ ∂ +i¯hq ~pb⊥ · + ·~pb⊥ ∂z ∂z ) 
∂ 2 2 2 2 − πb⊥ , i¯h q A⊥ + p˜ I, ∂z ( ! ~ ~   1 ∂ A ∂ A ⊥ ⊥ Ob(1) = − 2 πb⊥2 , qAz + i¯hq ~pb⊥ · + ·~pb⊥ ∂z ∂z 2p0 )
∂ 2 2 2 q A⊥ + p˜ iσy , + i¯h ∂z

(5.22)

219

Scalar Theory: Spin-0 and Spinless Particles

with the odd term Ob(1) of order 1/p20 . By another transformation of the same type as in (5.20) with Sb2 = (i/2p0 )σz Ob(1) , we can transform b H(1) into a b H(2) with an Ob(2) 3 (1) (1) b of order 1/p0 . Since E , the even part of b H , itself is a sufficiently good approximation for our purpose, we shall not continue the process of transformations further. Hence, we write (1)

∂ i¯h ∂z

ψ+ (1) ψ−

!

(1)

ψ+ (1) ψ−

b(1)

=H

! ,

b H(1) = −p0 σz + Eb(1) ,

(5.23)

dropping the odd term. !

(1)

Let us now look at

ψ+ (1) ψ−

corresponding to ψ(~r) = φ~p0 (~r), the plane wave, in

free space. We get, using (5.15) (1)

ψ+ (1) ψ−

2

! iSb1



=e

 ≈

1  2

ψ+ ψ−



 pb − ⊥2 σx 4p

=e

0

pb2

0

pb2

− 4p⊥2



  1+     1−

− 4p⊥2

1

ψ+ ψ−

1

0

   1+ 1 =   2  1−

(1)

p0z p0

p0z p0

p0z p0



p0z p0



 φ~p0    φ~p0

 φ~p0   ,    2   p p − 1 + p0z0 4p0⊥2 φ~p0  − 1−

p0z p0



p20⊥ 4p20



(5.24)

0

(1)

showing that ψ+  ψ− for a quasiparaxial beam. The result easily extends to any (1) (1) wave packet of the form in (5.16). Thus, in general, we can take ψ+  ψ− in (5.23) for the beam wave functions of interest to us. We can express this property (1) (1) (1) (1) that ψ+  ψ− , or essentially ψ− ≈ 0 compared to ψ+ , as (1)

σz

ψ+ (1) ψ−

!

(1)

≈I

ψ+ (1) ψ−

! ,

(5.25)

220

Quantum Mechanics of Charged Particle Beam Optics

where I is the 2 × 2 identity matrix. Then, we can approximate (5.23) further to read ! ! (1) (1) ψ+ ψ + =b H(1) , (1) (1) ψ− ψ− 
2
1 1 b πb⊥2 + p˜2 + 3 πb⊥2 + p˜2 H(1) ≈ −p0 − qAz + 2p0 8p0 (" !# ~A⊥ ∂ ~A⊥
 2  1 ∂ 2 2 − πb⊥ + p˜ , πb⊥ , qAz + i¯hq ~pb⊥ · + ·~pb⊥ ∂z ∂z 32p40  )!
∂ q2 A2⊥ + p˜2 − πb⊥2 , i¯h I. (5.26) ∂z ∂ i¯h ∂z

Now, we have two choices to proceed. If we want to study the z-evolution of the wave function of a system, we can specify it in the original representation as ψ(~r⊥ , z), get it transformed to the representation (1)

ψ+ (1) ψ−

! iSb1

=e M

ψ πbz p0 ψ

! ,

(5.27)

and study its z-evolution according to (5.26) with b H(1) as the optical Hamiltonian. In other words, we can model the wave function of the system directly in the representation (5.27). The Hermitian operators for the observables of interest, like~r⊥ and ~pb⊥ , also need to be changed to the representation (5.27) as b(1) = eiSb1 M OIM b −1 e−iSb1 , O

(5.28)

where I is the 2 × 2 identity matrix. The other choice, which we shall follow for the scalar theory, is to get the zevolution equation for the original wave function ψ(~r⊥ , z) by retracing the transfor  ψ1 mations and rewriting (5.26) in terms of in which ψ1 = ψ. In this case, ψ2 we can use the usual Hermitian operators for the observables. It may look like that retracing the transformations will get us only back to the original evolution equation with the old Hamiltonian. This is not so. Of course, we will get back to the original representation for the wave function. But, due to the transformation and truncation of the resulting series, what we get will be the series of paraxial and nonparaxial approximations of the optical Hamiltonian in the original representation. With this understanding, let us proceed as follows. Substituting in (5.26), 

ψ1 ψ2

 =M

−1



ψ+ ψ−

(1)

 =M

−1 −iSb1

e

ψ+ (1) ψ−

! ,

(5.29)

221

Scalar Theory: Spin-0 and Spinless Particles

we get ∂ i¯h ∂z



ψ1 ψ2



" !# (1) ∂ ψ b + M −1 e−iS1 = i¯h (1) ∂z ψ− " !# (1) ∂ ψ+ −iSb1 −1 e = M i¯h (1) ∂z ψ− !     (1) ∂ −iSb1 ψ+ −iSb1 b(1) −1 e +e H =M i¯h (1) ∂z ψ−       ∂ −iSb1 ψ1 b (1) b = M −1 i¯h e + e−iS1 b H eiS1 M ψ2 ∂z        ∂ ψ1 b b b (1) iSb1 e−iS1 eiS1 + e−iS1 b H e M = M −1 i¯h ψ2 ∂z        ∂ ψ1 b b (1) iSb1 b eiS1 + e−iS1 b H e M = M −1 −i¯he−iS1 ψ2 ∂z   ψ1 =b Ho , (5.30) ψ2

b o becomes where H    
1 1  2 2 2 b b π + p˜ Ho ≈ −p0 − qAz + I − 2 πb⊥ , qAz 2p0 ⊥ 4p0 ( ! )! 2 ~A⊥ ∂ ~A⊥ ∂ A ∂ + i¯hq ~pb⊥ · + ·~pb⊥ − i¯hq2 ⊥ σz ∂z ∂z ∂z (  2
2

  1 1 2 2 πb⊥ + p˜ − πb⊥ + p˜2 , πb⊥2 , qAz + 3 4 16p0 8p0 !#  )! ~A⊥ ∂ ~A⊥
∂ ∂ 2 2 2 2 + ·~pb⊥ − πb⊥ , i¯h q A⊥ + p˜ I. +i¯hq ~pb⊥ · ∂z ∂z ∂z (5.31) Since b Ho is diagonal, the equation (5.30) describes the z-evolution of ψ1 and ψ2 separately. We are interested in the z-evolution of ψ = ψ1 , and hence we write i¯h

∂ ψ (~r⊥ , z) b = Ho ψ (~r⊥ , z) , ∂z   2 
b o ≈ −p0 − qAz + 1 πb2 + p˜2 − 1 H πb⊥ , qAz ⊥ 2 2p0 4p0 ( ! )! 2 ∂ ~A⊥ ∂ ~A⊥ ~ 2 ∂ A⊥ ~ + i¯hq pb⊥ · + · pb⊥ − i¯hq ∂z ∂z ∂z

222

Quantum Mechanics of Charged Particle Beam Optics


2 1 1 + 3 πb⊥2 + p˜2 − 16p40 8p0

("
  πb⊥2 + p˜2 , πb⊥2 , qAz

!# ~A⊥ ∂ ~A⊥ ∂ + ·~pb⊥ +i¯hq ~pb⊥ · ∂z ∂z  
∂ − πb⊥2 , i¯h q2 A2⊥ + p˜2 . ∂z

(5.32)

We have derived the above z-evolution equation for ψ (~r⊥ , z) starting from the Klein– Gordon equation which, as we have seen, does not have a proper probability interpretation. However, let us look at the case of free space in which φ = 0 and ~A = (0, 0, 0). Then, the equation (5.32) reduces to   pb2⊥ ∂ ψ (~r⊥ , z) b b i¯h = Ho ψ (~r⊥ , z) , Ho = −p0 + , (5.33) ∂z 2p0 which has a Hermitian Hamiltonian. It is analogous to the nonrelativistic two dimensional Schr¨odinger equation for a free particle in which z is taken as time, p0 is taken as the mass of the particle, and the constant potential-like term −p0 is ignorable. In RR this case, the z-evolution of ψ (~r⊥ , z) is unitary, dxdy |ψ (~r⊥ , z)|2 is conserved, and hence |ψ (~r⊥ , z)|2 can be taken as the probability of finding the particle at the position (x, y) in the vertical xy-plane at the point z on the z-axis (optic axis). HowRR ever, in general, it is clear that dxdy |ψ (~r⊥ , z)|2 need not be conserved. Only RRR 2 dxdydz |ψ (~r⊥ , z)| = 1 has to be conserved as it represents the probability of finding the particle somewhere in space. Hence, it is not surprising that the Hamiltonian in (5.32) is not Hermitian and the corresponding z-evolution is not unitary. In fact, one should expect a loss of intensity of the forward propagating beam along the optic axis, since there will, in general, be reflections at the boundaries of the system as we have seen in the case of scattering by potential well and potential barrier. Actually, we are dealing with the scattering states of the beam particle. Thus, it is natural that there are non-Hermitian terms in Ho (terms ∝ 1/p20 ). In the above expansion scheme, in general, among the terms ∝ (even powers of 1/p0 ), alternate terms will be non-Hermitian. Of course, the effect of these non-Hermitian terms can b be expected to be quite small and negligible. Hence, we can approximate H   o , further, 1 b b† by Hermitianizing it, i.e., writing it as the sum of its Hermitian part 2 Ho + H o   1 b † b and the anti-Hermitian part 2 Ho − Ho and dropping the anti-Hermitian part. This leads to i¯h

∂ ψ (~r⊥ , z) co ψ (~r⊥ , z) =H ∂z

or,

∂ co |ψ(z)i, |ψ(z)i = H ∂z

2 1 πb⊥2 + p˜2 + 3 πb⊥2 + p˜2 8p0 i¯h

co = −p0 − qAz + 1 H 2p0 ("
  1 − πb⊥2 + p˜2 , πb⊥2 , qAz 4 16p0

223

Scalar Theory: Spin-0 and Spinless Particles

∂ ~A⊥ ∂ ~A⊥ ~ +i¯hq ~pb⊥ · + · pb⊥ ∂z ∂z with p20 =

!#


1 E 2 − m2 c4 , 2 c

 
∂ − πb⊥2 , i¯h q2 A2⊥ + p˜2 , ∂z   qφ 2qEφ 2 1− p˜ = , (5.34) c2 2E

co is the basic Hamiltonian of the scalar theory of quantum charged particle Thus, H beam optics applicable when the spin of the particle is zero or ignored. With the quantum charged particle beam optical Hamiltonian, or simply the quantum beam co , being Hermitian we can take ψ(~r⊥ , z) to be normalized optical Hamiltonian, H over the xy-plane at any z, i.e., for any z, we shall take hψ(z)|ψ(z)i =

Z Z

dxdy |ψ (~r⊥ , z) |2 = 1.

(5.35)

R RR Hereafter, we shall write d 2 r⊥ for dxdy. It may be noted that we have obtained

from the Kelin–Gordon a one-way propagation equation, along the +z-direction, for the wave function representing the charged particle beam using the Feshbach– Villars-like formalism and the Foldy–Wouthuysen-like transformation technique. Detailed and rigorous studies on one-way wave equation modeling in two-way wave propagation problems have been made by Fishman et al. (see Fishman, de Hoop, and van Stralen [51], Fishman [52], and references therein). Having obtained the basic quantum beam optical Hamiltonian operator, we can proceed to get the relation between ψ (~r⊥ , zi ), wave function in the plane vertical to the optic axis at the point zi where the beam enters the system, and ψ (~r⊥ , z) wave function in the plane vertical to the optical axis at any other point of interest (z). By formal integration of (5.34), using the analogous result (3.449) for the timedependent Schr¨odinger equation, we can write b (z, zi ) |ψ (zi )i , |ψ(z)i = U   R co (z) − h¯i zz dz H b i U (z, zi ) = P e  2 Z z Z z2 Z i z co (z2 ) H co (z1 ) co (z1 ) + − i dz1 H dz1 H dz2 h¯ zi h¯ zi zi   Z Z z2 Z z3 i 3 z co (z3 ) H co (z2 ) H co (z1 ) + · · · , dz1 H + − dz2 dz3 h¯ zi zi zi (5.36)

=I−

b (z, zi ) is seen to satisfy the with I as the identity operator. The z-evolution operator U relations ∂ b coU b (z, zi ) , i¯h U (z, zi ) = H ∂z ∂ b† co , b (z, zi )† H (z, zi ) = −U i¯h U ∂z b (z, zi )† U b (z, zi ) = I, b (zi , zi ) = I. U U (5.37)

224

Quantum Mechanics of Charged Particle Beam Optics

b (z, zi ) in the Dyson form (3.454), i.e., as an We can write the z-evolution operator U z-ordered exponential. For our purpose, the most convenient form of expression for b (z, zi ) is the equivalent Magnus form U b (z, zi ) = e− h¯i Tb(z,zi ) , U Z z

co (z1 ) dz1 H  Z z Z z2 h i i 1 co (z2 ) , H co (z1 ) − dz2 dz1 H + 2 h¯ zi zi 2 Z z  Z z3 Z z2 nhh i i 1 co (z3 ) , H co (z2 ) , dz3 − dz2 + dz1 H h¯ 6 z zi zi i i hh i io c c c co (z3 ) + · · · . Ho (z1 ) + Ho (z1 ) , Ho (z2 ) , H (5.38)

Tb (z, zi ) =

zi

co (z) is Hermitian, such that U b (z, zi ) is uniNote that Tb (z, zi ) is Hermitian, when H tary. Analogous to the semigroup property (3.167) of the time-evolution operator, we have the semigroup property for the z-evolution operator: for z > zn > zn−1 > · · · > z2 > z1 > zi , b (z, zi ) = U b (z, zn ) U b (zn , zn−1 ) . . . U b (z2 , z1 ) U b (z1 , zi ) . U

(5.39)

> z0

This follows from the observation that, for z > zi ,



 b z0 , zi |ψ (zi )i , b z, z0 ψ z0 , ψ z0 = U |ψ(z)i = U b (z, zi ) |ψ (zi )i , |ψ(z)i = U

(5.40)

The integral form of the scalar quantum charged particle beam optical Schr¨odinger equation (5.34) becomes Z

ψ (~r⊥ , z) =

d 2 r⊥i K (~r⊥ , z;~r⊥i , zi ) ψ (~r⊥i , zi ) ,

(5.41)

where the kernel of z-propagation, or the z-propagator, is given by b (z, zi ) |~r⊥i i K (~r⊥ , z;~r⊥i , zi ) = h~r⊥ | U Z

=



0 2 0 0 b (z, zi ) δ 2 ~r⊥ d 2 r⊥ δ ~r⊥ −~r⊥ U −~r⊥i .

(5.42)

We have seen from Ehrenfest’s theorem that quantum averages, or the expectation values, of observables behave like the corresponding classical observables. Hence, to study the quantum corrections to the classical transfer maps of the optical elements, we have to find the transfer maps for the corresponding quantum averages. When we want to relate the values of the quantum averages of the beam at two different points along the optic axis of the system, we can use the Heisenberg picture. For an b the average for the observable O associated with the Hermitian quantum operator O, state |ψ(z)i at the xy-plane at the point z is given by b hOi(z) = hψ(z)|O|ψ(z)i =

Z

b (~r⊥ , z) , d 2 r⊥ ψ ∗ (~r⊥ , z) Oψ

(5.43)

225

Scalar Theory: Spin-0 and Spinless Particles

b with hψ(z)|ψ(z)i = 1. Sometimes we may denote hOi(z) by hOi(z) also. Now, from (5.36), we have b hOi(z) = hψ(z)|O|ψ(z)i D E bU b † (z, zi ) O b (z, zi ) |ψ (zi ) = ψ (zi ) |U D E bU b † (z, zi ) O b (z, zi ) (zi ) , = U

(5.44)

leading to the required transfer map giving the expectation values of the observables in the plane at z in terms of their values in the plane at zi . We are considering only the single particle theory and assuming that all the particles of the beam are in the same quantum state, pure or mixed. If a particle of the monoenergetic quasiparaxial beam entering the optical system, from the free space outside it, is associated with a wave function as we have assumed above, then a representation for the wave function is q Ψ(~r,t) = e−iEt/¯h ψ (~r⊥ , z) , E = m2 c4 + c2 p20 , 1 ψ (~r⊥ , z) = (2π h¯ )3/2

Z Z

i

d px d py C (px , py ) e h¯ (~p⊥ ·~r⊥ +pz z) ,

p2⊥ + p2z = p20 , |~p⊥ |  pz ≈ p0 , 2 d px d py C (px , py ) = 1.

Z Z

(5.45)

When the particle is not in a pure state representable by a wave function as above, it has to be represented by a mixed state density operator 



ρb(z,t) = ∑ p j e−iEt/¯h ψ j (z) ψ j (z) eiEt/¯h = ∑ p j ψ j (z) ψ j (z) , j

j

∑ p j = 1,



 ψ j (z) ψ j (z) = 1,

(5.46)

j

where for each j

 ~r⊥ ψ j (z) = ψ j (~r⊥ , z) =

1 (2π h¯ )3/2

Z Z

i

d px d py C j (px , py ) e h¯ (~p⊥ ·~r⊥ +pz z) ,

p2⊥ + p2z = p20 , |~p⊥ |  pz ≈ p0 , 2 d px d py C j (px , py ) = 1.

Z Z

(5.47)

Note that ρb(z,t) is independent of time when the beam is monoenergetic which, of course, is an ideal situation never realized in practice. However, we shall assume the beam to be monoenergetic. In the position representation, the matrix elements of ρb in the xy-plane at z are given by 0
0 h~r⊥ | ρb(z) ~r⊥ = ∑ p j ψ j (~r⊥ , z) ψ ∗j ~r⊥ ,z . (5.48) j

226

Quantum Mechanics of Charged Particle Beam Optics

Note that at any z we have


Tr(ρb(z)) = ∑ p j Tr ψ j (z) ψ j (z) j



 = ∑ p j ψ j (z) ψ j (z) = ∑ p j = 1, j

(5.49)

j

as required. If we are representing the beam particle by such a density matrix, then the average of an observable in the xy-plane at z will be given by   b . (5.50) hOi(z) = Tr ρb(z)O It follows from (5.36) that the z-evolution of the density matrix will be given by b (z, zi ) ρb (zi ) U b † (z, zi ) . ρb(z) = U

(5.51)

This leads to the transfer map   b b (z, zi ) ρb (zi ) U b † (z, zi ) O hOi(z) = Tr U   bU b † (z, zi ) O b (z, zi ) . = Tr ρb (zi ) U

(5.52)

Note that, as follows from (5.44) and (5.52), this transfer map depends essentially on b (z, zi ). U The z-dependent operator bH (z) = U bU b † (z, zi ) O b (z, zi ) O

(5.53)

represents the observable O in the beam optical Heisenberg picture in which the initial state of the particle |ψ (zi )i, or ρb (zi ), remains constant and the observables evolve along the optic axis of the system. From (5.44) we have D E D E D E b (z) = ψ (zi ) O bH ψ (zi ) = O bH (zi ) O (5.54) From (5.37), it follows that     bH dO ∂ b† ∂ b † b b b b i¯h = i¯h U (z, zi ) OU (z, zi ) + U (z, zi ) O U (z, zi ) dz ∂z ∂z ! b b † (z, zi ) ∂ O U b (z, zi ) + i¯h U ∂z o n co O coU bU bH b † (z, zi ) H b (z, zi ) + U b † (z, zi ) O b (z, zi ) = −U ! b ∂ O b † (z, zi ) b (z, zi ) + i¯h U U ∂z n coU bU b † (z, zi ) H b (z, zi ) U b † (z, zi ) O b (z, zi ) = −U

227

Scalar Theory: Spin-0 and Spinless Particles

o coU bH b † (z, zi ) O b (z, zi ) U b † (z, zi ) U b (z, zi ) +U ! b ∂ O b (z, zi ) b † (z, zi ) U + i¯h U ∂z ! h i b ∂ O c bH + i¯h = −Ho,H , O . ∂z

(5.55)

H

or bH dO 1h c b i = −Ho,H , OH + dz i¯h

b ∂O ∂z

! .

(5.56)

H

This equation is the quantum beam optical equation h i of motion for observables and b in the classical limit, with the correspondence A, Bb /i¯h −→ {A, B}, it becomes the co need classical beam optical equation of motion (2.96). Note that a z-dependent H c c c b not commute with U (z, zi ), and hence Ho,H 6= Ho , unless Ho is z-independent. The equation (5.53) is the integral version of (5.56). Since h i h i co,H , O co , O bH = −H b −H (5.57) H

we can also write bH 1 h c bi dO = −Ho , O + dz i¯h H

b ∂O ∂z

! .

(5.58)

H

Correspondingly, D from E (5.54), we get the quantum beam optical equation of motion b for the average O (z) as D E b (z) d O dz

* + dO E d D b bH ψ (zi ) OH ψ (zi ) = ψ (zi ) = ψ (zi ) dz dz * ! + 1 h i b ∂ O c b + = ψ (zi ) −Ho , O ψ (zi ) i¯h ∂z H H * + b ∂O 1 Dh c biE −Ho , O (z) + (z). = i¯h ∂z

(5.59)

b does not have an explicit z-dependence, the equations of motion become When O bH dO 1 h c bi −Ho , O , = dz i¯h H and

D E b (z) d O dz

=

1 Dh c biE −Ho , O (z). i¯h

(5.60)

(5.61)

228

Quantum Mechanics of Charged Particle Beam Optics

Particularly, for~r⊥ and ~pb⊥ /p0 , in which we shall be primarily interested, we have d~r⊥,H ih c i 1h c i Ho ,~r⊥ , −Ho ,~r⊥ = = h¯ dz i¯h H H i h c~ i 1 d~pb⊥,H 1 h c~ i (5.62) −Ho , pb⊥ = Ho , pb⊥ , = p0 dz i¯h p0 h¯ p0 H H and d h~r⊥ i (z) 1 Dh c iE i Dh c iE Ho ,~r⊥ (z), = −Ho ,~r⊥ (z) = h¯ dz i¯h D E ~ 1 Dh c ~ iE 1 d pb⊥ (z) i Dh c ~ iE = −Ho , pb⊥ (z) = Ho , pb⊥ (z). (5.63) p0 dz i¯h p0 h¯ p0 D E To find the transfer maps for h~r⊥ i and ~pb⊥ , we can use, depending on convenience, either (5.63) or

 h~r⊥ i (z) = ~r⊥,H (zi ) D E b† b (z, zi ) ψ (zi ) = ψ (zi ) U (z, zi )~r⊥U i E D i b b = ψ (zi ) e h¯ T (z,zi )~r⊥ e− h¯ T (z,zi ) ψ (zi ) 1 D~ E 1 D~ E pb⊥ (z) = pb (zi ) p0 p0 ⊥,H E 1 D b† b (z, zi ) ψ (zi ) = ψ (zi ) U (z, zi )~pb⊥U p0 i E i b 1 D b = ψ (zi ) e h¯ T (z,zi )~pb⊥ e− h¯ T (z,zi ) ψ (zi ) . (5.64) p0 h i b , Bb , b : Bb = A Let us now recall the identities (3.142 and 3.143): with : A   b −A b b:+ 1 :A b :2 + 1 : A b :3 + · · · Bb = e:A: b b = I+ : A b eA Be B, 2! 3!

(5.65)

and e:aA: = ea:A: . (5.66) b = iTb/¯h and Bb =~r⊥ , and ~p⊥ , we have Using these identities in (5.64), with A b

i

b

h~r⊥ i (z) = hψ (zi )| e h¯ :T (z,zi ):~r⊥ |ψ (zi )i , ( *   1 i 2 b i b = ψ (zi ) I + : T (z, zi ) : + : T (z, zi ) :2 h¯ 2! h¯ ! ) +   1 i 3 b 3 + : T (z, zi ) : + · · · ~r⊥ ψ (zi ) 3! h¯ b

Scalar Theory: Spin-0 and Spinless Particles

229

(   i 1 i 2 b I + : Tb (z, zi ) : + : T (z, zi ) :2 ψ (zi ) h¯ 2! h¯ + ! )   1 i 3 b (5.67) + : T (z, zi ) :3 + · · · ~pb⊥ ψ (zi ) . 3! h¯

1 1 D~ E pb (z) = p0 ⊥ p0

*

b we will have In general, for any observable O, with the quantum operator O, D E D E b (z) = ψ (zi ) e h¯i :Tb(z,zi ): O b ψ (zi ) (5.68) O Note that i b b = e h¯i Tb(z,zi ) Oe b − h¯i Tb(z,zi ) = U bU bH b † (z, zi ) O b (z, zi ) = O e h¯ :T (z,zi ): O

(5.69)

b with no explicit zThe beam optical Heisenberg equation of motion for any O dependence is bH 1h c b i dO −Ho , OH . (5.70) = dz i¯h Directly integrating this equation, a´ la Magnus (see the Appendix at the end of ChapbH (zi ) = O, b we get an expression for O bH (z) equivalent to the one given ter 3), with O in (5.69): b i ) O, b bH (z) = eT(z,z (5.71) O where z co : b (z, zi ) = 1 T dz : −H i¯h zi  2 Z z Z z2 h i 1 1 co : (z2 ) , : −H co : (z1 ) dz2 + dz1 : −H i¯h 2 zi zi  3 Z z Z z3 Z z2 1 1 dz2 dz1 dz3 + 6 i¯h zi zi zi nhh i i co : (z3 ) , : −H co : (z2 ) , : −H co : (z1 ) : −H hh i io co : (z1 ) , : −H co : (z2 ) , : −H co : (z3 ) + : −H

Z

+··· ,

(5.72)

where h i co : (z1 ) , : −H co : (z2 ) : −H co : (z1 ) : −H co : (z2 ) − : −H co : (z2 ) : −H co : (z1 ) = : −H h i co (z1 ) , −H co (z2 ) :, = : −H

(5.73)

230

Quantum Mechanics of Charged Particle Beam Optics

as follows from (3.32). In the classical limit 1 co : O b −→: −Ho : O, : −H i¯h

(5.74)

the equation (5.72) is seen to become the classical equation O(z) = eT (z,zi ) O (zi ) ,

(5.75)

with T (z, zi ) given by (4.97). This shows that in the classical limit, the quantum equation (5.68), which we shall be using throughout in the quantum beam optics for getting the transfer maps, becomes the classical equation (5.75), which is the basis for the Lie transfer operator method for getting the transfer maps in classical beam optics. 5.2.2

FREE PROPAGATION: DIFFRACTION

Let us now consider the propagation of a quasiparaxial beam through free space, or between two optical elements, around the z-axis. The corresponding quantum beam optical Hamiltonian is co ≈ −p0 + 1 pb2 + 1 pb4 , H 2p0 ⊥ 8p30 ⊥

(5.76)

obtained from (5.34) by putting φ = 0 and ~A = (0, 0, 0), and consequently taking πb⊥2 = pb2⊥ and p˜2 = 0. For an ideal paraxial beam, we can take co = −p0 + 1 pb2 , H 2p0 ⊥

(5.77)

only terms up to second order in ~pb⊥ in view of the paraxiality condition retaining ~ pb⊥  p0 assumed to be valid for all particles of the beam. Actually, the free propagation can be treated exactly. The expression for the quantum q beam optical c Hamiltonian Ho in (5.76) is an approximation for the exact result − p2 − pb2 , cor0

⊥

responding to −pz , which will be obtained in the infinite series form if we continue the Foldy–Wouthuysen-like transformation process up to all orders starting with (5.13) corresponding to free propagation. First, let us look at free propagation of a paraxial beam in the Schr¨odinger picture. With the corresponding optical Hamiltonian given by (5.77), the Schr¨odinger equation for free propagation of a paraxial beam along the z-direction becomes  2  ∂ co |ψ(z)i = −p0 + pb⊥ |ψ(z)i. i¯h |ψ(z)i = H (5.78) ∂z 2p0 On integration, we have |ψ(z)i = e

  pb2 − i∆z −p0 + 2p⊥ h¯ 0

|ψ (zi )i ,

∆z = z − zi .

(5.79)

231

Scalar Theory: Spin-0 and Spinless Particles

In position representation, we can write Z

d 2 r⊥i K (~r⊥ , z;~r⊥i , zi ) ψ (~r⊥i , zi ) ,   + * pb2⊥ − i∆z −p + 0 2p h ¯ 0 K (~r⊥ , z;~r⊥i , zi ) = ~r⊥ e ~r⊥i . ψ (~r⊥ , z) =

(5.80)

The Schr¨odinger equation for free propagation of a paraxial beam (5.78), with z replaced by t, is the same as the nonrelativistic Schr¨odinger equation of a free particle of mass p0 moving in the xy-plane, in a constant potential −p0 . We can ignore the constant potential. We have already calculated in (3.313) the propagator for a free particle of mass m. It is straightforward to use (3.313) to derive the desired result:   n o ip0 2 i p0 h¯ p0 ∆z+ 2¯h∆z |~r⊥ −~r⊥i | K (~r⊥ , z;~r⊥i , zi ) = . (5.81) e 2πi¯h∆z Now, the equation (5.80) becomes   Z ip0 2 i p0 ψ (~r⊥ , z) = e h¯ p0 ∆z d 2 r⊥i e 2¯h∆z |~r⊥ −~r⊥i | ψ (~r⊥i , zi ) . 2πi¯h∆z

(5.82)

Substituting 2π h¯ /p0 = λ0 , the de Broglie wavelength of the particle, we get   i2π (z−z ) 1 i ψ(x, y, z) = e λ0 iλ0 (z − zi ) Z Z iπ [(x−xi )2 +(y−yi )2 ] ψ (xi , yi , zi ) , (5.83) × dxi dyi e λ0 (z−zi ) which is the well-known Fresnel diffraction formula. Here, the xy-plane at zi is the plane of the diffracting object and the plane at z is the observation plane. The wave function on the exit side of the diffracting object is ψ (~r⊥ , zi ), and |ψ (~r⊥ , z)|2 gives the intensity distribution of the diffraction pattern (probability distribution for the particle) at the observation plane. It is clear that the paraxial approximation used in deriving (5.77), dropping terms of order higher than second in ~pb⊥ , essentially corresponds to the traditional approximation used in deriving the Fresnel diffraction formula from the general Kirchchoff’s result. Equations (5.34), (5.36), (5.41), and (5.42), represent, in operator form, the general theory of charged particle diffraction in the presence of electromagnetic fields (for more details of the wave theory of electron diffraction, see Hawkes and Kasper [72]). Next, let us work out the transfer maps for the expectation values of ~r⊥ and ~pb⊥ . From (5.36) and (5.38), b (z, zi ) |ψ (zi )i , |ψ(z)i = U b (z, zi ) = e U

n o R co − h¯i zz dz H i

i

= e− h¯ T (z,zi ) , b

  co = ∆z −p0 + 1 pb2 , Tb (z, zi ) = ∆zH 2p0 ⊥

∆z = (z − zi ) .

(5.84)

232

Quantum Mechanics of Charged Particle Beam Optics

co is z-independent. Now, from b (z, zi ) is an ordinary exponential since H Note that U (5.67), with   h i   1 2 b b pb ,~r⊥ : T (z, zi ) :~r⊥ = T (z, zi ) ,~r⊥ = ∆z −p0 + 2p0 ⊥     1 2 pb⊥ = ∆z pb ,~r⊥ = −i¯h∆z , 2p0 ⊥ p0   h i   1 2 ~ ~ b b : T (z, zi ) : pb⊥ = T (z, zi ) , pb⊥ = ∆z −p0 + pb ,~p⊥ 2p0 ⊥     1 2 ~ pb , pb⊥ = 0, = ∆z 2p0 ⊥ h h ii : Tb (z, zi ) :2 ~r⊥ = Tb (z, zi ) , Tb (z, zi ) ,~r⊥ i ∆z h b T (z, zi ) ,~pb⊥ = 0, = −i¯h p0

(5.85)

we get     pb⊥ h~r⊥ i (z) = ψ (zi ) ~r⊥ + ∆z ψ (z ) , i p0 ∆z D~ E = h~r⊥ i (zi ) + pb⊥ (zi ) , p 0 E D E D E D ~pb (z) = ψ (zi ) ~pb ψ (zi ) = ~pb (zi ) . ⊥

⊥

(5.86)

⊥

Thus, the resulting transfer map, h~Dr⊥ i (z) E 1 ~ pb (z)

p0

⊥

!

 =

1 ∆z 0 1



h~Dr⊥ i (z Ei ) 1 ~ pb (zi )

p0

! ,

(5.87)

⊥

is exactly the same as the transfer map (4.16) for the classical ~r⊥ and ~p⊥ in accordance with Ehrenfest’s theorem. 5.2.3 5.2.3.1

AXIALLY SYMMETRIC MAGNETIC LENS: ELECTRON OPTICAL IMAGING Paraxial Approximation: Point-to-Point Imaging

Let us now consider a round magnetic lens. Let us recall that it comprises an axially symmetric magnetic field, and there is no electric field, i.e., φ = 0. Taking the z-axis as the optic axis, the vector potential can be taken, in general, as   ~A = − 1 yΠ (~r⊥ , z) , 1 xΠ (~r⊥ , z) , 0 , 2 2

(5.88)

Scalar Theory: Spin-0 and Spinless Particles

233

with  2 n r 1 − ⊥ B(2n) (z) Π (~r⊥ , z) = ∑ n!(n + 1)! 4 n=0 ∞

1 4 0000 1 2 00 B (z) + = B(z) − r⊥ r B (z) − · · · , 8 192 ⊥

(5.89)

2 = x2 + y2 , B(0) (z) = B(z), B0 (z) = dB(z)/dz, B00 (z) = d 2 B(z)/dz2 , B000 (z) = where r⊥ 3 d B(z)/dz3 , B0000 (z) = d 4 B(z)/dz4 , . . ., and B(2n) (z) = d 2n B(z)/dz2n . The corresponding magnetic field is given by   2 ~B⊥ = − 1 B0 (z) − 1 B000 (z)r⊥ + · · · ~r⊥ , 2 8 1 1 2 4 Bz = B(z) − B00 (z)r⊥ + B0000 (z)r⊥ −··· . (5.90) 4 64

The practical boundaries of the lens, say z` and zr , are determined by where B(z) becomes negligible, i.e., B (z < z` ) ≈ 0 and B (z > zr ) ≈ 0. First, we are concerned with the propagation of a monoenergetic paraxial beam with design momentum p0 such that |~p| = p0 ≈ pz for all its constituent particles. For a paraxial beam propagating through the round magnetic lens, along the optic axis, the vector potential ~A can be taken as   ~A ≈ ~A0 = − 1 B(z)y, 1 B(z)x, 0 , (5.91) 2 2 corresponding to the field   ~B ≈ ~B0 = − 1 B0 (z)x, − 1 B0 (z)y, B(z) . 2 2

(5.92)

In the paraxial approximation, only the lowest order terms in ~r⊥ are considered to contribute to the effective field felt by the particles moving close to the optic axis. Further, it entails approximating the quantum beam optical Hamiltonian by dropping ~ from it the terms of order higher than second in pb⊥ /p0 in view of the condition |~p⊥ | /p0  1. Thus, the quantum beam optical Hamiltonian of the round magnetic lens derived from (5.34) in the paraxial approximation is 2 2 co ≈ −p0 + 1 pb2 + q B(z) r2 − qB(z) L bz H 2p0 ⊥ 8p0 ⊥ 2p0   h¯ 2 q2 0 ~pb +~pb ·~r⊥ , + B(z)B (z) ~ r · ⊥ ⊥ ⊥ 16p40

where

 B(z)

6= 0 =0

in the lens region (z` ≤ z ≤ zr ) outside the lens region (z < z` , z > zr ) ,

(5.93)

(5.94)

234

Quantum Mechanics of Charged Particle Beam Optics

bz = (x pby − y pbx ) is the z-component of the orbital angular momentum operator. and L co , the last term is proportional to h¯ 2 and is very small compared to In the above H the other terms. We will call the terms proportional to powers of h¯ as h¯ -dependent since they contain h¯ explicitly such that their quantum averages, being proportional to powers of h¯ , would vanish in the classical limit h¯ −→ 0. Note that the components of ~pb = −i¯h~∇ also contain h¯ , but we are able to identify their quantum averages with the components of the classical momentum vector ~p. For the present, we shall omit co and take the optical Hamiltonian of the the last h¯ -dependent term from the above H round magnetic lens as 2 2 co = −p0 + 1 pb2 + q B(z) r2 − qB(z) L bz . (5.95) H 2p0 ⊥ 8p0 ⊥ 2p0   Note that the replacement of the quantum operators ~r⊥ ,~pb⊥ by the classical varico exactly to the classical Ho in (4.24). As a result, without the ables (~r⊥ ,~p⊥ ) takes H co leads exactly to the same performance of the lens h¯ -dependent term, the paraxial H as in the classical paraxial theory, except for the replacement of the classical (~r⊥ ,~p⊥ ) by their quantum averages. Later we shall see the effect of the h¯ -dependent nonclassical term at the paraxial level which, of course, will be very small. Introducing the notation qB(z) α(z) = , (5.96) 2p0

we shall write co = H co,L + H co,R , H co,L = −p0 + 1 pb2 + 1 p0 α 2 (z)r2 , H ⊥ 2p0 ⊥ 2 co,R = −α(z)L bz . H

(5.97)

Let us consider the propagation of a paraxial beam from the xy-plane at zi < z` to the xy-plane at z > zr . We have from (5.36) and (5.38) bp (z, zi ) |ψ (zi )i , |ψ(z)i = U   R co i b − h¯i zz dz H b i U p (z, zi ) = P e = e− h¯ T (z,zi ) , co is z-dependent. Now, using (5.38), we have where it has to be noted that H Z z   co,L (z1 ) + H co,R (z1 ) + Tb (z, zi ) = dz1 H z   Zi z Z z2 h  1 i co,L (z2 ) + H co,R (z2 ) , − dz2 dz1 H 2 h¯ zi zi  i co,L (z1 ) + H co,R (z1 ) + H

(5.98)

235

Scalar Theory: Spin-0 and Spinless Particles

  Z Z z3 Z z2 nhh  i 2 z 1 co,L (z3 ) + H co,R (z3 ) , − dz3 dz2 dz1 H 6 h¯ zi zi z  i i i co,L (z2 ) + H co,R (z2 ) , H co,L (z1 ) + H co,R (z1 ) + H   i hh co,L (z1 ) + H co,R (z1 ) , H co,L (z2 ) + H co,R (z2 ) , H  io co,L (z3 ) + H co,R (z3 ) H +··· .

(5.99)

Note that h

i bz = 0, since p0 = constant, p0 , L h i       2 b r⊥ , Lz = x2 + y2 , (x pby − y pbx ) = y2 , x pby − x2 , y pbx h

bz pb2⊥ , L

i

= 2i¯h(xy − yx) = 0,       = pb2x + pb2y , (x pby − y pbx ) = pb2x , x pby − pb2y , y pbx = 2i¯h( pbx pby − pby pbx ) = 0.

(5.100)

This makes   h



i co,L z0 , H co,R z00 = −p0 + 1 pb2 + 1 p0 α 2 z0 r2 , α z00 L b H z ⊥ 2p0 ⊥ 2 = 0, for all z0 , z00 , h i h



i co,R z0 , H co,R z00 = α z0 L bz , α z00 L bz H

h
co,L z0 , H co,L H

= 0, for all z0 , z00 , 
2
i 1 2 1 pb + p0 α 2 z0 r⊥ , z00 = −p0 + 2p0 ⊥ 2 
2 1 2 1 pb + p0 α 2 z00 r⊥ −p0 + 2p0 ⊥ 2 

 i¯h 2 0
~r⊥ ·~pb⊥ +~pb⊥ ·~r⊥ = α z − α 2 z00 2 6= 0, in general. (5.101)

As a result, we get Tb (z, zi ) = TbR (z, zi ) + TbL (z, zi ) ,

(5.102)

with Z z

TbR (z, zi ) = θ (z, zi ) =

z Z iz

co,L (z) = −θ (z, zi ) L bz , dz H dz α(z) =

zi

q 2p0

Z z

dz B(z), zi

(5.103)

236

Quantum Mechanics of Charged Particle Beam Optics

and  Z z Z z2 h i i 1 co,L (z2 ) , H co,L (z1 ) + − dz2 dz1 H h¯ 2 zi zi zi  2 Z z Z z3 Z z2 nhh   i 1 i co,L (z3 ) , H co,L (z2 ) , − dz3 dz2 dz1 H 6 h¯ zi z zi  i hhi   i  io c c co,L (z2 ) , H co,L (z3 ) Ho,L (z1 ) + Ho,L (z1 ) , H +··· . Z z

TbL (z, zi ) =

co,L (z) + dz H

(5.104) Since

h i TbR (z, zi ) , TbL (z, zi ) = 0,

(5.105)

we have bp (z, zi ) = e− h¯i Tb(z,zi ) = e− h¯i (TbR (z,zi )+TbL (z,zi )) U i

i

i

i

= e− h¯ TR (z,zi ) e− h¯ TL (z,zi ) = e− h¯ TL (z,zi ) e− h¯ TR (z,zi ) bR (z, zi ) U bL (z, zi ) = U bL (z, zi ) U bR (z, zi ) . =U b

b

b

b

(5.106)

Thus, in the Schr¨odinger picture, we have Z



d 2 r⊥,i K ~r⊥ , z;~r⊥,i , zi ψ ~r⊥,i , zi Z D E
b = d 2 r⊥,i ~r⊥ , z U p (z, zi ) ~r⊥,i , zi ψ ~r⊥,i , zi Z Z ED D E
b 0 0 0 b ~r⊥ U ~r⊥ = d 2 r⊥,i d 2 r⊥ UL (z, zi ) ~r⊥,i ψ ~r⊥,i , zi . R (z, zi ) ~r⊥

ψ (~r⊥ , z) =

(5.107) bR through Let us first compute the required matrix element of the rotation operator U an angle θ around the z-axis. To this end, let us choose the cylindrical coordinate system: x = ρ cos ϑ , y = ρ sin ϑ , z = z. Then, i

i

e h¯ θ Lz ψ(x, y, z) = e h¯ θ (x pby −y pbx ) ψ(x, y, z) b

= eθ (∂ /∂ ϑ ) ψ(ρ, ϑ , z) = ψ(ρ, ϑ + θ , z) = ψ(x(θ ), y(θ ), z),      x(θ ) cos θ − sin θ x with = . y(θ )

sin θ

cos θ

y

Using this result, we get E D i E D bz 00 00 0 b 0 h¯ θ (z,zi )L ~r⊥ = ~r⊥ ~r⊥ e UR (z, zi ) ~r⊥ Z

=


i θ (z,z )Lb 2
0 00 i zδ d 2 r⊥ δ 2 ~r⊥ −~r⊥ e h¯ ~r⊥ −~r⊥

(5.108)

237

Scalar Theory: Spin-0 and Spinless Particles Z



2 00 0 δ ~r⊥ (θ (z, zi )) −~r⊥ d 2 r⊥ δ 2 ~r⊥ −~r⊥
00 0 (θ (z, zi )) −~r⊥ , (5.109) = δ 2 ~r⊥ =

0 (θ (z, z )) is~r (θ (z, z )) with (x, y) replaced by (x0 , y0 ) (see (5.108)). Thus, where~r⊥ i i ⊥ substituting the required matrix element,

E D
b 0 2 0 ~r⊥ U R (z, zi ) ~r⊥ = δ ~r⊥ (θ (z, zi )) −~r⊥ ,

(5.110)

in (5.107), we get Z

d 2 r⊥,i

Z

Z

d 2 r⊥,i KL ~r⊥ θ (z, zi ) , z;~r⊥,i , zi


0 0 d 2 r⊥ δ 2 ~r⊥ (θ (z, zi )) −~r⊥ D E
0 b × ~r⊥ UL (z, zi ) ~r⊥,i ψ ~r⊥,i , zi Z E D
b = d 2 r⊥,i ~r⊥ (θ (z, zi )) U (z, z ) r ~ L i ⊥,i ψ ~r⊥,i , zi

ψ (~r⊥ , z) =

=






ψ ~r⊥,i , zi .

(5.111)



The exact expression for the propagator KL ~r⊥ θ (z, zi ) , z;~r⊥,i , zi can be written co,L , except for the additive constant −p0 , is exactly like the Hamildown. Note that H tonian of a two-dimensional isotropic harmonic oscillator with time-dependent frequency. Such a connection between optics and harmonic oscillator is of course well known (see, e.g., Agarwal and Simon [2] in the context of light optics and Dattoli, Renieri, and Torre [31] in the context of charged particle optics). For such quantum systems with time-dependent Hamiltonians quadratic in ~r and ~pb, the exact propagators have been found using the Feynman path integral method (see, e.g., Khandekar and Lawande [121], and Kleinert [122]). We shall closely follow Wolf [190] who has given a method, using techniques of canonical transformations and Lie algebra, to get the propagator for any quantum system with a time-dependent Hamiltonian quadratic in~r and ~pb (see the Appendix at the end of this chapter for an outline of the method applied in our case). The resulting expression for KL is

KL ~r⊥ θ (z, zi ) , z;~r⊥,i , zi  i p0 = exp p0 ∆z i2π h¯ h (z, zi ) h¯    ip0 2 2 + g (z, zi ) r⊥,i − 2~r⊥,i ·~r⊥ (θ (z, zi )) + h0 (z, zi ) r⊥ , 2¯hh (z, zi ) if h (z, zi ) 6= 0,

(5.112)

238

Quantum Mechanics of Charged Particle Beam Optics

and

KL ~r⊥ θ (z, zi ) , z;~r⊥,i , zi     i ip0 g0 (z, zi ) 2 ~r⊥ (θ (z, zi )) 1 2 exp p0 ∆z + r δ ~r⊥,i − , = g (z, zi ) h¯ 2¯hg (z, zi ) ⊥ g (z, zi ) if h (z, zi ) = 0, (5.113) where g (z, zi ), h (z, zi ), g0 (z, zi ), and h0 (z,zi ) arethe elements of  the classical trans~ ~ fer matrix (4.53) for ~r⊥ (zi ) , P⊥ (zi ) /p0 −→ ~r⊥ (z), P⊥ (z)/p0 , the two linearly independent solutions of the classical paraxial equation (4.51), satisfying the initial conditions g (zi , zi ) = h0 (zi , zi ) = 1,

h (zi , zi ) = g0 (zi , zi ) = 0,

(5.114)

and the relation g (z, zi ) h0 (z, zi ) − h (z, zi ) g0 (z, zi ) = 1,

for any z ≥ zi .

(5.115)


Note that when z = zi , KL = δ 2 ~r⊥,i −~r⊥ as required. Now, from (5.111)–(5.113), it follows that    i2π h0 (z, zi ) 2 1 exp (z − zi ) + r ψ (~r⊥ , z) = iλ0 h (z, zi ) λ0 2h (z, zi ) ⊥   Z   iπ 2 2 × d r⊥,i exp g (z, zi ) r⊥,i − 2~r⊥,i ·~r⊥ (θ (z, zi )) , λ0 h (z, zi ) if h (z, zi ) 6= 0,   i2π g0 (z, zi ) 2 1 exp (z − zi ) + r ψ (~r⊥ , z) = g (z, zi ) λ0 2g (z, zi ) ⊥   ~r⊥ (θ (z, zi )) ×ψ , zi , g (z, zi ) 

if h (z, zi ) = 0.

(5.116)

This equation represents the well-known general law of propagation of the paraxial wave function in the case of a round magnetic lens (see Glaser [59, 62]), Glaser and Schiske [60, 61], and Hawkes and Kasper [72]). The phase factor exp {i2π (z − zi ) /λ0 } would not be there if we drop the constant term −p0 in the optical Hamiltonian. Usually, a rotated coordinate frame is introduced, as we have done in the classical treatment, to avoid the rotation factor θ (z, zi ) in the final z-plane. Equation (5.116) is the basis for the development of Fourier transform methods in the electron optical imaging techniques (for details, see Hawkes and Kasper [72]). Let us take zi = zob j < z` , where the object plane is, and z = zimg >
zr , where the image of the object is formed.
As is clear from (5.116), if h zimg
, zob j = 0, then we have, with M = g zimg , zob j and θL = θ (z` , zr ) = θ zimg , zob j ,

239

Scalar Theory: Spin-0 and Spinless Particles

#!
g0 zimg , zob j 2 ψ ~r⊥,img , zimg zimg − zob j + r⊥,img 2M   1 ×ψ ~r⊥,img (θL ) , zob j , M   2
ψ ~r⊥,img , zimg 2 = 1 ψ 1 ~r⊥,img (θL ) , zob j . (5.117) 2 M M


1 = exp M

i2π λ0

"





This equation shows that at the image plane at z = zimg , where h zimg , zob j = 0, the intensity distribution at the object
plane is reproduced exactly, point-to-point, with the magnification M = g zimg , zob j and a rotation through the angle
Z θL = θ zob j , zimg =

zimg

dz α(z)

zob j

=

q 2p0

Z zimg

dz B(z) = zob j

q 2p0

Z zr

dz B(z).

(5.118)

z`

Note that B(z) = 0 outside the lens region (z < z` , z > zr ). This image rotation in a single-stage electron optical imaging using a round magnetic lens is the effect of Larmor rotation of any charged particle in a magnetic field. Let us observe that, as should be, the total intensity is conserved:   2 Z Z
2 1 1 2 2 d r⊥,img ψ ~r⊥,img , zimg = 2 d r⊥,img ψ ~r⊥,img (θL ) , zob j M M   2 Z 1 1 2 = 2 d r⊥,img (θL ) ψ ~r⊥,img (θL ) , zob j M M   2 Z 1 1 2 d r⊥,ob j ψ = ~r⊥,ob j , zob j 2 M M Z
2 2 = d r⊥,ob j ψ ~r⊥,ob j , zob j . (5.119) Let us now understand the Gaussian, point-to-point, or stigmatic imaging D E of an object by the round magnetic lens through the transfer maps for h~r⊥ i and ~pb⊥ /p0 . To this D E end, we shall use the z-evolution equations in (5.63) for h~r⊥ i and ~pb⊥ /p0 . We get d i Dh c iE hxi (z) = Ho , x (z) dz h¯    i 1 2 1 2 2 b = −p0 + pb + p0 α (z)r⊥ − α(z)Lz , x (z) h¯ 2p0 ⊥ 2 1 h pbx i (z) + α(z)hyi(z), = p0 i Dh c iE d hyi (z) = Ho , y (z) dz h¯

240

Quantum Mechanics of Charged Particle Beam Optics

   i 1 2 1 2 bz , y (z) pb⊥ + p0 α 2 (z)r⊥ −p0 + − α(z)L h¯ 2p0 2 1  pby (z) − α(z)hxi(z), = p0 iE i Dh c 1 d h pbx i (z) = Ho , pbx (z) p0 dz h¯ p  0   i 1 2 1 2 2 b = −p0 + pb + p0 α (z)r⊥ − α(z)Lz , pbx (z) h¯ p0 2p0 ⊥ 2 1 = −α 2 (z)hxi(z) + α(z)h pby i(z) p0 iE i Dh c 1 d  pby (z) = Ho , pby (z) h¯ p p0 dz  0   i 1 2 1 2 2 b = −p0 + pb + p0 α (z)r⊥ − α(z)Lz , pby (z) h¯ p0 2p0 ⊥ 2 1 (5.120) = −α 2 (z)hyi(z) − α(z)h pbx i(z). p0 =

We can write these equations of motion as   hxi(z) 0 α(z) 1 0 hyi(z)   −α(z) d  0 0 1  1 = 2 (z)    b h i p (z) −α 0 0 α(z) x dz p0  1 0 −α 2 (z) −α(z) 0 by (z) p0 p 

Let us write this as  hxi(z) hyi(z) d   1 dz  p0 h pbx i (z) 1 by (z) p0 p

 hxi(z)   hyi(z)  i 1    p h pbx i (z)  . 0

 1 by (z) p0 p (5.121)  

 hxi(z)   hyi(z)   = [µ(z) ⊗ I + I ⊗ ρ(z)]  1    p h pbx i (z)  , 0

 1 by (z) p0 p 



(5.122)

with  µ(z) =

0 1 −α 2 (z) 0

 ,

(5.123)

.

(5.124)

and  ρ(z) = α(z)

0 −1

1 0



It may be noted that these equations are exactly the same as in the classical theory, except for the replacement of the classical~r⊥ and ~p⊥ by their corresponding quantum D E ~ averages h~r⊥ i (z) and pb⊥ (z). So, to make the present chapter on quantum theory self-contained, we shall be essentially repeating the same equations and statements in the chapter on classical theory.

241

Scalar Theory: Spin-0 and Spinless Particles

Let us now integrate (5.122). Noting that µ(z) ⊗ I and I ⊗ ρ(z) commute with each other, let  hxi (zi ) hyi (zi )   1 h pbx i (zi )  p0

 1 by (zi ) p0 p

  hxi(z)   hyi(z)     1  p h pbx i (z)  = M (z, zi ) R (z, zi )  0

 1 by (z) p0 p 

for any z ≥ zi ,

(5.125)

where dM (z, zi ) = (µ(z) ⊗ I) M (z, zi ) , dz dR (z, zi ) = (I × ρ(z)) R (z, zi ) , dz

M (zi , zi ) = I,

(5.126)

R (zi , zi ) = I.

(5.127)

dM (z, zi ) dM (z, zi ) = ⊗ I, dz dz

(5.128)

It is clear that we can write M (z, zi ) = M (z, zi ) ⊗ I, with dM (z, zi ) = µ(z)M (z, zi ) , dz

(5.129)

and R((z, zi )) = I ⊗ R (z, zi ) ,

dR (z, zi ) dR (z, zi ) =I⊗ , dz dz

(5.130)

with dR (z, zi ) = ρ(z)R (z, zi ) . dz

(5.131)

Note that M (z, zi ) and R (z, zi ) commute with each other, dM (z, zi ) /dz commutes with R((z, zi )), and dR (z, zi ) /dz commutes with M ((z, zi )). Then,  hxi(z) hyi(z)  d   1   h pbx i (z)  dz p0

 1 by (z) p0 p 

 hxi (zi ) hyi (zi )  dR (z, zi )  dM (z, zi )  1  R (z, zi ) + M (z, zi )  h pbx i (zi )  dz dz p0

 1 by (zi ) p0 p 

 =



242

Quantum Mechanics of Charged Particle Beam Optics

 hxi (zi )  hyi (zi )   = [µ(z) ⊗ I + I ⊗ ρ(z)]M (z, zi ) R (z, zi )   p1 h pbx i (zi )  0

 1 by (zi ) p0 p   hxi(z)  hyi(z)   = [µ(z) ⊗ I + I ⊗ ρ(z)]  (5.132)  p1 h pbx i (z)  , 0

 1 by (z) p0 p 

as required by (5.122). The equation for R (z, zi ), (5.127), can be readily integrated by ordinary exponentiation, since [ρ (z0 ) , ρ (z00 )] = 0 for any z0 and z00 . Thus, we have Rz

R (z, zi ) = e

zi

dz ρ(z)

    0 1 = exp I ⊗ θ (zi , z) −1 0   cos θ (zi , z) sin θ (zi , z) =I⊗ − sin θ (zi , z) cos θ (zi , z) = I ⊗ R (z, zi ) , with

(5.133)

Z z

θ (zi , z) =

dz α(z) = zi

q 2p0

Z z

(5.134)

dz B(z). zi

Let us now introduce an XY z-coordinate system with its X and Y axes rotating along the z-axis at the rate dθ (z)/dz = α(z) = qB(z)/2p0 , such that we can write     x(z) X(z) = R (z, zi ) , (5.135) y(z) Y (z) or 

X(z) Y (z)





−1

x(z) y(z)



= R (z, zi )    cos θ (zi , z) − sin θ (zi , z) x(z) = . sin θ (zi , z) cos θ (zi , z) y(z)

(5.136)

Then, we have  or



PX (z) PY (z)



 =

px (z) py (z)



 = R (z, zi )

PX (z) PY (z)

cos θ (zi , z) − sin θ (zi , z) sin θ (zi , z) cos θ (zi , z)





, px (z) py (z)

(5.137)  .

(5.138)

where PX and PY are the components of momentum in the rotating coordinate system. Note that in the vertical plane at z = zi , where the particle enters the system,

243

Scalar Theory: Spin-0 and Spinless Particles

XY z coordinate system coincides with the xyz coordinate system. Thus, we can write (5.125) as, for any z ≥ zi ,     hXi (zi ) hXi(z)  hY i (zi )    hY     D i(z) E D E   1 1 R (z, zi )  bX (z)  = M (z, zi ) R (z, zi )  bX (zi )  P P    p0 D E    p0 D E 1 1 bY (z) bY (zi ) P P p0 p0   hXi (zi )  hY i (zi )    D E  = R (z, zi ) M (z, zi )   p10 PbX (zi )  .   D E 1 bY (zi ) P p0 (5.139) In other words, the quantum averages of position and momentum of the particle with reference to the rotating coordinate system evolve along the optic axis according to     hXi(z) hXi (zi )   hY i (zi )   hY     D i(z) E D E  1 b  = M (z, zi )  1 b  (5.140)  p0 PX (z)   p0 PX (zi )  , for any z ≥ zi . D E  D E    1 1 b b p0 PY (z) p0 PY (zi ) From (5.123) and (5.126), the corresponding equations of motion follow:  D E   D E  ~ ~ R (z) R (z) ⊥ ⊥ d   = µ(z)   D E D E 1 ~b 1 ~b dz p0 P⊥ (z) p0 P⊥ (z)   D E   ~R⊥ (z) 0 1  , D E = 1 ~b −α 2 (z) 0 P⊥ (z)

(5.141)

p0

with   D E ~R⊥ (z) = hXi(z) , hY i(z)

 D E  bX (z) D E P ~Pb (z) =  D E . ⊥ PbY (z)

From this, it follows that    D E   ~ 2 2 R (z) ⊥ d  −α (z) 0 =  D E 0 (z) −α 2 (z) 1 ~b −2α(z)α dz2 P⊥ (z) p0

(5.142)

D E  ~R⊥ (z) , D E 1 ~b P (z)

p0

⊥

(5.143)

244

Quantum Mechanics of Charged Particle Beam Optics

or, D E00 D E ~R⊥ (z) + α 2 (z) ~R⊥ (z) = 0, D E D E 1 D~b E00 1 ~ P⊥ (z) + 2α(z)α 0 (z) ~R⊥ (z) + α 2 (z) Pb⊥ (z) = 0. p0 p0

(5.144)

First of the equations (5.144) is the quantum paraxial equation of motion in the rotating coordinate system. The second equation is not independent of (5.144), since it is just the consequence of the relation ~P⊥ (z)/p0 = d ~R⊥ (z)/dz. Equation (5.141) cannot be integrated by ordinary exponentiation since µ (z0 ) and µ (z00 ) do not commute when z0 6= z00 if α 2 (z0 ) 6= α 2 (z00 ). So, to integrate (5.141), we have to follow the same method used to integrate the time-dependent Schr¨odinger b (t,t0 ) given by the series expression in (3.449). equation (3.439) as in (3.440) with U b Thus, replacing t by z, and −iH(t)/¯h by µ(z), we get D E  D E     ~R⊥ (z) ~R⊥ (zi ) g (z, z ) h (z, z ) i i   =  D E D E 1 ~b 1 ~b g0 (z, zi ) h0 (z, zi ) P P (z) (z ) i ⊥ ⊥ p0 p0  D E  ~R⊥ (zi ) , D E = M (z, zi )  (5.145) 1 ~b p0 P⊥ (zi ) 

where Z z

M (z, zi ) = I +

z

Z z

dz1 µ (z1 ) +

Z zi

+ zi

dz3

dz2

z

zi

Z z2 i

Z z3 zi

Z z2

dz2

zi

dz1 µ (z2 ) µ (z1 )

dz1 µ (z3 ) µ (z2 ) µ (z1 )

+ ··· .

(5.146)

It can be directly verified that dM (z, zi ) = µ(z)M (z, zi ) , dz

(5.147)

and hence D E   ~ R (z) ⊥ d   = dM (z, zi )  D E 1 ~b dz dz P⊥ (z) 

p0

D E   ~R⊥ (zi )  = µ(z)  D E 1 ~b P (z )

p0

⊥

i

D E  ~R⊥ (zi ) , D E 1 ~b P (z )

p0

⊥

i

(5.148)

245

Scalar Theory: Spin-0 and Spinless Particles

as required in (5.141). Explicit expressions for the elements of M can be written down from (5.146) as follows: g (z, zi ) = 1 −

Z z zi

Z z2

dz2

Z z

Z z4

+

dz4

zi

zi

h (z, zi ) = (z − zi ) − Z z z Z iz zi

h0 (z, zi ) = 1 −

dz4

Z z zi

dz3 α 2 (z3 )

zi z4

zi

Z z2

dz2

zi

zi

Z z3 zi

Z z2

dz2

zi

dz1 α 2 (z1 ) − · · · ,

dz1 α 2 (z1 ) (z1 − zi )

dz3 α 2 (z3 )

dz1 α 2 (z1 ) +

Z z

+

dz1 α 2 (z1 )

Z z

Z

+ g0 (z, zi ) = −

zi

Z z zi

Z z3 zi

Z z2

dz2

dz3 α 2 (z3 )

zi

dz1 α 2 (z1 ) (z1 − zi ) − · · · ,

Z z3 zi

Z z2

dz2

zi

dz1 α 2 (z1 ) − · · · ,

dz1 α 2 (z1 ) (z1 − zi )

dz3 α 2 (z3 )

Z z3 zi

Z z2

dz2

zi

dz1 α 2 (z1 ) (z1 − zi ) − · · · .

(5.149)

Note that

dh (z, zi ) dg (z, zi ) = g0 (z, zi ) , = h0 (z, zi ) , (5.150) dz dz D E D E ~ consistent with the relation d ~R⊥ (z)/dz = Pb⊥ (z)/p0 . From the differential equation (5.147) for M, we know that we can write n M (z, zi ) = lim lim e∆zµ(zi +N∆z) e∆zµ(zi +(N−1)∆z) e∆zµ(zi +(N−2)∆z) · · · N→∞ ∆z→0 o · · · e∆zµ(zi +2∆z) e∆zµ(zi +∆z) , (5.151) with N∆z = (z − zi ). Since the trace of µ(z) is zero, each of the factors in the above product expression, of the type e∆zµ(zi + j∆z) , has unit determinant. Thus, the matrix M has unit determinant, i.e., g (z, zi ) h0 (z, zi ) − h (z, zi ) g0 (z, zi ) = 1,

(5.152)

forany (z, zi ). Note that the solution for the transfer matrix for the quantum averages  ~ of ~r⊥ , pb⊥ /p0 is exactly same as the transfer matrix for the corresponding classical variables, since the quantum paraxial equation of motion (5.144) is same as the classical paraxial equation of motion (4.51), except for the replacement of classical (~r⊥ ,~p⊥ /p0 ) by their quantum averages. In a simplified picture of electron microscope, let us consider the monoenergetic paraxial beam comprising of electrons scattered elastically from the specimen (the object to be imaged) being illuminated. The beam transmitted by the specimen carries the information about the structure of the specimen. This beam going through the magnetic lens gets magnified and is recorded at the image plane. Let us take the

246

Quantum Mechanics of Charged Particle Beam Optics

positions of the object plane and the image plane in the z-axis, the optic axis, as zob j and zimg respectively. As already mentioned, we shall take the practical boundaries of the lens to be at z` and zr . The magnetic field of the lens is practically confined to the region between z` and zr and zob j < z` < zr < zimg . The region between zob j and z` and the region between zr and zimg are free spaces. Thus, the electron beam transmitted by the specimen (object) travels first through the free space between zob j and z` , then the lens between z` and zr , and finally, the free space between zr and zimg where it is recorded (image). Hence, the z-evolution matrix, or the transfer matrix,
M zimg , zob j can be written as

M zimg , zob j = MD (zimg , zr ) ML (zr , z` ) MD z` , zob j , (5.153) where subscript D indicates the drift region and L indicates the lens region. Since α 2 (z) = 0 in the free space regions, or drift regions, we get from (5.149)
   
1 (zimg − zr ) 1 z` − zob j MD (zimg , zr ) = , MD z` , zob j = . 0 1 0 1 (5.154) For the lens region, we have from (5.149)     g (zr , z` ) h (zr , z` ) gL hL ML (zr , z` ) = = . (5.155) g0 (zr , z` ) h0 (zr , z` ) g0L h0L Thus, we get




M zimg , zob j =



1 (zimg − zr ) 0 1



gL g0L

hL h0L



1 0

z` − zob j 1





gL z` − zob j + hL
+g0L (zimg − zr ) z` − zob j + h0L (zimg − zr )]

 gL + g0 (zimg − zr ) L  =  
g0L g0L z` − zob j + h0L

  g zimg , zob j
h zimg , zob j
= g0 zimg , zob j h0 zimg , zob j

      (5.156)


Note that MD z` , zob j and MD (zimg , zr ) are of unit determinant. From the general theory, we know that ML (zr , z` ) should also be of unit determinant, i.e., gL h0L − hL g0L = 1.

(5.157)
Since zimg is the position of the image plane h zimg , zob j , the 12-element of the
matrix M zimg , zob j should vanish such that D E D E
~R⊥ (zimg ) ∝ ~R⊥ zob j . (5.158)

247

Scalar Theory: Spin-0 and Spinless Particles

This means that we should have

gL z` − zob j + hL + g0L (zimg − zr ) z` − zob j + h0L (zimg − zr ) = 0. Now, let


h0 − 1 u = z` − zob j + L 0 , gL

v = (zimg−zr ) +

and f =−

gL − 1 , g0L

1 . g0L

(5.159)

(5.160)

(5.161)

We are to interpret u as the object distance, v as the image distance, and f as the focal length of the lens. Using straightforward algebra, and the relation (5.157), one can verify that the familiar lens equation, 1 1 1 + = , u v f

(5.162)

implies (5.159). From (5.160) it is seen that the principal planes from which u and v are to be measured in the case of a thick lens are situated at
h0L − 1 = z` + f 1 − h0L , 0 gL gL − 1 = zr − f (1 − gL ) , = zr − g0L

zPob j = z` + zPimg

(5.163)

such that u = zPob j − zob j ,

v = zimg − zPimg .

(5.164)

The explicit expression for the focal length f follows from (5.161) and (5.149): 1 = f =

Z zr

dz α 2 (z) −

z` q2

Z zr

4p20

z`

Z zr z`

dz B2 (z) −

dz2 α 2 (z2 ) q4 16p40

Z zr z`

Z z2 z`

Z z1

dz1

dz α 2 (z) + · · ·

z`

dz2 B2 (z2 )

Z z2 z`

Z z1

dz1

dz B2 (z) + · · · .

z`

(5.165) To understand the behavior of the lens, let us consider the idealized model in which B(z) = B is a constant in the lens region and zero outside. Then,   1 qB qBw = sin , (5.166) f 2p0 2p0 where w = (zr − z` ) is the width, or thickness, of the lens. This shows that the focal length is always positive to start with and is then periodic with respect to the variation of the field strength. Thus, the round magnetic lens is convergent up to a certain strength of the field. Round magnetic lenses commonly used in electron microscopy

248

Quantum Mechanics of Charged Particle Beam Optics

are always convergent. Note that zz`r dz α 2 (z) has the dimension of reciprocal length. The round magnetic lens is considered weak when R

Z zr

dz α 2 (z) 

z`

1 . w

(5.167)

For such a weak lens, the expression for the focal length (4.74) can be approximated by 1 ≈ f

Z zr z`

dz α 2 (z) =

q2 4p20

Z zr z`

dz B2 (z) =

q2 4p20

Z ∞

dz B2 (z),

(5.168)

−∞

the Busch formula (Busch [17]). A weak lens is said to be thin, since in this case 1/ f  1/w, or f  w. Thus, we now understand how quantum mechanics leads to the classical Busch formula for an axially symmetric thin magnetic lens. The paraxial z-evolution matrix from the object plane to the image plane (5.156) becomes, taking into account (5.159), ! z −z gL − imgf r 0 M(zimg , zob j ) = z −z − 1f h0L − ` f ob j   0 gL − v+ f (gf L −1)  = u+ f (h0L −1) − 1f h0L − f !   0 1 − vf − uv 0 = = − 1f − uv − 1f 1 − uf   M 0 = 1 − 1f M
  g zimg , zob j
0
= (5.169) g0 zimg , zob j 1/g zimg , zob j
where g zimg , zob j = M is the magnification, as we have identified (see (5.117)) while understanding imaging in the Schr¨odinger picture. Further, we find that
g0 zimg , zob j = −1/ f , and hence we can write, from (5.117),   

i2π 1 2 1 zimg − zob j − r ψ ~r⊥,img , zimg = exp M λ0 2 f M ⊥,img   1 ×ψ ~r⊥,img (θL ) , zob j . (5.170) M In our notation, both u and v are positive and hence M = −v/u is negative, showing the inverted nature of the image, as should be in the case of imaging by a convergent lens. For a thin lens with f  w, we can take w ≈ 0, and the two principal planes collapse into a single plane at the center of the lens, i.e., we can take zP = (z` + zr ) /2.

249

Scalar Theory: Spin-0 and Spinless Particles

Then, from (5.149) it is clear that for the thin lens, we can make the approximation   1 0 MT L (zr , z` ) ≈ , (5.171) − 1f 1 where the subscript TL stands for thin lens. Thus, corresponding to (5.156), we have for the thin lens
   
1 0 1 (zimg − zP ) 1 zP − zob j M zimg , zob j ≈ − 1f 1 0 1 0 1   1 − 1 (zimg − zP ) f  =   − 1f




zP − zob j
− 1f (zimg − zP ) zP − zob j + (zimg − zP )] 1 − 1f zP − zob j

   .  


(5.172)




Since zP − zob j = u and (zimg − zP ) = v, and 1/u + 1/v = 1/ f , we get !
1 − vf u − uvf + v M zimg , zob j = − 1f 1 − uf  v    −u 0 M 0 = = , 1 − 1f − uv − 1f M

(5.173)

where M = −v/u is the magnification. Note thatthe effect ofpassage through a thin lens of negligible width with the transfer matrix will be to have

1 −1/ f

0 1

on the wave function



− iπ r2 (5.174) ψ ~r⊥,zr , zr = e λ0 f ⊥,zr ψ ~r⊥,zr (θL ) , z` ,   2 , as is well i.e., multiplication of the wave function by the phase factor exp − λiπf r⊥ 0 known (see Hawkes and Kasper [72] for more details of electron optical imaging). So far, we have been describing the behavior of the lens in the rotated XY zcoordinate system. Now, if we return to the original xyz-coordinate system, we would have, as seen from (5.125), (5.128), and (5.130),
! ! h~Dr⊥ i (z h~Dr⊥ i Ezob j

Eimg )
, = M zimg , zob j ⊗ R zimg , zob j 1 ~ 1 ~ b⊥ (zimg ) b⊥ zob j p0 p p0 p (5.175)   D E  h pb i  hxi x with h~r⊥ i = and ~pb⊥ =  . Since the rotation of the image is the hyi pby effect of only the magnetic field of the lens  
cos θL sin θL R zimg , zob j = R (z` , zr ) = , (5.176) − sin θL cos θL

250

Quantum Mechanics of Charged Particle Beam Optics

where θL = θ (z` , zr ) =

q 2p0

Z zr

(5.177)

dz B(z). z`

From (5.172) and (5.175), it is clear that apart from rotation and drifts through fieldfree regions in the front and back of the lens,the effect of a thin convergent lens is essentially described by the transfer matrix

1 −1/ f

0 1

. This can also be seen

simply as follows. If a particle of a paraxial beam


enters a thin lens in the transverse xy-plane with (hxi, hyi) = (hxi, 0) and h pbx i , pby = (0, 0), i.e., in the xz-plane and parallel to the optic (z) axis, then we have !     0 1 0 hxi 1 f . (5.178) = − 1f 1 0 0 1 − hxi f This means that all the particles of the paraxial beam hitting the lens parallel to the optic axis in the xz-plane meet the optic axis at a distance f from the lens independent of the hxi with which it enters the lens. This implies that the lens is a convergent lens with the focal length f . If the lens is a thin divergent lens of focal length − f , then correspondingly we would have !     0 1 0 1 −f hxi , (5.179) = hxi 1 1 0 1 0 f f implying that the particles appear virtually to meet behind the lens at a distance f . Thus, the effectof a thin divergent lens is essentially described by the transfer matrix  g is 0 g

1 1/ f

0 1

. Note that, in general, if the transfer matrix of a lens system  h , then it will be focusing or defocusing depending on whether g0 is 0

h

negative or positive, respectively. This is so because if g0 is negative then h pbx i /p0 decreases proportional to the hxi of the beam particle driving it towards the optic axis, and if g0 is positive then h pbx i /p0 increases proportional to the hxi of the beam particle driving it away from the optic axis. 5.2.3.2

Going Beyond the Paraxial Approximation: Aberrations

Paraxial beam is an idealization. Let us now turn to the realistic case of quasiparaxial beam that would lead to due to the necessity to include terms of order  aberrations  ~ co . higher than quadratic in ~r⊥ , pb⊥ in the quantum beam optical Hamiltonian H co , which will be anyway small comWe have to treat the nonparaxial terms in H pared to the paraxial terms, as perturbations and use the well-known techniques of time-dependent perturbation theory of quantum mechanics utilizing the interaction picture approach. In the classical limit, the formalism of quantum theory of paraxial and aberrating charged particle optical systems presented here tends to the similar approach to the classical theory of charged particle optics, including accelerator

Scalar Theory: Spin-0 and Spinless Particles

251

optics, based on Lie transfer operator methods developed extensively by Dragt et al. (see Dragt [34], Dragt and Forest [35], Dragt et al. [36], Forest, Berz, and Irwin [54], Rangarajan, Dragt, and Neri [156], Forest and Hirata [55], Forest [56], Radli˘cka [154], and references therein; see also Berz [11], Mondragon and Wolf [135], Wolf [191], Rangarajan and Sachidanand [157], Lakshminarayanan, Sridhar, and Jagannathan [124], and Wolski [192]). When the beam deviates from the ideal paraxial condition, as is always the case in practice, going beyond the paraxial approximation,  the next approximation entails ~ c retaining in Ho terms of order up to fourth in ~r⊥ , pb⊥ . To this end, we substitute in co obtained in (5.34) H φ (~r) = 0,   ~A = − 1 yΠ (~r⊥ , z) , 1 xΠ (~r⊥ , z) , 0 , 2 2 1 2 00 with Π (~r⊥ , z) = B(z) − r⊥ B (z), (5.180) 8 expand, and approximate as desired. Then, we get, with α(z) = qB(z)/2p0 , co = H co,p + H c0 H o 1 1 2 2 co,p = bz pb + p0 α 2 (z)r⊥ − α(z)L H 2p0 ⊥ 2 2 α 2 (z)  ~ c0 = 1 pb4 − α(z) pb2 L b ~r⊥ · pb⊥ +~pb⊥ ·~r⊥ H o ⊥ ⊥ z− 3 2 8p0 2p0 8p0 2
2
1 00 3α (z) 2 2 2 bz r⊥ r⊥ pb⊥ + pb2⊥ r⊥ + + α (z) − 4α 3 (z) L 8p0 8
4 p0 4 + α (z) − α(z)α 00 (z) r⊥ , (5.181) 8 in which we have dropped unimportant constant terms, giving rise only to multiplicative phase factors, and retained only the Hermitian part without the h¯ -dependent co , H co,p is the paraxial term, same as in (5.97) except for the Hermitian terms. In H c0 represents the lowest order nonparaxial, or peradditive constant term −p0 , and H o   co,p is a homogeneous quadratic polynomial in ~r⊥ ,~pb turbation, term. Note that H ⊥   c0 is a homogeneous fourth-degree polynomial in ~r⊥ ,~pb . Since the system and H ⊥ o is rotationally symmetric aboutthe z-axis, the optical Hamiltonian does not contain  any odd-degree polynomial in ~r⊥ ,~pb⊥ . Now the z-evolution equation for the system is ∂ |ψ(z)i co |ψ(z)i. =H ∂z Following the time-dependent perturbation theory, let    i Rz ψ (z) = U bp† (z, zi ) |ψ(z)i, bp (z, zi ) = P e− h¯ zi dz Hco,p . U i i¯h

(5.182)

(5.183)

252

Quantum Mechanics of Charged Particle Beam Optics

Note that  ψ (zi ) = U bp† (zi , zi ) |ψ (zi )i = |ψ (zi )i . i

(5.184)

Then, we have i¯h

 ∂ b† ∂ bp† (z, zi ) i¯h ∂ |ψ(z)i ψ (z) = i¯h U (z, zi ) |ψ(z)i + U ∂z i ∂z p ∂z † † c co |ψi b b = −U p (z, zi ) Ho,p |ψ(z)i + U p (z, zi ) H   co,pU coU bp† (z, zi ) H bp (z, zi ) ψ (z) + U bp† (z, zi ) H bp (z, zi ) ψ (z) = −U i i    co − H co,p U bp† (z, zi ) H bp (z, zi ) ψ (z) =U i  c0U bp† (z, zi ) H b =U (5.185) o p (z, zi ) ψi (z) .

Defining c0U c0 = U bp† (z, zi ) H b H o p (z, zi ) , o,i

(5.186)

  ∂ c0 ψ (z) , ψ (z) = H i o,i ∂z i

(5.187)

we have i¯h

where the subscript i denotes the interaction picture. Integrating (5.187), we get   ψ (z) = U b (z, zi ) |ψ (zi )i , b (z, zi ) ψ (zi ) = U i i i i − h¯i Tb (z,zi ) b i Ui (z, zi ) ≈ e , Z z Z z c0U c0 = bp† (z, zi ) H b dz U Tbi (z, zi ) = dz H o p (z, zi ) , o,i zi zi

(5.188)

b (z, zi ) by an ordinary integral disregarding all the where we have approximated U i commutator terms in (5.38), since they lead to polynomials of degree higher than   ~ four in ~r⊥ , pb⊥ . From the paraxial theory, we know   hxi(z) hxi (zi )  hyi(z)   hyi (zi )  1    p h pbx i(z)  = M (z, zi ) ⊗ R (θ (z, zi ))  p1 h pbx i (zi ) 0 0 1 1 by i(z) by i (zi ) p0 h p p0 h p   g (z, zi ) h (z, zi ) M (z, zi ) = g0 (z, zi ) h0 (z, zi )   cos θ (z, zi ) sin θ (z, zi ) R (θ (z, zi )) = . − sin θ (z, zi ) cos θ (z, zi ) 

   

(5.189)

253

Scalar Theory: Spin-0 and Spinless Particles

Since   hxi(z)  hyi(z)    1    p h pbx i(z)  =  0 1 by i(z) p0 h p 

   = 

  hxi (zi )  hyi (zi )    1    p h pbx i (zi )  =  0 1 by i (zi ) p0 h p 

 hψ(z)|x|ψ(z)i hψ(z)|y|ψ(z)i   1 bx |ψ(z)i  p0 hψ(z)| p 1 by |ψ(z)i p0 hψ(z)| p bp† (z, zi ) xU bp (z, zi ) |ψ (zi )i hψ (zi ) |U bp† (z, zi ) yU bp (z, zi ) |ψ (zi )i hψ (zi ) |U 1 b† bp (z, zi ) |ψ (zi )i bxU p0 hψ (zi ) |U p (z, zi ) p 1 † b bp (z, zi ) |ψ (zi )i byU p0 hψ (zi ) |U p (z, zi ) p

   , 

 hψ (zi ) |x|ψ (zi )i hψ (zi ) |y|ψ (zi )i   1 bx |ψ (zi )i  , p0 hψ (zi ) | p 1 by |ψ (zi )i p0 hψ (zi ) | p

(5.190)

we have     

bp† (z, zi ) xU bp (z, zi ) U † b b U p (z, zi ) yU p (z, zi ) 1 b† bp (z, zi ) bxU p0 U p (z, zi ) p 1 b† bp (z, zi ) byU p0 U p (z, zi ) p



 x   y     = M (z, zi ) ⊗ R (θ (z, zi ))   p1 pbx  .  0 1 by p0 p 

(5.191) c0 , Tb (z, zi ), and U b b (z, zi ). Note that for any O Using this result, we can compute H i o,i i   n bnU bU bp† (z, zi ) O bp (z, zi ) = U bp† (z, zi ) O bp (z, zi ) . After a considerable, but straightforU ward algebra, we obtain i

b (z, zi ) = e− h¯ Tbi (z,zi ) U i   n o i 1 1 = exp − C (z, zi ) pb4⊥ + 2 K (z, zi ) pb2⊥ , ~pb⊥ ·~r⊥ +~r⊥ ·~pb⊥ 3 h¯ 4p0 4p0  2 1 bz + 1 A (z, zi ) ~pb⊥ ·~r⊥ +~r⊥ ·~pb⊥ + 2 k (z, zi ) pb2⊥ L 4p0 p0  
1 2 2 2 bz + 1 F (z, zi ) pb2⊥ r⊥ + a (z, zi ) ~pb⊥ ·~r⊥ +~r⊥ ·~pb⊥ L + r⊥ pb⊥ 2p0 4p0 n o 1 2 2b + d (z, zi ) r⊥ Lz + D (z, zi ) ~pb⊥ ·~r⊥ +~r⊥ ·~pb⊥ , r⊥ 4 io p0 4 + E (z, zi ) r⊥ , (5.192) 4

254

Quantum Mechanics of Charged Particle Beam Optics

n o b , Bb = A bBb + BbA b and where A Z o
1 z n 4 2 4 C (z, zi ) = dz α − αα 00 h4 + 2α 2 h2 h0 + h0 , 2 zi Z o
1 z n 4 3 K (z, zi ) = dz α − αα 00 gh3 + α 2 (gh)0 hh0 + g0 h0 , 2 zi    Z z 1 00 1 3 2 1 0 2 α − α h − αh , k (z, zi ) = dz 8 2 2 zi Z z n o
1 2 2 A (z, zi ) = dz α 4 − αα 00 g2 h2 + 2α 2 gg0 hh0 + g0 h0 − α 2 , 2 zi    Z z 1 00 α − α 3 gh − αg0 h0 , a (z, zi ) = dz 4 zi Z
1 z  4 F (z, zi ) = dz α − αα 00 g2 h2 2 zi   o 2 2 2 2 +α 2 g2 h0 + g0 h2 + g0 h0 + 2α 2 , Z o
1 z n 4 3 D (z, zi ) = dz α − αα 00 g3 h + α 2 gg0 (gh)0 + g0 h0 , 2 zi    Z z 1 00 1 3 2 1 0 2 d (z, zi ) = dz α − α g − αg , 8 2 2 zi Z o
1 z n 4 2 4 dz α − αα 00 g4 + 2αg2 g0 + g0 , E (z, zi ) = (5.193) 2 zi with g = g (z, zi ), h = h (z, zi ), g0 = g0 (z, zi ), and h0 = h0 (z, zi ). From (5.183) and (5.188), we have bp (z, zi ) U b (z, zi ) |ψ (zi )i , |ψ(z)i = U (5.194) i representing the generalization of the paraxial propagation law (5.116) including the lowest order aberrations. Now, the transfer map becomes E D bpU b (zi ) , b †U p†~r⊥U h~r⊥ i (z) = U i i D E D E † † ~pb (z) = U bpU b (zi ) , b U p~pb⊥U (5.195) ⊥ i i bp = U bp (z, zi ) and U b =U b (z, zi ). Explicitly writing, with U i i  D E b † xU b (zi ) U   i i  hxi(z)     D E    b † yU b (zi ) U  hyi(z)   i i      = (M (z, zi ) ⊗ R(θ (z, zi ))   1   D E  p h pbx i (z)   1 b† b  0   p0 Ui pbxUi (zi )    

 1 D E  b p (z) y p0 1 b † byU b (zi ) i p0 Ui p

       .      

(5.196)

255

Scalar Theory: Spin-0 and Spinless Particles

We shall write, with Tbi = Tbi (z, zi ),

             

E D b † xU b (zi ) U i i



D ib E T − i Tb e h¯ i xe h¯ i (zi )



      E D  † b (zi )  b yU U   i i   =   D E  1 b † pbxU b (zi )  U   i p0 i     E D   † 1 b (zi ) b pbyU U i p0 i





      E D ib iT b   T − e h¯ i ye h¯ i (zi )     =   D ib E i b   − h¯ T 1 h¯ Ti p i   b e (z ) e i x p0       E D ib iT b T − 1 h h ¯ ¯ i i pby e (zi ) p0 e

D i b E :T : e h¯ i x (zi )



   D i b E  : T : e h¯ i y (zi )   .  D i b E  1 h¯ :Ti : p  b e (z ) x i p0    D i b E : T : 1 h ¯ i pby (zi ) p0 e (5.197)

Now, let us make the approximation

i

:T : e h¯ i ≈ I + b

i b : T :, h¯ i

(5.198)

such that we retain only the single commutator terms since multiple  the remaining  commutator terms lead to polynomials of degree ≥5 in ~r⊥ ,~pb⊥ , which are to be ignored consistent with the approximation we are considering. Then, we have

         

hxi(z)



     ≈ (M (z, zi ) ⊗ R (θ (z, zi ))  1  b h i p (z) x p0  

 1 by (z) p0 p hyi(z)

         ×      

hxi (zi )





     hyi (zi )       + 1   b h p i (z ) x i p0       1   b h p i (z ) y i p0 

D h iE i b (zi ) h¯ Ti , x



        D h iE  1 i b bx (zi )  p0 h¯ Ti , p   D h iE  1 i b b T , p (z ) y i i p0 h¯ D h iE i b T , y (zi ) i h¯

256

Quantum Mechanics of Charged Particle Beam Optics

= (M (z, zi ) ⊗ R (θ (z, zi ))     hxi (zi ) (δ x) (zi )  hyi (zi )   (δ y) (zi )      ×  p1 h pbx i (zi )  +  p1 (δ px ) (zi )  0 0 1 1 by i (zi ) p0 h p p0 (δ py ) (zi )     hxi p (z) (∆x)(z)  hyi p (z)   (∆y)(z)      =   p1 h pbx i p (z)  +  p1 (∆px )(z)  , (5.199) 0

0  1 1 by p (z) p0 p p0 (∆py )(z) where (∆x)(z), (∆y)(z), p10 (∆px )(z), and p10 (∆py )(z) are third-order aberrations, deviations  from the paraxial results involving the averages of third-order polynomials in ~r⊥ ,~pb⊥ . Obviously, the plane at which the influence of aberrations is to be known is the image plane at z = zimg :   hxi (zimg )     hyi (zimg )  M 0 =  1 ⊗ R (θL ) × 1  p h pbx i (zimg )  − 1f M 0

 1 by (zimg ) p0 p
 
  hxi zob j
(δ x) zob j
 hyi zob j
  (δ y) zob j
  1     p h pbx i zob j  +  p1 (δ px ) zob j  0

0 

1 1 by zob j p0 p p0 (δ py ) zob j   hxi p (zimg )  hyi p (zimg )   =   p1 h pbx i p (zimg )  0

 1 by p (zimg ) p0 p
  (δ x) zob j
    (δ y) zob j
 M 0  + ⊗ R (θL ) ×  1  p1 (δ px ) zob j  , − 1f M 0
1 p0 (δ py ) zob j (5.200) with
Cs


(δ x) zob j = 3 pbx pb2⊥ zob j p0  o  E
K Dn + 2 pbx , ~pb⊥ ·~r⊥ +~r⊥ ·~pb⊥ + x , pb2⊥ zob j 2p0   o 1
k n 2 b + 2 y , pb⊥ pbx , Lz − zob j 2 p0

Scalar Theory: Spin-0 and Spinless Particles

257

oE
A Dn ~ zob j x , pb⊥ ·~r⊥ +~r⊥ ·~pb⊥ 2p0 oE
a Dn b o n ~ zob j + x , Lz − y , pb⊥ ·~r⊥ +~r⊥ ·~pb⊥ 2p0 


2


2
F  2 + pbx , r⊥ zob j + D xr⊥ zob j − d yr⊥ zob j , (5.201) 2p0 +


Cs


(δ y) zob j = 3 pby pb2⊥ zob j p0  o  E
K Dn pby , ~pb⊥ ·~r⊥ +~r⊥ ·~pb⊥ + y , pb2⊥ zob j + 2 2p0   o 1
k n 2 b + 2 pby , Lz + x , pb⊥ zob j 2 p0 oE
A Dn ~ + y , pb⊥ ·~r⊥ +~r⊥ ·~pb⊥ zob j 2p0 oE
a Dn b o n ~ y , Lz − x , pb⊥ ·~r⊥ +~r⊥ ·~pb⊥ zob j + 2p0 


2


2
F  2 pby , r⊥ + zob j + D yr⊥ zob j + d xr⊥ zob j , (5.202) 2p0

K

(δ px ) zob j = − 2 pbx pb2⊥ zob j p0 
k

− 2 pby pb2⊥ zob j p0  oE
A Dn pbx , ~pb⊥ ·~r⊥ +~r⊥ ·~pb⊥ zob j − 2p0 o n  oE
a Dn bz + pby , ~pb⊥ ·~r⊥ +~r⊥ ·~pb⊥ − pbx , L zob j 2p0 
F  x , pb2⊥ zob j − 2p0 oE n 
D D 2 − pbx , r⊥ + x , ~pb⊥ ·~r⊥ +~r⊥ ·~pb⊥ zob j 2  n o



2  2 bz + 1 pby , r⊥ zob j , (5.203) zob j − E p0 xr⊥ −d x, L 2

K

(δ py ) zob j = − 2 pby pb2⊥ zob j p0 
k

+ 2 pbx pb2⊥ zob j p0  oE
A Dn − pby , ~pb⊥ ·~r⊥ +~r⊥ ·~pb⊥ zob j 2p0

258

Quantum Mechanics of Charged Particle Beam Optics

oE o n 
a Dn bz + pbx , ~pb⊥ ·~r⊥ +~r⊥ ·~pb⊥ zob j pby , L 2p0 
F  − y , pb2⊥ zob j 2p0 oE
n  D D 2 zob j pby , r⊥ + y , ~pb⊥ ·~r⊥ +~r⊥ ·~pb⊥ − 2  n o


2
 2 bz − 1 pbx , r⊥ −d y, L zob j − E p0 yr⊥ zob j , (5.204) 2 −

with


Cs = C zimg , zob j , K = K zimg , zob j , k = k zimg , zob j ,


A = A zimg , zob j , a = a zimg , zob j , F = F zimg , zob j ,


D = D zimg , zob j , d = d zimg , zob j , E = E zimg , zob j .

(5.205)

With reference to the aberrations of position, (5.201) and (5.202), the constants Cs , K, k, A, a, F, D, and d, respectively, are known as the aberration coefficients corresponding to spherical aberration, coma, anisotropic coma, astigmatism, anisotropic astigmatism, curvature of field, distortion, and anisotropic distortion. The gradient aberrations, (5.203) and (5.204), do not affect the single-stage image, but should be taken into account as the input to the next stage when the lens forms a part of a complex imaging system. It should be noted that the expressions for the various aberration coefficients obtained above are exactly same as the classical expressions for them. For a detailed picture of the effects of these aberrations on the quality of the image and the classical methods of computation of these aberrations, see Hawkes and Kasper [70]. Ximen [194] has given a treatment of the classical theory of aberrations using position, momentum, and Hamilton’s equations of motion. Introducing the notations u = x + iy,

υ=

1 ( pbx + i pby ) , p0

(5.206)

the transfer map including the third-order aberrations, (5.200–5.204) can be written in a compact matrix form (see Hawkes and Kasper [70] for details) as follows: 

hui (zimg ) hυi (zimg )



−iθL



=e

 ×

1 0

M − 1f

0



1 M

0 Cs 1 ik − K D + id −E

2K 2k ia − 2A −a

2A + ia −a −2D 2d

F id − D  K + ik −F

259

Scalar Theory: Spin-0 and Spinless Particles

        ×       


hui zob j


hυi zob j
†υ υυ zob  j


1 †υ + υ †u υ , u zob j
4   1 υ  , u† υ − υ † u z
ob j 4i 1 †u υ , u z 2


ob j uu† u zob  j


1 † † 4 u , υ u + u υ  zob j
1 †υ u  , υ † u − u  z
ob j 4i 1 † u , υ υ z ob j 2

        .       

(5.207)

The wave function at the image plane is


Z

Z

D
E b d r⊥,zob j d 2 r⊥ ~r⊥,zimg U p zimg , zob j ~r⊥ D E  
b × ~r⊥ U z , z r ψ ~ r , z ~ ⊥,zob j ob j i img ob j ⊥,zob j ! 2 iπr⊥,zimg 1 ∼ exp − M λ0 f   Z Z 1 × d 2 r⊥,zob j d 2 r⊥ δ 2 ~r⊥ − ~r⊥,zimg (θL ) M D E  
b × ~r⊥ Ui zimg , zob j ~r⊥,zob j ψ ~r⊥,zob j , zob j ! 2 iπr⊥,z 1 img = exp − M λ0 f Z D E
b × d 2 r⊥,zob j ~r⊥,zimg (θL ) /M U z , z r ~ img ob j ⊥,z ob j i   × ψ ~r⊥,zob j , zob j . (5.208)

ψ ~r⊥,img , zimg =

2

When there are no aberrations   D E
1 b 2 ~r⊥,zimg (θL ) /M Ui zimg , zob j ~r⊥,zob j = δ ~r⊥,zob j − ~r⊥,zimg (θL ) , (5.209) M and hence one obtains the point-to-point imaging as seen earlier. It is clear from (5.208) that, when aberrations are present, the intensity distribution in the image plane will represent only a blurred and distorted version of the image. Note that in
b zimg , zob j the most dominant aberration term is the spherical aberration term that U i is independent of the position of the object point, as seen from (5.201) and (5.202) (for details on the practical aspects of electron optical imaging, see Hawkes and Kasper [70, 71, 72]).

260

5.2.3.3

Quantum Mechanics of Charged Particle Beam Optics

Quantum Corrections to the Classical Results

Before closing the discussion on the quantum mechanics of round magnetic lens, comments on the effects of h¯ -dependent Hermitian and anti-Hermitian terms, which have been dropped in deriving the final quantum charged particle beam optical Hamiltonian for the axially symmetric magnetic lens (5.181) are in order. Let us first consider, for example, the h¯ -dependent anti-Hermitian term:   b = i¯h α(z)α 0 (z)r2 − p0 α 0 (z)L b , A z ⊥ 2p20

(5.210)

that has got dropped in (5.181) in the process of making the quantum beam optical Hermitian. This term is a paraxial term, since it is quadratic in  Hamiltonian  ~ ~r⊥ , pb⊥ . If we retain any such anti-Hermitian term in the paraxial Hamiltonian, the resultant z-evolution operator will be nonunitary and will have the form   i b th (z, zi ) + b ta (z, zi ) T p (z, zi ) = exp − Tb (z, zi ) + b h¯       i i i ≈ exp − Tb (z, zi ) exp − b th (z, zi ) exp − b ta (z, zi ) , (5.211) h¯ h¯ h¯ where b th (z, zi ) and b ta (z, zi ) are, respectively, the Hermitian and anti-Hermitian correction terms to the main part Tb (z, zi ). It may be noted that any term of the type h i b , Bb /¯h is Hermitian when both A b and Bb are Hermitian or anti-Hermitian, and i A   b Bb is Hermitian and the such a term is anti-Hermitian if one of the two operators A, other is anti-Hermitian. We can use b T p (z, zi ) to calculate the transfer maps as h~r⊥ i (zi ) −→ h~r⊥ i (z) hψ(z) |~r⊥ | ψ(z)i hψ(z)|ψ(z)i †

 ψ (zi ) b T p (z, zi ) ψ (zi ) T p (z, zi )~r⊥ b D E = , † ψ (zi ) b T p (z, zi ) b T p (z, zi ) ψ (zi )

=

D E D E ~pb (zi ) −→ ~pb (z) ⊥ ⊥

=

D E ψ(z) ~pb⊥ ψ(z)

hψ(z)|ψ(z)i D E † ψ (zi ) b T p (z, zi )~pb⊥ b T p (z, zi ) ψ (zi ) E . = D † ψ (zi ) b T p (z, zi ) b T p (z, zi ) ψ (zi )

(5.212)

261

Scalar Theory: Spin-0 and Spinless Particles

Note that since the z-evolution of |ψ(z)i is not unitary, for calculating the average values, p we have to take the normalized state at the plane at any z as T p (z, zi ) |ψ (zi )i. It is seen that the Hermi|ψ(z)i/ hψ(z)|ψ(z)i, where |ψ(z)i = b tian correction term modifies the paraxial term leads



map while
the anti-Hermitian ta /¯h ψ zob j , affectto an overall real scaling factor ∼ 1/ ψ zob j exp −2ib ing the image magnification as a consequence of nonconservation of intensity, and
†
contributes to aberrations since the terms like exp −ib ta /¯h ~r⊥ exp −ib ta /¯h and
†
exp −ib ta /¯h ~pb⊥ exp −ib ta /¯h leadon expansion, respectively, to Hermitian   terms  ~ ~ of the form~r⊥ + nonlinear terms in ~r⊥ , pb and pb + nonlinear terms in ~r⊥ ,~pb ⊥

⊥

⊥

b in (5.210) does not lead to any only. In this present case, the anti-Hermitian term A h i 0 00 b b Hermitian correction term (note that A (z ) , A (z ) = 0 for any z0 and z00 ), and its contribution to the functioning of the lens is only through the anti-Hermitian correction term affecting the conservation of intensity and adding to the aberrations. Let us now turn to the h¯ -dependent Hermitian terms   2 b p = h¯ α(z)α 0 (z) ~r⊥ ·~pb⊥ +~pb⊥ ·~r⊥ H 4p20   2 n 000 b 0 = h¯ bz ~r⊥ ·~pb⊥ +~pb⊥ ·~r⊥ α (z) L H 32p30 oo
n 2  −p0 α 0 (z)α 00 (z) + α(z)α 000 (z) r⊥ , ~r⊥ ·~pb⊥ +~pb⊥ ·~r⊥ .

(5.213)

which have also been dropped in (5.181). Taking into account the influence of these co,L b p is a paraxial term and should be added to H terms is straightforward. Note that H while studying the functioning of the lens in the paraxial approximation. The rotation b p . In the rotated coordinate system, does not get affected by the addition of the term H the equation of motion (5.141) gets replaced by  d  dz

D E   ~R⊥ (z)  = µ(z) D E ¯  1 ~b P (z)

p0

⊥

 =

D E  ~R⊥ (z)  D E 1 ~b P (z)

p0

⊥

h¯ 2 α(z)α 0 (z) 2p20 −α 2 (z)



1 2

h¯ 0 − 2p 2 α(z)α (z)



0

D E  ~R⊥ (z) , D E × 1 ~b P (z) 

p0

⊥

(5.214)

262

Quantum Mechanics of Charged Particle Beam Optics

This equation can be integrated to give the transfer map  D E   D E  ~R⊥ (z) ~R⊥ (zi )   = M¯ (z, zi )   D E D E 1 ~b 1 ~b P (z) P (z ) p0

⊥

p0

 =

⊥

g¯ (z, zi ) h¯ (z, zi ) g¯0 (z, zi ) h¯ 0 (z, zi )

i



D E  ~R⊥ (zi ) ,  D E 1 ~b P (z ) 

p0

⊥

i

(5.215) where the matrix elements of M¯ (z, zi ) can be obtained from (5.146) by replacing ¯ µ(z) by µ(z). The resulting expression for g¯0 (z, zi ) shows that the focal length gets an additive contribution ∼ h¯ 2 vanishing in the classical limit. The other Hermitian term c0 in the quantum b 0 , has to be added to the perturbation term H mentioned above, H o beam optical Hamiltonian (5.181). The corresponding computation of the transfer maps, using the z-dependent perturbation theory in terms of the interaction picture, leads to the modification of aberration coefficients. For example, the modified spherical aberration coefficient turns out to be  Z
1 zimg C¯s = α 4 − αα 00 dz 2 zob j  h¯ 4  4 0 2 0 2 00 2 0 000 + 4 α α + αα α + α α α h¯ 4 2p0  2 
h¯ 4 h¯ 2 + 2 2αα 0 − α 0 α 00 − αα 000 − 4 α 4 α 0 h¯ 3 h¯ 0 p0 p0   4 3¯h 2 + 2α 2 + 4 α 2 α 0 h¯ 2 h¯ 02 2p0  2  
h¯ + 2 2αα 0 − α 0 α 00 − αα 000 h¯ h¯ 03 + h¯ 04 , (5.216) p0 where h¯ = h¯ (z, zi ) and h¯ 0 = h¯ 0 (z, zi ) in (5.215). Again, the tiny h¯ -dependent contributions to Cs , and similarly to other aberrations, vanish in the classical limit. The positivity of the spherical aberration coefficient Cs is a celebrated theorem of Scherzer [167], and the quantum corrections do not affect this result. This is why the classical theory of charged particle optics is working so well! 5.2.4

NORMAL MAGNETIC QUADRUPOLE

Let us now look at the quantum mechanics of a normal magnetic quadrupole lens. Let a quasiparaxial beam of particles of charge q and rest mass m be propagating through a normal magnetic quadrupole lens. We shall take the optic axis of the lens to be along the z-direction and the practical boundaries of the lens to be at z` and zr with z` < zr . The field of the normal magnetic quadrupole is ~B(~r) = (−Qn y, −Qn x, 0) ,

(5.217)

263

Scalar Theory: Spin-0 and Spinless Particles

where

 Qn =

constant in the lens region (z` < z < zr ) 0 outside the lens region (z < z` , z > zr )

(5.218)

There is no electric field and hence φ (~r) = 0. The vector potential of the field can be taken as  
~A(~r) = 0, 0, 1 Qn x2 − y2 . (5.219) 2 Now, from (5.34), the quantum beam  optical  Hamiltonian operator is seen to be, ~ keeping terms up to fourth power in ~r⊥ , pb⊥ , co = H co,p + H c0 + H bp +A b H o 2
co,p = −p0 + pb⊥ − 1 p0 Kn x2 − y2 , H 2p0 2 2
b p = h¯ Kn pb2 − pb2 , H x y 3 4p0

c0 = 1 pb4 , H o 8p30 ⊥

b = i¯hKn (x pbx − y pby ) , A 2p0

(5.220)

where Kn = qQn /p0 . Let us consider the propagation of a paraxial beam from the transverse xy-plane at z = zi < z` to the transverse plane at z > zr . The z-evolution of the beam is governed by the paraxial quantum beam optical Hamiltonian 2
co,p = −p0 + pb⊥ − 1 p0 Kn x2 − y2 , H 2p0 2

(5.221)

co by dropping the nonparaxial perturbation term H c0 , h¯ obtained from H o b dependent paraxial Hermitian term H p , and the h¯ -dependent anti-Hermitian term b The beam propagates in free space from zi to z` , passes through the lens from A. z` to zr , and propagates through free space again from zr to z. Following (5.44), the transfer map for the quantum averages of transverse coordinates and momenta of a beam particle is given by  D  E b (z, zi ) (zi ) b † (z, zi ) xU U     D E hxi(z)  b † (z, zi ) pbxU b (z, zi ) (zi )   p1 h pbx i (z)   p10 U   0   D  E (5.222)  hyi(z)  =  † b (z, zi ) (zi )  b (z, zi ) yU U  

 1 D E   by (z) p0 p 1 b † (z, zi ) pbyU b (z, zi ) (zi ) U p0

co,p . b (z, zi ) is the z-evolution operator corresponding to the Hamiltonian H where U Using the semigroup property of the z-evolution operator (5.39), we can write b (z, zi ) = U b (z, zr ) U b (zr , z` ) U b (z` , zi ) U bD (z, zr ) U bL (zr , z` ) U bD (z` , zi ) , =U

(5.223)

264

Quantum Mechanics of Charged Particle Beam Optics

where the subscripts D and L stand for drift and propagation through the lens, respectively, and the intervals (zi , z` ), (z` , zr ), and (zr , z) correspond to free space, lens, and co,p is independent of z, free space, respectively. Since within each of these regions H we get   2  bD (z, zr ) = exp − i (z − zr ) −p0 + pb⊥ , U h¯ 2p0   
pb2⊥ 1 i 2 2 b − p0 Kn x − y , UL (zr , z` ) = exp − (zr − z` ) −p0 + 2p0 2 h¯    2 bD (z` , zi ) = exp − i (z` − zi ) −p0 + pb⊥ . (5.224) U h¯ 2p0 b (z, zi ) in (5.222), we can write Substituting the above expression for U * hxi(z) =

e

1 1 h pbx i (z) = p0 p0 * hyi(z) =

e

1  1 pby (z) = p0 p0

pb2 i ⊥ h¯ (z−zr ): 2p0 :

* e

e

e

pb2 i ⊥ h¯ (z−zr ): 2p0 :

pb2 i ⊥ h¯ (z−zr ): 2p0 :

*

 2 pb

i h¯ (zr −z` ):

1 ⊥ 2p0 − 2 p0 Kn

 2 pb

e

1 ⊥ 2p0 − 2 p0 Kn

i h¯ (zr −z` ):

 2 pb

e

i h¯ (zr −z` ):

pb2 i ⊥ h¯ (z−zr ): 2p0 :

 2 : i (z −z ): pb⊥ : i 2p h¯ `

(x2 −y2 )

1 ⊥ 2p0 − 2 p0 Kn

e

1 ⊥ 2p0 − 2 p0 Kn

0

x (zi ) ,

 2 : i (z −z ): pb⊥ : i 2p h¯ `

(x2 −y2 )

e

0

 2 : i (z −z ): pb⊥ : i 2p h¯ `

(x2 −y2 )

 2 pb

i h¯ (zr −z` ):

e

+

e

0

e

pbx (zi ) ,

+ y (zi ) ,

 2 : i (z −z ): pb⊥ : i 2p h¯ `

(x2 −y2 )

+

0

+ pby (zi ) . (5.225)

From the results  i pb2⊥ : h¯ 2p0

  : 

x





0 pbx   0 p0  = y    0 pby 0 p0

1 0 0 0

0 0 0 0

 x 0  pbx 0    p0 1  y pby 0

   , 

(5.226)

p0

and 

x





0  2  pbx 
 pb⊥ 1 i Kn  p0   2 2  : − p0 Kn x − y : = h¯ 2p0 2  y   0 pby 0 p0

1 0 0 0 0 0 0 −Kn

 x  0  pbx  0     p0  , 1  y  pby 0 p0 (5.227)

265

Scalar Theory: Spin-0 and Spinless Particles

we get   hxi(z) hxi (zi )   p1 h pbx i (zi ) h pbx i (z)  0     hyi(z)  = TD (z, zr ) TL (zr , z` ) TD (z` , zi )  hyi (zi )



 1 1 by (zi ) by (z) p0 p p0 p   1 (z − zr ) 0 0  0  1 0 0  TD (z, zr ) =   0 0 1 (z − zr )  0 0 0 1 √
√
√1 sinh w Kn cosh w Kn Kn √ √
√
TL (zr , z` ) = Kn sinh w Kn cosh w Kn √
√
√1 sin w Kn cos w Kn K n √
√
√ cos w Kn − Kn sin w Kn   1 (z` − zi ) 0 0   0 1 0 0  TD (z` , zi ) =   0 0 1 (z` − zi )  0 0 0 1 

1 p0

with w = (zr − z` ) .

  , 

! ⊕ !!

(5.228)


This transfer map for the quantum averages (hxi(z) , h pbx i (z)/p0 , hyi(z), pby (z)/p0 is seen to be exactly the same as the transfer map (4.142) for the classical variables (x(z), px (z)/p0 , y(z), py (z)/p0 ) in the paraxial approximation. Thus, the properties of the normal magnetic quadrupole lens we have derived earlier, in the context of the classical transfer map (4.142), are valid in the present context of the quantum transfer map (5.228). Let us recall. When Kn > 0, the lens is divergent in the xz-plane and convergent in the yz-plane. In other words, the normal magnetic quadrupole lens produces a line focus. When Kn < 0, the lens is convergent in the xz-plane and diver−2 gent in the yz-plane. p It is seen that Kn has the dimensions of length . In the weak field case, when w |Kn |  1, the lens can be considered as a thin lens with the focal lengths 1 1 = −wKn , = wKn . (5.229) (x) f f (y) A lens of focal length ± f can be considered effectively to be two lenses of focal length ±2 f , respectively, joined together without any gap between them. A doublet of focusing and defocusing lenses of focal lengths f and − f with a distance d between them would have the effective focal length f 2 /d > 0 and hence would behave as a focusing lens. A triplet of quadrupoles can be arranged in such a way to get net focusing effect in both the transverse planes. Series of such quadrupole triplets are the main design elements of long beam transport lines or circular accelerators to provide a periodic focusing structure called a FODO-channel. FODO stands for F(ocusing)O(nonfocusing)D(efocusing)O(nonfocusing), where O can be a drift space or a bending magnet.

266

Quantum Mechanics of Charged Particle Beam Optics

b p , we have the paraxial quanNow, if we retain the h¯ -dependent Hermitian term H tum beam optical Hamiltonian as     2 2
co,p = −p0 + 1 1 + h¯ Kn pb2 + 1 1 − h¯ Kn pb2 − 1 p0 Kn x2 − y2 . H x y 2 2 2p0 2p0 2 2p0 2p0 (5.230) This modifies the equation (5.227) as      1 h¯ 2 Kn 1 h¯ 2 Kn i 2 : 1+ pbx + 1− pb2y h¯ 2p0 2p0 2p20 2p20   x   pbx 
 1  − p0 Kn x2 − y2 :  p0  2  y  pby p0



0

  Kn =   0 0

2

1 + h¯2pK2n

0

0



0 0

0 0

0 1 − h¯2pK2n

0

−Kn

0

  pbx  p  0  y  pb

0

2

0

x

y

   , 

(5.231)

p0

leading to the transfer map     hxi(z) hxi (zi )  p1 h pbx i (z)   1 h pb i (z )   0  = TD (z, zr ) TL (zr , z` ) TD (z` , zi )  p0 x i  ,  hyi(z)   hyi (zi ) 



 1 1 b by (zi ) p (z) y p0 p0 p   1 (z − zr ) 0 0   0 1 0 0   TD (z, zr ) =  0 0 1 (z − zr )  0 0 0 1  q

 √ √ 1+ + cosh w Kn 1+ sinh w K 1 n K n  TL (zr , z` ) =  q

√ √ Kn + + sinh w K 1 cosh w K 1 n n + 1  q

 √ √ 1− − cos w Kn 1− sin w K 1 n K n  , ⊕ q

√ √ − 1K−n sin w Kn 1− cos w Kn 1− h¯ 2 Kn h¯ 2 Kn − with 1+ = 1 + , 1 = 1 − , 2p20 2p20   1 (z` − zi ) 0 0   0 1 0 0 . TD (z` , zi ) =   0 0 1 (z` − zi )  0 0 0 1

(5.232)

Scalar Theory: Spin-0 and Spinless Particles

267

p For a thin lens, with w |Kn |  1 and w ≈ 0, the above transfer map reduces to     hxi(z) hxi (zi )  p1 h pbx i (z)   1 h pb i (z )   0  = TD (z, zr ) TL (zr , z` ) TD (z` , zi )  p0 x i  ,  hyi(z)   hyi (zi ) 



 1 1 b by (zi ) p (z) y p0 p0 p   1 (z − zr ) 0 0  0  1 0 0 , TD (z, zr ) =   0 0 1 (z − zr )  0 0 0 1  2  1 2 + 1 + 2 w Kn 1 w1+ + 3!1 w3 Kn (1+ ) TL (zr , z` ) ≈ wKn + w3 Kn2 1+ 1 + 21 w2 Kn 1+  2  1 − 21 w2 Kn 1− w1− − 3!1 w3 Kn (1− ) ⊕ , −wKn + w3 Kn2 1− 1 − 12 w2 Kn 1−   1 0 0 0  wKn 1 0 0  , ≈  0 0 1 0  0 0 −wKn 1   1 (z` − zi ) 0 0   0 1 0 0 , (5.233) TD (z` , zi ) =   0 0 1 (z` − zi )  0 0 0 1 showing that the two focal lengths of the thin normal magnetic quadrupole lens remain the same, 1 1 = −wKn , = wKn , (5.234) (x) f f (y) without any appreciable change due to the h¯ -dependent Hermitian term in the optical Hamiltonian. As we have already mentioned, the h¯ -dependent anti-Hermitian term in the optical Hamiltonian also does not affect the lens properties in any appreciable way. This is why accelerator optics works so well without any need to consider quantum mechanics in its design and operation. 5.2.5

SKEW MAGNETIC QUADRUPOLE

Quantum mechanics of the skew magnetic quadrupole can be analysed in a similar way. As seen in the classical theory, the skew magnetic quadrupole will be equivalent to the normal magnetic quadrupole rotated by an angle π4 about the optic axis. A skew magnetic quadrupole is associated with the magnetic field ~B(~r) = (−Qs x, Qs y, 0) ,

(5.235)

corresponding to the vector potential ~A(~r) = (0, 0, −Qs xy) .

(5.236)

268

Quantum Mechanics of Charged Particle Beam Optics

Let z` and zr > z` be the boundaries of the quadrupole. Then, Qs is a nonzero constant for z` ≤ z ≤ zr and Qs = 0 outside the quadrupole (z < z` , z > zr ). For a paraxial beam propagating through this skew magnetic quadrupole along its optic axis, z-axis, the quantum beam optical Hamiltonian will be   −p + pb2⊥ , for z < z` , z > zr , 0 2p0 c (5.237) Ho =  −p0 + pb2⊥ + p0 Ks xy, for z ≤ z ≤ zr , `

2p0

where Ks = qQs /p0 . Now, if we make the transformation 

  x 1 0 1 0  pbx   0 1 1 0 1     y  = √2  −1 0 1 0 pby 0 −1 0 1

  x0    pb0x   0  = R   y  pb0y 

 x0 pb0x  , y0  pb0y

the Hamiltonian is transformed to  02  −p0 + pb⊥ , for z < z` , z > zr , 2p0 0 c   Ho = 02 b p 2 2 1 0 0  −p0 + ⊥ − p0 Ks x − y , for z` ≤ z ≤ zr , 2p0 2

(5.238)

(5.239)

which is same as for the propagation of the paraxial beam through a normal magnetic quadrupole (5.220) in the new (x0 , y0 ) coordinate system, with Kn replaced by Ks . The above transformation corresponds to a clockwise rotation of the (x, y) coordinate axes by π4 about the optic axis. Hence, the transfer map for the skew magnetic quadrupole can be obtained from the transfer map for the normal magnetic quadrupole as Tsq = RTnq R−1 ,

(5.240)

where the subscripts sq and nq stand for the skew magnetic quadrupole and the normal magnetic quadrupole, respectively. Using the same arguments as in the classical theory of the skew magnetic quadrupole, we find that we can write Tsq (z, zi ) = TD (z, zr ) Tsq,L (zr , z` ) TD (z` , zi ) = TD (z, zr ) RTnq,L (zr , z` ) R−1 TD (z` , zi ) ,

(5.241)

where R is as in (5.238), TD (z, zr ) and TD (z` , zi ) are the same as in (5.228), and Tnq,L (zr , z` ) is the same as TL (zr , z` ) in (5.228), with Kn replaced by Ks . One can get Tsq,L (zr , z` ) directly as follows. Observe that

i : h¯



   x   pbx   pb2⊥   + p0 Ks xy :  p0  =  2p0  y   pby p0

0 0 0 −Ks

1 0 0 −Ks 0 0 0 0

 x  0  pbx  0     p0  . (5.242) 1  y  pby 0 p0

269

Scalar Theory: Spin-0 and Spinless Particles

Then, one gets 

hxi (zr )

 h pbx i(zr )  p0  hyi (z r)  h pby i(zr )

hxi (z` )



 h pbx i(z` )    p0  = Tsq,L (zr , z` )  hyi (z `)   h pby i(z` )

   





p0

p0

 

0   0  = exp  w  0 −Ks  C+ √  1  − Ks S− =  C− 2 √ − Ks S+  hxi (z` )  h pbx i(z` )  p0 ×  hyi (z `)  h pby i(z` )

1 0 0 −Ks 0 0 0 0 √1 S+ Ks +

C

√1 S− Ks −

C

  hxi (z` ) 0 h pb i(z )   xp0 ` 0     1   hyi (z` ) h pby i(z` ) 0 C− √ − Ks S+ C+ √ − Ks S−

p0 √1 S− Ks −

C

√1 S+ Ks +

         

C

   , 

p0

with w (zr − z` ) ,

(5.243)

√ √ C± = cos(w Ks ) ± cosh(w Ks ), √ √ S± = sin(w Ks ) ± sinh(w Ks ).

(5.244)

where

It can be verified that Tsq,L = RTnq,L (zr , z` ) R−1 . 5.2.6

AXIALLY SYMMETRIC ELECTROSTATIC LENS

An axially symmetric electrostatic lens, or a round electrostatic lens, with the axis along the z-direction consists of the electric field corresponding to the potential ∞

φ (~r⊥ , z) =

(−1)n

∑ (n!)2 4n φ (2n) (z)r⊥2n

n=0

1 1 2 4 = φ (z) − φ 00 (z)r⊥ + φ 0000 (z)r⊥ −··· , 4 64

(5.245)

inside the lens region (z` < z < zr ). Outside the lens, i.e., (z < z` , z > zr ), φ (z) = 0. And, there is no magnetic field, i.e., ~A (~r) = (0, 0, 0). To get the paraxial quantum beam optical Hamiltonian of the round electrostatic lens, we have to start with (5.34) and make the paraxial approximation by dropping

270

Quantum Mechanics of Charged Particle Beam Optics

  terms of order higher than quadratic in ~r⊥ ,~pb⊥ . We also drop the h¯ -dependent terms. Then, we get     2 2 2 co = − 1 − qE φ (z) − q E φ 2 (z) p0 + 1 + qE φ (z) pb⊥ H 2p0 c2 p20 2c4 p40 c2 p20   qE qE 2 − 2 1 + 2 2 φ (z) φ 00 (z)r⊥ . (5.246) 4c p0 c p0 This quantum beam optical Hamiltonian of the round electrostatic lens is seen to be very similar to the quantum beam optical Hamiltonian of the round magnetic lens (5.95), except for the absence of the rotation term and the presence of an z-dependent coefficient for the drift term pb2⊥ /2p0 . The first constant term, though  z-dependent,  can be ignored as before since it has vanishing commutators with ~r⊥ ,~pb . The ⊥

~ above paraxial quantum beam optical Hamiltonian, being  quadratic  in ~r⊥ and pb⊥ , leads to a linear transfer map for the quantum averages of ~r⊥ ,~pb⊥ , and it is straightforward to calculate the map following the same procedure used in the case of round magnetic lens. When the beam is not paraxial, there will be aberrations and we have to go beyond the paraxial approximation. We shall not pursue the quantum mechanics of this lens further. As already mentioned, electrostatic lenses are used in the extraction, preparation, and initial acceleration of electron and ion beams in a variety of applications. 5.2.7

ELECTROSTATIC QUADRUPOLE LENS

The electric field of an ideal electrostatic quadrupole lens with the optic axis along the z-direction corresponds to the potential  1
in the lens region (z` < z < zr ) , Qe x2 − y2 2 (5.247) φ (~r⊥ , z) = 0 outside the lens (z < z` , z > zr ) , where Qe is a constant and zr − z` = w is the width of the lens. Starting with (5.34), using (5.247) and ~A = (0, 0, 0) since there is no magnetic field, and making  the parax ial approximation by dropping terms of order higher than quadratic in ~r⊥ ,~pb⊥ , we get 2
co ≈ −p0 + pb⊥ + 1 p0 Ke x2 − y2 , H 2p0 2

with Ke =

qEQe , c2 p20

(5.248)

as the paraxial quantum beam optical Hamiltonian operator for the electrostatic quadrupole lens. Actually, no h¯ -dependent terms appear up to this approximation. Simply by comparing the quantum beam optical Hamiltonian of the normal magnetic quadrupole lens (5.220) with the quantum beam optical Hamiltonian of the electrostatic quadrupole lens (5.248), it is readily seen that the electrostatic quadrupole lens is convergent in the xz-plane and divergent in the yz-plane when

271

Scalar Theory: Spin-0 and Spinless Particles

Ke > 0. When Ke < 0, this lens is divergent in the xz-plane and convergent in the yz-plane. At the paraxial level, there is no difference between the classical and quantum theories of the electrostatic quadrupole lens, except for the replacement of the classical variables by their quantum averages. Let us recall from the earlier discussion of the classical theory of electrostatic quadrupole lens. Ke has the dimension of length−2 . In the weak field case, when w2  1/ |Ke |, the lens can be considered as thin and has focal lengths 1 f (x)

= wKe ,

1 f (y)

= −wKe .

(5.249)

Deviations from the ideal behavior result from nonparaxial conditions, and to treat these deviations, we have to go beyond the paraxial approximation. We shall not pursue the quantum mechanics of this lens further here. 5.2.8

BENDING MAGNET

We have already discussed the classical mechanics of bending the charged particle beam by a dipole magnet. A constant magnetic field in the vertical direction produced by a dipole magnet bends the beam along a circular arc in the horizontal plane. The circular arc of radius of curvature ρ, the design trajectory of the particle, is the optic axis of the bending magnet. It is natural to use the arclength, say S, measured along the optic axis from some reference point as the independent coordinate, instead of z. Let the reference particle moving along the design trajectory carry an orthonormal XY -coordinate frame with it. The X-axis is taken to be perpendicular to the tangent to the design orbit and in the same horizontal plane as the trajectory, and the Y -axis is taken to be in the vertical direction perpendicular to both the X-axis and the trajectory. The curved S-axis is along the design trajectory and perpendicular to both the X and Y axes at any point on the design trajectory. The instantaneous position of the reference particle in the design trajectory at an arclength S from the reference point corresponds to X = 0 and Y = 0. Let any particle of the beam have coordinates (x, y, z) with respect to a fixed right-handed Cartesian coordinate frame, with its origin at the reference point on the design trajectory from which the arclength S is measured. Then, the two sets of coordinates of any particle of the beam, (X,Y, S) and (x, y, z), will be related as follows:     S S x = (ρ + X) cos − ρ, z = (ρ + X) sin , y = Y. (5.250) ρ ρ To understand the quantum mechanics of bending of the beam particle by the dipole magnet, we have to change the corresponding Klein–Gordon equation to the curved (X,Y, S)-coordinate system and find the quantum beam optical Hamiltonian governing the S-evolution of the beam variables. To this end, we proceed as follows. The free particle Klein–Gordon equation is    mc 2 1 ∂2 2 ∇ − 2 2 Ψ(~r,t) = Ψ(~r,t). (5.251) c ∂t h¯

272

Quantum Mechanics of Charged Particle Beam Optics

Multiplying throughout by c2 h¯ 2 , we can write this equation as −¯h2


∂ 2 Ψ(~r,t) = −c2 h¯ 2 ∇2 + m2 c4 Ψ(~r,t). 2 ∂t

(5.252)

In the curved (X,Y, S) coordinate frame, the line element ds, distance between two infinitesimally close points (X,Y, S) and (X + dX,Y + dY, S + dS), is given by ds2 = dx2 + dy2 + dz2 = dX 2 + dY 2 + ζ 2 dS2 ,

(5.253)

where ζ = 1 + κX and κ = 1/ρ is the curvature of the design orbit. Correspondingly, we know (see e.g., Arfken, Weber, and Harris [3]) that the Laplacian ∇2 in terms of (X,Y, S) coordinates is given by   1 ∂2 1 ∂ ∂ ∂2 + 2 2 ∇2 = ζ + 2 ζ ∂X ∂X ∂Y ζ ∂S =

∂2 ∂2 1 ∂2 κ ∂ + + 2 2+ . 2 2 ∂X ∂Y ζ ∂S ζ ∂X

(5.254)

  If we substitute this expression for ∇2 in (5.252) and change Ψ(~r,t) to Ψ ~R⊥ , S,t , we get     2 ∂ 2 Ψ ~R⊥ , S,t ∂2 1 ∂2 ∂ 2 2 −¯h2 = −c + + h ¯ ∂t 2 ∂ X 2 ∂Y 2 ζ 2 ∂ S2     κ ∂ + + m2 c4 Ψ ~R⊥ , S,t . (5.255) ζ ∂X
where ~R⊥ = (X,Y ). While on the left-hand side −¯h2 ∂ 2 /∂t 2 is Hermitian, the righthand side is not Hermitian because of the last term in the expression for ∇2 in (5.254). So, let us replace this term by the Hermitian term "    #   κ ∂ κ ∂ † 1 κ ∂ ∂ κ 1 + = − 2 ζ ∂X ζ ∂X 2 ζ ∂X ∂X ζ   κ2 1 κ ∂ , = 2, (5.256) = 2 ζ ∂X 2ζ where we have to remember that ζ is a function of X. Thus, we shall take the free particle Klein–Gordon equation in the curved (X,Y, S) coordinate system to be    2  ∂ 2 Ψ ~R⊥ , S,t ∂ ∂2 1 ∂2 2 2 2 h −¯h = −c + + ¯ ∂t 2 ∂ X 2 ∂Y 2 ζ 2 ∂ S2     κ2 2 4 + 2 + m c Ψ ~R⊥ , S,t . (5.257) 2ζ

273

Scalar Theory: Spin-0 and Spinless Particles

Let us now rewrite this equation as    ∂ 2 ~ i¯h Ψ R⊥ , S,t ∂t ( "      # ∂ 2 1 ∂ 2 ∂ 2 2 + −i¯h + 2 −i¯h = c −i¯h ∂X ∂Y ζ ∂S    2 2 2 c h¯ κ − + m2 c4 Ψ ~R⊥ , S,t . 2ζ 2

(5.258)

This suggests that in the (X,Y, S) coordinate system, we can take the Klein–Gordon equation for a charged particle in an electromagnetic field as  2 ∂ i¯h − qφ Ψ(~R⊥ , S,t) ∂t ( " 2  2 ∂ ∂ − qAX + −i¯h − qAY = c2 −i¯h ∂X ∂Y )  2 #   1 ∂ c2 h¯ 2 κ 2 2 4 ~R⊥ , S,t , + m c Ψ − qζ AS − + 2 −i¯h ζ ∂S 2ζ 2 (5.259) where (AX , AY , AS ) are the magnetic vector potentials and φ is the electric scalar potential, and using the principle of minimal coupling, we have made the replacements ∂ ∂ −→ −i¯h − qAX , ∂X ∂X ∂ ∂ −i¯h −→ −i¯h − qAY , ∂Y ∂Y     ∂ 1 ∂ 1 −i¯h −→ −i¯h − qAS ζ ∂S ζ ∂S   1 ∂ = −i¯h − qζ AS . ζ ∂S −i¯h

(5.260)

Let us now consider the beam to be monoenergetic and a beam particle to be associated with the wave function q     Ψ ~R⊥ , S,t = e−iEt/¯h ψ ~R⊥ , S , with E = m2 c4 + c2 p20 , (5.261) where p0 is the design momentum with which the particle enters the bending magnet from the free space outside. The time-independent Klein–Gordon equation becomes

274

Quantum Mechanics of Charged Particle Beam Optics

  (E − qφ )2 ψ ~R⊥ , S ( " =

2  2 ∂ ∂ − qAX + −i¯h − qAY c −i¯h ∂X ∂Y )  2 #   −i¯h ∂ c2 h¯ 2 κ 2 2 4 ~R⊥ , S , + + m c ψ − qAS − ζ ∂S 2ζ 2 2

(5.262) Let   ∂ −i¯h − qAX = PbX − qAX = πbX , ∂X   ∂ − qAY = PbY − qAY = πbY , −i¯h ∂Y   −i¯h ∂ − qAS = PbS − qAS = πbS , ζ ∂S

(5.263)

Then, by dividing throughout by c2 and rearranging the terms in (5.262), we can rewrite the time-independent Klein–Gordon equation as   
 πbS2 ψ ~R⊥ , S = p20 − πbX2 − πbY2 − p2 ψ ~R⊥ , S , (5.264) where p2 =

2qEφ c2

  qφ h¯ 2 κ 2 1− − . 2E 2ζ 2

(5.265)

For the dipole magnet, we have to take φ = 0,

AX = 0,

AY = 0,

  κX 2 AS = −B0 X − , 2ζ

(5.266)

so that the magnetic field is BX = 0,

BY = B0 ,

BS = 0,

(5.267)

as we have already seen in the classical theory of bending magnet. Then, we have       (5.268) πbS2 ψ ~R⊥ , S = p20 − Pb⊥2 + e p2 ψ ~R⊥ , S , with

h¯ 2 κ 2 2ζ 2 This equation can be equivalently written as ! ! 0 ψ πbS   1 = πbS 1 p20 − Pb⊥2 + e p2 0 p0 p0 ψ p2 e p2 =

0

(5.269)

ψ πbS p0 ψ

! .

(5.270)

275

Scalar Theory: Spin-0 and Spinless Particles

Let us introduce the two-component wave function !     ψ 1 1 1 ψ+ = =M πbS ψ− 2 1 −1 p0 ψ ! b ψ + πpS0 ψ 1 = . b 2 ψ − πpS0 ψ

ψ πbS p0 ψ

!

(5.271)

Now, πbS p0



ψ+ ψ−



1 = p0



−i¯h ∂ − qAS ζ ∂S



ψ+ ψ−



!   0 ψ+  1 −1 M =M 1 p20 − Pb⊥2 + e p2 0 ψ− p20         p2 − 2p1 2 Pb⊥2 − e p2 1 − 2p1 2 Pb⊥2 − e 0 0     ψ+ . = 1 ψ− Pb⊥2 − e p2 −1 + 2p1 2 Pb⊥2 − e p2 2p2 

0

0

(5.272) Rearranging this equation, we get     ∂ ψ+ ψ+ b i¯h =H , ψ− ∂ S ψ− b b H = −p0 σz + Eb + O,    ζ  b2 P⊥ − e p2 − p0 κX σz , Eb = −qζ AS I + 2p0   iζ Pb⊥2 − e p2 σ y . Ob = 2p0 Let us now apply a Foldy–Wouthuysen-like transformation !   (1) i ψ+ ψ+ iSb1 b = e , with Sb1 = σz O. (1) ψ− 2p0 ψ−

(5.273)

(5.274)

The result is ∂ i¯h ∂S

(1)

ψ+ (1) ψ−

!

!  (1)   ψ+ iSb1 b −iSb1 iSb1 ∂ −iSb1 = e He − i¯he e (1) ∂S ψ− ! ! (1) (1) ψ+ ψ+ b b = H(1) , = eiS1 b He−iS1 (1) (1) ψ− ψ−

(5.275)

since Sb1 is independent of the coordinate S. Calculating H(1) we have, up to the paraxial approximation required for our purpose,  
1 b2 (1) 2 H ≈ −qζ AS I + P − ζ p0 + e p σz . (5.276) 2p0 ⊥

276

Quantum Mechanics of Charged Particle Beam Optics

As we have done previously for deriving the general quantum beam optical Hamiltonian (5.34) for any system with a straight optic axis, we can retrace the above transformation and identify the upper component of the resulting two-component wave function with the wave function of the particle moving forward along the curved path. Then, we get the required S-evolution equation for the wave function as     1 b2 1 h¯ 2 κ 2 ∂  + P + qB0 κX 2 i¯h ψ ~R⊥ , S = − p0 + ∂S 4p0 2p0 ⊥ 2      h¯ 2 κ 3 X ψ ~R⊥ , S , (5.277) + qB0 − κ p0 − 4p0 where we have taken κX  1, reasonably, and hence 1/ζ ≈ (1 − κX). With the curvature of the design trajectory matched to the dipole magnetic field as p0 B0 ρ = , or, (qB0 − κ p0 ) = 0, (5.278) q we have i¯h

   ∂ ~ f co,d ψ ~R⊥ , S , ψ R⊥ , S = H ∂S     1 b2 1 h¯ 2 κ 2 f 2 2 c H o,d = −p0 + P⊥ + p0 κ X − (1 + κX) ψ ~R⊥ , S . 2p0 2 4p0 (5.279)

f co,d reproduces the classical beam optical Hamiltonian of the dipole Note that H fo,d in (4.185) when ~Pb⊥ and ~R⊥ are taken as the corresponding classical magnet H variables, and the h¯ -dependent term is dropped. f co,d , does not depend on The quantum beam optical Hamiltonian for the dipole, H S. Thus, the quantum transfer map for the dipole becomes, with ∆S = S − Si ,     f i c h¯ ∆S:H o,d : X e (S ) i         hXi(S)  1  f D E i ∆S:H co,d :    1 b PbX (Si )   p0 PX (S)   p0 e h¯  =   .  (5.280)     f i hY i(S) co,d : ∆S: H    D E  h¯ e Y (S ) i   1 b     p0 PY (S) f  1  i ∆S:H co,d : b h¯ PY (Si ) p0 e With   2 3 b i f co,d : X = i 1 Pb2 + 1 p0 κ 2 X 2 − h¯ κ X , X = PX , :H ⊥ h¯ h¯ 2p0 2 4p0 p0 " # 2 b 1 b2 1 i f h¯ κ 3 PbX h¯ 2 κ 3 co,d : PX = i :H P⊥ + p0 κ 2 X 2 − X, = −κ 2 X + , h¯ h¯ 2p0 p0 2 4p0 p0 4p20

277

Scalar Theory: Spin-0 and Spinless Particles

  h¯ 2 κ 3 PbY i f i 1 b2 1 2 2 c X,Y = , : H o,d : Y = P⊥ + p0 κ X − 2 4p0 p0 h¯ h¯ 2p0 " # 2 3 b b i f 1 b2 1 h¯ κ PY co,d : PY = i :H X, = 0, P + p0 κ 2 X 2 − h¯ h¯ 2p0 ⊥ 2 p0 4p0 p0

(5.281)

we get      

  hXi(S) D E cos(κ∆S)  1 b  p0 PX (S)   =    hY D i(S)  E 1 −κ sin(κ∆S) b p0 PY (S)   ⊕     +  

1 κ

sin(κ∆S)

   

cos(κ∆S)   hXi D (S Ei ) 1 ∆S  1 b   p0 PX (Si )     hY i (Si )  D E 1 0 1 b p0 PY (Si )  ¯ 2κ − h4p 2 (cos(κ∆S) − 1) 0  2 2  − h¯4pκ2 sin(κ∆S) .  0  0 0

     

(5.282)

This shows that a particle the dipole magnet the design trajectory, D entering E D along E i.e., with (hXi (Si ) = 0, PbX (Si ) = 0, hY i (Si ) = 0, PbY (Si ) = 0), will follow the curved design trajectory, except for some tiny quantum kicks in the X coordinate (∼ λ02 /ρ) and the X-gradient (∼ λ02 /ρ 2 ), where λ0 is the de Broglie wavelength! This again proves the remarkable effectiveness of classical mechanics in the design and operation of accelerator optics.

5.3

EFFECT OF QUANTUM UNCERTAINTIES ON ABERRATIONS IN ELECTRON MICROSCOPY AND NONLINEARITIES IN ACCELERATOR OPTICS

Now we have to emphasize an important aspect of phase space transfer maps, as revealed by the quantum theory in contrastDto the E classical theory. We have ~ identified the quantum averages h~r⊥ i (z) and pb⊥ (z)/p0 as the classical ray coordinates corresponding to the position and the slope of the ray intersecting the xy-plane at z. As we have already noted, for the round magnetic lens, central to electron microscopy, the expressions we have derived from quantum theory for the various aberration coefficients are the same as their respective classical expressions, of course, under the approximations considered. However, the quantum expressions in (5.201–5.204) would correspond exactly to the

278

Quantum Mechanics of Charged Particle Beam Optics

classical expressions foraberrations of position provided we can oE and gradient,   Dn

2 2 ~ ~ , x , pb⊥ , etc., respectively, by replace pbx pb⊥ , pbx , pb⊥ ·~r⊥ +~r⊥ · pb⊥    

2 

 2 2 2 h pbx i h pbx i + pby , etc. But, , 4 hxi h pbx i + hyi h pbx i pby , 2hxi h pbx i2 + pby that   Emechanics, in general, for any observable O, DIn quantum   beE done. D cannot b b unless |ψi is an eigenstate of O. And, for any ψ f O ψ 6= f ψ O ψ D E D E  E D  b b b b two observables O1 and O2 , ψ f O1 , O2 ψ 6= f ψ O 1 ψ , ψ O2 ψ b1 and O b2 . It is thus clear that for unless |ψi is a simultaneous eigenstate of both O the wave packets involved in electron optical imaging the replacement as mentioned

 
above is not allowed. As an illustration, consider the term ∼ ~r⊥ , pb2⊥ zob j , one of the terms contributing to coma (see (5.201) and (5.202)) which, being linear in position, is the dominant aberration next
to the spherical aberration. The corresponding classical term, (dx/dz)2 + (dy/dz)2 ~r⊥ at zob j , vanishes obviously for an object
h~r⊥ i zob j = (0, 0) the value point on the axis. But, for a quantum wave packet with

 

of ~r⊥ , pb2⊥ zob j need not be zero since it is not linear in h~r⊥ i zob j . This can be seen more explicitly as follows. Let δcx = x − hxi, δcpx = pbx − h pbx i,

δcy = y − hyi, δcpy = pby − h pby i.

(5.283)

Then,   2 cr⊥ , h pbx i + δcp h~r⊥ i + δ~ x   2 
zob j + pby + δcpy   2   2 2 cr⊥ , h pbx i2 + δcp h~r⊥ i + δ~ = + pby + δcpy x  


+ 2 h pbx i δcpx + pby δcpy zob j
D E
2 = 2 h~r⊥ i zob j ~pb⊥ zob j  
 c 2  c 2
+ 2 h~r⊥ i zob j δ px + δ py zob j   2  2 
c c c + δ~r⊥ , δ px + δ py zob j Dn oE

cr⊥ , δcp + 2 δ~ zob j h pbx i zob j x Dn oE

cr⊥ , δcp + 2 δ~ zob j pby zob j , (5.284) y

 
~r⊥ , pb2⊥ zob j =

showing clearly that this coma term is not necessarily zero for an object point on the axis, i.e., when h~r⊥ i = (0, 0). The above equation also shows how this coma term for

279

Scalar Theory: Spin-0 and Spinless Particles

off-axis points (h~r⊥ i 6= (0, 0)) depends also on the higher order central of D Emoments 


~ the wave packet besides the position h~r⊥ i zob j and the slope pb⊥ zob j /p0 of the corresponding classical ray. When an aperture is introduced in the path of the beam to limit the transverse momentum spread, one will be introducing uncertainties in position coordinates s  s  2 2 c δx , δcy , ∆y = (5.285) ∆x = and hence the corresponding momentum uncertainties s s   2 2 c c ∆px = δp , ∆py = δp , x

y

(5.286)

in accordance with the Heisenberg uncertainty principle, and this would influence the aberrations. However, in practice, such tiny quantum effects might be masked by classical uncertainties caused by instrumental imperfections. In the context of accelerator optics, let us consider a sextupole magnet normally used for the control of chromaticity arising due to the beam being not monoenergetic. Sextupoles are used in electron microscopy for the correction of spherical aberration. A sextupole magnet, with its straight optic axis along the z-direction, is associated with the magnetic field  
~B = Qsx xy, 1 Qsx x2 − y2 , 0 , (5.287) 2 corresponding to the vector potential  
~A = 0, 0, − 1 Qsx x3 − 3xy2 , 6

(5.288)

where Qsx is a constant in the sextupole region z` ≤ z ≤ zr and zero outside. For a quasiparaxial beam propagating in the +z-direction through the sextupole, the quantum beam optical Hamiltonian is seen to be, from (5.34),   −p + pb2⊥ + 1 p K x3 − 3xy2
, z ≤ z ≤ z , r 0 ` 2p0 6 0 sx c (5.289) Ho =  −p0 + pb2⊥ , z < z , z > zr , 2p0

`

co , we have to keep terms of with Ksx = qQsx /p0 . Note that in deriving the above H order up to third power in (x, y), even for a paraxial beam, since the paraxial approximation is meaningless in this case; in the paraxial approximation, the sextupole will disappear. Let us assume the sextupole to be thin such that w = zr − z` ≈ 0. Then, for the propagation of the beam through the sextupole, from z` to zr , the phase space transfer map is seen to be

280

Quantum Mechanics of Charged Particle Beam Optics



   

      hxi (zr )   1  bx i (zr ) p0 h p =   hyi 

(zr )  1 b p (z )  y r p0        ≈  

* e *

i h¯ w:

  pb2 −p0 + 2p⊥ + 61 p0 Ksx (x3 −3xy2 ) : 0

+



x (z` ) +

     1 0 pbx (z` )  p0 e    +  * , 2 b p i w: −p + ⊥ + 1 p K 3 2  0 sx (x −3xy ) : 0 2p 6 h¯ 0 y (z` )  e      * + 2  i w: −p + pb⊥ + 1 p K 3 2 sx (x −3xy ) : 0 0  2p 6 h ¯ 1 0 b e p (z ) y ` p0   pb2 i 1 3 2 ⊥ h¯ w: −p0 + 2p + 6 p0 Ksx (x −3xy ) :

`) hxi (z` ) + w hpxpi(z 0




1 bx i (z` ) − 21 wKsx x2 (z` ) − y2 (z` ) p0 h p h py i(z ) hyi (z` ) + w p0 `

 1 by (z` ) + wKsx hxyi (z` ) p0 p

   .   (5.290)

Let us look at the nonlinear maps for the momentum components




1 h pbx i (zr ) ≈ h pbx i (z` ) − wp0 Ksx x2 (z` ) − y2 (z` ) , 2



 pby (zr ) ≈ pby (z` ) + wp0 Ksx hxyi (z` ) .

(5.291)

It is clear that in the classical theory, one would have the corresponding maps as   1 px (zr ) ≈ px (z` ) − wp0 Ksx x (z` )2 − y (z` )2 , 2 py (zr ) ≈ py (z` ) + wp0 Ksx x (z` ) y (z` ) . (5.292) Substituting x = hxi + δcx and y = hxi + δcy, the quantum maps (5.291) become     2 1 2 c h pbx i (zr ) ≈ h pbx i (z` ) − wp0 Ksx hxi (z` ) + δ x (z` ) 2     2 2 c − hyi (z` ) + δ y (z` ) , h



 pby (zr ) ≈ pby (z` ) + wp0 Ksx hxi (z` ) hyi (z` ) D   E i + δcx δcy (z` ) . (5.293) This shows that, generally, the leading quantum effects on the nonlinear accelerator optics can be expected to be due to the uncertainties in the position coordinates and the momentum components of the particles of the beam entering the optical elements. Such quantum effects involve h¯ only through the uncertainties and not explicitly. This has already been pointed out by Heifets and Yan [73], who have shown

281

Scalar Theory: Spin-0 and Spinless Particles

that the quantum effects due to such uncertainties affect substantially the classical results of tracking for trajectories close to the separatrix, and hence the quantum maps can be useful in quick findings of the nonlinear resonances. Accelerator beams are complicated nonlinear, stochastic, many-particle systems. For details on the nonlinear optics of accelerator beams, see, e.g., the references on accelerator physics mentioned earlier (Berz, Makino, and Wan [12], Conte and MacKay [27], Lee [127], Reiser [158], Seryi [168], Weidemann [187], Wolski [192], Chao, Mess, Tigner, and Zimmermann [20]), and also Mais [131], Todesco [184], and references therein.

5.4

NONRELATIVISTIC QUANTUM CHARGED PARTICLE BEAM OPTICS: SPIN-0 AND SPINLESS PARTICLES

The nonrelativistic Schr¨odinger equation for a particle of charge q and mass m moving in the time-independent electromagnetic field of an optical system is   2 1 ~ ∂ Ψ (~r,t) ~ pb − qA (~r) + qφ (~r) Ψ (~r,t) , (5.294) = i¯h ∂t 2m where ~A (~r) and φ (~r) are the magnetic vector potential and the electric scalar potential. Let us consider a particle of a nonrelativistic monoenergetic quasiparaxial beam with total energy E = p20 /2m, where p0 is the design momentum with which it enters the optical system from the free space outside and |~p0⊥ |  p0 . We assume that the total energy E is conserved when the beam propagates through any optical system we are considering. Choosing the wave function of the particle as Ψ (~r,t) = e−iEt/¯h ψ (~r) , we get the time-independent Schr¨odinger equation obeyed by ψ (~r⊥ , z):  2  πb + qφ ψ (~r⊥ , z) = Eψ (~r⊥ , z) , 2m

(5.295)

(5.296)

where ~πb = ~pb − q~A. Multiplying the equation on both sides by 2m, taking 2mE = p20 ,

2mqφ = p˜ 2 ,

(5.297)

and rearranging the terms, we have
πbz2 ψ = p20 − p˜ 2 − πb⊥2 ψ.

(5.298)

Note that this equation is identical, except for the expressions of the symbols p20 and p˜ 2 , to the time-independent Klein–Gordon equation (5.7) which is the starting point for our derivation of the relativistic quantum beam optical Hamiltonian (5.34). Now, starting with (5.298), we can follow the same procedure, going through the Feshbach–Villars-like formalism and the Foldy–Wouthuysen-like transformations, and finally arrive at the nonrelativistic quantum beam optical Schr¨odinger equation,

282

Quantum Mechanics of Charged Particle Beam Optics

i¯h

∂ ψ (~r⊥ , z) co,NR ψ (~r⊥ , z) , =H ∂z

co,NR = −p0 − qAz + 1 πb2 + p˜ 2 + 1 πb2 + p˜ 2 2 H 2p0 ⊥ 8p30 ⊥ ("
  1 − πb⊥2 + p˜ 2 , πb⊥2 , qAz 16p40 !# ∂ ~A⊥ ∂ ~A⊥ ~ ~ +i¯hq pb⊥ · + · pb⊥ ∂z ∂z  
∂ 2 2 2 2 q A⊥ + p˜ , − πb⊥ , i¯h ∂z with p20 = 2mE,

p˜ 2 = 2mqφ ,

(5.299)

which is identical to the relativistic quantum beam optical equation (5.34), except for the expressions of p20 and p˜ 2 . This shows that one can approximate the expressions and formulae derived from the relativistic quantum beam optical equation (5.34) using the nonrelativistic approximation


1 1 1 E 2 − m2 c4 = 2 E + mc2 E − mc2 ≈ 2 2mc2 E = 2mE, 2 c c c   2φ 2qEφ qφ 2qEφ 2qmc p˜2 = 1− ≈ ≈ = 2mqφ , (5.300) c2 2E c2 c2

p20 =

as is appropriate for the charged particle beam optical systems for which E = mc2 + E, E  mc2 and qφ  mc2 . Conversely, since the nonrelativistic and the relativistic quantum beam optical equations of motion, (5.299) and (5.34) are identical, except for the expressions of the symbols E, p20 , and p˜ 2 , it is possible to convert the expressions and formulae derived from the nonrelativistic equation to relativistic expressions and formulae by the replacement √ p0 = 2mE = mv0 1p 2 −→ p0 = E − m2 c4 c v   u u 1u 2 4  1 mv0 = tm c − 1 = q , (5.301) v20 c v2 1 − c2 1 − c02 or, m −→ q

m v2

,

(5.302)

1 − c02

i.e., replacement of the rest mass by the so-called relativistic mass, as has been the common practice in electron optics.

Scalar Theory: Spin-0 and Spinless Particles

5.5

283

APPENDIX: PROPAGATOR FOR A SYSTEM WITH TIME-DEPENDENT QUADRATIC HAMILTONIAN

Let it be required to find the propagator for a one-dimensional system obeying the Schr¨odinger equation ∂ |Ψ(t)i b i¯h = H(t)|Ψ(t)i, (5.303) ∂t where the time-dependent Hamiltonian has the form b = A(t) pb2x + B(t) (x pbx + pbx x) +C(t)x2 . H(t) with A(t), B(t), and C(t) being real functions of t. Using the Magnus formula (3.768), we can write

b t,t 0 |Ψ t 0 i |Ψ(t)i = U

(5.304)

(5.305)

with  Z
i t 0 b b (t1 ) U t,t = exp − dt1 H h¯ t 0   Z Z t2 h i i 2 t 1 b (t2 ) , H b (t1 ) − dt2 dt1 H + 2 h¯ t0 t0  3 Z t Z t3 Z t2 hh i i i 1 b (t3 ) , H b (t2 ) , H b (t1 ) − dt3 + dt2 dt1 H 6 h¯ t0 t0 t0 hh i i o b (t1 ) , H b (t2 ) , H b (t3 ) + · · · . . + H (5.306) The required propagator is given by
D
E b t,t 0 x0 , K x,t; x0 ,t 0 = x U such that

Z

Ψ(x,t) =



dx K x,t; x0 ,t 0 Ψ x0 ,t 0 .

(5.307)

(5.308)


Note that the operators pb2x , (x pbx + pbx x) , x2 are closed under commutation leading to the Lie algebra  2  pbx , (x pbx + pbx x) = −4i¯h pb2x ,  2 2 pbx , x = −2i¯h (x pbx + pbx x) ,   (x pbx + pbx x) , x2 = −4i¯hx2 . (5.309) Then, it is clear that we can write  
2


 i 0 0 0 2 0 b , U t,t = exp − a t,t pbx + b t,t (x pbx + pbx x) + c t,t x h¯

(5.310)

284

Quantum Mechanics of Charged Particle Beam Optics

where a (t,t 0 ), b (t,t 0 ), and c (t,t 0 ) are infinite series expressions in terms of A(t), B(t), and C(t). To obtain the precise form of the equation (5.310) we proceed as follows. b (t,t 0 ) |Ψ (t 0 )i in (5.303), it is seen that Substituting |Ψ(t)i = U i¯h

b (t,t 0 )
∂U b U b t,t 0 , = H(t) ∂t


b t 0 ,t 0 = I, U

(5.311)

where I is the identity operator. Following Wolf [190], let us write a=

ϕβ , 2 sin ϕ

b=

ϕ(α − δ ) , 4 sin ϕ

c=−

ϕγ , 2 sin ϕ

1 cos ϕ = (α + δ ). 2

(5.312)

Then, we have   

ϕ (t,t 0 ) b t,t 0 = exp − i U β t,t 0 pb2x 0 h¯ 2 sin ϕ (t,t ) 



1 + α t,t 0 − δ t,t 0 (x pbx + pbx x) − γ t,t 0 x2 . 2 (5.313) b (t,t 0 ) in (5.311), with H(t) given by (5.304), shows Substituting this expression for U 0 0 0 that α (t,t ), β (t,t ), γ (t,t ), and δ (t,t 0 ) satisfy the following equations: 
 ˙ − 2AB˙ α = 0, Aα¨ − A˙ α˙ + 4A AC − B2 + 2AB


α t 0 ,t 0 = 1, α˙ t 0 ,t 0 = 2B t 0 , (5.314)


0 0 0 0 0 α β¨ − β α¨ − 2A˙ = 0, β t ,t = 0, β˙ t ,t = 2A t , (5.315) α˙ − 2αB = 2Aγ, αδ − β γ = 1,

(5.316) (5.317)

where α˙ = dα/dt, α¨ = d 2 α/dt 2 , etc. It is thus clear that, given A(t), B(t), and C(t) in b H(t), the above equations provide α (t,t 0 ), β (t,t 0 ), γ (t,t 0 ), and δ (t,t 0 ), and ϕ (t,t 0 ) b 0 ) is now completely determined through is given by 2 cos ϕ = α + δ . Hence, U(t,t (5.313). To find the required propagator, observe that      b † (t,t 0 ) xU b (t,t 0 ) x U α (t,t 0 ) β (t,t 0 ) . (5.318) b † (t,t 0 ) pbxU b (t,t 0 ) = γ (t,t 0 ) δ (t,t 0 ) pbx U b (t,t 0 ) generates a real linear canonical transformation of the conjugate Note that U pair (x , pbx ), i.e., if Xb = αx + β pbx , Pb = γx + δ pbx , (5.319) then

h i Xb , Pb = (αδ − β γ) [x , pbx ] = i¯h.

(5.320)

285

Scalar Theory: Spin-0 and Spinless Particles

We can rewrite (5.318) as




b t,t 0 |ψi = U b t,t 0 α t,t 0 x + β t,t 0 pbx |ψi xU




b t,t 0 |ψi = U b t,t 0 γ t,t 0 x + δ t,t 0 pbx |ψi, pbxU

(5.321)

for any |ψi. Writing out E explicitly in terms of the matrix elements, it is D this equation b 0 possible to solve for x U (t,t ) x0 = K (x,t; x0 ,t 0 ), up to a multiplicative constant phase factor (see Wolf [189, 190] for details of the solution). The result is h

0

K x,t; x ,t

0




i

i 0 02 0 0 2 1 0 α (t,t )x −2xx +δ (t,t )x =p e 2¯hβ (t,t ) , 0 2πi¯hβ (t,t )
if β t,t 0 6= 0,

(5.322)

and 0

K x,t; x0 ,t


0

iγ (t,t )   0 e 2¯hα (t,t ) x 0 , =p δ x− α (t,t 0 ) α (t,t 0 )


if β t,t 0 = 0.

(5.323)

In the two-dimensional case, if the Hamiltonian is of the form   2 b = A(t) pb2⊥ + B(t) ~r⊥ ·~pb⊥ +~pb⊥ ·~r⊥ + c(t)r⊥ H(t) ,

(5.324)

the variables x and y are independent and separable, and the above results can be 2 2 2 extended in a straightforward manner with   the replacements x −→ r⊥ , pbx −→
pb2 , and (x pbx + pbx x) −→ ~r⊥ ·~pb +~pb ·~r⊥ . Then, we can write K ~r⊥ ,t;~r0 ,t 0 = ⊥

⊥

⊥

⊥

K (x,t; x0 ,t 0 ) K (y,t; y0 ,t 0 ). In the case of the round magnetic lens taking (t,t 0 ) as (z, zi ), one has in (5.97) A(z) =

1 , 2p0

C(z) =

1 p0 α 2 (z). 2

(5.325)

β (z, zi ) =

1 h (z, zi ) , p0

(5.326)

B(z) = 0,

Then, writing α (z, zi ) = g (z, zi ) , we have from (5.314)–(5.317) g00 (z, zi ) + α 2 (z)g (z, zi ) = 0,

g0

g (zi , zi ) = 1,

g0 (zi , zi ) = 0,

h00 (z, zi ) + α 2 (z)h (z, zi ) = 0, h (zi , zi ) = 0, 0 γ (z, zi ) = p0 g (z, zi ) ,
1 δ (z, zi ) = 1 + h (z, zi ) g0 (z, zi ) , g (z, zi ) α (z, zi ) δ (z, zi ) − β (z, zi ) γ (z, zi ) = 1,

h0 (zi , zi ) = 1,

g00

d 2 g/dz2 ,

(5.327)

where = dg/dz, = etc. showing that g (z, zi ) and h (z, zi ) are two linearly independent solutions of the paraxial equation (4.51) satisfying the initial conditions (5.114) and the relation (5.115).

Charged Particle 6 Quantum Beam Optics Spinor Theory for Spin- 21 Particles 6.1 6.1.1

RELATIVISTIC QUANTUM CHARGED PARTICLE BEAM OPTICS BASED ON THE DIRAC–PAULI EQUATION GENERAL FORMALISM

For electrons, or any spin- 12 particle, the proper equation to be the basis for the theory of beam optics is the Dirac equation. Sudarshan, Simon, and Mukunda have shown that there is a simple rule by which paraxial scalar wave optics based on the Helmholtz equation can be generalized consistently to paraxial Maxwell wave optics, i.e., to Fourier optics for the Maxwell field (Sudarshan, Simon, and Mukunda [175], Mukunda, Simon, and Sudarshan [137, 138], Simon, Sudarshan, and Mukunda [170, 171]). Searching for a similar passage from the scalar electron optics based on the nonrelativistic Schr¨odinger equation to the spinor electron optics based on the Dirac equation, a formalism of quantum electron beam optics was initiated with application to round magnetic lens by Jagannathan, Simon, Sudarshan, and Mukunda [79]. A more comprehensive formalism of quantum charged particle beam optics, with applications to several other electromagnetic optical systems, was developed later by Jagannathan [80], Khan and Jagannathan [90], Jagannathan and Khan [82], Conte, Jagannathan, Khan and Pusterla [26], and Khan [91]. This chapter on the spinor theory of quantum charged particle beam optics follows mostly the work by Jagannathan, Simon, Sudarshan, and Mukunda [79], Jagannathan [80], Khan and Jagannathan [90], Jagannathan and Khan [82], Conte, Jagannathan, Khan, and Pusterla [26], and Khan [91] (see also Jagannathan [83, 84, 85], Khan [92, 93, 95, 99]. There were some preliminary studies by Rubinowicz [161, 162, 163, 164], Durand [38], Phan-Van-Loc [144, 145, 146, 147, 148, 149] on the use of the Dirac equation in electron optics. In 1986, Ferwerda, Hoenders, and Slump [48, 49] concluded that the use of the scalar Klein–Gordon equation in electron microscopy could be vindicated because a scalar approximation of the Dirac spinor theory would be justifiable under the conditions obtaining in electron microscopy (see also Lubk [129]). However, with the technological developments in electron microscopy, like the Low Energy Electron Microscopy (LEEM) (see Bauer [7]) and Spin Polarized Low Energy Electron Microscopy (SPLEEM) (see Rougemaille and 287

288

Quantum Mechanics of Charged Particle Beam Optics

Schmid [160]), the need for a proper quantum theory of electron beam optics based on the Dirac equation is imminent. As we have already seen, the Dirac equation is the relativistic equation linear in the first derivative with respect to time, possessing proper probability and current densities obeying a continuity equation, and naturally incorporating an intrinsic angular momentum, or spin, h¯ /2 for the particle described by it. Being linear in all the first derivatives with respect to the space and time coordinates (x, y, z,t), it is, in a way, already in a quantum beam optical form. Let us recall. The Dirac equation for a free particle of mass m is i¯h

∂ |Ψ(t)i b |Ψ(t)i, =H D,f ∂t b = mc2 β + c~α ·~pb, H D,f   I O β= , O −I

 ~α =

O ~σ ~σ O

 ,

(6.1)

where the state vector |Ψ(t)i is the four-component Dirac spinor,     1 0 0 0 I= , O= , 0 1 0 0 and

 σx =

0 1

1 0



 ,

σy =

0 −i i 0



 ,

σz =

1 0 0 −1

(6.2)  ,

(6.3)

are the Pauli matrices. The matrices αx , αy , αz , and β are known as the Dirac matrices. For a particle of charge q, mass m, and  anomalous  magnetic moment µa , in a time-independent electromagnetic field ~E((~r) , ~B (~r) , with the magnetic vector potential ~A (~r) and the electric scalar potential φ (~r), the Dirac–Pauli equation is   ∂ bDP |Ψ(t)i, − qφ ) |Ψ(t)i = H i¯h ∂t   bDP = mc2 β + c~α · ~pb − q~A − µa β~Σ · ~B, H (6.4) obtained from the free particle equation (6.1) by the principle of electromagnetic minimal coupling and adding the Pauli term to take into account the anomalous magnetic moment. In general, for any spin- 21 particle, with charge q and mass m, we can write the total spin magnetic moment as ~µ = (gq/2m)~S = (gq¯h/4m)~σ = µ ~σ . Writing g = 2(1 + a), we have µ = [(1 + a)q¯h/2m] with q¯h/2m as the Dirac magnetic moment corresponding to g = 2 and µa = aq¯h/2m as the anomalous magnetic moment corresponding to the magnetic anomaly a = (g/2) − 1. First, we shall get the general formalism to study the propagation of a quasiparaxial Dirac particle beam through electromagnetic optical systems with straight optic axis along the z-direction. To this end, we proceed as follows. Let us take Ψ(~r,t) = e−iEt/¯h ψ (~r⊥ , z) ,

(6.5)

289

Spinor Theory for Spin-1/2 Particles

as the four-component wave function of a particle of the monoenergetic quasiparaxial beam moving forward along the optic axis in which ψ (~r⊥ , z) is the time-independent four-component wave function. With ~p0 = p0~k as the design momentum, and ~p0,⊥  p0 , we shall take the energy of the particle to be positive given by q E = + m2 c4 + c2 p20 . Substituting the expression for Ψ(~r,t) in (6.4) we get h   i (E − qφ )ψ (~r⊥ , z) = mc2 β + c~α · ~pb − q~A − µa β~Σ · ~B ψ (~r⊥ , z) .

(6.6)

Multiplying this equation throughout from left by αz /c and rearranging the terms, we can write i¯h

∂ |ψ(z)i ∂z

c|ψ(z)i, =H

c= −p0 β χαz − qAz I + (q/c)φ αz + αz~α⊥ ·~πb⊥ + µa β αz~Σ · ~B, H s   E + mc2 ξI O , (6.7) with χ = , ξ= −1 O −ξ I E − mc2 c is not Hermitian. This is where I is the 4 × 4 identity matrix. It is seen that H related to the fact that, as we have already noted in the case of the scalar theory, the probability of finding the particle in the vertical plane at the point z, R 2 d r⊥ ψ † (~r⊥ , z) ψ (~r⊥ , z), need not be conserved along the z-axis. Noting that, with 1 M = √ (I + χαz ) , 2

1 M−1 = √ (I − χαz ) , 2

(6.8)

one has we define

Mβ χαz M−1 = β ,

(6.9)

E E 0 ψ (z) = M ψ(z) .

(6.10)

Then, i¯h

E ∂ 0 E c0 ψ 0 (z) , ψ (z) = H ∂z c0 = MH cM−1 = −p0 β + Eb + O, b H
qφ Eb = −qAz I + ξ + ξ −1 β 2c i

µa h − ξ + ξ −1 ~B⊥ ·~Σ⊥ + ξ − ξ −1 β Bz Σz , n2c
µa  Ob = χ ~α⊥ ·~πb⊥ + ξ − ξ −1 (σy ⊗ (Bx σy − By σx )) 2c
 + ξ + ξ −1 Bz (σx ⊗ I) .

(6.11)

290

Quantum Mechanics of Charged Particle Beam Optics

To see the effect of the transformation (6.10), let us apply it to a free timeindependent paraxial positive-energy Dirac plane wave propagating in the +zdirection:   1 ψ10 (~r⊥ , z)   ψ 0 (~r⊥ , z)   = √1  0−1  20  ψ3 (~r⊥ , z)  2  −ξ 0 ψ4 (~r⊥ , z) 0  s 1 cp0 ξ   4 π 3 h¯ 3 E  

0 1 0 ξ −1

1 p0 ξ 1 p0 ξ

ξ 0 1 0

 0 −ξ  × 0  1 

a+  i (~p ·~r +p z) a−  h¯ ⊥ ⊥ z , (a+ pz + a− p− )  e (a+ p+ − a− pz )

with |~p| = p0 , p+ = px + ipy , p− = px − ipy , |~p⊥ |  pz ≈ p0 ,

|a+ |2 + |a− |2 = 1,

(6.12)

where we have used the positive-energy solutions of the free-particle Dirac equation (see (3.637), (3.640), and (3.641)). The result is  s ψ10 (~r⊥ , z)  ψ 0 (~r⊥ , z)  1 cp0 ξ   20  ψ3 (~r⊥ , z)  = 4 2π 3 h¯ 3 E ψ40 (~r⊥ , z) 

    

1 p0 1 p0 1 −p ξ 0 1 p0 ξ

[a+ (p0 + pz ) + a− p− ] [a− (p0 + pz ) − a+ p+ ] [a+ (p0 − pz ) − a− p− ] [a− (p0 − pz ) + a+ p+ ]

  i  h¯ (~p⊥ ·~r⊥ +pz z) . e  (6.13)

It is seen that the upper pair of components of |ψ 0 (z)i are very large compared to the lower pair of components since pz ≈ p0 for any particle of the paraxial beam. The wave packet representing any particle of a positive-energy monoenergetic paraxial beam will be of the form Ψ(~r,t) = e−iEt/¯h ψ (~r⊥ , z) , Z

ψ (~r⊥ , z) =

i

|~p|=p0

d 2 p⊥ u (~p) e h¯ (~p⊥ ·~r⊥ +pz z) ,

 a+ (~p)  a− (~p) cp0 ξ  1  1 , u (~p) = 4 π 3 h¯ 3 E  p0 ξ (a+ (~p) pz + a− (~p) p− )  1 (a+ (~p) p+ − a− (~p) pz ) p0 ξ Z   with d 2 p⊥ |a+ (~p)|2 + |a− (~p)|2 = 1, |~p|=p0 q |~p⊥ |  pz ≈ p0 , E = c2 p20 + m2 c4 . 

s

(6.14)

291

Spinor Theory for Spin-1/2 Particles

E E From (6.13), it is clear that for any such paraxial wave packet, ψ 0 (z) = M ψ(z) will have its upper pair of components large compared to the lower pair of compoc0 in (6.11) does not nents. Thus, in the paraxial situation, the even operator Eb of H E couple the large upper components and small lower components of ψ 0 (z) while the c0 does couple them. This is exactly the same as in the nonrelaodd operator Ob of H tivistic situation obtained in the standard Dirac theory with respect to time evolution. This and the striking resemblance of (6.11) to the standard Dirac equation (3.711) make us turn to the Foldy–Wouthuysen transformation technique to analyse (6.11) further. Let us recall that the Foldy–Wouthuysen technique is useful in analysing the Dirac equation systematically as a sum of nonrelativistic part and a series of relativistic correction terms. It is essentially based on the fact that β commutes with any even operator that does not couple the upper and lower pairs of components of the Dirac spinor and anticommutes with any odd operator that couples the upper and lower pairs of components of the Dirac spinor. So, applying a Foldy–Wouthuysenlike transformation to (6.11) should help us analyse it as a sum of the paraxial part and a series of nonparaxial, or aberration, correction terms. To this end, we make the first transformation E E b (1) = eiS1 ψ 0 , ψ

i b Sb1 = β O. 2p0

(6.15)

E The resulting equation for ψ (1) is i¯h

E ∂ (1) E c(1) ψ (1) , =H ψ ∂z   1 b − 2p1 β Ob c0 2p1 β Ob − 2p1 β Ob ∂ (1) 2p0 β O c 0 0 0 e H =e H e − i¯he ∂z = −p0 β + Eb(1) + Ob (1) , E

b(1)

" 1 1 2 ≈ Eb − β Ob − 2 Ob , 2p0 8p0

1 Ob (1) ≈ − β 2p0

h i ∂ Ob Ob , Eb + i¯h ∂z

h i ∂ Ob Ob , Eb + i¯h ∂z ! −

1 b3 O . 3p20

!# −

1 β Ob 4 , 8p30 (6.16)

E Before proceeding further, let us find out the nature of ψ (1) by looking at the free particle case. From (6.13), for the free particle positive-energy plane wave spinor, we get

292

Quantum Mechanics of Charged Particle Beam Optics − 2p1 β χ~α⊥ ·~pb⊥

ψ (1) (~r⊥ , z) = e

1 ≈ 4

s

0

ψ 0 (~r⊥ , z)

cp0 ξ 2π 3 h¯ 3 E

  1 I− β χ~α⊥ ·~pb⊥ 2p0



1 p0 1 p0 1 −p ξ 0 1 p0 ξ



 h  i  p2 a+ 1 + pp0z − 2p⊥2 + 12 a− pp−0 1 + pp0z 0 i  h  p2 a− 1 + pp0z + 2p⊥2 − 12 a+ pp+0 1 + pp0z 0 n   h  io p2 − ξ1 a+ 1 − pp0z − 2p⊥2 + 12 a− pp−0 1 − pp0z 0 n  h  io p2⊥ p+ pz pz 1 1 a 1 − − + a 1 − − + 2 p0 2 p p ξ 2p 0 0

 [a+ (p0 + pz ) + a− p− ]  [a− (p0 + pz ) − a+ p+ ]    h¯i (~p⊥ ·~r⊥ +pz z) × e [a (p − p ) − a p ]  + z − −  0 [a− (p0 − pz ) + a+ p+ ]   1 0 0 − ξ2pp−0 s   0  0 1 − ξ2pp+0 1 cp0 ξ    = 4 2π 3 h¯ 3 E  − ξ2pp−0 1 0   0  0 0 1 − ξ2pp+0   1 p0 [a+ (p0 + pz ) + a− p− ] 1    p0 [a− (p0 + pz ) − a+ p+ ]  h¯i (~p⊥ ·~r⊥ +pz z) × e  − p01ξ [a+ (p0 − pz ) − a− p− ]  1 [a (p0 − pz ) + a+ p+ ] p0 ξ − s 1 cp0 ξ = 4 2π 3 h¯ 3 E

    ×   

        

0

i

× e h¯ (~p⊥ ·~r⊥ +pz z) ,

(6.17)

showing clearly that the transformation (6.15) preserves the largeness of the upper pair of components of the Dirac spinor compared to its lower pair of components. It is seen that the effect of the transformation (6.15) is to eliminate from the odd c0 , the terms of zeroth order in 1/p0 . The odd part of H c(1) , Ob (1) , conpart of H tains only terms of first and higher orders in 1/p0 . By a series of successive transformations with the same recipe (6.15), one can eliminate the odd parts up to any desired order in 1/p0 . This process will lead to the required quantum beam optical Hamiltonian necessary for finding the transfer maps for observables of interest. In the following section, we shall consider some examples of this general formalism.

Spinor Theory for Spin-1/2 Particles

6.1.1.1

293

Free Propagation: Diffraction

Let us now consider the propagation of a monoenergetic beam of Dirac particles of mass m through free space along the +z-direction. Let us start with the equation (6.11) in the field-free case. Substituting φ = 0 and ~A = (0, 0, 0) in (6.11), we have E E ∂ c0 ψ 0 (z) , c0 = −p0 β + χ~α⊥ ·~pb . (6.18) H i¯h ψ 0 (z) = H ⊥ ∂z c0 is not Hermitian since there is no conservation of probability along It is seen that H the z-axis. Note that !2  2 ~pb
~ −p I ξ σ · 0 ⊥ 0 ⊥ c = H = p20 − pb2⊥ I. (6.19) −1 ~ −ξ ~σ⊥ · pb⊥ p0 I c0 can be identified with the classical beam optical Hamiltonian, indicating that H q − p20 − p2⊥ , for free propagation of a monoenergetic beam with the square root taken in the Dirac way. Although in the present case it may look like as if one can take such a square root using only the 2 × 2 Pauli matrices, it is necessary to use the 4 × 4 Dirac matrices to take into account the two-component spin and the propagations in the forward and backward directions along the z-axis considered separately. The spinor wave function   a1 q   i (p z+~p ·~r ) a2  e h¯ z ⊥ ⊥ , p0 = p2z + p2 , a2 p+ (6.20) ψ 0 (~r⊥ , z) =  ⊥   ξ (p0 +pz ) a1 p− ξ (p0 +pz )

with arbitrary coefficients a1 and a2 including normalization constants is a general solution of (6.18) corresponding to a particle moving in the +z-direction with q

pz = p20 − p2⊥ . It is seen that this spinor wave function has large upper pair of components compared to the lower pair of components when the motion is paraxial, i.e., |~p⊥ |  pz ≈ p0 . This is analogous to the nonrelativistic positive-energy solutions of the free-particle Dirac equation having large upper pair of components compared to the lower pair of components. In the same way as the free-particle Dirac Hamiltonian can be diagonalized by c0 can be diagonalized exactly by a Foldy– a Foldy–Wouthuysen transformation, H Wouthuysen-like transformation. Define  ~pb⊥ E E  ~ (1) . (6.21) ψ (z) = e−θ β χ~α⊥ · pb⊥ ψ 0 (z) , with tanh 2 ~pb⊥ θ = p0 Then, we have E E ∂ ψ (1) (z) c(1) ψ (1) (z) i¯h =H ∂z c(1) = e−θ β χ~α⊥ ·~pb⊥ H c0 eθ β χ~α⊥ ·~pb⊥ . H (6.22)

294

Quantum Mechanics of Charged Particle Beam Optics

Let us write θ β χ~α⊥ ·~pb⊥

e

     β χ~α ·~pb ⊥ = exp  ⊥  ~pb⊥ θ ,   ~ b p ⊥

2 ~α⊥ ·~pb⊥ β χ with   = I. (6.23) ~pb 

⊥

Then, we have e

θ β χ~α⊥ ·~pb⊥

      β χ~α ·~pb ~ ~  ⊥ ⊥  = cosh pb⊥ θ I + sinh pb⊥ θ ~ pb⊥

Thus,       β χ~α ·~pb ⊥ ⊥ c(1) = cosh ~pb θ I − sinh ~pb θ   H ⊥ ⊥ ~ pb⊥   × −p0 β + χ~α⊥ ·~pb⊥        β χ~α ·~pb ⊥ × cosh ~pb⊥ θ I + sinh ~pb⊥ θ  ⊥  ~ pb⊥        β χ~α ·~pb ⊥ = cosh ~pb⊥ θ I − sinh ~pb⊥ θ  ⊥  ~pb⊥    β χ~α⊥ ·~pb β −p0 + ~pb⊥ ×  ⊥  ~pb⊥        β χ~α ·~pb ⊥ × cosh ~pb⊥ θ I + sinh ~pb⊥ θ  ⊥  ~ pb⊥    β χ~α ·~pb ⊥ = β −p0 + ~pb⊥  ⊥  ~ pb⊥ 2       β χ~α ·~pb ⊥ × cosh ~pb⊥ θ I + sinh ~pb⊥ θ  ⊥  ~ pb⊥    β χ~α ·~pb ~ ⊥ = β −p0 + ~pb⊥  ⊥  e2θ β χ~α⊥ · pb⊥ ~pb⊥    β χ~α ·~pb ⊥ = β −p0 + ~pb⊥  ⊥  ~ pb 

⊥

(6.24)

295

Spinor Theory for Spin-1/2 Particles

    β χ~α ·~pb   ⊥ × cosh 2 ~pb⊥ θ I + sinh 2 ~pb⊥ θ  ⊥  ~ pb⊥      = −β p0 cosh 2 ~pb⊥ θ − ~pb⊥ sinh 2 ~pb⊥ θ       ~α⊥ ·~pb⊥  ~ β χ +   pb⊥ cosh 2 ~pb⊥ θ − p0 sinh 2 ~pb⊥ θ ~pb⊥ q    ~pb⊥ =− p20 − pb2⊥ β , with tanh 2 ~pb⊥ θ = . (6.25) p0 

This shows that in the ψ (1) -representation the quantum beam optical Dirac Hamiltonian for free propagation takes the classical form for each component of the spinor wave function. In this representation, the four-component Dirac spinor representing the beam moving in the forward z-direction has the lower pair of components as zero and the four-component spinor representing the beam moving in the backward z-direction has the upper pair of components as zero. We are interested in the propagation of a monoenergetic quasiparaxial beam for which |~p⊥ |  pz ≈ p0 , where p0 is the design momentum. In this case,   ~pb⊥ tanh 2 ~pb⊥ θ =  1, p0

(6.26)

and therefore, we have ~   pb⊥ ~ ~ tanh 2 pb⊥ θ ≈ 2 pb⊥ θ = , p0

or, θ ≈

1 . 2p0

(6.27)

Let us make the first Foldy–Wouthuysen-like transformation, as in (6.16), with the result E  − 1 β χ~α⊥ ·~pb⊥ 0 (1) ψ (z) , ψ (z) = e 2p0 E E ∂ c(1) ψ (1) (z) i¯h ψ (1) (z) = H ∂z 1

~

1

~

c0 e 2p0 β χ~α⊥ · pb⊥ c(1) = e− 2p0 β χ~α⊥ · pb⊥ H H   1 − 1 β χ~α⊥ ·~pb⊥ β χ~α⊥ ·~pb⊥ = e 2p0 −p0 β + χ~α⊥ ·~pb⊥ e 2p0   1 2 1 4 ≈ −p0 + pb − pb β , 2p0 ⊥ 8p0 3 ⊥

(6.28)

296

Quantum Mechanics of Charged Particle Beam Optics

where we have omitted the odd term of second order in 1/p0 . The expression side gives the first three terms of the series expansion of  qon the right-hand −

p20 − pb2⊥ β . For an ideal paraxial situation, we have

  1 2 (1) c H ≈ −p0 + pb β . (6.29) 2p0 ⊥ E Since the lower pair of components of ψ (1) are too small, in fact almost zero, c(1) by I. Hence, compared to the upper pair of components, we can replace β in H we can take   1 2 (1) c pb . H ≈ −p0 + (6.30) 2p0 ⊥ Let us now retrace the transformations to get back to the original Dirac representation. We get E E 1 β χ~ α⊥ ·~pb⊥ (1) ψ(z) = M−1 e 2p0 ψ (z) , E E ∂ co ψ(z) , i¯h ψ(z) = H ∂z ~

1

~

1

co = M−1 e 2p0 β χ~α⊥ · pb⊥ H c(1) e− 2p0 β χ~α⊥ · pb⊥ M H   1 ~ 1 2 − 1 β χ~α⊥ ·~pb⊥ −1 2p0 β χ~α⊥ · pb⊥ =M e −p0 + M pb⊥ e 2p0 2p0   1 2 pb . = −p0 + 2p0 ⊥

(6.31)

Thus, we arrive at the quantum beam optical form of the free particle Dirac equation   ∂ ψ (~r⊥ , z) 1 2 ≈ −p0 + pb ψ (~r⊥ , z) , (6.32) i¯h ∂z 2p0 ⊥ which is same as in the scalar relativistic Klein–Gordon or nonrelativistic Schr¨odinger theory, except for the wave function having four components. Let us look at the transfer maps for transverse position coordinates and momentum components. Recall that the position operator ~r of the nonrelativistic Schr¨odinger theory has the zitterbewegung motion in the Dirac theory. In the Foldy–Wouthuysen representation of the Dirac theory, ~r behaves as a normal position operator and has been interpreted as the mean position operator. In the Dirac representation, the Newton–Wigner position operator behaves as a normal position operator, and it transforms to~r, the mean position operator, in the Foldy–Wouthuysen representation. Now, let us observe that in the ψ (1) -representation~r⊥ behaves as the normal transverse position operator. To this end, let us define the Heisenberg picture operator corresponding to~r⊥ as ~r(z)⊥ = e h¯ H i

~r⊥ e− h¯ H

c(1) z

i

c(1) z

.

(6.33)

297

Spinor Theory for Spin-1/2 Particles

Then, it follows that i i c(1) d~r(z)⊥ i i c(1) h c(1) = e h¯ H z H ,~r⊥ e− h¯ H z dz h¯    i c(1) 1 2 i i Hc(1) z pb⊥ ,~r⊥ e− h¯ H z −p0 + = e h¯ 2p0 h¯ 1 i Hc(1) z~ − i Hc(1) z pb⊥ e h¯ = e h¯ , p0

(6.34)

as should be. This also shows that the transverse momentum operator is ~pb⊥ . So, we can take ~r⊥ and ~pb⊥ to be the mean transverse position operator and transverse momentum operator, respectively, in the ψ (1) -presentation. The corresponding mean transverse position operator in the quantum beam optical Dirac representation, or ψ-representation, would be 1

~

1

~

~r˜⊥ = M−1 e 2p0 β χ~α⊥ · pb⊥~r⊥ e− 2p0 β χ~α⊥ · pb⊥ M  i 1 h −1 ~ ≈M ~r⊥ + β χ~α⊥ · pb⊥ ,~r⊥ M 2p0 i¯h =~r⊥ − β χ~α⊥ . 2p0

(6.35)

Since ~r˜⊥ is non-Hermitian, we can take its Hermitian part   ~r⊥ = 1 ~r˜⊥ +~r˜† ⊥ 2
i¯h =~r⊥ − ξ − ξ −1 β ~α⊥ 4p0 iλ 2 imc¯h β ~α⊥ =~r⊥ − 0 β ~α⊥ , =~r⊥ − 2 4πλc 2p0

(6.36)

as the mean transverse position operator, where λ0 is the de Broglie wavelength and λc is the Compton wavelength. Noting that in~r⊥ the constant term added to~r⊥ is tiny, we can as well drop it and take the mean transverse position operator in the quantum beam optical Dirac representation to be~r⊥ itself. The transverse momentum operator remains the same, ~pb⊥ , in the quantum beam optical Dirac representation, since it 1

β χ~α ·~pb⊥

⊥ commutes with M−1 e 2p0 (1) from the ψ -representation.

and hence is unchanged under the transformation

co is exactly same Since the quantum beam optical paraxial Dirac Hamiltonian H as the quantum beam optical paraxial Hamiltonian of the scalar theory (5.77), the transfer map is same as in (5.87), !  !  h~Dr⊥ i (z) h~Dr⊥ i (z 1 ∆z E Ei ) = , (6.37) 1 ~ 1 ~ 0 1 b⊥ (z) b⊥ (zi ) p0 p p0 p

298

Quantum Mechanics of Charged Particle Beam Optics

where the beam travels from the vertical plane at zi to the vertical plane at z. The difference now is that E Z D h~r⊥ i (z) = ψ(z) |~r⊥ | ψ(z) = d 2 r⊥ ψ † (~r⊥ , z)~r⊥ ψ (~r⊥ , z) 4

=

Z

∑

j=1

d 2 r⊥ ψ ∗j (~r⊥ , z)~r⊥ ψ j (~r⊥ , z) ,

E Z D E D ~pb (z) = ψ(z) ~pb ψ(z) = d 2 r⊥ ψ † (~r⊥ , z)~pb ψ (~r⊥ , z) ⊥ ⊥ ⊥ 4

=

Z

∑

j=1

d 2 r⊥ ψ ∗j (~r⊥ , z)~pb⊥ ψ j (~r⊥ , z) .

(6.38)

co is Hermitian, we have Since the quantum beam optical free-particle Hamiltonian H E taken ψ(z) to be normalized, i.e., D E Z ψ(z) ψ(z) = d 2 r⊥ ψ † (~r⊥ , z) ψ (~r⊥ , z) 4

=

Z

∑

j=1

d 2 r⊥ ψ ∗j (~r⊥ , z) ψ j (~r⊥ , z) = 1.

(6.39)

If it is desired to keep any non-Hermitian terms in Ea quantum beam optical Dirac Hamiltonian, the four-component state vector ψ(z) will have to be normalized at each z since the normalization will not be conserved. Then, the average value of any b corresponding to an observable O, b which may be a 4 × 4 matrix with operator O operator entries like, say, ~α ·~pb, can be defined by D E b D E ψ(z) O ψ(z) b (z) = D E O ψ(z) | ψ(z) R 2 b (~r⊥ , z) d r⊥ ψ † (~r⊥ , z) Oψ = R 2 † d r⊥ ψ (~r⊥ , z) ψ (~r⊥ , z) R 2 4 ∗

=

b jk ψk (~r⊥ , z) ∑ j,k=1 d r⊥ ψ j (~r⊥ , z) O ∑4j=1 d 2 r⊥ ψ ∗j (~r⊥ , z) ψ j (~r⊥ , z) R

.

(6.40)

The quantum beam optical Dirac Hamiltonian for free propagation is same as in the scalar theory, except that it now acts on a four-component wave function. Integrating (6.32), we have E − i∆z h¯ ψ(z) = e

  pb2 −p0 + 2p⊥ I 0

E ψ (zi ) ,

∆z = z − zi .

(6.41)

299

Spinor Theory for Spin-1/2 Particles

In position representation, Z

d 2 r⊥i [K (~r⊥ , z;~r⊥i , zi ) I] ψ (~r⊥i , zi ) ,   * + pb2 − i∆z −p0 + 2p⊥ h ¯ 0 K (~r⊥ , z;~r⊥i , zi ) = ~r⊥ e ~r⊥i . ψ (~r⊥ , z) =

(6.42)

The expression for K (~r⊥ , z;~r⊥i , zi ) is already known to us from the scalar theory:   n o i p ∆z+ ip0 |~r −~r |2 p0 K (~r⊥ , z;~r⊥i , zi ) = (6.43) e h¯ 0 2¯h∆z ⊥ ⊥i . 2πi¯h∆z Using this result, we have     Z ip0 2 i p0 ψ (~r⊥ , z) = e h¯ p0 ∆z d 2 r⊥i e 2¯h∆z |~r⊥ −~r⊥i | I ψ (~r⊥i , zi ) . 2πi¯h∆z

(6.44)

With 2π h¯ /p0 = λ0 , the de Broglie wavelength, we get   i2π (z−z ) 1 i ψ j (x, y, z) = e λ0 iλ0 (z − zi ) Z Z iπ [(x−xi )2 +(y−yi )2 ] × dxi dyi e λ0 (z−zi ) ψ j (xi , yi , zi ) , j = 1, 2, 3, 4,

(6.45)

which is the generalization of the Fresnel diffraction formula for a paraxial beam of free Dirac particles. Thus, it is obvious that the diffraction pattern due to a paraxial beam of free Dirac particles will be the of the patterns due to the four superposition E individual components of the spinor ψ(z) representing the beam and the intensity distribution of the diffraction pattern at the xy-plane at z will be given by Z Z 2 iπ [(x−xi )2 +(y−yi )2 ] λ0 (z−zi ) I(x, y, z) ∝ ∑ dxi dyi e ψ j (xi , yi , zi ) 4

(6.46)

j=1

where the plane of the diffracting object is at zi and the plane at z is the observation co plane. When the presence of a field makes the quantum beam optical Hamiltonian H acquire a matrix component, the propagator K (~r⊥ , z;~r⊥i , zi ) would have nontrivial matrix structure, instead of being E just ∝ I, leading to interference between the four diffracted components of ψ(z) . When the monoenergetic beam is not ideally paraxial to allow the relevant approximations to be made, one can directly use the free z-evolution equation, i¯h

E E  ∂ ψ(z) = − p0 β χαz + i (Σx pby − Σy pbx ) ψ(z) , ∂z

(6.47)

300

Quantum Mechanics of Charged Particle Beam Optics

obtained by setting φ = 0 and ~A = (0, 0, 0) in (6.7). On integration, we get   E E i b b = exp ∆z (p β χα + i (Σ p − Σ p )) ψ (zi ) , ψ(z) z x y y x 0 h¯

(6.48)

the general law of propagation of the free Dirac wave function in the +z-direction, showing the subtle way in which the spinor components are mixed up. For some detailed studies on the optics of general free Dirac waves, particularly diffraction, see Rubinowicz ([161, 162, 163, 164], Durand [38], and Phan-Van-Loc [144, 145, 146, 147, 148, 149], Boxem, Partoens, and Verbeeck [15], and references therein). A path integral approach to the optics of Dirac particles has been developed by Li˜nares [128]. 6.1.1.2

Axially Symmetric Magnetic Lens

We shall study the propagation of a quasiparaxial charged Dirac particle beam through a round magnetic lens without taking into account the anomalous magnetic moment of the particle. We shall take the z-axis to be the optic axis of the lens. Since there is no electric field, φ = 0. The vector potential can be taken, in general, as   ~A = − 1 yΠ (~r⊥ , z) , 1 xΠ (~r⊥ , z) , 0 , (6.49) 2 2 with 1 2 00 1 4 0000 Π (~r⊥ , z) = B(z) − r⊥ B (z) + r B (z) − · · · . 8 192 ⊥ The corresponding magnetic field is given by   2 ~B⊥ = − 1 B0 (z) − 1 B000 (z)r⊥ + · · · ~r⊥ , 2 8 1 1 2 4 + B0000 (z)r⊥ −··· . Bz = B(z) − B00 (z)r⊥ 4 64

(6.50)

(6.51)

The practical boundaries of the lens, say z` and zr , are determined by where B(z) becomes negligible, i.e., B (z < z` ) ≈ 0 and B (z > zr ) ≈ 0. As we have already seen, E the z-evolution equation for the time-independent spinor 0 wave function in the ψ -representation (6.11) reads i¯h

E ∂ 0 E c0 ψ 0 (z) , ψ (z) = H ∂z c0 = −p0 β + Eb + O, b H Eb = −qAz I,   Ob = χ~α⊥ ·~πb⊥ = χ~α⊥ · ~pb⊥ − q~A⊥ .

(6.52)

301

Spinor Theory for Spin-1/2 Particles

E E Let us recall that the ψ 0 is related to the Dirac ψ by E E 0 ψ = M ψ(z) , 1 M = M = √ (I + χαz ) , 2   ξI O χ= , O −ξ −1 I

s with ξ =

E + mc2 , E − mc2

(6.53)

q where E is the conserved total energy m2 c4 + c2 p20 of a particle of the beam with the design momentum p0 . To obtain the desired quantum beam optical representation of the z-evolution equation, we have to perform a series of successive Foldy– Wouthuysen-like transformations outlined earlier. The first transformation is E E b (1) = eiS1 ψ 0 , ψ i i Sb1 = β Ob = β χ~α⊥ ·~πb⊥ . 2p0 2p0

(6.54)

E The resulting equation for ψ (1) is

i¯h

E ∂ (1) E c(1) ψ (1) , =H ψ ∂z   1 1 1 1 b b b b c(1) = e− 2p0 β O H c0 e 2p0 β O − i¯he− 2p0 β O ∂ e 2p0 β O H ∂z (1) (1) = −p0 β + Eb + Ob , " !# h i b 1 1 ∂ O 1 (1) 2 Eb ≈ Eb − β Ob − 2 Ob , Ob , Eb + i¯h − 3 β Ob 4 , 2p0 ∂z 8p0 8p0 ! h i 1 ∂ Ob 1 (1) b b b O ≈− β O , E + i¯h − 2 Ob 3 , (6.55) 2p0 ∂z 3p0

in which we have to substitute Eb = −qAz I and Ob = χ~α⊥ · ~πb⊥ and calculate the expressions for Eb(1) and Ob (1) . The second transformation will have the same prescription E E b (2) = eiS2 ψ (1) , ψ

i Sb2 = β Ob (1) , 2p0

(6.56)

302

Quantum Mechanics of Charged Particle Beam Optics

leading to the equation i¯h

E ∂ (2) E c(2) ψ (2) , =H ψ ∂z c(1) e 2p0 β O c(2) = e− 2p0 β O H H 1

b (1)

1

b (1)

− 2p1 β Ob (1)

− i¯he

0

  1 βO b (1) ∂ e 2p0 ∂z

= −p0 β + Eb(2) + Ob (2) , 1  b (1) 2 Eb(2) ≈ Eb(1) − β O 2p0 " !# h i 1 ∂ Ob (1) (1) (1) b(1) b b + i¯h − 2 O , O ,E ∂z 8p0 −

1  b (1) 4 β O , 8p30

h i ∂ Ob (1) Ob (1) , Eb(1) + i¯h ∂z   3 1 − 2 Ob (1) , 3p0

1 β Ob (2) ≈ − 2p0

!

(6.57)

in which we have to get the expressions for Eb(2) and Ob (2) by substituting the expressions for Eb(1) and Ob (1) obtained from the first transformation. We shall stop with the third transformation in this case. With the same prescription, i Sb3 = β Ob (2) , 2p0

E E b (3) = eiS3 ψ (2) , ψ

(6.58)

we get i¯h

E ∂ (3) E c(3) ψ (3) , =H ψ ∂z c(3) = e− 2p0 β O H c(2) e 2p0 β O H 1

b (2)

1

b (2)

− 2p1 β Ob (2)

− i¯he

0

  1 b (2) ∂ 2p0 β O e ∂z

≈ −p0 β + Eb(3) , 1  b (2) 2 Eb(3) ≈ Eb(2) − β O 2p0 " !# h i 1 ∂ Ob (2) (2) (2) b(2) b b − 2 O , O ,E + i¯h ∂z 8p0   4 1 − 3 β Ob (2) , 8p0

(6.59)

303

Spinor Theory for Spin-1/2 Particles

in which we have omitted the odd term, and we have to get the expression for Eb(3) by substituting the expressions for Eb(2) and Ob (2) obtained from the second transformation. The above transformations make the lower components of the spinor successively smaller and smaller compared to the upper components for a quasiparaxial beam moving in the +z-direction. In other words, one has E E (6.60) β ψ (3) ≈ I ψ (3) . c(3) , without an odd term, is found to be of the form Now, H c(3) = H

c(3) + H c(3) H 1 2 O

O (3) c c(3) H1 − H 2

! .

c(3) further and write In view of (6.60), we can approximate H E E ∂ c ψ (3) , i¯h ψ (3) ≈ H ∂z ! c(3) + H c(3) H O 1 2 c= . H c(3) + H c(3) O H 1

(6.61)

(6.62)

2

To enable using directly the original Dirac |ψ(z)i, satisfying (6.7), let us retrace the transformations: E E E c b b b (3) −→ |ψ(z)i = M−1 e−iS1 e−iS2 e−iS3 ψ (3) , ≈ M−1 e−iS ψ (3) , (6.63) ψ where i h i h i h hh i i c≈ Sb1 + Sb2 + Sb3 − i Sb1 , Sb2 + Sb1 , Sb3 + Sb2 , Sb3 − 1 Sb1 , Sb2 , Sb3 · · · , S 2 4 (6.64) obtained by applying the Baker–Campbell–Hausdorff formula b b 1 b b

eA eB ≈ eA+B+ 2 [A , B] ,

(6.65) h i b , Bb 6= 0 (see Appendix A for details). Implementing the inverse transforwhen A mation leads to E E ∂ co ψ(z) , i¯h ψ(z) = H ∂z    −1 −iS c iS −iS ∂ c M. (6.66) Ho = M e H e − i¯he eiS ∂z   co , using (3.706), up to fourth order in ~r⊥ ,~pb , we have Calculating H ⊥ b b

h) co = H co,p + H c0 + H co (¯h) + H b (¯ H o , o

(6.67)

304

Quantum Mechanics of Charged Particle Beam Optics

where  1 2 1 2 bz I, pb⊥ + p0 α 2 (z)r⊥ − α(z)L 2p0 2  2 1 4 α(z) 2 b α 2 (z)  ~ c0 = H pb⊥ − pb⊥ Lz − ~r⊥ · pb⊥ +~pb⊥ ·~r⊥ o 3 2 8p0 2p0 8p0

co,p = H




1 00
2 3α 2 (z) 2 2 2 bz r⊥ + α (z) − 4α 3 (z) L r⊥ pb⊥ + pb2⊥ r⊥ 8p0 8
4 p0 4 + α (z) − α(z)α 00 (z) r⊥ I, 8  2   2 2 co (¯h) ≈ h¯ α 0 (z) − 2α(z)α 00 (z) r⊥ H 8p0 
4 h¯ 2 00 2 0 000 α (z) − α (z)α (z) r⊥ I, + 32p0

(6.68)

+

(6.69)

(6.70)

and, retaining only the Hermitian part, 3¯h2 0 h) 2 b (¯ H α (z)α 00 (z)yr⊥ (yΣx − xΣy ) o ≈ 64p0 
2 h¯ h¯ α 00 (z) − 2α(z)3 r⊥ + − 2 α(z) pb2⊥ 4 2p0  h¯ h¯ 0  ~ bz α (z) ~r⊥ · pb⊥ +~pb⊥ ·~r⊥ + α(z)2 L + 4p0 p0
3¯h h¯ 2 2 2 2 bz r⊥ + α 00 (z) pb2⊥ r⊥ α(z)α 00 (z)L + r⊥ pˆ⊥ − 16p0 8p0   h¯ 3¯h 4 2 − α 000 (z) ~r⊥ ·~pb⊥ +~pb⊥ ·~r⊥ r⊥ + α(z)2 α 00 (z)r⊥ 16 64p0   
h¯ 2 2 α 000 (z) pbx , xr⊥ + pby , yr⊥ − h¯ α(z) Σz − 64p0 n  oi i¯h 0 h ~pb +~pb ·~r⊥ ~ b − α (z) β χ , α · p , ~ r · ⊥ ⊥ ⊥ ⊥ ⊥ 48p20 oi h n  i¯h − α(z)α 0 (z) β χ , ~α⊥ ·~r⊥ , ~r⊥ ·~pb⊥ +~pb⊥ ·~r⊥ 32p0 i¯h + α 0 (z) [β χ , (αx y − αy x)] 4   i¯h 2 + (¯hα(z) − p0 ) α 000 (z) β χ , (αx y − αy x) r⊥ 32p0   h¯ 2 1 0 0 2 2 000 4 + 2α (z) r⊥ − α (z)α (z)r⊥ {β χ , αz } 32p0 2    h¯ 2 2 + α 000 (z) β χ , αx pby − αy pbx , r⊥ , 2 128p0 with α(z) = qB(z)/2p0 .

(6.71)

305

Spinor Theory for Spin-1/2 Particles

It should be noted that we are interested in the transfer map for the transverse coordinates and momenta of the beam particle from the field-free input region to the field-free output region. Hence, we can take the transverse position operator to be~r⊥ and the transverse momentum operator to be ~pb⊥ , as was found earlier in the discussion of free propagation. Calculation of the transfer maps using the above quantum beam optical Dirac Hamiltonian is exactly along the same lines as in the scalar theco , ory, except that now we have to use (6.40) for the definition of averages. In H c the quantum beam optical Dirac Hamiltonian for the magnetic round lens, Ho,p is c0 is the perturbation the paraxial term leading to the point-to-point imaging, and H o co (¯h) are term responsible for the main third-order aberrations. These two terms and H E essentially scalar terms, ∝ I, which act on each component of the Dirac spinor ψ(z) (¯h)

b individually without mixing any Eof them. The h¯ -dependent Ho is a matrix term that mixes the components of ψ(z) and carries the signature of spin explicitly. Both h) co (¯h) and H b (¯ the h¯ -dependent terms, H o , being proportional to the de Broglie wavelength, might be negligible under the conditions in electron microscopy. This justifies ignoring the electron spin and approximating the Dirac theory by the scalar Klein– Gordon theory in electron microscopy (see, e.g., Hawkes and Kasper [72], Ferwerda, Hoenders, and Slump [48, 49], and Lubk [129] for details of such an approximation procedure different from our analysis). However, the following should be pointed out: If the Dirac theory is approximated by a scalar theory, with the inclusion of the h¯ -dependent scalar terms, it will differ from the Klein–Gordon theory in the co (¯h) in (6.70) above and the h¯ -dependent scalar terms, as is seen by comparing H h¯ -dependent scalar Hermitian terms of the Klein–Gordon theory (5.213). co in the Dirac theory, H b o(¯h) , adds to the deviation from the The matrix part in H Klein–Gordon theory. further ado, let us just note that the aberrations of
Without

position (δ x) zob j , (δ y) zob j , given in (5.201) and (5.202), get additional contributions of every type from the matrix part.
For example, the
additional contributions of spherical aberration type to (δ x) zob j and (δ y) zob j are, respectively,

Z zimg   
h¯ 1 (¯h)

2 2 b b C p p Σ z = dz 6α 2 α 00 h4 − α 000 h3 h0 + 4α 00 h2 h0 s x ⊥ z ob j 3 4 8p zob j p0 0





2 × ψ1 zob j pbx pb⊥ ψ1 zob j − ψ2 zob j pbx pb2⊥ ψ2 zob j








, (6.72) + ψ3 zob j pbx pb2⊥ ψ3 zob j − ψ4 zob j pbx pb2⊥ ψ4 zob j

and Z zimg   
1 (¯h)

h¯ 2 2 00 4 000 3 0 00 2 0 2 b b C p p Σ z = dz 6α α h − α h h + 4α h h s y ⊥ z ob j 8p40 zob j p30







 × ψ1 zob j pby pb2⊥ ψ1 zob j − ψ2 zob j pby pb2⊥ ψ2 zob j








+ ψ3 zob j pby pb2⊥ ψ3 zob j − ψ4 zob j pby pb2⊥ ψ4 zob j , (6.73)

306

Quantum Mechanics of Charged Particle Beam Optics


with α = α(z), h = h z, zob j . Obviously, such contributions, with unequal weights
E for the four spinor components, would depend on the nature of ψ zob j , the spinor wave function at the object plane, with respect to spin. co in (6.67), It is to be noted that in the quantum beam optical Dirac Hamiltonian H 0 c c the paraxial part Ho,p and the main perturbation part Ho are scalar, i.e., ∝ I, and are identical to the respective paraxial part and the main perturbation part of the scalar quantum beam optical Hamiltonian (5.181) . The additional h¯ -dependent scalar and matrix terms of the quantum beam optical Dirac Hamiltonian, though different from the h¯ -dependent terms of the scalar quantum beam optical Hamiltonian, are also very small, like the h¯ -dependent terms of the scalar quantum beam optical Hamiltonian, compared to the main paraxial and perturbation parts of the Hamiltonian which are responsible for the focusing and main aberration aspects of the magnetic round lens. Thus, we understand the behavior of a quasiparaxial beam of spin-1/2 particles propagating through an axially symmetric magnetic lens on the basis of the Dirac equation. 6.1.1.3

Bending Magnet

We have already discussed the classical mechanics and the scalar Klein–Gordon theory of the bending of a charged particle beam by a dipole magnet. Let us recall. A constant magnetic field in the vertical direction produced by a dipole magnet bends the beam along a circular arc in the horizontal plane. The circular arc of radius of curvature ρ, the design trajectory of the particle, is the optic axis of the bending magnet. It is natural to use the arclength, say S, measured along the optic axis from some reference point as the independent coordinate, instead of z. Let the reference particle moving along the design trajectory carry an orthonormal XY -coordinate frame with it. The X-axis is taken to be perpendicular to the tangent to the design orbit and in the same horizontal plane as the trajectory, and the Y -axis is taken to be in the vertical direction perpendicular to both the X-axis and the trajectory. The curved S-axis is along the design trajectory and perpendicular to both the X- and Y -axes at any point on the design trajectory. The instantaneous position of the reference particle in the design trajectory at an arclength S from the reference point corresponds to X = 0 and Y = 0. Let any particle of the beam have coordinates (x, y, z) with respect to a fixed right-handed Cartesian coordinate frame with its origin at the reference point on the design trajectory from which the arclength S is measured. Then, the two sets of coordinates of any particle of the beam, (X,Y, S) and (x, y, z), will be related as follows:     S S x = (ρ + X) cos − ρ, z = (ρ + X) sin , y=Y (6.74) ρ ρ To understand the bending of the path of a Dirac particle by a dipole magnet, we have to use the Dirac equation written in the curved (X,Y, S)-coordinate system. To this end, following Jagannathan [80], we borrow the formalism from the theory of general relativity (see, e.g., Brill and Wheeler [16]). Formalism of general relativity has been

307

Spinor Theory for Spin-1/2 Particles

adopted in particle beam optics by Wei, Li, and Sessler [186] also in the context of studying crystalline beams. The Dirac equation for a free particle of mass m is i¯h

h  i ∂ ψ (~r,t) = β mc2 + c~α · −i¯h~∇ ψ (~r,t) . ∂t

(6.75)

Multiplying both sides from left by −iβ /c, this equation can be written in the relativistically covariant form as "   #  3 

∂ ∂ γ 0 i¯h 0 − ∑ γ j −i¯h j ψ x0 , x j = −imcψ x0 , x j , (6.76) ∂x ∂ x j=1 or ! 
∂ mc ∑ γ ∂ x j + h¯ ψ x j = 0, j=0
where x0 = ct, x1 = x, x2 = y, x3 = z and the γ-matrices are defined by 3

j

   −iI O O −iσx 1 γ = −iβ = , γ = −iβ αx = , O iI iσx O     O −iσy O −iσz 2 3 γ = −iβ αy = , γ = −iβ αz = . iσy O iσz O

(6.77)



0

(6.78)

Note that γ0


2

= −I,

n o γ j , γ k = 0, or we can write

γ1


2

= I,

γ2


2

= I,

γ3

for j 6= k,

o n γ j , γ k = 2η jk I,


2

= I, (6.79)

j, k = 0, 1, 2, 3. ,

(6.80)

where  −1 h i  0 jk  = η 0 0

 0 0 0 1 0 0  , 0 1 0  0 0 1

(6.81)

is the contravariant metric tensor of the flat space-time. The corresponding covariant metric tensor is   −1 0 0 0   h jk i−1 h jk i  0 1 0 0   η jk = η = η = (6.82)  0 0 1 0 , 0 0 0 1

308

Quantum Mechanics of Charged Particle Beam Optics

   such that η jk η jk = I, or ∑3`=0 ηk` η ` j = δkj . Defining 3

γj =

∑ η jk γ k ,

j = 0, 1, 2, 3. ,

(6.83)

k=0

we find  γ j , γk = 2η jk I,

j, k = 0, 1, 2, 3.

(6.84)
The coordinates of a space-time point form a contravariant four-vector x j =
x0 , x1 , x2 , x3 . The components of the corresponding covariant four-vector (x j ) = (x0 , x1 , x2 , x3 ) are given by 4

xj =

∑ η jk xk ,

j = 0, 1, 2, 3, .

(6.85)

k=0

or x0 = −x0 ,

x1 = x1 ,

x2 = x2 ,

x3 = x3 .

(6.86)

Using the Einstein summation convention, in which it is understood that repeated indices are to be summed over, we can write ∑4k=0 η jk xk = η jk xk . Thus, we have ηk` η ` j = δkj . The relativistically
invariant distance ds between two points
in flat space-time, say, x0 , x1 , x2 , x3 and x0 + dx0 , x1 + dx1 , x2 + dx2 , x3 + dx3 , is given by ds2 = η jk dx j dxk = η jk dx j dxk = dx j dx j
2
2
2
2 = − dx0 + dx1 + dx2 + dx3 = − (dx0 )2 + (dx1 )2 + (dx2 )2 + (dx3 )2 .

(6.87)

Note that writing equation (6.77) as    mc  

∂ ψ xj , γ j j ψ xj = − ∂x h¯

(6.88)

implies immediately, from the algebra of γ-matrices (6.80),   ∂ 2 
 mc 2  j
, γ j j ψ xj = − ψ x ∂x h¯

(6.89)

   mc 2 1 ∂2 2 ∇ − 2 2 ψ (~r,t) = ψ (~r,t) , c ∂t h¯

(6.90)

or,

the free-particle Klein–Gordon equation for each component of the free-particle Dirac spinor.

309

Spinor Theory for Spin-1/2 Particles


µ 0 1 2 3 Let the
curved space-time, in which we are interested, have (x˜ ) = x˜ , x˜ , x˜ , x˜ and x˜µ = (x˜0 , x˜1 , x˜2 , x˜3 ) as the contravariant and covariant coordinates, respectively, and the metric given by ds2 = g˜µν d x˜µ d x˜ν = g˜µν d x˜µ d x˜ν ,

(6.91)

with g˜µν = g˜ν µ ,

g˜µν = g˜ν µ ,

µ

g˜µλ g˜λ ν = δν .

(6.92)

Let dx j = bµj d x˜µ ,

d x˜µ = ak dxk , µ

with ak bkν = δν . µ

µ

(6.93)

Since we should have η jk dx j dxk = η jk bµj bkν d x˜µ d x˜ν ,

(6.94)

η jk bµj bkν = g˜µν .

(6.95)

γ j = η jk γ k ,

(6.96)

 γ j , γk = 2η jk .

(6.97)

γ˜µ = bµj γ j ,

(6.98)

we get the relation With we see from (6.80) that Defining we get   γ˜µ , γ˜ν = bµj bkν γ j , γk = 2η jk bµj bkν = 2g˜µν . (6.99)  Corresponding to the covariant γ˜µ , we have the contravariant {γ˜µ } given by γ˜µ = g˜µν γ˜ν .

(6.100)

Now, the free-particle Dirac equation in the curved space-time is taken as     mc ∂ µ ψ ({x˜µ }) = 0, − Γµ + γ˜ ∂ x˜µ h¯  where the spin connection matrices Γµ are given by Γµ = g˜αβ

∂bj aαj µν − Γαµν ∂x

(6.101)

! σβν,

(6.102)

with Γαµν

1 = g˜αβ 2



∂ g˜νβ ∂ g˜µβ ∂ g˜µν + − ∂ xµ ∂ xν ∂ xβ

 ,

(6.103)

310

Quantum Mechanics of Charged Particle Beam Optics

the Christoffel symbols, and σβν =

1 h β νi γ˜ , γ˜ . 2

(6.104)

 Note that Γαµν are symmetric in µ and ν, and hence, there are 40 such symbols.  In the presence of an electromagnetic field, associated with the four-potential Aµ , the equation for the Dirac particle of charge q and mass m in the curved space-time becomes     iq mc ∂ − Γ − A + ψ ({x˜µ }) = 0, (6.105) γ˜µ µ µ ∂ x˜µ h¯ h¯ using the principle of minimal coupling. For the curved space we are interested in x˜0 = ct,

x˜1 = X,

x˜2 = Y,

x˜3 = S,

(6.106)

with the relation between (X,Y, S) and (x, y, z) given by (6.74). Thus, from (6.74) we find that, with ζ = 1 + (X/ρ) = 1 + κX,   dx0 1  dx1   0  2 =  dx   0 0 dx3 

 0 d x˜ 0 0  d x˜1 0 −ζ sin(κS)   2   d x˜ 1 0 0 ζ cos(κS) d x˜3

0 cos(κS) 0 sin(κS)

  , 

(6.107)

defining the relation dx j = bµj d x˜µ . The inverse relation, d x˜µ = ak dxk , is given by µ

  1 0 d x˜0  d x˜1   0 cos(κS)  2 =  d x˜   0 0 0 − ζ1 sin(κS) d x˜3 

0 0 1 0

 0 0 dx  dx1 sin(κS)   2   dx 0 1 cos(κS) dx3 ζ

  . 

(6.108)

g˜33 = ζ 2 ,

g˜µν = 0 for µ 6= ν,

(6.109)

1 , ζ2

g˜µν = 0 for µ 6= ν.

(6.110)

From (6.95), we get g˜00 = −1,

g˜11 = g˜22 = 1,

and from (6.92), it follows that g˜00 = −1,

g˜11 = g˜22 = 1,

g˜33 =

 Among the 40 Christoffel symbols Γαµν , only three are nonvanishing: Γ313 =

κ , ζ

Γ133 = −κζ ,

Γ331 =

κ . ζ

(6.111)

311

Spinor Theory for Spin-1/2 Particles

Calculating the spin connection matrices, using (6.102), we find all of them vanishing: Γ0 = Γ1 = Γ2 = Γ3 = 0. (6.112) From (6.83), (6.98), and (6.100), we have γ˜0 = γ 0 , γ˜1 = cos(κS)γ 1 + sin(κS)γ 3 , γ˜2 = γ 2 ,  1 − sin(κS)γ 1 + cos(κS)γ 3 . γ˜3 = ζ

(6.113)

Thus, the equation for a free Dirac particle in the curved space of interest to us becomes   ∂ ∂ ∂ ∂ mc γ˜0 0 + γ˜1 1 + γ˜2 2 + γ˜3 3 + ψ ({x˜µ }) = 0. (6.114) h¯ ∂ x˜ ∂ x˜ ∂ x˜ ∂ x˜ Writing explicitly, this equation is   ∂ ∂ 1 ∂  + cos(κS)γ 1 + sin(κS)γ 3 + γ2 γ0 c ∂t ∂X ∂Y        1 ∂ mc + − sin(κS)γ 1 + cos(κS)γ 3 + ψ ~R⊥ , S,t = 0. ζ ∂S h¯

(6.115)

In the case of the dipole magnet used for bending the beam, the required magnetic field is given by BX = 0, BY = B0 , BS = 0. (6.116) The corresponding vector potential has the components AX = 0,

AY = 0,

  κX 2 AS = −B0 X − , 2ζ

(6.117)

such that   1 ∂ (ζ AS ) ∂ AY ~ ~ BX = (∇ × A)X = − = 0, ζ ∂Y ∂S   1 ∂ AX ∂ (ζ AS ) BY = (~∇ × ~A)Y = − = B0 , ζ ∂S ∂X ∂ AY ∂ AX BS = (~∇ × ~A)S = − = 0. ∂X ∂Y

(6.118)

Now, following (6.105), we get the equation for the Dirac particle moving in the field of the dipole magnet as

312

Quantum Mechanics of Charged Particle Beam Optics

  ∂ 1 ∂  ∂ γ0 + cos(κS)γ 1 + sin(κS)γ 3 + γ2 c ∂t ∂X ∂Y      1 ∂ iq + − sin(κS)γ 1 + cos(κS)γ 3 − ζ AS ζ ∂ S h¯    mc ψ ~R⊥ , S,t = 0. + h¯

(6.119)

Multiplying throughout from the left by −i¯hcγ 0 and rearranging the terms, we can rewrite this equation as       ∂ ~ ∂ ∂ 2 i¯h ψ R⊥ , S,t = mc β + cαX −i¯h + cαY −i¯h ∂t ∂X ∂Y     i¯h ∂ − qAS ψ ~R⊥ , S,t , +cαS − ζ ∂S

(6.120)

where αX = cos(κS)αx + sin(κS)αz  O = cos(κS)σx + sin(κS)σz   O σy αY = αy = , σy O αS = − sin(κS)αx + cos(κS)αz  O = − sin(κS)σx + cos(κS)σz

cos(κS)σx + sin(κS)σz O

 ,

− sin(κS)σx + cos(κS)σz O

 .

(6.121)

Note that αX , αY , and αS are Hermitian, odd operators, and anticommute with each other and with β . Further, they obey αX2 = αY2 = αS2 = I. In other words, αX , αY , and αS obey exactly the same properties as the standard Dirac matrices αx , αy , and αz . Let us now consider a monoenergetic paraxial Dirac particle beam passing through the bending magnet. Taking     i Ψ ~R⊥ , S,t = ψ ~R⊥ , S e− h¯ Et ,

(6.122)

q with E = m2 c4 + c2 p20 as the total conserved energy of the beam particle with design momentum p0 , we get the time-independent equation      ∂ ∂ 2 mc β + cαX −i¯h + cαY −i¯h ∂X ∂Y   E E i¯h ∂ +cαS − − qAS ψ(S) = E ψ(S) . ζ ∂S

(6.123)

313

Spinor Theory for Spin-1/2 Particles

Multiplying this equation on both sides from the left by αS /c and rearranging the terms, we get E E i¯h ∂ f cζ ψ(S) , ψ(S) = H ζ ∂S   f cζ = −p0 β χαS − qAS I + αS αX PbX + αY PbY , H where PbX = −i¯h(∂ /∂ X), PbY = −i¯h(∂ /∂Y ), and χ is same as in (6.7): s   E + mc2 ξI O . , with ξ = χ= −1 O −ξ I E − mc2

(6.124)

(6.125)

We can now proceed exactly in the same way as we handled (6.7). Observing that, with 1 e M = √ (I + χαS ) , 2

1 e M−1 = √ (I − χαS ) , 2

(6.126)

one has we define

e Mβ χαS e M−1 = β ,

(6.127)

E E 0 M ψ(S) . ψ (S) = e

(6.128)

Then, E 0 i¯h ∂ 0 E f c ψ 0 (S) , ψ (S) = H ζ ζ ∂S   −1  0 M i¯h e ∂ e f f −1 c =e cζ e H M M H M − ζ ζ ∂S e eb = −p0 β + Eb + O, h¯ κ e β Σy , Eb = −qAS I + i [cos(κS)Σx + sin(κS)Σz ] PbY − 2ζ     i e Ob = χ αX PbX − h¯ κ + αY PbY , (6.129) 2 e e where the even term Eb commutes with β and the odd term Ob anticommutes with β . To obtain the desired quantum beam optical representation of the S-evolution equation, we have to perform the Foldy–Wouthuysen-like transformations. The first transformation is E E b (1) = eiS1 ψ 0 , ψ     i i i eb b b b S1 = βO = β χ αX PX − h¯ κ + αY PY . (6.130) 2p0 2p0 2

314

Quantum Mechanics of Charged Particle Beam Optics

E The resulting equation for ψ (1) is E (1) i¯h ∂ (1) E f c ψ (1) , =H ψ ζ ζ ∂S    (1) 1 1 1 1 bf0 b b b f c = e− 2p0 β O H c e 2p0 β O − i¯h e− 2p0 β O ∂ e 2p0 β O H ζ ζ ζ ∂S (1) (1) b b = −p0 β + E + O , 1 β Ob 2 − · · · , Eb(1) = Eb − 2p0 ! h i i¯h ∂ Ob 1 (1) b b b β O,E + −··· . O =− 2p0 ζ ∂S

(6.131)

(1) f c . Let us stop at this first Foldy–Wouthuysen-like transformation and calculate H ζ The result is   (1) f c ≈ −p0 β − qAS I + 1 Pb2 + Pb2 − 1 h¯ 2 κ 2 − i¯hκ PbX β H ζ X Y 2p0 4 h¯ κ + i [cos(κS)Σx + sin(κS)Σz ] PbY − β Σy , (6.132) 2ζ (1) f c in (6.131), replacing β by I in the omitting the odd term. Substituting this H ζ E resulting equation since ψ (1) will have the lower pair of components very small compared to the upper pair of components, and returning to the original Dirac representation, we get    E  h¯ 2 κ 2 1  b2 b2 i¯h ∂ b + P + PY − i¯hκ PX I ψ(S) ≈ −qAS − p0 + ζ ∂S 8p0 2p0 X  E h¯ κ − Σy ψ(S) . (6.133) 2ζ

Multiplying both sides of this equation from the left by ζ , we can write    E   ∂ h¯ 2 κ 2 1  b2 b2 i¯h ψ(S) ≈ ζ −qAS − p0 + + PX + PY − i¯hκ PbX I ∂S 8p0 2p0  E 1 − h¯ κΣy ψ(S) . (6.134) 2 Following (6.117), we take   κX 2 AS = −B0 X − . 2ζ

(6.135)

315

Spinor Theory for Spin-1/2 Particles

Further, let the curvature of the design trajectory be matched to the dipole magnetic field such that qB0 = κ p0 .

(6.136)

Then, simplifying and keeping only terms of order up to quadratic in (X,Y ) and   b b PX , PY , we have

i¯h

 E   h¯ 2 κ 2 1  b2 b2  1 ∂ + P + PY + p0 κ 2 X 2 ψ(S) = − p0 + ∂S 8p0 2p0 X 2   E 2 3 i¯hκ 1 h¯ κ X− (1 + κX)PbX I − h¯ κΣy ψ(S) . − 8p0 2p0 2

(6.137)

Now, let us replace the non-Hermitian term i(1 + κX)PbX on the right-hand side by its Hermitian part,    †  1  i(1 + κX)PbX + i(1 + κX)PbX 2 o 1 1n i(1 + κX)PbX − iPbX (1 + κX) = − h¯ κ. = 2 2

(6.138)

This leads to i¯h

E E ∂ f cD,d ψ(S) ψ(S) = H ∂S  1 b2 1 f c P + p0 κ 2 X 2 H D,d = −p0 + 2p0 ⊥ 2   1 h¯ 2 κ 2 (1 − κX) I − h¯ κΣy . + 8p0 2

(6.139)

f cD,d , like the scalar quanNote that this quantum beam optical Dirac Hamiltonian H f co,d in (5.279), reproduces the classical beam optitum beam optical Hamiltonian H f in (4.185), when ~Pb and ~R are taken cal Hamiltonian for the dipole magnet H o,d

⊥

⊥

as the corresponding classical variables, and the h¯ -dependent terms are dropped. It differs from the scalar quantum beam optical Hamiltonian in the h-dependent scalar term ∝ I and has an extra h¯ -dependent matrix term that will have different effects on particles in different spin states. However, these h¯ -dependent terms are very small compared to the main classical part of the Hamiltonian, which leads to the bending of the particle trajectory by the dipole. Thus, we understand the bending action of the dipole magnet on a paraxial beam of spin- 12 particles on the basis of the Dirac equation.

316

Quantum Mechanics of Charged Particle Beam Optics

6.1.2

BEAM OPTICS OF THE DIRAC PARTICLE WITH ANOMALOUS MAGNETIC MOMENT

6.1.2.1

General Formalism

The main framework for studying the spin dynamics and beam polarization in accelerator physics is essentially based on the quasiclassical Thomas–Frenkel– Bargmann–Michel–Telegdi equation, or the Thomas–BMT equation as is commonly called (see Thomas [182, 183], Frenkel [57, 58], Bargmann, Michel, and Telegdi [6]). The Thomas–BMT equation has been understood on the basis of the Dirac equation, independent of accelerator beam optics, in different ways (see Sokolov and Ternov [173], Ternov [180], Corben [29], and references therein). As shown by Derbenev and Kondratenko [32], it is possible to obtain from the Dirac Hamiltonian, using the Foldy–Wouthuysen representation, a quasiclassical effective Hamiltonian accounting for the orbital motion, Stern–Gerlach effect, and the Thomas–BMT spin evolution (see Barber, Heinemann, and Ripken [4, 5]). The same quasiclassical effective Hamiltonian has also been justified by reducing the Dirac theory to the Pauli theory (see Jackson [77]). Based on such a quasiclassical Hamiltonian, a completely classical approach to accelerator beam dynamics has also been developed (see Barber, Heinemann, and Ripken [4, 5]) in which an extended classical canonical formalism is used by adding to the classical phase space two new real canonical variables describing all the three components of spin. Following Conte, Jagannathan, Khan, and Pusterla [26], we present here the spinor quantum charged particle beam optical formalism which, at the level of single particle theory, gives a unified account of orbital motion, Stern–Gerlach effect, and the Thomas–BMT spin evolution for a paraxial beam of Dirac particles with anomalous magnetic moment. Let us consider the propagation of a paraxial beam of charged spin- 12 particles with an anomalous magnetic moment through an optical element with straight optic axis. We shall take the optic axis of the system to be along the z-direction. As we have already seen, the z-evolution equation for the time-independent spinor wave E 0 function of the beam particle in the ψ -representation, (6.11), is i¯h

E ∂ 0 E c0 ψ 0 (z) , ψ (z) = H ∂z c0 = −p0 β + Eb + O, b H
qφ Eb = −qAz I + ξ + ξ −1 β 2c h i

µa − ξ + ξ −1 ~B⊥ ·~Σ⊥ + ξ − ξ −1 β Bz Σz , 2c n
µa  ~ b O = χ ~α⊥ · πb⊥ + ξ − ξ −1 (σy ⊗ (Bx σy − By σx )) 2c
 + ξ + ξ −1 Bz (σx ⊗ I) , s   E + mc2 ξI O with χ = , ξ= , (6.140) −1 O −ξ I E − mc2

317

Spinor Theory for Spin-1/2 Particles

q m2 c4 + c2 p20 as the conserved total energy of a particle of the beam with E E the design momentum p0 . Let us recall that ψ 0 is related to the Dirac ψ by and E =

E E 0 ψ = M ψ(z) ,

1 M = √ (I + χαz ) . 2

(6.141)

To obtain the desired quantum beam optical representation of the z-evolution equation, we have to perform the Foldy–Wouthuysen-like transformations outlined earlier. The first transformation is E E b (1) = eiS1 ψ 0 , ψ

i b Sb1 = β O. 2p0

(6.142)

E The resulting equation for ψ (1) is i¯h

E ∂ (1) E c(1) ψ (1) , =H ψ ∂z    1 1 1 1 b b b b c0 e 2p0 β O − i¯h e− 2p0 β O ∂ e 2p0 β O c(1) = e− 2p0 β O H H ∂z (1) (1) b b = −p0 β + E + O , 1 β Ob 2 − · · · , Eb(1) ≈ Eb − 2p0 ! h i 1 ∂ Ob (1) b b b O ≈− β O , E + i¯h −··· . 2p0 ∂z

(6.143)

b and ξ from (6.140), we have, Substituting the expressions for Eb, O,    2 2 2 c(1) = −p0 − qAz + 1 πb2 + m µa B2 + E B2 I H ⊥ 2p0 ⊥ m2 c4 z 2p30   qEφ m¯hµa + 2 β+ ∇ × ~B β c p0 z 2p20   1 Eµa − (¯hq + 2mµa ) Bz Σz + 2 ~B⊥ ·~Σ⊥ 2p0 c       mµa E ~ ~ ~ ~ ~ ~ ~ ~ b b b b + 2 B Σ · π + Σ · π B − B · π + π · B β Σ z ⊥ z . ⊥ ⊥ ⊥ z ⊥ ⊥ ⊥ ⊥ 2p0 mc2 (6.144) c(1) keeping only terms of order 1/p0 , consistent with the Let us now approximate H paraxial approximation. Then, with 2mµa = aq¯h, E = γmc2 , and ~S = 2h¯ ~Σ, we have

318

Quantum Mechanics of Charged Particle Beam Optics

E ∂ (1) E c(1) ψ (1) , =H ψ ∂z   1 2 qγmφ (1) c H ≈ −p0 − qAz + β πb⊥ I + 2p0 p0   q − (1 + a)Bz Sz + γa~B⊥ ·~S⊥ . (6.145) p0 E As we have seen already in (6.17), for a paraxial plane wave, ψ (1) has almost vanishing lower pair of components compared to the upper pair of components. Up to now all the observables, the field components, time, etc., are defined with reference to the laboratory frame. But, in the covariant description, the spin of the Dirac particle has simple operator representation in terms of the Pauli matrices only in a frame in which the particle is at rest. So, it is usual in accelerator physics to define spin with reference to the instantaneous rest frame of the particle while keeping the other observables, field components, etc., defined with reference to the laboratory frame. In the Dirac representation, the operator for spin as defined in the instantaneous rest frame of the particle is given by (see Sokolov and Ternov [173] for details)      2 ~Σ · ~ ~πb ·~Σ b π + c ~πb 1 c ~SR = h¯ β ~Σ −  + αx  . (6.146) 2 2E (E + mc2 ) E i¯h

E E The relation between the Dirac ψ -representation and the ψ (1) -representation being E E − 1 β Ob (1) = e 2p0 M ψ , (6.147) ψ E the operator for spin in the rest frame of the particle in the ψ (1) -representation is ~S(1) = e− 2p0 β O M~SR M−1 e 2p0 β O , R 1

b

1

b

(6.148)

as follows from (3.134) since in this case the transformation is not unitary. Let us now make another transformation, E E i b (A) ψ (1) , b (A) = e− 2p0 (πbx Σy −bπy Σx ) . (6.149) U ψ A = U Note that this transformation preserves the smallness ofEthe lower pair of components compared to the upper pair of components. In the ψ A -representation, the operator for spin in the instantaneous rest frame of the particle becomes  † ~S(A) = U b (A)~S(1) U b (A) ≈ h¯ ~Σ = ~S, (6.150) R R 2 E as we desire, considering the paraxial approximation and the effect of β on ψ A being the same as I. Using the transformation (6.149) in (6.145), and up calculating E to the paraxial approximation, we get the z-evolution equation for ψ A as

Spinor Theory for Spin-1/2 Particles

i¯h

319

E ∂ E cA ψ , ψ A = H A ∂z    † †  (A) c(1) b (A) (A) ∂ (A) c b b b HA = U H U − i¯hU U ∂z   1 2 qγmφ ≈ −p0 − qAz + β πb⊥ I + 2p0 p0 n  o q ~ ~ − B · S + a Bz Sz + γ ~B⊥ ·~S⊥ . p0

(6.151)

In the previous treatments of optical elements we used to return to the original Dirac representation by retracing the transformations made on it and the Dirac Hamiltonian. This led to the expansion of the Dirac Hamiltonian as a paraxial part plus a series of nonparaxial aberration parts so that we can work with the Dirac Hamiltonian approximated up to any desired level of accuracy. In the present case, we shall follow an equivalent procedure. We shall notEreturn to the original Dirac representation. We shall continue to work with the ψ A -representation. Thus, we have to model the E E wave function of any system in the ψ A -representation. Since ψ A has the lower pair of components vanishing compared to the upper pair of components, we almost E cA is even, i.e., blockcan as well take ψ A to be a two-component spinor. Since H diagonal with the odd part in E zero, its 11-block element becomes the Hamiltonian E the two-component ψ A -representation. We shall denote the two-component ψ A E as ψ (A) , which satisfies the z-evolution equation i¯h

E ∂ (A) E c(A) ψ (A) , =H ψ ∂z   c(A) ≈ −p0 − qAz + 1 πb2 + qγmφ I H 2p0 ⊥ p0 n  o q ~ ~ − B · S + a Bz Sz + γ ~B⊥ · ~S⊥ , p0

where ~S = (¯h/2)~σ . We shall now rewrite this equation as E E ∂ c(A) ψ (A) , i¯h ψ (A) = H ∂z   c(A) ≈ −p0 − qAz + 1 πb2 + qγmφ I + γm ~Ω · ~S, H 2p0 ⊥ p0 p0   q with ~Ω = − (1 + a)~Bk + (1 + γa)~B⊥ , γm

(6.152)

(6.153)

where ~Bk and ~B⊥ are the components of ~B along the optic axis and perpendicular E (A) to it. We shall refer to the ψ -representation as the quantum accelerator optical representation.

320

Quantum Mechanics of Charged Particle Beam Optics

c(A) , the quantum accelerator optical Hamiltonian, is Hermitian. In Note that H E other words, ψ (A) has unitary z-evolution, and hence it can be normalized as D E ψ (A) (z) ψ (A) (z) =

2

∑

j=1

Z

2 (A) d 2 r⊥ ψ j (~r⊥ , z) = 1,

at any z. When the beam is described by a 2 × 2 density matrix ! (A) (A) ρ (z) ρ (z) 11 12 ρ (A) (z) = , (A) (A) ρ21 (z) ρ22 (z)

(6.154)

(6.155)

at any z, the quantum accelerator optical z-evolution equation is i¯h

∂ ρ (A) (z) h c(A) (A) i = H , ρ (z) . ∂z

(6.156)

(A) If the beam E D can be described as a pure state, we would have ρ (z) = (A) ψ (z) ψ (A) (z) .

6.1.2.2

Lorentz and Stern–Gerlach Forces, and the Thomas–Frenkel–BMT Equation for Spin Dynamics

b in the quantum accelThe average of any observable, represented by the operator O erator optical representation, is given by D E   b (z) = Tr ρ (A) (z)O b O 2

=

∑ j,k=1

Z Z

ED E D (A) 0 0 0 b d 2 r⊥ d 2 r⊥ ~r⊥ ρ jk (z) ~r⊥ ~r⊥ Ok j ~r⊥ ,

(6.157)

in the D Etransverse plane at z. It follows from (6.156) that the equation for z-evolution b (z) is given by of O i Dh c(A) biE d D bE O (z) = H , O (z) + dz h¯

*

b ∂O ∂z

+ (z).

(6.158)

Thus, iE d h~r⊥ i i Dh c(A) 1 hπb⊥ i . H ,~r⊥ = = dz h¯ p0 Quasiclassically, for any observable O, we can write D E D E D E b b b d O d O p0 d O ≈ vz ≈ . dt dz γm dz

(6.159)

(6.160)

Spinor Theory for Spin-1/2 Particles

321

For example, it follows from the above two equations that p0 d h~r⊥ i 1 D~ E d h~r⊥ i πb⊥ , ≈ = dt γm dz γm

(6.161)

as should be. For ~πb⊥ we have, with πbz ≈ p0 , D E * + d ~πb⊥ i Dh c(A) ~ iE ∂~πb⊥ = H , πb⊥ + dz ∂z h¯ + * i Dh c(A) ~ iE ∂ ~A⊥ = H , πb⊥ − q ∂z h¯ D E D E  E γm D  qγm q ~πb × ~B − ~B ×~πb ~∇⊥ ~Ω · ~S −~∇⊥ φ + ≈ − p0 2p0 p0 ⊥ qγm D~ E q D~ ~ ~ ~  E γm D~ ~ ~ E = E⊥ + ∇⊥ Ω · S . (6.162) πb × B − B × πb − p0 2p0 p0 ⊥ Hence, D E d ~πb⊥ dt

D E q D~ ~ ~ ~  E D~ ~ ~ E − ∇⊥ Ω · S . πb × B − B × πb ≈ q ~E⊥ + 2γm ⊥

(6.163)

c(A) corresponds to i¯h(∂ /∂ z) = − pbz . The quantum accelerator optical Hamiltonian H Hence, we have to take the operator corresponding to πz , the kinetic momentum in the longitudinal direction, as   c(A) + qAz . πbz = pbz − qAz = − H (6.164) Then, we have  iE  ∂   d hπbz i i Dh c(A) c(A) + qAz c(A) + qAz = H ,− H − H dz h¯ ∂z   Dh iE i c(A) , −qAz − qγm ∂ φ H = h¯ p0 ∂z      γm ∂ ~ ~  1 ∂ 2 − πb⊥ − Ω·S 2p0 ∂ z p0 ∂ z D   E γm  ∂  q qγm ~πb × ~B − ~B ×~πb ~Ω · ~S , hEz i + = − p0 2p0 p0 ∂ z z

(6.165)

and   d hπbz i q D~ ~ ~ ~  E γm ∂ ~ ~  ≈ q hEz i + πb × B − B × πb − Ω·S . dt 2γm p0 ∂ z z

(6.166)

322

Quantum Mechanics of Charged Particle Beam Optics

Combining the two equations (6.163) and (6.166), we get D E d ~πb dt

* " =

1 q ~E + 2

~πb ~πb × ~B − ~B × γm γm

!#+

D  E − ~∇ ~Ω · ~S .

(6.167)

h  i The first term of this equation represents the Lorentz force, q ~E + ~v × ~B . The second term represents the Stern–Gerlach force. In the instantaneous rest frame of the particle where γ = 1, the second term is seen to reduce to the familiar Stern– Gerlach force ~FSG = −~∇V, V = −µ ~σ · ~B, (6.168) in which µ is the total spin magnetic moment of the particle, and apart from the spin, the field components, the coordinates, etc. are also defined in the rest frame of the particle. Equations (6.162) and (6.165), respectively, correspond to the transverse and longitudinal Stern–Gerlach kicks. For a discussion of relativistic Stern–Gerlach force using the Derbenev–Kondratenko semiclassical Hamiltonian, see Heinemann [74]. In the case of spin, we have D E d ~S dz

i Dh c(A) ~ iE i γm Dh~ ~ ~ iE Ω·S, S H ,S = h¯ h¯ p0 D E γm ~ ~ Ω×S . = p0 =

(6.169)

As a result, we can write D E d ~S dt

D E ≈ ~Ω × ~S ,

(6.170)

which corresponds to the classical equation for spin dynamics, d~S ~ ~ = Ω × S, dt

(6.171)

the Thomas–Frenkel–Bargmann–Michel–Telegdi equation, or the Thomas–BMT equation. For more details on the Thomas–BMT equation, see, e.g., Conte and MacKay [27], Lee [127], and Montague [136]. Thus, Ω in the quantum accelerc(A) is the Thomas–BMT spin precession frequency. ator optical Hamiltonian H ~ ~ The polarization D E of the beam is characterized by the vector P defined by (¯h/2)P = (¯h/2) h~σ i = ~S . Thus, the quantum accelerator optical z-evolution equation (6.153) accounts for the orbital and spin motions of the particle, up to the paraxial approximation. To get higher order corrections, one has to go beyond the paraxial approximation by continuing the process of Foldy–Wouthuysen-like transformations up to the desired order.

323

Spinor Theory for Spin-1/2 Particles

6.1.2.3

Phase Space and Spin Transfer Maps for a Normal Magnetic Quadrupole

Let us now consider a paraxial beam of Dirac particles of charge q, mass m, and anomalous magnetic moment µa , propagating through a normal magnetic quadrupole with the optic axis along the z-direction. The practical boundaries of the lens can be taken to be at z` and zr with z` < zr . Since there is no electric field, φ = 0. The field of the normal magnetic quadrupole is ~B(~r) = (−Qn y, −Qn x, 0) , where

 Qn =

(6.172)

constant in the lens region (z` ≤ z ≤ zr ) 0 outside the lens region (z < z` , z > zr )

The corresponding vector potential can be taken as  
~A(~r) = 0, 0, 1 Qn x2 − y2 . 2

(6.173)

(6.174)

Now, from (6.153), the quantum accelerator optical Hamiltonian for the normal magnetic quadrupole is seen to be, with w = zr − z` , Kn = qQn /p0 , and ηn = q(1 + γa)Qn w/p20 ,  pb2⊥   for z < z` and z > zr ,  −p0 + 2p0 , 2
c b p (6.175) H (z) = −p0 + 2p⊥0 − 12 p0 Kn x2 − y2    + ηn p0 (yS + xS ) , for z ≤ z ≤ z . w

x

y

`

r

c(A) simply as H c, and we shall do so in the rest of this Here, we have denoted H section also. Similarly, we will write ρb(z) instead of ρb(A) (z). Let us consider the propagation of the beam from the transverse xy-plane at z = zi < z` to the transverse plane at z > zr . The beam propagates in free space from zi to z` , passes through the lens from z` to zr , and propagates through free space again c(z) as a core part H c0 (z) plus a from zr to z. Let us now write the Hamiltonian H 0 c perturbation part H (z): c(z) = H c0 (z) + H c0 (z) H   −p + pb2⊥ , for z < z` and z > zr , 0 2p0 c0 (z) = H  −p0 + pb2⊥ − 1 p0 Kn x2 − y2
, for z ≤ z ≤ zr , ` 2p0 2  for z < z` and z > zr , c0 (z) = 0, H (6.176) ηn p0 (xS + yS ) , for z` ≤ z ≤ zr . y x w c(z) is Hermitian. A formal integration of (6.156), the z-evolution equaNote that H tion for ρb(z), gives b (z, zi ) ρb (zi ) U b † (z, zi ) , ρb(z) = U (6.177)

324

Quantum Mechanics of Charged Particle Beam Optics

with

  R c(z) − h¯i zz dzH b i , U (z, zi ) = P e

b (zi , zi ) = I, U

(6.178)

and ∂ b c(z)U b (z, zi ) , U (z, zi ) = H ∂z ∂ b† c(z). b † (z, zi ) H i¯h U (z, zi ) = −U ∂z i¯h

(6.179)

b (z, zi ) is given by the Magnus formula: As we know, a convenient expression for U b (z, zi ) = e− h¯i Tb(z,zi ) , U  Z z Z z2 i h c(z1 ) + 1 − i c(z2 ) , H c(z1 ) dz2 dz1 H dz1 H 2 h¯ zi zi zi  2 Z z Z z3 Z z2 nhh i i i 1 c(z3 ) , H c(z2 ) , H c(z1 ) − + dz3 dz2 dz1 H 6 h¯ zi z zi hh ii io c c c + H (z1 ) , H (z2 ) , H (z3 ) + · · · . (6.180) Z z

Tb (z, zi ) =

Going to the interaction picture, let us define

with

b † (z, zi ) ρb(z)U b0 (z, zi ) , ρbi (z) = U 0

(6.181)

  R c0 (z) − h¯i zz dzH b i . U0 (z, zi ) = P e

(6.182)

It follows that i¯h

    b ∂ ρbi (z) ∂ b† b0 (z, zi ) + U b † (z, zi ) i¯h ∂ ρ (z) U b0 (z, zi ) b = i¯h U (z, z ) ρ (z) U i 0 ∂z ∂z 0 ∂z   b † (z, zi ) ρb(z) i¯h ∂ U b0 (z, zi ) +U 0 ∂z h i † c0 (z)ρb(z)U c(z) , ρb(z) U b (z, zi ) H b0 (z, zi ) + U b † (z, zi ) H b0 (z, zi ) = −U 0 0 c0 (z)U b † (z, zi ) ρb(z)H b0 (z, zi ) +U 0 n   o c(z) − H c0 (z) ρb(z) − ρb(z) H c(z) − H c0 (z) b † (z, zi ) H =U 0 h i c0 (z) , ρb(z) U b † (z, zi ) H b0 (z, zi ) =U 0 h i c0 (z) , ρb (z) = H (6.183) i i

where c0 (z) = U c0 (z)U b0 (z, zi ) . b † (z, zi ) H H 0 i

(6.184)

325

Spinor Theory for Spin-1/2 Particles

Formally integrating (6.183), the z-evolution equation for ρbi (z), we get b 0† (z, zi ) , b 0 (z, zi ) ρbi (zi ) U ρbi (z) = U i i

(6.185)

  i Rz c0 b 0 (z, zi ) = P e− h¯ zi dz Hi (z) U i    R c0 (z)U b † (z,zi )H b0 (z,zi ) − i z dz U 0 = P e h¯ zi .

(6.186)

where

From (6.181), we find that ρbi (zi ) = ρb (zi ) ,

(6.187)

and hence the equation (6.185) becomes b 0 (z, zi ) ρb (zi ) U b 0† (z, zi ) . ρbi (z) = U i i

(6.188)

b0 (z, zi ) ρbi (z)U b † (z, zi ) . ρb(z) = U 0

(6.189)

b0 (z, zi ) U b 0 (z, zi ) ρb (zi ) U b 0† (z, zi ) U b † (z, zi ) . ρb(z) = U 0 i i

(6.190)

From (6.181), we have

Thus,

The transfer maps for the observables of the system are to be calculated using the formula for the average  D E  b b (z) = Tr ρb(z)O O   b b0 (z, zi ) U b 0 (z, zi ) ρb (zi ) U b 0† (z, zi ) U b † (z, zi ) O = Tr U 0 i i i h  0† † bU b (z, zi ) U b (z, zi ) O b0 (z, zi ) U b 0 (z, zi ) . = Tr ρb (zi ) U (6.191) 0 i i Note that in the field-free regions (z < z` , z > zr ) and within the lens region c(z) is z-independent. In view of this and the semi(z` ≤ z ≤ zr ), the Hamiltonian H b we have group property of the z-evolution operator U, b0 (z, zi ) = U b0 (z, zr ) U b0 (zr , z` ) U b0 (z` , zi ) U    pb2⊥ i = exp − (z − zr ) −p0 + h¯ 2p0   
pb2 1 i × exp − (zr − z` ) −p0 + ⊥ − p0 Kn x2 − y2 h¯ 2p0 2    pb2⊥ i × exp − (z` − zi ) −p0 + , h¯ 2p0 b 0 (z, zi ) = U b 0 (z, zr ) U b 0 (zr , z` ) U b 0 (z` , zi ) U i

i

i

i

326

Quantum Mechanics of Charged Particle Beam Optics

    iηn p0 b † b0 (z, zi ) = exp − U0 (z, zi ) (xSy + ySx ) U h¯ w   √
iηn 1 √ sinh w Kn p0 x = exp − h¯ w Kn 
√
1 cosh w Kn − 1 pbx Sy + wKn  √
1 √ sin w Kn p0 y + w Kn  
√
1 cos w Kn − 1 pby Sx . + wKn

(6.192)

It is straightforward to calculate the required transfer maps using (6.191) and (6.192). For the transverse position coordinates and momentum components, we get   (X) (X) T11 T12 hxi(z)  1 (X) (X)  p h pbx i (z)   T T22  ≈  21  0  hyi(z)   0 0

 1 by (z) p0 p 0 0  1 

  +ηn  

 

 hxi (zi ) h pbx i (zi )   (Y ) (Y )  hyi (z ) i T11 T12

 1 (Y ) (Y ) by (zi ) p0 p T21 T22
 √
 n − 1 Sy (zi ) wKn cosh w K√


  − w√1K sinh w Kn Sy (zi )   n

√   , (6.193) 1 − wK cos w Kn − 1 hSx i (zi )  n
√ − w√1K sin w Kn hSx i (zi ) 0 0

0 0

      

1 p0

n

where (X)

T11 (X) T21

(X)

T12 (X) T22

!

 =

1 z − zr 0 1

×  × (Y )

T11 (Y ) T21

(Y )

T12 (Y ) T22

!

 =

1 0

×  ×



√
cosh w Kn √ √
Kn sinh w Kn  1 z` − zi , 0 1  z − zr 1 √
cos w Kn √ √
− Kn sin w Kn  1 z` − zi . 0 1

√
! sinh w Kn √
cosh w Kn

√1 Kn

√
! sin w Kn √
cos w Kn

√1 Kn

(6.194)

Spinor Theory for Spin-1/2 Particles

For the components of spin, we have  √
1 √ sinh w Kn hxSz i (zi ) hSx i (z) ≈ hSx i (zi ) + p0 ηn w Kn 
√
1 + cosh w Kn − 1 h pbx Sz i (zi ) , p0 wKn 



 √
1 √ sin w Kn hySz i (zi ) Sy (z) ≈ Sy (zi ) − p0 ηn w Kn 


 √
1 cos w Kn − 1 pby Sz (zi ) , − p0 wKn  √
1 √ sinh w Kn hxSx i (zi ) hSz i (z) ≈ hSz i (zi ) − p0 ηn w Kn √
 1 − √ sin w Kn ySy (zi ) w Kn
√
1 + cosh w Kn − 1 h pbx Sx i (zi ) p0 wKn  


√
1 cos w Kn − 1 pby Sy (zi ) . + p0 wKn

327

(6.195)

Thus we have derived the transfer maps for the quantum averages of transverse position and momentum components, and the spin components, for a monoenergetic paraxial beam of Dirac particles propagating through a normal magnetic quadrupole lens with a straight optic axis. It is evident that the transfer maps for position and momentum components include the Stern–Gerlach effect. The lens is focusing (defocusing) in the yz-plane and defocusing (focusing) in the xz-plane when n > 0 (Kn zr ,

  

for z` ≤ z ≤ zr .

pb2 −p0 + 2p⊥0 + p0 Ks xy + ηswp0 (xSx − ySy ) ,

(6.198)

Now, let us make the following transformations: 

 x  pbx     y = pby   Sx = Sy

 1 0 1 0  1  0 1 0 1   √    −1 0 1 0  2 0 −1 0 1   0  1 Sx 1 1 √ . Sy0 2 −1 1 

 x0 pb0x  , y0  pb0y (6.199)

cs (z) is seen to become Then, the Hamiltonian H  pb02 ⊥   for z < z` and z > zr ,  −p0 + 2p0 ,   02 b p c 2 2 1 0 0 ⊥ , Hs (z) =  −p0 + 2p0 − 2 p0 Ks
x − y   ηs p0 + w y0 Sx0 + x0 Sy0 , for z` ≤ z ≤ zr .

(6.200)

same as for a normal magnetic quadrupole (6.175) with Qs , Ks , and ηs instead of Qn , Kn , and
ηn , respectively,
and (x, y), ( pbx , pby ), and (Sx , Sy ) replaced by (x0 , y0 ), pb0x , pb0y , and Sx0 , Sy0 , respectively. Note that Sz is unchanged and
σx0 = 2Sx0 /¯h, σy0 = 2Sy0 /¯h, σz = 2Sz /¯h obey the same algebra as (σx , σy , σz ), i.e., 2

σx0 = I,

2

σy0 = I,

σz σx0 = −σx0 σz ,

σx0 σy0 = −σy0 σx0 , σz σy0 = −σy0 σz .

(6.201)

329

Spinor Theory for Spin-1/2 Particles




This shows that the transfer maps for hx0 i , h pb0x i /p0 , hy0 i , pb0y /p0 , and hSx0 i , Sy0 , hSz i) will be the same as in (6.193 and  for

and (6.195), respectively, except

6.194) the replacement: hxi , h pbx i /p0 , hyi , pby /p0 −→ hx0 i , h pb0x i /p0 , hy0 i , pb0y /p0 ,
(Sx , Sy ) −→ Sx0 , Sy0 , Qn −→ Qs , Kn −→ Ks , ηn −→ ηs . Once the transfer maps



are obtained in terms of hx0 i , h pb0x i /p0 , hy0 i , pb0y /p0 , and Sx0 , Sy0 , Sz , the required


transfer maps for hxi , h pbx i /p0 , hyi , pby /p0 , and (Sx , Sy , Sz ) can be obtained using the inverse of the transformations (6.199). The resulting transfer maps are as follows:   hxi(z)   h pbx i (z)      hyi(z)  ≈ Tsq (z, zi ) 

 1 by (z) p0 p 

1 p0

 +

ηs    2 

 hxi (zi ) h pbx i (zi )   hyi (zi ) 

1 by (zi ) p0 p 1 p0



−C− hSx i (zi ) + C+ − 2 Sy (zi )


√1 −S+ hSx i (zi ) + S− Sy (zi ) w Ks



+ − 1 i) wKs − C − 2 hSx i (zi ) + C Sy (z
√1 −S− hSx i (zi ) + S+ Sy (zi ) w K 1 wKs

    , 

s

(6.202) and 
1 1 hSx i (z) ≈ hSx i (zi ) + p0 ηs − √ S− hxSz i (zi ) + S+ hySz i (zi ) 2 w Ks 



1 C+ − 2 h pbx Sz i + C− pby Sz + p0 wKs 




 1 1 S+ hxSz i (zi ) + S− hySz i (zi ) Sy (z) ≈ Sy (zi ) + p0 ηs − √ 2 w Ks 




1 + C− h pbx Sz i + C+ − 2 pby Sz p0 wKs 

 1 1 hSz i (z) ≈ hSz i (zi ) + p0 ηs √ S− hxSx i (zi ) + S+ xSy (zi ) 2 w Ks


+S+ hySx i (zi ) + S− ySy (zi ) −

1 p0 wKs −

+C






 C+ − 2 h pbx Sx i (zi ) + C− pbx Sy (zi )

 



+ b b py Sx (zi ) + C − 2 py Sy (zi )

(6.203)

330

Quantum Mechanics of Charged Particle Beam Optics

where 

1 z − zr  0 1 Tsq (z, zi ) =   0 0 0 0  1  ×  2 

 0 0 0 0   1 z − zr  0 1

C+ √ − Ks S− C− √ − Ks S+

1 z` − zi  0 1 ×  0 0 0 0

√1 S+ Ks +

C

√1 S− Ks −

C

C− √ − Ks S+ C+ √ − Ks S− 

√1 S− Ks −

C

√1 S+ Ks +

    

C

0 0 0 0  , 1 z` − zi  0 1

(6.204)

and √ √ C± = cos(w Ks ) ± cosh(w Ks ), √ √ S± = sin(w Ks ) ± sinh(w Ks ).

6.2

(6.205)

NONRELATIVISTIC QUANTUM CHARGED PARTICLE BEAM OPTICS: SPIN- 12 PARTICLES

The nonrelativistic Schr¨odinger–Pauli equation for a spin- 12 particle of charge q and mass m moving in the time-independent electromagnetic field of an optical system is    2 ∂ Ψ (~r,t) 1 ~ i¯h pb − q~A (~r) + qφ (~r) I − µ ~σ · ~B Ψ (~r,t) , (6.206) = ∂t 2m where ~A (~r) and φ (~r) are the magnetic vector potential and the electric scalar potential of the field, Ψ (~r,t) is the two-component wave function of the particle, and µ is the total magnetic moment of the particle. Let us consider a particle of a nonrelativistic monoenergetic quasiparaxial beam with total energy E = p20 /2m, where p0 is the design momentum with which it enters the optical system from the free space outside and |~p0⊥ |  p0 . The total energy E is conserved when the beam propagates through any optical system with time-independent electromagnetic field we are considering. Choosing the wave function of the particle as Ψ (~r,t) = e−iEt/¯h ψ (~r) ,

(6.207)

we get the time-independent Schr¨odinger–Pauli equation obeyed by ψ (~r⊥ , z): 

  πb2 ~ + qφ (~r) I − µ ~σ · B ψ (~r) = Eψ (~r) , 2m

(6.208)

331

Spinor Theory for Spin-1/2 Particles

where ~πb = ~pb − q~A. Multiplying the equation on both sides by 2m, taking 2mE = p20 ,

2mqφ = p˜ 2 ,

(6.209)

and rearranging the terms, we have   πbz2 ψ = p20 − p˜ 2 − πb⊥2 + 2mµ ~σ · ~B ψ,

(6.210)

in which we have dropped the identity matrix I as to be understood. Note that this equation is identical, except for the additional term 2mµ ~σ · ~B and ψ being a two-component wave function, to the time-independent nonrelativistic Schr¨odinger equation (5.298), which is the starting point for our derivation of the nonrelativistic quantum beam optical Hamiltonian (5.299). Now, starting with (6.210), we can follow the same procedure, going through the Feshbach–Villars-like formalism and the Foldy–Wouthuysen-like transformations. So, let us introduce a four-component wave function !   ψ ψ1 . (6.211) = πbz ψ2 p0 ψ Then, we can write (6.210) equivalently as !    0 ψ1 ψ1 πbz   1 = . 1 p20 − p˜2 − πb⊥2 + 2mµ ~σ · ~B 0 ψ2 ψ2 p0 p2

(6.212)

0

Now, let ψ+ ψ−

!

 =M

=

1 2

   ψ1 1 1 1 = ψ2 2 1 −1 ! b ψ + pπ0z ψ . b ψ − pπ0z ψ ψ1 ψ2



(6.213)

Then, πbz p0

ψ+ ψ−

!

1 = p0 =M

  ∂ −i¯h − qAz ∂z

ψ+ ψ−

!

0   1 1 2 2 2 p0 − p˜ − πb⊥ + 2mµ ~σ · ~B 0 p2

! M −1

0

1 − 2p12 πb⊥2 + p˜2 0    ~B ~ −2mµ σ ·   =  1 πb⊥2 + p˜2  2p20   −2mµ ~σ · ~B 

− 2p12 πb⊥2 + p˜2 0  −2mµ ~σ · ~B

ψ+ ψ−

!



      −1 + 2p12 πb⊥2 + p˜2  0   −2mµ ~σ · ~B

ψ+ ψ−

! .

(6.214)

332

Quantum Mechanics of Charged Particle Beam Optics

Rearranging this equation, we get ! ! ψ+ ψ+ ∂ i¯h =b H , ψ− ψ− ∂z b b H = −p0 β + Eb + O, h  i 1 σz ⊗ πb⊥2 + p˜2 − 2mµ ~σ · ~B , Eb = −qAz I + 2p0 h  i i σy ⊗ πb⊥2 + p˜2 − 2mµ ~σ · ~B . Ob = 2p0

(6.215)

Note that b H has the same structure as the Dirac equation, with the leading term −p0 β , an even term Eb which does not couple ψ + and ψ − and an odd term Ob which couples ψ + and ψ − . Further, the odd term anticommutes with β . It is also clear that the lower pair of components ψ − will be small compared to the upper pair of components ψ + . Foldy–Wouthuysen-like transformations will further make the lower pair of components smaller compared to the upper pair of components. If we stop with one Foldy–Wouthuysen-like transformation, we can get the quantum beam optical Hamiltonian up to paraxial approximation. If we do this following the same procedure as done earlier, and approximate the resulting four-component wave function by a two-component wave function consisting of only the upper pair of components, we get the nonrelativistic quantum beam optical Hamiltonian for a spin- 12 particle, in the paraxial approximation, as  
mµ ~ 1 2 2 c ~σ · B πb⊥ + p˜ I− Ho,NR = −p0 − qAz + 2p0 p0   πb2 mqφ mµ ~ ~σ · B. = −p0 − qAz + ⊥ + I− (6.216) 2p0 p0 p0 Thus, the nonrelativistic quantum beam optical z-evolution equation for the twocomponent wave function of a spin- 12 particle is, in the paraxial approximation,    ∂ ψ (~r⊥ , z) πb2 mqφ mµ ~ ~σ · B ψ (~r⊥ , z) , (6.217) ı¯h = −p0 − qAz + ⊥ + I− ∂z 2p0 p0 p0 to which the relativistic paraxial quantum beam optical equation (6.153) reduces while taking the nonrelativistic limit γ −→ 1,

p0 −→ p0 ,

(6.218)

and noting that m q(1 + a) ~ ~ mq(1 + a)¯h ~ ~σ · B = B·S = p0 2mp0 p0



 qg¯h mµ ~ ~σ · ~B = ~σ · B. 4m p0

(6.219)

This shows that one can get the nonrelativistic expressions and formulae by taking the nonrelativistic limit of the results derived from the quantum beam optical Dirac

Spinor Theory for Spin-1/2 Particles

333

equation exactly like in the scalar theory based on the Klein–Gordon equation. However, it is not possible to get the relativistic quantum beam optical equation (6.153) by replacing m by γm and p0 by p0 in the nonrelativistic quantum beam optical equation (6.217). Thus, it should be noted that converting the expressions and formulae derived in the nonrelativistic theory to relativistic expressions and formulae by replacing the rest mass m by γm, the so-called relativistic mass, as has been the common practice in electron optics, is misleading in the spinor theory of quantum charged particle beam optics.

Remarks and 7 Concluding Outlook on Further Development of Quantum Charged Particle Beam Optics In conclusion, we have to make several remarks. Analogy between optics and mechanics and its role in progress of physics is well known (for historical details see, e.g., Hawkes and Kasper [70], Lakshminarayanan, Ghatak, and Thyagarajan [125], Khan [108, 113, 115]). In the early days of electron optics, analogy with light optics provided much guidance. In the development of the formalism of spinor theory of charged particle beam optics presented here, the inspiration came from the work of Sudarshan, Simon, and Mukunda in light optics (see Sudarshan, Simon, and Mukunda [175], Mukunda, Simon, and Sudarshan [137, 138], Simon, Sudarshan, and Mukunda [170, 171], Khan [120]). The formalism of quantum charged particle beam optics in turn has led to a similar approach, called quantum methodology, to the Helmholtz scalar and the Maxwell vector optics. In analogy with the derivation of the scalar theory of quantum charged particle beam optics starting with the Klein–Gordon equation, scalar theory of light beam optics is derived starting with the Helmholtz equation and using the Feshbach–Villars-like form and the Foldy– Wouthuysen-like transformations. In analogy with the derivation of the spinor theory of quantum charged particle beam optics starting with the Dirac equation, the vector theory of light beam optics, including polarization, is derived starting with the Maxwell equations expressed in a matrix form and using the Foldy–Wouthuysenlike transformations. This methodology leads to scalar and vector theories of light beam optics expressed as paraxial approximation followed by nonparaxial aberrations (for details, see Khan, Jagannathan, and Simon [100], and Khan [101, 102, 103, 104, 105, 106, 107, 109, 110, 111, 112, 114, 115, 116, 117, 118, 119]). It is certain that at some point in future, quantum mechanics would become important in accelerator beam optics. For example, Hill has discussed in [76] how quantum mechanics places limits on the achievable transverse beam spot sizes in accelerators, Kabel has considered in [89] how the Pauli exclusion principle will limit the minimum achievable emittance for a beam of fermions in a circular accelerator, Venturini and Ruth have suggested in [185] a framework to compute the wave function of a single charged particle confined to a magnetic lattice transport line or storage ring, Chao and Nash have examined in [19] the possible measurable macroscopic effect 335

336

Quantum Mechanics of Charged Particle Beam Optics

of quantum mechanics on beams through its echo effect using the Wigner phase space distribution function, and Heifets and Yan have pointed out in [73] that quantum uncertainties affect substantially the classical results of tracking for trajectories close to the separatrix and hence the quantum maps can be useful in quick findings of the nonlinear resonances. It is clear that several quantum mechanical problems related to the beam optics, particularly the nonlinear beam optics, will have to be handled in the future particle accelerators built to explore particle physics beyond the standard model (For an overview of the currently envisaged landscape of future particle accelerators, see, e.g., Syphers and Chattopadhyay [176]). In the field of small-scale electron beam devices, electron beam lithography is the primary tool in micro- and nanofabrications, particularly in the fabrication of semiconductor devices and mesoscopic devices. In electron beam lithography for nanoscale fabrication, it is required to generate high-resolution electron beams to facilitate high-speed drawing of fine patterns on material substrates (see, e.g., Manfrinato et al. [132], Wu, Makiuchi, and Chen [193]). Though classical electron beam optics may be successful in designing the beam optical systems of the present-day electron beam lithography, future advances in the technology would certainly require the quantum theory of electron beam optics. Advancements in high-resolution electron microscopy take place using various aspects of quantum mechanics. Scanning tunneling microscopy was invented using the quantum tunneling effect to image directly the atoms of a sample (see, e.g., Chen [21]). Recently, Kruit et al. [123] had a proposal to implement the suggestion of the so-called quantum electron microscope (Putnam and Yanik [153]) that would use the principle of interaction-free measurement based on the quantum Zeno effect (Misra and Sudarshan [133]). Another suggestion is electron microscopy based on quantum entanglement (Okamoto and Nagatani [140]). Highest ever resolution in transmission electron microscopy has been reported by Jiang et al. [87] using ptychographic analysis of the diffraction patterns of low-energy electrons by a two-dimensional specimen. It should be emphasized that the formalism of quantum charged particle beam optics presented here might not be related to the quantum aspects of the operational principles of such novel electron microscopes, but will benefit the designing of electron beam optical elements like the electron lenses used in the system. It may be noted that the quantum charged particle beam optics presented here is only at a preliminary stage of development with the quantum theory of several concepts and topics of classical charged particle beam dynamics, particularly the accelerator beam dynamics, waiting to be developed. The single most important task would be to develop the quantum theory of particle acceleration and the longitudinal beam dynamics and integrate it with the formalism of quantum beam optics to help the treatment of energy spread, energy loss, etc. To this end, one has to find the quantum mechanical analog of the classical six-dimensional phase space in which the relative energy deviation and the longitudinal distance relative to the reference particle are taken as the longitudinal phase space coordinates in addition to the four transverse phase space coordinates. Next, one has to extend the theory beyond the single particle dynamics with the generalization of the concepts like emittances and

Concluding Remarks

337

the Twiss (or Courant-Snyder) parameters. In this respect, phenomenological models of quantum theory of charged particle beam dynamics based on Schr¨odinger-like equations, often with the Planck constant replaced by an emittance, could provide some guidance. Fedele et al. have developed extensively such quantum-like theories of charged particle beam physics, called thermal wave model and quantum wave model, to deal with different situations, including collective effects, space charge, etc. (see, e.g., Fedele and Miele [40], Fedele and Shukla [41], Fedele et al. [42], Fedele, Man’ko, and Man’ko [43], Fedele et al. [45, 46, 47] and references therein). A model similar to the thermal wave model of Fedele et al. has been considered also by Dattoli et al. [30]. Quantum-like approach has been used to study the phenomenon of halo formation in accelerator beams by Pusterla [152], and Khan and Pusterla [94, 96, 97, 98]. Similarly, a semiclassical stochastic model for the collective dynamics of charged particle beams has been developed by Petroni et al. [143]. May be it will be profitable to formulate the quantum charged particle beam optics using the Wigner function (see, e.g., Chao and Nash [19], Dragt and Habib [37], Jagannathan and Khan [81], Fedele, Man’ko, and Man’ko [44]). Heinemann and Barber have studied the dynamics of spin- 21 polarized beams using the Wigner function in the context of accelerator physics (see [75]; see also Mitra and Ramanathan [134] for a discussion of the Wigner function for the Klein–Gordon and Dirac particles). In fine, we have presented the elements of Quantum Charged Particle Beam Optics showing why and how classical charged particle beam optics works very successfully in the design and operation of charged particle beam devices from low-energy electron microscopes to high-energy particle accelerators. In future, developments in technology would certainly require quantum theory for the purpose of design and operation of various kinds of charged particle beam devices. We hope that the formalism of quantum charged particle beam optics presented here would be the basis for the development of the quantum theory of future charged particle beam devices.

Bibliography 1. Acharya, R. and Sudarshan, E. C. G., Front description in relativistic quantum mechanics, J. Math. Phys., 1 (1960) 532–536. 2. Agarwal, G. S. and Simon, R., A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function, Optics Comm., 110 (1994) 23–26. 3. Arfken, G. B., Weber, H. J., and Harris, F. E., Mathematical Methods for Physicists: A Comprehensive Guide, (7th Edition, Academic Press, Waltham, MA, 2012). 4. Barber, D. P., Heinemann, K., and Ripken, G., A canonical 8-dimensional formalism for classical spin-orbit motion in storage rings, Z. Phys. C, 64(1) (1994) 117–142. 5. Barber, D. P., Heinemann, K., and Ripken, G., A canonical 8-dimensional formalism for classical spin-orbit motion in storage rings, Z. Phys. C, 64(1) (1994) 143–167. 6. Bargmann, V., Michel, L., and Telegdi, V. L., Precession of the polarization of particles moving in a homogeneous electromagnetic field, Phys. Rev. Lett., 2 (1959) 435–436. 7. Bauer, E., LEEM basics, Surface Rev. and Lett., 5(6) (1998) 1275–1286. 8. Bell, J. S., in: Progress in Particle Physics: Proceedings of the XIII. Internationale Universit¨atswochen f¨ur Kernphysik 1974 der Karl-Franzens-Universit¨at Graz at Schladming, Austria, 1974, Ed. P. Urban, Acta Physica Austriaca, 13 (1974) 395. 9. Bell, J. S. and Leinass, J. M., The Unruh effect and quantum fluctuations of electrons in storage rings, Nucl. Phys. B, 284 (1987) 488–508. 10. Bellman, R. and Vasudevan, R., Wave Propagation: An Invariant Imbedding Approach, (D. Reidel, Dordrecht, 1986). 11. Berz, M., Modern map methods in particle beam physics, in Advances in Imaging and Electron Physics, Vol. 108, Ed. P. W. Hawkes, (Academic Press, San Diego, 1999) pp. 1–318. 12. Berz, M., Makino, K., and Wan, W., An Introduction to Beam Physics, (Taylor & Francis, Boca Raton, FL, 2016). 13. Bjorken, J. D. and Drell, S. D., Relativistic Quantum Mechanics, (McGraw-Hill, New York, 1994). 14. Blanes, S., Casas, F., Oteo, J. A., and Ros, J., The Magnus expansion and some of its applications, Phys. Rep., 470 (2009) 151–238. 15. Boxem, R. V., Partoens, B., and Verbeeck, J., Dirac Kirchhoff diffraction theory, arXiv:1303.0954[quant-ph]. 16. Brill, D. R. and Wheeler, J. A., Interaction of neutrinos and gravitational fields, Rev. Mod. Phys., 29(3) (1957) 465–479. 17. Busch, H., On the action of the concentration coil in a Braun tube, Arch. Elektrotech. 18 (1927) 583–594. 18. Byron, F. W. and Fuller, R. W., Mathematics of Classical and Quantum Physics, (Dover, New York, 1992). 19. Chao, A. and Nash, B., Possible quantum mechanical effect on beam echo, in Proceedings of the 18th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics—2000, Italy, Ed. P. Chen, (World Scientific, 2002), pp. 90–94. 20. Chao, A. W., Mess, K.H., Tigner, M., and Zimmerman, F., (Eds.), Handbook of Accelerator Physics and Engineering, (2nd Edition, World Scientific, Singapore, 2013).

339

340

Bibliography

21. Chen, C. J., Introduction to Scanning Tunneling Microscopy, (2nd Edition, Oxford University Press, New York, 2008). 22. Chen, P., (Ed.), Proceedings of the 15th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics—1998, USA, (World Scientific, 1999). 23. Chen, P., (Ed.), Proceedings of the 18th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics—2000, Italy, (World Scientific, 2002). 24. Chen, P., and Reil, P., (Eds.), Proceedings of the 28th ICFA Advanced Beam Dynamics and Advanced & Novel Accelerators Workshop—2003, Japan, (World Scientific, 2004). 25. Cohen-Tannoudji, C., Diu, B., and Lolo¨e, F., Quantum Mechanics—Vols. 1&2, (Wiley, New York, 1992). 26. Conte, M., Jagannathan, R., Khan, S. A., and Pusterla, M., Beam optics of the Dirac particle with anomalous magnetic moment, Part. Accel., 56 (1996) 99–126. 27. Conte, M. and MacKay, W. W., An Introduction to the Physics of Particle Accelerators, (2nd Edition, World Scientific, Singapore, 2008). 28. Corben, H. C. and Stehle, P., Classical Mechanics, (2nd Edition, Wiley, New York, 1964). 29. Corben, H. C., Classical and Quantum Theories of Spinning Particle, (Holden-Day, San Francisco, CA, 1968). 30. Dattoli, G., Giannessi, L., Mari, C., Rihetta, M., and Torre, A., Formal quantum theory of electronic rays, Opt. Commun. 87 (1992) 175–180. 31. Dattoli, G., Renieri, A., and Torre, Lectures on the Free Electron Laser Theory and Related Topics, (World Scientific, Singapore, 1993). 32. Derbenev, Ya. S. and Kondratenko, A. M., Polarization kinetics of particles in storage rings, Sov. Phys. JETP, 37(6) (1973) 968–973. 33. Dragt, A. J., A Lie Algebraic theory of geometrical optics and optical aberrations, J. Opt. Soc. America, 72(3) (1982) 372–379. 34. Dragt, A. J., Lie Methods for Nonlinear Dynamics with Applications to Accelerator Physics, (University of Maryland, 2018); https://physics.umd.edu/dsat/dsatliemeth 35. Dragt, A. J. and Forest, E., Lie algebraic theory of charged particle optics and electron microscopes, in Advances in Electronics and Electron Physics, Vol. 67, Ed. P. W. Hawkes, (Academic Press, Cambridge, MA, 1986) pp. 65–120. 36. Dragt, A. J., Neri, F., Rangarajan, G., Healy, L. M., and Ryne, R. D., Lie algebraic treatment of linear and nonlinear beam dynamics, Ann. Rev. Nucl. Part. Sci., 38 (1988) 455–496. 37. Dragt, A. J. and Habib, S., How Wigner functions transform under symplectic maps, in Proceedings of the 15th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics—1998, USA, Ed. P. Chen (World Scientific, 1999), pp. 651–669. 38. Durand, E., Le principe de Huygens et la diffraction de l’´electron en th´eorie de Dirac, Comptes rendus de l’Acad´emie des Sciences, Paris, 236 (1953) 1337–1339. 39. Esposito, G., Marmo, G., Miele, G., and Sudarshan, E. C. G., Advanced Concepts in Quantum Mechanics, (Cambridge University Press, Cambridge, 2014). 40. Fedele, R. and Miele, G., A thermal-wave model for relativistic-charged-particle beam propagation, Il Nuovo Cimento D, 13(12) (1991) 1527–1544. 41. Fedele, R. and Shukla, P. K., Self-consistent interaction between the plasma wake field and the driving relativistic electron beam, Phys. Rev. A, 45(6) (1992) 4045–4049. 42. Fedele, R., Miele, G., Palumbo, L., and Vaccaro, V. G., Thermal wave model for nonlinear longitudinal dynamics in particle accelerators, Phys. Lett. A, 179(6) (1993) 407–413. 43. Fedele, R., Man’ko M. A., and Man’ko, V.I., Wave-optics applications in chargedparticle-beam transport, J.Russ. Laser Res., 21(1) (2000) 1–33.

Bibliography

341

44. Fedele, R., Man’ko M. A., and Man’ko, V.I., Charged-particle-beam propagator in wave-electron optics: Phase-space and tomographic pictures, J. Opt. Soc. America A, 17 (2000) 2506–2512. 45. Fedele, R., Tanjia, F., De Nicola, S., Jovanovi´c, D., and Shukla, P. K., Quantum ring solitons and nonlocal effects in plasma wake field excitations, Phys. Plasmas, 19(10) (2012) 102106. 46. Fedele, R., Tanjia, F., Jovanovic, D., De Nicola, S., and Ronsivalle, C., Wave theories of non-laminar charged particle beams: From quantum to thermal regime, J. Plasma Phys., 80 (2014) 133–145. 47. Fedele, R., Jovanovi, D., De Nicola, S., Mannan, A., and F. Tanjia, F., Self-modulation of a relativistic charged-particle beam as thermal matter wave envelope, J. Phy. Conf. Ser., 482 (2014) 012014. 48. Ferwerda, H. A., Hoenders, B. J., and Slump, C. H., Fully relativistic treatment of electron-optical image formation based on the Dirac equation, Opt. Acta, 33 (1986) 145–157. 49. Ferwerda, H. A., Hoenders, B. J., and Slump, C. H., The fully relativistic foundation of linear transfer theory in electron optics based on the Dirac equation, Opt. Acta, 33 (1986) 159–183. 50. Feshbach, H. and Villars, F., Elementary relativistic wave mechanics of spin 0 and spin 1 2 particles, Rev. Mod. Phys., 30 (1958) 24–45. 51. Fishman, L., de Hoop, M. V., and van Stralen, M. J. N., Exact constructions of squareroot Helmholtz operator symbols: The focusing quadratic profile, J. Math. Phys. 41 (2000) 4881–4938. 52. Fishman, L., One-way wave equation modeling in two-way wave propagation problems, in Mathematical Modeling of Wave Phenomena 2002, Mathematical Modeling in Physics, Engineering, and Cognitive Sciences, Vol. 7, Eds. B. Nilsson and L. Fishman, (V¨axj¨o University Press, V¨axj¨o, Sweden, 2004) pp. 91–111. 53. Foldy, L. L. and Wouthuysen, S. A., On the Dirac theory of spin 12 particles and its non-relativistic limit, Phys. Rev., 78 (1950) 29–36. ´ Berz, M., and Irwin, J., Normal form methods for computational periodic 54. Forest, E., systems, Part. Accel., 24 (1989) 91–97. ´ and Hirata, K., A contemporary guide to beam dynamics, KEK Report 92-12 55. Forest, E. (KEK, Tsukuba, 1992). ´ Beam Dynamics: A New Attitude and Framework, (Taylor & Francis, 56. Forest, E., New York, 1998). 57. Frenkel, J., Die Elektrodynamik des rotierenden Elektrons, Z. Phys., 37(4-5) (1926) 243–262. 58. Frenkel, J., Spinning Electrons, Nature, 117 (1926) 653–654. 59. Glaser, W., Grundlagen der Elecktronenoptik (Springer, New York, 1952). 60. Glaser, W. and Schiske, P., Elektronenoptische Abbildung auf Grund der Wellenmechanik. I, Annalen der Physik, 12 (1953) 240–266. 61. Glaser, W. and Schiske, P., Elektronenoptische Abbildungen auf Grund der Wellenmechanik. II, Annalen der Physik, 12 (1953) 267–280. 62. Glaser, W., Elektronen und Ionenoptik in Handbuch der Physik, Vol. 33, Ed. S. Fl¨ugge, (Springer, Berlin, 1956) 123–395. 63. Goldstein, H., Poole, C. P., and Safko, J. L., Classical Mechanics, (3rd Edition, Addison Wesley, Boston, MA, 2002). 64. Greiner, W., Quantum Mechanics, (4th Edition, Springer, New York, 2001).

342

Bibliography

65. Greiner, W., Relativistic Quantum Mechanics: Wave Equations, (3rd Edition, Springer, New York, 2000). 66. Griffiths, D. J., Introduction to Electrodynamics, (4th Edition, Cambridge University Press, Cambridge, 2017). 67. Griffiths, D. J. and Schroeter, D. F., Introduction to Quantum Mechanics, (3rd Edition, Cambridge University Press, Cambridge, 2018). 68. Groves, T. R., Charged Particle Optics Theory: An Introduction, (Taylor & Francis, Boca Raton, FL, 2015). 69. Hand, I. N. and Skuja, A., in Proceedings of the Conference on Spin and Polarization Dynamics in Nuclear and Particle Physics, Trieste, 1988, Eds. A. O. Barut, Y. Onel, and A. Penzo, (World Scientific, 1990) p.185. 70. Hawkes, P. W. and Kasper, E., Principles of Electron Optics—Vol. 1: Basic Geometrical Optics, (2nd Edition, Elsevier, San Diego, CA, 2017). 71. Hawkes, P. W. and Kasper, E., Principles of Electron Optics—Vol. 2: Applied Geometrical Optics, (2nd Edition, Elsevier, San Diego, CA, 2017). 72. Hawkes, P. W. and Kasper, E., Principles of Electron Optics—Vol. 3: Wave Optics, (3rd Edition, Elsevier, San Diego, CA, 1994). 73. Heifets, S. and Yan, Y. T., Quantum effects in tracking, in Proceedings of the 15th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics— 1998, USA, Ed. P. Chen (World Scientific, 1999), pp. 121–130. 74. Heinemann, K., On Stern–Gerlach force allowed by special relativity and the special case of the classical spinning particle of Derbenev and Kondratenko, arXiv:physics/9611001. 75. Heinemann, K. and Barber, D. P., The semiclassical FoldyWouthuysen transformation and the derivation of the Bloch equation for spin- 12 polarised beams using Wigner functions, in Proceedings of the 15th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics—1998, USA, Ed. P. Chen (World Scientific, 1999), pp. 695–701. 76. Hill, C. T., The diffractive quantum limits of particle colliders, in Proceedings of the 18th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics— 2000, Italy, Ed. P. Chen, (World Scientific, 2002), pp. 3–18. 77. Jackson, J. D., On understanding spin-flip synchrotron radiation and the transverse polarization of electrons in storage rings, Rev. Mod. Phys., 48(3) (1976) 417–434. 78. Jackson, J. D., Classical Electrodynamics, (3rd Edition, Wiley, New York, 1998). 79. Jagannathan, R., Simon, R., Sudarshan, E. C. G., and Mukunda, N. Quantum theory of magnetic electron lenses based on the Dirac equation, Phys. Lett. A, 134 (1989) 457–464. 80. Jagannathan, R., Quantum theory of electron lenses based on the Dirac equation, Phys. Rev. A, 42 (1990) 6674–6689. 81. Jagannathan, R. and Khan, S. A., Wigner functions in charged particle optics, in Selected Topics in Mathematical Physics—Professor R. Vasudevan Memorial Volume, Eds. R. Sridhar, K. Srinivasa Rao, and V. Lakshminarayanan, (Allied Publishers, New Delhi, 1995), pp. 308–321. 82. Jagannathan, R. and Khan, S. A., Quantum theory of the optics of charged particles, in Advances in Imaging and Electron Physics, Vol. 97, Eds. P. W. Hawkes, B. Kazan, and T. Mulvey, (Academic Press, San Diego, CA, 1996), pp. 257–358. 83. Jagannathan, R., The Dirac equation approach to spin- 12 particle beam optics, in Proceedings of the 15th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics—1998, USA, Ed. P. Chen (World Scientific, 1999), pp. 670–681.

Bibliography

343

84. Jagannathan, R., Quantum mechanics of Dirac particle beam optics: Single-particle theory, in Proceedings of the 18th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics—2000, Italy, Ed. P. Chen, (World Scientific, 2002), pp. 568–577. 85. Jagannathan, R., Quantum mechanics of Dirac particle beam transport through optical elements with straight and curved axes, in Proceedings of the 28th ICFA Advanced Beam Dynamics and Advanced & Novel Accelerators Workshop—2003, Japan, Eds. P. Chen and K. Reil, (World Scientific, 2003), pp. 13–21. 86. Jagannathan, R., On generalized clifford algebras and their physical applications, in The Legacy of Alladi Ramakrishnan in the Mathematical Sciences, Eds. K. Alladi, J. Klauder, and C. R. Rao (Springer, New York, 2010) pp. 465–489. 87. Jiang, Y., Chen, Z., Han, Y., Deb, P., Gao, H., Xie, S., Purohit, P., Tate, M. W., Park, J., Gruner, S. M., Elser, V., and Muller, D. A., Electron ptychography of 2D materials to ˚ deep sub-Angstr¨ om resolution, Nature, 559 (2018) 343–349. 88. Johnson, M. H. and Lippmann, B. A., Motion in a constant magnetic field, Phys. Rev. 76 (1949) 828–832. 89. Kabel, A. C., Quantum ground state and minimum emittance of a fermionic particle beam in a circular accelerator, in Proceedings of the 18th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics—2000, Italy, Ed. P. Chen, (World Scientific, 2002), pp. 67–75. 90. Khan, S. A. and Jagannathan, R., Quantum mechanics of charged-particle beam transport through magnetic lenses, Phys. Rev. E 51(3) (1995) 2510–2515. 91. Khan, S. A., Quantum Theory of Charged-Particle Beam Optics, Ph.D Thesis, (University of Madras, Chennai, India, 1997). https://imsc.res.in/xmlui/handle/123456789/75 92. Khan, S. A., Quantum theory of magnetic quadrupole lenses for spin- 21 particles, in Proceedings of the 15th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics - 1998, USA, Ed. P. Chen (World Scientific, 1999), pp. 682–694. 93. Khan, S. A., Quantum aspects of accelerator optics, in Proceedings of the 1999 Particle Accelerator Conference (PAC99), March 1999, New York, Eds. A. Luccio and W. MacKay, (1999), pp. 2817–2819. 94. Khan, S. A., and Pusterla, M., Quantum mechanical aspects of the halo puzzle, in Proceedings of the 1999 Particle Accelerator Conference (PAC99), March-April, 1999, New York, Eds. A. Luccio and W. MacKay, pp. 3280–3281. 95. Khan, S. A., and Pusterla, M., On the form of Lorentz-Stern-Gerlach force, arXiv:physics/9910034 [physics.class-ph] (1999). 96. Khan, S. A. and Pusterla, M., Quantum-like approach to the transversal and longitudinal beam dynamics: The halo problem, Eur. Phys. J. A, 7 (2000) 583–587. 97. Khan, S. A., and Pusterla, M., Quantum-like approaches to the beam halo problem, in Proceedings of the 6th International Conference on Squeezed States and Uncertainty Relations (ICSSUR’99), May 1999, Napoli, Italy, Eds. D. Han, Y.S Kim, and S. Solimeno, NASA Conference Publication Series, 2000-209899 (2000) pp. 438–441. 98. Khan, S. A., and Pusterla, M., Quantum approach to the halo formation in high current beams, Nucl. Instrum. Methods Phys. Res. Sect. A, 464 (2001) 461–464. 99. Khan, S. A., Quantum formalism of beam optics, in Proceedings of the 18th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics—2000, Italy, Ed. P. Chen, (World Scientific, 2002), pp. 517–526. 100. Khan, S. A., Jagannathan, R., and Simon, R., Foldy–Wouthuysen transformation and a quasiparaxial approximation scheme for the scalar wave theory of light beams, arXiv:physics/0209082 [physics.optics], (2002).

344

Bibliography

101. Khan, S. A., Wavelength-dependent modifications in Helmholtz optics, Int. J. Theor. Phys., 44 (2005) 95–125. 102. Khan, S. A., An exact matrix representation of Maxwell’s equations, Phys. Scr., 71(5) (2005) 440–442. 103. Khan, S. A., The Foldy–Wouthuysen transformation technique in optics, Optik, 117(10) (2006) 481–488. 104. Khan, S. A., Wavelength-dependent effects in light optics, in New Topics in Quantum Physics Research, Eds. V. Krasnoholovets and F. Columbus, (Nova Science Publishers, New York, 2006), pp. 163–204. 105. Khan, S. A., The Foldy–Wouthuysen transformation technique in optics, in Advances in Imaging and Electron Physics, Vol. 152, Ed. P. W. Hawkes, (Academic Press, San Diego, CA, 2008) pp. 49–78. 106. Khan, S. A., Maxwell optics of quasiparaxial beams, Optik, 121 (2010) 408–416. 107. Khan, S. A., Aberrations in Maxwell optics, Optik, 125 (2014) 968–978, 108. Khan, S. A., International Year of Light and Light-based Technologies, (LAP LAMBERT Academic Publishing, Germany, 2015). 109. Khan, S. A., Passage from scalar to vector optics and the Mukunda–Simon–Sudarshan theory for paraxial systems, J. Mod. Opt., 63 (2016) 1652–1660. 110. Khan, S. A., Quantum aspects of charged-particle beam optics, in: Proceedings of the Fifth Saudi International Meeting on Frontiers of Physics 2016 (SIMFP 2016), February 2016, Gizan, Saudi Arabia, Eds. A. Al-Kamli, N. Can, G. O. Souadi, M. Fadhali, A. Mahdy and M. Mahgoub, AIP Conference Proceedings, 1742 (2016) pp. 030008-1– 030008-4. 111. Khan, S. A., Quantum methodologies in Helmholtz optics, Optik, 127 (2016) 9798–9809. 112. Khan, S. A, Quantum methods in light beam optics, Opt. Photonics News, 27 (2016) 47. 113. Khan, S. A., International year of light and history of optics, in Advances in Photonics Engineering, Nanophotonics and Biophotonics, Ed. T. Scott, (Nova Science Publishers, New York, 2016), pp. 1–56. 114. Khan, S. A., Quantum methodologies in Maxwell optics, in Advances in Imaging and Electron Physics, Vol. 201, Ed. P. W. Hawkes, (Academic Press, New York, 2017) pp. 57–135. 115. Khan, S. A., Hamilton’s optical-mechanical analogy in the wavelength-dependent regime, Optik, 130 (2017) 714–722. 116. Khan, S. A., Linearization of wave equations, Optik, 131C (2017) 350–363. 117. Khan, S. A., Polarization in Maxwell optics, Optik, 131 (2017) 733–748. 118. Khan, S. A., Aberrations in Helmholtz optics, Optik, 153C (2018) 164–181. 119. Khan, S. A., Quantum mechanical techniques in light optics, in Proceedings of the Sixth Saudi International Meeting on Frontiers of Physics 2018 (SIMFP 2018), February– March 2018, Gizan, Saudi Arabia, Eds. Al-Kamli, A., Souadi, G. O., Hakami, J., Can, N., Mahdy, A., Mahgoub, M., and Mujahid, Z., AIP Conference Proceedings, 1976 (American Institute of Physics, 2018) 020016. 120. Khan, S. A., E. C. G. Sudarshan and the quantum mechanics of charged-particle beam optics, Curr. Sci., 115(9) (2018) 1813–1814. 121. Khandekar, D. C. and Lawande, S. V., Path Integral Methods and Their Applications, (World Scientific, Singapore, 1993). 122. Kleinert, H., Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, (5th Edition, World Scientific, Singapore, 2009).

Bibliography

345

123. Kruit, P., Hobbs, R. G., Kim, C.-S., Yang, Y., Manfrinato, V. R., Hammer, J., Thomas, S., Weber, P., Klopfer, B., Kohstall, C., Juffmann, T., Kasevich, M. A., Hommelhoff, P., and Berggren, K. K., Designs for a quantum electron microscope, Ultramicroscopy, 164 (2016) 31–45. 124. Lakshminarayanan, V., Sridhar, R., and Jagannathan, R., Lie algebraic treatment of dioptric power and optical aberrations, J. Opt. Soc. Am. A, 15(9) (1998) 2497–2503. 125. Lakshminarayanan, L., Ghatak, A., and Thyagarajan, K., Lagrangian Optics, (Springer, New York, 2002). 126. Lakshminarayanan, V. and Varadharajan, L. S., Special Functions for Optical Science and Engineering, (SPIE: The International Society for Optics and Photonics, 2015). 127. Lee, S. Y., Accelerator Physics, (2nd Edition, World Scientific, Singapore, 2004) 128. Li˜nares, J., Optical propagators in vector and spinor theories by path integral formalism, in Lectures on Path Integration:Trieste, Ed. A. Cerdeira, et al., (World Scientific, Singapore, 1993) 378–397. 129. Lubk, A., Paraxial quantum mechanics, in Advances in Imaging and Electron Physics, Vol. 206, Ed. P. W. Hawkes, (Academic Press, London, 2018). 130. Magnus, W., On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math., 7 (1954) 649–673. 131. Mais, H., Nonlinear Problems in Accelerator Physics, (DESY Report, 93-058). 132. Manfrinato, V. R., Stein, A., Zhang, L., Nam, C.-Y., Yager, K. G., Stach, E. A., and Black, C. T., Aberration-corrected electron beam lithography at the one nanometer length scale, Nano Lett. 17 (2017) 4562–4567. 133. Misra, B. and Sudarshan, E. C. G., The Zeno’s paradox in quantum theory, J. Math. Phys., 18 (1977) 756–763. 134. Mitra, A. N. and Ramanathan, R., On simulating Liouvillian flow from quantum mechanics via Wigner functions, J. Math. Phys., 39(9) (1998) 4492–4498. 135. Mondrag´on, J. S. and Wolf, K. B., Lie Methods in Optics: Proceedings of the Workshop—1985, Mexico, Eds. J. S. Mondrag´on and K. B. Wolf (Springer, New York, 1986). 136. Montague, B. W., Polarized beams in high energy storage rings, Phys. Rep., 113(1) (1984) 1–96. 137. Mukunda, N., Simon, R., and Sudarshan, E. C. G., Paraxial-wave optics and relativistic front description II: The vector theory, Phys. Rev. A, 28 (1983) 2933–2942. 138. Mukunda, N., Simon, R., and Sudarshan, E. C. G., Fourier optics for the Maxwell field: Formalism and applications, J. Opt. Soc. Am. A, 2 (1985) 416–426. 139. Newton, T. D. and Wigner, E. P., Localized states for elementary systems, Rev. Mod. Phys., 21(3) (1949) 400–405. 140. Okamoto, H. and Nagatani, Y., Entanglement-assisted electron microscopy based on a flux qubit, Appl. Phys. Lett., 104 (2014) 062604. 141. Orloff, J., (Ed.), Handbook of Charged Particle Optics, (2nd Edition, Taylor & Francis, NewYork, 2009) 142. Parthasarathy, R., Relativistic Quantum Mechanics, (Narosa (India), 2011). 143. Petroni, N. C., De Martino, S., De Siena, S., and Illuminati, F., Stochastic collective dynamics of charged-particle beams in the stability regime,Phys. Rev. E, 63 (2000) 016501. 144. Phan-Van-Loc, Sur le principe de Huygens en Th´eorie de l’´electron de Dirac, Comptes rendus de l’Acad´emie des Sciences, Paris, 237 (1953) 649–651. 145. Phan-Van-Loc, Principe de Huygenes Th´eorie de l’´electron de Dirac et int´egrales de contour, Comptes rendus de l’Acad´emie des Sciences, Paris, 238 (1954) 2494–2496.

346

Bibliography

146. Phan-Van-Loc, Diffractions des ondes Ψn de l’´electron de Dirac, Ann. Fac. Sci. Univ. Toulouse, 18 (1955) 178–192. 147. Phan-Van-Loc, Interpr´etation physique de I’expression math´ematique du principe de Huygens en th´eorie de I’´electron de Dirac, Cahiers Physique, No. 97 12 (1958) 327–340. 148. Une nouvelle mani`ere d’etablir l’expression math´ematique du principe de Huygens en th´eorie de l’electron de Dirac, C. R. Acad. Sci. Paris, 246 (1958) 388–390. 149. Phan-Van-Loc, Principes de Huygens en theorie de l’electron de Dirac, (Thesis, Toulouse, 1960). 150. Pozzi, G., Particles and waves in electron optics and microscopy, in Advances in Imaging and Electron Physics, Vol. 194, Ed. P. W. Hawkes, (Academic Press, New York, 2016) pp. 1–234. 151. Pryce, M. H. L., The mass-centre in the restricted theory of relativity and its connexion with the quantum theory of elementary particles, Proc. Roy. Soc. Ser. A, 195 (1948) 62–81. 152. Pusterla, M., Quantum-like approach to beam dynamics: Application to the LHC and HIDIF projects, in Proceedings of the 18th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics—2000, Italy, Ed. P. Chen, (World Scientific, Singapore, 2002), pp. 561–567. 153. Putnam, W. P. and Yanik, M. F., Noninvasive electron microscopy with interaction-free quantum measurements, Phys. Rev. A 80 (2009) 040902. 154. Radli˘cka, T., Lie algebraic methods in charged particle optics, in Advances in Imaging and Electron Physics, Vol. 151, Ed. P.W. Hawkes, (Academic Press, New York, 2008) pp. 241–362. 155. Ramakrishnan, A., L-Matrix Theory or the Grammar of Dirac Matrices, (Tata-McGraw Hill, New Delhi, 1972). 156. Rangarajan, G., Dragt, A. J., and Neri, F., Solvable map representation of a nonlinear symplectic map, Part. Accel., 28 (1990) 119–124. 157. Rangarajan, G. and Sachidanand, M., Spherical aberrations and its correction using Lie algebraic methods, Pramana, 49(6) (1997) 635–643. 158. Reiser, M., Theory and Design of Charged Particle Beams, (2nd Edition, Wiley, New York, 2008). 159. Rosenzweig, J. B., Fundamentals of Beam Physics, (Oxford University Press, Oxford, 2003). 160. Rougemaille, N. and Schmid, A., Magnetic imaging with spin-polarized low-energy electron microscopy, Europ. Phys. J.: Appl. Phys., 50(2) (2010) 20101. ¨ 161. Rubinowicz, A., Uber das Kirchhoffsche Beugungsproblem f¨ur Elektronenwellen, Acta. Phys. Polon., 3 (1934) 143–163. 162. Rubinowicz, A., Die Beugungswelle in der Kirchhoffschen Theorie der Beugung, Acta. Phys. Polon., 23 (1963) 727–744. 163. Rubinowicz, A., The Miyamoto–Wolf diffraction wave, Progress in Optics, Vol. 4, Ed. E. Wolf, (Elsevier, 1965) pp. 199–240. 164. Rubinowicz, A., Die Beugungswelle in der Kirchhoffschen Theorie der Beugung, (2nd Edition, Panstwowe Wydawnictwo Naukowe, Warsaw; Springer, Berlin, 1966). 165. Ryne, R. D. and Dragt, A. J., Magnetic optics calculations for cylindrically symmetric beams, Part. Accel., 35 (1991) 129–165. 166. Sakurai, J. J. and Napolitano, J., Modern Quantum Mechanics, (2nd Edition, Cambridge University Press, Cambridge, 2017).

Bibliography

347

¨ 167. Scherzer, O., Uber einige Fehler von Elektronenlinsen, Z. Physik, 101(9-10) (1936) 593–603. 168. Seryi, A., Unifying Physics of Accelerators, Lasers, and Plasma, (Taylor & Francis, Boca Raton, FL, 2016). 169. Shankar, R., Principles of Quantum Mechanics, (2nd Edition, Springer, New York, 1994). 170. Simon, R., Sudarshan, E. C. G., and Mukunda, N., Gaussian-Maxwell beams, J. Opt. Soc. Am. A 3 (1986) 536–540. 171. Simon, R., Sudarshan, E. C. G., and Mukunda, N., Cross polarization in laser beams, Appl. Opt., 26 (1987) 1589–1593. 172. Singh, A. and Carroll, S. M., Modeling position and momentum in finite-dimensional Hilbert spaces via generalized Clifford algebra, arXiv:1806.10134 [quant-ph] 173. Sokolov, A. A. and Ternov, I. M., Radiation from Relativistic Electrons, Translated by S. Chomet, Ed. C. W. Kilmister, (American Institute of Physics, New York, 1986). 174. Sudarshan, E. C. G. and Mukunda, N., Classical Dynamics: A Modern Perspective, (Wiley, New York, 1974). 175. Sudarshan, E. C. G., Simon, R., and Mukunda, N., Paraxial-wave optics and relativistic front description I: The scalar theory, Phys. Rev. A, 28 (1983) 2921–2932. 176. Syphers, M. J. and Chattopadhyay, S., Landscape of future accelerators at the energy and intensity frontiers, in Proceedings of the 38th International Conference on High Energy Physics, USA, 2016, Proceedings of Science: PoS(ICHEP2016) 043. 177. Tani, S., Connection between particle models and field theories. I. The case spin 12 , Prog. Theor. Phys. 6 (1951) 267–285. 178. Tekumalla, A. R., A class of transformations on the Dirac Hamiltonian, Physica 78 (1974) 191–197. 179. Tekumalla, A. R. and Santhanam, T. S., Unitary Foldy–Wouthuysen transformations for particles of arbitrary spin, Lett. Nuovo Cimento 10 (1974) 737–740. 180. Ternov, I. M., Evolution equation for the spin of a relativistic electron in the Heisenberg representation, Sov. Phys. JETP, 71(4) (1990) 654–656. 181. Ternov, I. M., Synchrotron radiation, Phys. Uspekhi, 38(4) (1995) 409–434. 182. Thomas, L. H., The motion of the spinning electron, Nature, 117 (1926) 514–514. 183. Thomas, L. H., The kinematics of an electron with an axis, Phil. Mag., 3 (1927) 1–22. 184. Todesco, E., Overview of single particle nonlinear dynamics, (CERN-LHC-99-001). AIP Conference Proceedings 468 (1999), pp. 157–172. 185. Venturini, M. and Ruth, R. D., Single-particle quantum dynamics in a magnetic lattice, in Proceedings of the 18th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics—2000, Italy, Ed. P. Chen, (World Scientific, Singapore, 2002), pp. 551–560. 186. Wei, J., Li, X. P., and Sessler, A. M., Low energy states of circulating stored ion beams: Crystalline beams, Phys. Rev. Lett., 73 (1994) 3089–3092. 187. Weidemann, H., Particle Accelerator Physics, (4th Edition, Springer, Berlin, 2015). 188. Wilcox, R. M., Exponential operators and parameter differentiation in quantum physics, J. Math. Phys., 8 (1967) 962–982. 189. Wolf, K. B., Integral Transforms in Science and Engineering, (Plenum, New York, 1979). 190. Wolf, K. B., On time-dependent quadratic quantum hamiltonians, SIAM J. Appl. Math., 40(3) (1981) 419–431.

348

Bibliography

191. Wolf, K. B., Lie Methods in Optics—II: Proceedings of the Second Workshop—1988, Mexico, Ed. K. B. Wolf (Springer, 1989). 192. Wolski, A., Beam Dynamics in High Energy Particle Accelerators, (Imperial College Press, London, 2014). 193. Wu, C. S., Makiuchi, Y., and Chen, C., High energy electron beam lithography for nanoscale fabrication, in Lithography, Ed. M. Wang, (InTech Publ., 2010) pp. 241–266. 194. Ximen, J., Canonical theory in electron optics, in Advances in Electronics and Electron Physics, Vol. 81, Ed. P. W. Hawkes, (Academic Press, London, 1991) pp. 231–277.

Index A Aberrations coefficients, 7, 258, 262, 277 effect of quantum uncertainties on, 277–281 paraxial approximation, 250–259 Anomalous magnetic moment beam optics of the Dirac particle general formalism, 316–320 Lorentz and Stern–Gerlach forces, 320–322 phase space and spin transfer maps, 323–330 Thomas–Frenkel–BMT equation, 320–322 Anti-Hermitian term, 46, 222, 260, 261, 263, 267 Axially symmetric electrostatic lens, 204–205, 269–270 Axially symmetric magnetic lens, 178–197, 300–306 aberrations, 250–259 point-to-point imaging, 232–250 quantum corrections, 260–262

B Baker–Campbell–Hausdorff (BCH) formula, 112, 303 Beam optical Hamiltonian classical, 5, 7, 174, 179, 197, 202, 204, 205, 207–211, 276, 293, 315 quantum, 5–8, 172, 223, 230, 233, 250, 262, 263, 266, 268–271, 276, 279, 281, 292, 306, 315, 331, 332 Beam optics anomalous magnetic moment general formalism, 316–320

Lorentz and Stern–Gerlach forces, 320–322 phase space and spin transfer maps, 323–330 Thomas–Frenkel–BMT equation, 320–322 Bending magnet, 206–210, 271–277, 306–315 Bohr magneton, 117 Bound states, 94, 110 Busch formula, 187, 248

C Canonical transformations electromagnetic optical element, 29–30 symplecticity Poisson bracket, 35–36 time-dependent, 32–35 time-independent, 31–32 theory, 25–29 Cauchy–Schwarz inequality, 83 Christoffel symbols, 310 Classical charged particle beam optics curved optic axis, optical elements, 206–210 free propagation, 174–178 nonrelativistic, 210–212 relativistic, 173–174 straight optic axis, optical elements axially symmetric electrostatic lens, 204–205 axially symmetric magnetic lens, 178–197 electrostatic quadrupole, 205–206 normal magnetic quadrupole, 197–201 skew magnetic quadrupole, 202–204 349

350

Classical Hamiltonian, 51, 78, 90, 139, 213 Constant magnetic field charged particle in, 18–20, 22–25 Dirac equation, 154–156 Klein-Gordon particle, charged, 139–141 quantum mechanics, 103–106 Curved optic axis, optical elements, 206–210 Cyclic property, 127 Cyclotron frequency, 20, 104–106

D de Broglie wavelength, 84, 87, 231, 277, 297, 299, 305 Density operator, 126–132 Derbenev–Kondratenko semiclassical Hamiltonian, 322 Diffraction Dirac-Pauli equation, 293–300 Klein–Gordon equation, 230–232 Dipole magnet, 6, 173, 206, 208–210, 271, 274, 276, 277, 306, 311, 315 Dirac delta function, 43, 48 Dirac equation constant magnetic field, 154–156 Dirac-Pauli equation, 153–154 Foldy–Wouthuysen transformation, 156–167 free-particle equation, 141–148 spin and helicity, 150–153 spin magnetic moment, 153–154 spin-1/2 particle, 217 Zitterbewegung, 149–150 Dirac Hamiltonian, 4, 8, 143, 150, 152, 154, 158, 160, 162, 165, 168, 217, 218, 293, 295, 297, 298, 305, 306, 315, 316, 319

Index

Dirac–Pauli equation, 4, 7, 8, 153–154 quantum charged particle beam optics, 287–292 axially symmetric magnetic lens, 300–306 bending magnet, 306–315 diffraction, 293–300 general formalism, 316–320 Lorentz and Stern–Gerlach forces, 320–322 normal magnetic quadrupole, 330–333 phase space and spin transfer maps, 323–330 Thomas–Frenkel–BMT equation, 320–322 Dirac probability current density, 145 Dirac probability density, 145 Dirac’s classical–quantum correspondence rule, 68, 227 Dirac spinor, 144, 145, 154, 156, 162, 287, 288, 291, 292, 295, 305, 308 Dynamics constant magnetic field, 18–20, 22–25 system of particles, 36–39 Dyson time-ordered exponential, 114

E Ehrenfest’s theorem, 4, 68–70, 87, 88, 224, 232 Electromagnetic field, 173–174 Hamiltonian formalism, 14–15 Lagrangian formalism, 10–12 Electromagnetic optical element, canonical transformations, 29–30 Electron in constant magnetic field, 154–156 Schr¨odinger–Pauli equation for, 117–119

351

Index

spin magnetic moment of, 153–154 Electron microscopy axially symmetric magnetic lens, 178–197 effect of quantum uncertainties, 277–281 Electron optical imaging aberrations, 250–259 point-to-point imaging, 232–250 quantum corrections, 260–262 Electron optical technology, 201 Electrostatic quadrupole lens, 205–206, 270–271 Euler–Lagrange equations, 10, 36 Extended canonical transformation, 32, 35

F Fermi–Dirac statistics, 77 Feshbach–Villars formalism, 4, 5, 215, 223, 281, 335 Feshbach–Villars representation, 137–139, 168–170 Feynman path integral method, 237 F(ocusing)O(nonfocusing)D(efocusing) O(nonfocusing) (FODO-channel), 201, 265 Foldy–Wouthuysen representation, 4, 165–167, 169, 170, 296, 316 Foldy–Wouthuysen theory, 218 Foldy–Wouthuysen transformation technique, 4–7, 217, 218, 223, 275, 281, 293–295, 301, 313–314, 317, 322, 335 Dirac equation, 156–167 Klein–Gordon equation, Feshbach–Villars form of, 168–170 Free-particle equation, 78–90, 141–148 Free propagation, 174–178 Dirac–Pauli equation, 293–300

quantum charged particle beam optics, 230–232 Frequency of gyration, 20 Fresnel diffraction formula, 231, 299

G Gaussian function, 84, 94 Green’s function, 67

H Hamiltonian formalism in electromagnetic field, 14–15 optical, 29–30 Poisson brackets in constant magnetic field, 18–20 theory, 16–18 theory, 12–13 time-dependent, 32–35, 114–115, 283–285 Hamiltonian operator, 51, 78, 92, 213, 223, 263, 270 Hamilton’s stationary action, 9 h¯ −dependent anti-Hermitian terms, 260, 261, 263, 267 h¯ −dependent Hermitian terms, 251, 260, 261, 266, 267 Heisenberg canonical commutation relation, 45, 47 commutation relation, 47 equation of motion, 4, 67, 68, 90, 150, 229 operator, 67 picture, 4, 67, 68, 81–82, 115, 116, 127, 130, 149, 165, 166, 170, 172, 224, 226, 296 uncertainty principle, 52, 84, 86–89, 98, 279 Helicity operator, 152 Helicity, spin and, 150–153 Helmholtz scalar optics, 335 Hermite polynomials, 93–94

352

Hermitian conjugate, 42, 144, 149, 160 Hermitian Hamiltonian, 139, 222 Hermitian operator, 4, 42, 43, 46, 47, 49, 52, 57–59, 67, 70, 73, 76, 82, 83, 95, 115, 126, 220 Hermiticity, 44, 45, 55, 144, 159, 165

I Identical particles system, 119–125 Independent variable in constant magnetic field, 22–25 theory, 21–22 Interaction picture, 115–116

J Jacobian matrix, 31–35, 38 Jacobi identity, 17, 46 Jeffreys–Wentzel–Kramers–Brillouin (JWKB) approximation technique, 111

K Kinetic energy, 211 Klein–Gordon equation, 6, 7, 84, 155, 213, 333 aberrations, 250–259 constant magnetic field, 139–141 diffraction, 230–232 Feshbach–Villars representation, 137–139, 168–170 formalism, 214–230 free particle, 271–273 free-particle equation and difficulties, 132–137 point-to-point imaging, 232–250 quantum corrections, 260–262 time-independent, 273–274 Klein–Gordon–Fock equation, 133 Klein–Gordon theory, 305, 306

Index

L Ladder operators, 95 Lagrangian formalism in electromagnetic field, 10–12 theory, 9–10 Laguerre polynomial, 102 Landau levels, 106 Land´e g-factor, 117 Larmor radius, 20 Lie operator, 17 Lie transfer operator method, 3, 18, 189, 197, 230, 251 Linear differential equation, 170–172 Linear harmonic oscillator, 90–100 Lorentz force, 320–322

M Magnetic quadrupole, 5, 8, 174 normal, 197–201, 262–267 phase space and spin transfer maps, 323–327 skew, 202–204, 267–269 phase space and spin transfer maps, 327–330 Magnetic rigidity, 209 Magnus formula, 114, 191, 283 linear differential equation, 170–172 Maxwell vector optics, 335 Minimal coupling principle, 273, 288, 310 Mixed state density operator, 126–132

N Negative-energy states, 153 Newton–Wigner position operator, 296 Nodes, 94 Non-Hermitian terms, 138, 222, 297, 298, 315 Nonrelativistic classical charged particle beam optics, 210–212

353

Index

Nonrelativistic probability current density, 134, 135 Nonrelativistic probability density, 134, 135 Nonrelativistic quantum charged particle beam optics spin-0 and spinless particles, 281–282 spin-1/2 particles, 330–333 Nonrelativistic Schr¨odinger equation, 103 Nonrelativistic single particle quantum mechanics constant magnetic field, 103–106 free particle, 78–90 interaction picture, 115–116 JWKB approximation technique, 111 linear harmonic oscillator, 90–100 scattering states, 107–111 Schr¨odinger–Pauli equation, 117–119 time-dependent systems, 111–116 two-dimensional isotropic harmonic oscillator, 100–103 Normal magnetic quadrupole, 197–201, 262–267, 323–327 Number operator, 96

O Optical Hamiltonian, 3, 5–8, 29–30, 34, 172, 174, 179, 189, 197, 200, 202, 204–213, 220, 223, 230, 233, 234, 238, 250, 251, 260, 262, 263, 266–271, 276, 279, 281, 292, 293, 299, 306, 315, 320–323, 328, 331, 332 Orbital angular momentum quantum number, 75–76

Oscillator linear harmonic, 90–100 two-dimensional isotropic harmonic, 100–103 Oscillatory motion, 91–92, 150

P Paraxial approximation, 232–259 Paraxial beam, 1, 174, 177, 178, 180, 184, 188, 189, 197, 202, 204, 230, 231, 233, 234, 245, 250, 268, 299, 316, 323, 327, 328 Pauli–Dirac equation, 154 Pauli equation, 117–119 Pauli exclusion principle, 77, 335 Phase space and spin transfer maps normal magnetic quadrupole, 323–327 skew magnetic quadrupole, 327–330 Point-to-point imaging, 232–250 Poisson brackets canonical transformations, 35–36 Hamiltonian formalism in constant magnetic field, 18–20 theory, 16–18 Position probability density, 42, 84, 92 Positive operator, 95 Potential energy, 90, 91, 104, 110, 154, 211 Propagator, 66, 89, 90, 99, 231, 237, 283–285, 299 Pryce–Tani–Foldy–Wouthuysen transformation, 157 Ptychographic analysis, 336 Pure state density operator, 126–132

Q Quantum corrections, 7, 224, 260–262 dynamics, 4, 42, 51–68, 214

354

Quantum (cont.) electrodynamics, 117 kinematics, 42–50 uncertainties, 277–281, 336 Quantum charged particle beam optics axially symmetric electrostatic lens, 269–270 bending magnet, 271–277 electron microscopy and nonlinearities, 277–281 electrostatic quadrupole lens, 270–271 formalism, 213–214 Klein–Gordon Equation aberrations, 250–259 diffraction, 230–232 formalism, 214–230 point-to-point imaging, 232–250 quantum corrections, 260–262 normal magnetic quadrupole, 262–267 skew magnetic quadrupole, 267–269 spin-0 and spinless particles, 281–282 time-dependent quadratic Hamiltonian, 283–285 Quantum mechanics density operator, 126–132 nonrelativistic single particle constant magnetic field, 103–106 free particle, 78–90 interaction picture, 115–116 JWKB approximation technique, 111 linear harmonic oscillator, 90–100 scattering states, 107–111 Schr¨odinger–Pauli equation, 117–119

Index

time-dependent systems, 111–116 two-dimensional isotropic harmonic oscillator, 100–103 relativistic (see relativistic quantum mechanics) single particle Ehrenfest’s theorem, 68–70 quantum dynamics, 51–68 quantum kinematics, 42–50 spin, 70–77 system of identical particles, 119–125 Quasiclassical Hamiltonian, 316

R Ray coordinates, 178, 277 Reference particle, 1, 2, 30, 206, 271, 306, 336 Relativistic classical charged particle beam optics, 173–174 Relativistic mass, 2, 5, 212, 282, 333 Relativistic quantum charged particle beam optics Dirac–Pauli equation, 287–292 axially symmetric magnetic lens, 300–306 bending magnet, 306–315 diffraction, 293–300 general formalism, 316–320 Lorentz and Stern–Gerlach forces, 320–322 normal magnetic quadrupole, 330–333 phase space and spin transfer maps, 323–330 Thomas–Frenkel–BMT equation, 320–322 Relativistic quantum mechanics, 137 Dirac equation constant magnetic field, 154–156 Dirac–Pauli equation, 153–154

Index

free-particle equation, 141–148 spin and helicity, 150–153 spin magnetic moment, 153–154 Zitterbewegung, 149–150 Foldy–Wouthuysen transformation Dirac equation, 156–167 Feshbach–Villars form of the Klein–Gordon equation, 168–170 Klein–Gordon equation constant magnetic field, 139–141 Feshbach–Villars representation, 137–139 free-particle equation and difficulties, 132–137 Rest energy, 12, 14, 78, 133, 136, 141, 154, 161, 165, 211 Right handed, 10, 153 Round electrostatic lens, 204–205 Round magnetic lenses, 186–187, 247–248

S Scanning tunneling microscopy, 110, 336 Scattering states, 107–111, 110 Schr¨odinger equation, 4, 7, 51, 231 energy eigenvalues of the system, 107, 110 nonrelativistic, 2–4, 6, 7, 84, 103, 132, 136, 213–214, 231, 281, 287, 296, 331 time-dependent, 4, 81, 93, 94, 183, 223, 244 Schr¨odinger–Pauli equation, 117–119, 130, 154, 156, 330 Schwarz inequality, 83 Semigroup property, 65, 191, 198, 224, 263, 325 S-evolution equation, 276

355

Sextupoles, 279 Single particle quantum mechanics Ehrenfest’s theorem, 68–70 quantum dynamics, 51–68 quantum kinematics, 42–50 spin, 70–77 Skew magnetic quadrupole, 202–204, 267–269, 327–330 Slater determinant, 125 Spin, 70–77 and helicity, 150–153 Spin-0 and spinless particles, 281–282 Spin magnetic moment, 153–154 Spinor theory for spin-1/2 particles Dirac–Pauli equation, quantum charged particle beam optics, 287–292 axially symmetric magnetic lens, 300–306 bending magnet, 306–315 diffraction, 293–300 general formalism, 316–320 Lorentz and Stern–Gerlach forces, 320–322 normal magnetic quadrupole, 330–333 phase space and spin transfer maps, 323–330 Thomas–Frenkel–BMT equation, 320–322 nonrelativistic quantum charged particle beam optics, 330–333 Stern–Gerlach effect, 316, 327 Stern–Gerlach force, 320–322 Straight optic axis, optical elements axially symmetric electrostatic lens, 204–205 axially symmetric magnetic lens, 178–197 electrostatic quadrupole, 205–206

356

Straight optic axis, optical elements (cont.) normal magnetic quadrupole, 197–201 skew magnetic quadrupole, 202–204 Symplectic matrix, 32, 34 System of identical particles, 119–125 System of particles, dynamics, 36–39

T Taylor series expansion, 61 Thomas–BMT equation, 316, 322, 327 Thomas–Frenkel–BMT equation (Thomas–Frenkel– Bargmann–Michel–Telegdi equation), 316, 320–322 Time-dependent canonical transformations, 32–35 Dirac equation, 127, 144, 153 perturbation theory, 4, 114, 116, 250–252 quadratic Hamiltonia, 283–285 systems, 111–116 Time-independent canonical transformations, 31–32 Dirac equation, 146, 154–155 free particle, 149

Index

Hamiltonian, 65, 81, 93, 111, 123 perturbation theory, 111, 124 Schr¨odinger equation, 92, 93, 101–104, 107, 124, 281, 330–331 Two-component spinor theory, 4, 8, 117, 165, 293, 319 Two-dimensional isotropic harmonic oscillator, 100–105, 237

U Uncertainty principle, 4, 52, 82, 84, 86–89, 98, 279 Unitary basis, 53, 55, 58 Unitary matrix, 54, 56, 59

V Variance, 82 von Neumann equation, 126, 129

W Wave function, 6, 41, 43–44, 47–51, 57, 66, 79, 81, 83–85, 92, 94, 98, 106, 109, 110, 125, 130, 131, 215, 220, 223, 225, 249, 276, 289, 319, 330, 335 Wigner function, 336, 337

Z Zitterbewegung phenomenon, 149–150, 165–167, 296