Discrete Quantum Mechanics 9781681741253, 9781681740614, 9781681742533

Table of contents :
Preface
Acknowledgments
Author biography
H Thomas Williams
ch1.pdf
Chapter 1 Postulates
1.1 State space
1.2 Time evolution
1.3 Quantum measurement
1.4 Composite systems
1.5 The genie
Exercises
Bibliography
ch2.pdf
Chapter 2 Two-state systems
2.1 Schrödinger’s cat
2.2 Expectation value; energy operator
2.3 Spin one-half
2.3.1 Spin orientation measurements
2.3.2 Quantum mathematics of spin one-half
2.3.3 Spin precession
2.3.4 Heisenberg picture
2.4 Ammonia molecule
2.5 Photons
ch3.pdf
Chapter 3 Entanglement
3.1 Entangled qubit pairs
3.2 Quantum gates
3.3 Utilizing entanglement
3.3.1 Superdense coding
3.3.2 Teleportation
3.3.3 Bell's inequality
3.4 Large-scale quantum algorithms
3.4.1 Classical algorithm for Simon's problem
3.4.2 Quantum algorithm for Simon's problem
Exercises
Bibliography
ch4.pdf
Chapter 4 Quantum angular momentum
4.1 Operators for orbital angular momentum
4.2 Operators for spin one-half
4.3 Generalized angular momentum theory
4.4 Angular momentum addition
4.5 Interaction operators
4.6 Isospin
4.7 And…
Exercises
Bibliography
ch5.pdf
Chapter 5 Quantum many-body problem
5.1 The general case: Heisenberg XYZ spin chain
5.2 Ising model
5.3 Heisenberg XXX spin chain
5.3.1 N = 4 case: diagonalization
5.3.2 N=4 case: Bethe ansatz
Exercises
Bibliography
ch6.pdf
Chapter 6 Infinity, and beyond
6.1 Schrödinger equation
6.2 Schrödinger equation in one dimension
6.2.1 Free particle
6.2.2 Infinitely deep square well
6.2.3 Harmonic oscillator in one dimension
6.3 Schrödinger equation in three dimensions
6.3.1 Free particle
6.3.2 Infinitely deep well
6.3.3 Harmonic oscillator
6.3.4 Spherically symmetric potentials
6.4 Defending the delta
Exercises
Bibliography
ch7.pdf
Chapter
A.1 Properties of linear operators
A.1.1 Unitary operators
A.1.2 Hermitian operators
Two elements of a Hilbert space have an inner product that is less than or equal to the product of their norms
For a Hermitian operator (Rˆ) all eigenvalues (r) are real
Eigenvectors of a Hermitian operator Rˆ corresponding to distinct eigenvalues are orthogonal
The expectation value of a Hermitian operator, with respect to any vector, is real
The expectation value of the square of a Hermitian operator, with respect to any vector, is non-negative
A.1.3 Positive operators
If an operator Qˆ on a complex vector space is positive, it is also Hermitian
If (v,Oˆv)=0 for all vectors ∣v〉 in , then Oˆ is the null operator
A.1.4 Projection operators, subspaces
A.1.5 Normal operators; spectral decomposition
Every operator on a finite-dimensional vector space has at least one eigenvalue
If ∣v〉 is an eigenstate of a normal matrix Nˆ with eigenvalue λ, it is also an eigenstate of Nˆ†
A.2 Simultaneous eigenvectors
A complete set of eigenvalues of Aˆ can be found that also consists of eigenvalues of Bˆ, when the two operators are Hermitian and commute with one another.
A.3 Operator functions
A.4 Trace
The trace is independent of the choice of orthonormal basis
A.5 Vector space of operators
A.6 Spin one-half component matrices and the SU(2) group
Bibliography
ch8.pdf
Chapter
B.1 Notation
B.2 Definitions

Citation preview

This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details: IP Address: 100.38.68.48 This content was downloaded on 07/08/2023 at 02:44 Please note that terms and conditions apply.

You may also like: INVERSE PROBLEMS NEWSLETTER

Discrete Quantum Mechanics

Discrete Quantum Mechanics H Thomas Williams Professor Emeritus, Washington and Lee University

Morgan & Claypool Publishers

Copyright ª 2015 Morgan & Claypool Publishers All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher, or as expressly permitted by law or under terms agreed with the appropriate rights organization. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency, the Copyright Clearance Centre and other reproduction rights organisations. Rights & Permissions To obtain permission to re-use copyrighted material from Morgan & Claypool Publishers, please contact [email protected]. ISBN ISBN ISBN

978-1-6817-4125-3 (ebook) 978-1-6817-4061-4 (print) 978-1-6817-4253-3 (mobi)

DOI 10.1088/978-1-6817-4125-3 Version: 20151201 IOP Concise Physics ISSN 2053-2571 (online) ISSN 2054-7307 (print) A Morgan & Claypool publication as part of IOP Concise Physics Published by Morgan & Claypool Publishers, 40 Oak Drive, San Rafael, CA, 94903, USA IOP Publishing, Temple Circus, Temple Way, Bristol BS1 6HG, UK

To Lynn, for her patience with and perseverance through this obsession.

Contents Preface

ix

Acknowledgments

xi

Author biography

xii

1

Postulates

1-1

1.1 1.2 1.3 1.4 1.5

State space Time evolution Quantum measurement Composite systems The genie Exercises Bibliography

2

Two-state systems

2.1 2.2 2.3

2.4 2.5

Schrödinger’s cat Expectation value; energy operator Spin one-half 2.3.1 Spin orientation measurements 2.3.2 Quantum mathematics of spin one-half 2.3.3 Spin precession 2.3.4 Heisenberg picture Ammonia molecule Photons

3

Entanglement

3.1 3.2 3.3

Entangled qubit pairs Quantum gates Utilizing entanglement 3.3.1 Superdense coding 3.3.2 Teleportation 3.3.3 Bell’s inequality Large-scale quantum algorithms 3.4.1 Classical algorithm for Simon’s problem 3.4.2 Quantum algorithm for Simon’s problem Exercises Bibliography

3.4

1-2 1-5 1-7 1-12 1-14 1-15 1-16 2-1 2-2 2-6 2-7 2-9 2-12 2-14 2-15 2-16 2-19 3-1

vii

3-2 3-5 3-7 3-7 3-8 3-10 3-14 3-15 3-16 3-20 3-21

Discrete Quantum Mechanics

4

Quantum angular momentum

4.1 4.2 4.3 4.4 4.5 4.6 4.7

Operators for orbital angular momentum Operators for spin one-half Generalized angular momentum theory Angular momentum addition Interaction operators Isospin And… Exercises Bibliography

5

Quantum many-body problem

5.1 5.2 5.3

The general case: Heisenberg XYZ spin chain Ising model Heisenberg XXX spin chain 5.3.1 N = 4 case: diagonalization 5.3.2 N = 4 case: Bethe ansatz Exercises Bibliography

6

Infinity, and beyond

6.1 6.2

Schrödinger equation Schrödinger equation in one dimension 6.2.1 Free particle 6.2.2 Infinitely deep square well 6.2.3 Harmonic oscillator in one dimension Schrödinger equation in three dimensions 6.3.1 Free particle 6.3.2 Infinitely deep well 6.3.3 Harmonic oscillator 6.3.4 Spherically symmetric potentials Defending the delta Exercises Bibliography

6.3

6.4

4-1 4-2 4-3 4-4 4-6 4-10 4-14 4-17 4-17 4-17 5-1 5-1 5-2 5-5 5-11 5-14 5-15 5-15 6-1 6-2 6-2 6-2 6-4 6-5 6-8 6-9 6-10 6-10 6-11 6-13 6-14 6-15

Appendix A

Relevant results from linear algebra

A-1

Appendix B

Directory of definitions and notation

B-1

viii

Preface ‘Common sense is what tells us the earth is flat.’ (Stuart Chase, American writer). Most of us are best at understanding and believing things that align with our experience. We – at least most of ‘we’ – have nonetheless proven capable of buying into propositions that defy everyday experience, common sense. There are few who nowadays do not believe the surface of the earth is curved, even though no-one reports a sensation that the ground is constantly moving under his or her feet and few have watched a ship slowly dip below the horizon as it sails into the distance. Within a paltry handful of centuries since it was proposed, the notion of the earth as bumpy ball has become commonplace. Quantum mechanics, just out of its first century, has not received such uniform acceptance. Some are barely aware of what it is (despite living surrounded by quantum-enabled devices), others know a bit about it (e.g. can use the phrase ‘uncertainty principle’ in a complete sentence) and a tiny few have studied it and can make use of it. If Richard Feynman is correct, however, nobody understands it. To the general public, a cat both dead and alive and instantaneous teleportation are interesting bits of fantasy rather than things made possible by the reality of quantum behavior. Much more serious detractors exist and need to be taken seriously. The most respected scientist of the twentieth century, Albert Einstein, never believed quantum mechanics to be a proper representation of physical reality. Writing in 1936 he declared all science to be ‘nothing more than a refinement of everyday thinking.’ Quantum mechanics does not come close to that standard, explaining why Einstein spent considerable time and effort towards proving it wrong. Although his specific objections have, by now, been countered by both reasoning and experimentation, there remains a small cadre of serious physicists who continue to try to create an explanation of physical behavior that is as predictive as quantum theory but lacks its confounding implications. Without the guidance of common sense, those who wish to pursue quantum explanations of phenomena need a map of another kind, and this is given by mathematics. The cleanest such map, in my opinion, is that provided by John von Neumann. In his 1932 volume Mathematical Foundations of Quantum Mechanics he proved the equivalence of Heisenberg’s and Schrödinger’s approaches by showing that each followed from a small number of postulates: theoretical explanations that do not follow from these postulates are not quantum explanations, but something else. This set of postulates has held up for the better part of a century, and has provided valuable guidance in the progression of basic physical science. In my career of teaching and research I have presented and used quantum mechanics in both its wave-equation and matrix formulations. Over time I have been drawn towards the discrete, matrix and vector forms whenever possible. This personal preference is clearly reflected in this volume. It begins with postulates and follows with a set of quantum applications that employ these postulates in ways that allow use of discretely-numbered quantum states. The applications are mostly

ix

Discrete Quantum Mechanics

disjoint, each presented with little reliance on those that come before. The logic of putting them together comes from my personal preferences and experience. My hope is that readers will find at least some of the topics herein of use in their own quantum pursuits. Teachers of the subject may find this provides some excursions outside the more usual topics presented in introductory quantum mechanics courses. Perhaps students could independently use chapters in this book as launch-pads for investigations of their own. In any case, I hope this volume will help some with the challenge of coming to grips with the fascinating realm of quantum theory. Tom Williams Lexington, Virginia, USA November 2015

x

Acknowledgments I wish to express my profound gratitude to Paul Bourdon, Dan Mazilu and Irina Mazilu for the many hours they spent reading and commenting on drafts of this material. It has been vastly improved by their close attention to it. Errors that remain are my own.

xi

Author biography H Thomas Williams A native of Hampton, Virginia, Tom Williams was educated in the public schools of tidewater Virginia and enrolled in the University of Virginia from which he received a BS in Physics (1963) and a PhD in Physics (1967). He spent the following two years on a National Research Council/National Science Foundation Postdoctoral Research Fellowship at the National Bureau of Standards (Washington, DC), and subsequently spent 1970 as Gastdozent at the Universität Erlangen-Nūrnberg, Germany. Following this he served as a staff scientist at Kaman Sciences, Colorado Springs, Colorado for three years, working primarily in the area of electromagnetic field propagation. In 1974 he joined the faculty of Washington and Lee University where he spent the remainder of his working career. But for brief administrative stints, including that of University Provost from 2003 until 2007, the majority of his thirty-six years at W&L were spent teaching and conducting research. During that time he also provided consultant services to the National Bureau of Standards (now NIST) and Los Alamos National Laboratory. Currently, primary research interests are quantum information theory and non-equilibrium statistical mechanics.

xii

IOP Concise Physics

Discrete Quantum Mechanics H Thomas Williams

Chapter 1 Postulates

The development of quantum mechanics into a consistent theory took place over about three decades. Its birth was in Planck’s revolutionary proposal regarding quantization of blackbody radiation [Planck1900] at the start of the twentieth century. Its coming of age was signaled by Heisenberg’s matrix mechanics [Heisenberg1925], Schrödinger’s wave mechanics [Schrödinger1926] and the efforts of Schrödinger, Dirac and von Neumann to show the equivalence of these two approaches. Starting in the early 1930s, John von Neumann set about putting quantum theory on a coherent mathematical footing [vonNeumann1932], and in so doing he established a series of postulates from which would follow the theories of Schrödinger and Heisenberg. As postulates, these notions were not derived from more basic principles, nor were they considered self-evident. Their validity was found in the fact that their consequences aligned with experimental results. Since von Neumann’s work, the postulatory basis of quantum theory has been debated, refined and expanded. There is currently no agreed-upon wording for these postulates, indeed even the number of postulates varies from three to as many as six, depending on the interests and tastes of particular authors. In this brief exposition on the topic of quantum theory, we begin with a set of postulates in the spirit mentioned above: they are neither self-evident, nor are they to be justified from more basic ideas. They are self-consistent, and are the strongest statements that lead to all currently recognized consequences of quantum theory. They will be referred to as appropriate in the chapters that develop the theory more fully, and in examples. The postulates will be stated in the language of density functions rather than state vectors, since the former is a more general formalism, from which the latter follows easily. The language of each postulate in terms of state vectors will be explained in every case. Our goal is to explore the consequences of the postulates to illuminate the beauty, mystery and power of quantum mechanics in a reasonably compact presentation.

doi:10.1088/978-1-6817-4125-3ch1

1-1

ª Morgan & Claypool Publishers 2015

Discrete Quantum Mechanics

In doing so, we restrict ourselves to finite and countably infinite dimensions, avoiding the conceptual and computational challenges that lurk in continuous infinitedimensional Hilbert spaces. The cost of this restriction is the omission of some important classic examples, such as the solutions to the continuous-space problems of the Coulomb and harmonic oscillator potentials. We thus avoid long diversions into Sturm–Liouville theory, but miss the elegance of this area of mathematics. Our goal is to prepare readers to address problems of this nature as well as issues now current in quantum information theory.

1.1 State space For a classical system, we seldom deal with a complete description of its properties. In many contexts, for instance, knowledge of the mass of the system is sufficient, in others mass and moment of inertia suffice, and in others a complete description of mass distribution is required. For other situations, charge or charge distribution are also relevant to the system’s behavior. By contrast, a quantum system is described as fully as possible by way of its state vector, purportedly carrying a complete description of all its relevant and knowable properties. We may often suppress explicit reference to particular parts of the state function in certain contexts, but understand them from the start as part of the total story.

Postulate To each isolated physical system, quantum mechanics assigns a Hilbert space called the state space of the system. A particular configuration of the system is described as fully as possible by a positive operator on the state space with trace one, called the density operator of the system configuration. A Hilbert space is a complete complex vector space with an inner product. We will denote non-zero Hilbert space vectors using the Dirac ket notation ∣·〉 and the zero vector simply as 0. The symbol within the ket is typically an integer or a symbol that suggests the specific vector being represented1. As a vector space, the space and its elements must have the properties: • commutivity under addition, e.g.

x + y = y + x ; • associativity under addition, e.g.

x +(y + z )=(x + y )+ z ; • the space contains an additive identity 0 such that

x +0=0+ x = x ; 1

Here and henceforth, we will usually adopt the convention that lower case roman letters represent complex scalars, upper case roman letters denote operators and, in situations where context does not make it clear, operators will be further distinguished by the use of a caret, e.g. Aˆ .

1-2

Discrete Quantum Mechanics

• each element (e.g. ∣x〉) has its additive inverse ( −∣x〉) within the space, obeying

( − x ) + x = 0; • associativity of scalar products

a(b x ) = (ab) x ; • distributivity of scalar sums

(a + b ) x = a x + b x ; • distributivity of vector sums

a( x + y ) = a x + a y ; • scalar multiplication identity

1x = x . Completeness of a vector space is the requirement that every Cauchy sequence of vectors in the space converges to a vector that is also in the space. We will not make explicit use of this condition in what follows. The inner product of a unit vector (∣x〉) with another (∣y〉) is denoted by 〈x∣y〉, and yields a complex scalar result. The inner product has the following properties: • it is distributive, i.e. if ∣w〉 = ∣y〉 +∣z〉 then 〈x∣w〉 = 〈x∣y〉 +〈x∣z〉; • 〈x∣y〉 = 〈y∣x〉, where the overbar represents complex conjugation; • if ∣y〉 = ∣y′〉, then 〈x∣y′〉 = 〈x∣y〉; • the inner product of ∣x〉 with itself is greater than or equal to zero, equality holding only when ∣x〉 = 0. We will restrict our attention to finite-dimensional Hilbert spaces. Any vector in a d-dimensional Hilbert space can be represented by a d-term sum d −1

x =

∑ cj bj j=0

when the set ∣bj 〉 comprise a basis for the space. Elements of a basis are linearly independent, that is d −1

∑ dj bj

=0

if and only if dj = 0 for all j .

j=0

We work most often with orthonormal bases, for which we reserve the notation ∣ej 〉. Elements of an orthonormal basis further satisfy 〈ej ∣ek〉 = δj,k . Whenever possible, we will write expressions entirely in terms of Dirac notation, but at times it will be useful to make explicit reference to vector and matrix representations of the Hilbert space vectors and the operators that act upon them. Each vector ∣x〉 can be represented by a d-component column vector with complex components. One useful such representation allows the elements of member ∣ej 〉 of an 1-3

Discrete Quantum Mechanics

orthonormal basis to be represented by a column vector of zeros but for a single 1 in the jth slot. In Dirac notation, the symbol 〈x∣ is itself a vector (in a dual vector space) that is the conjugate transpose of the vector ∣x〉. 〈x∣ can be represented by a row of complex numbers which are the complex conjugates of the corresponding elements of the column that represents ∣x〉. The inner product 〈x∣y〉 is evaluated in the vector/matrix representation, following the rules of matrix multiplication, as a complex constant. The construction ∣x〉〈y∣ is an operator on a Hilbert space vector, such that (∣x〉〈y∣)∣z〉 = (〈y∣z〉)∣x〉 and similarly 〈z∣(∣x〉〈y∣) = (〈z∣x〉)〈y∣. Such entities in the vector/matrix representation, using matrix multiplication rules, evaluate to a d × d complex matrix. General operators on Hilbert spaces can always be written as a sum of such expressions, particularly useful when using orthonormal basis vectors such as: d −1

Aˆ =

∑ aj,k ej

ek .

j, k = 0

The complex constants aj,k in this expression are the matrix elements of a matrix representation of the operator Aˆ . † The Hermitian conjugate of an operator Aˆ is denoted by Aˆ . It is defined by the fact that for arbitrary Hilbert space unit vectors ∣x〉 and ∣y〉, the inner product of ∣x〉 with Aˆ ∣y〉, x Aˆ y , † is equal to the inner product of vector Aˆ ∣x〉 with ∣y〉. Given the elements ai, j of a † matrix representation of Aˆ , the corresponding matrix elements of Aˆ in the same basis are aj ,i . For this reason, the Hermitian conjugate of an operator is also referred to as its conjugate transpose. The density operator is a particularly important object in this presentation of quantum mechanics, for which we reserve the symbol ρ. The properties that distinguish it from a general operator in Hilbert space are: • it is a positive operator, that is, for an arbitrary vector in its Hilbert space ∣ϕ〉, 〈ϕ∣ρ∣ϕ〉 ⩾ 0; • it has trace 1. The trace of an operator is the sum of values along the principal diagonal of any of its matrix representations.

Since any density operator ρ is a positive operator, it can be shown that ρ† = ρ. Operators which have this property are called Hermitian or self-adjoint. Positive operators thus have all real eigenvalues, and a complete set of orthogonal eigenvectors, properties shared with all Hermitian operators. Proof of these properties can be found in appendix A. A particularly significant subset of density operators is the collection of those that can be represented in the form ρps = ∣ϕ〉〈ϕ∣ for a unit vector ∣ϕ〉 in the Hilbert space. Quantum states that can be so represented are referred to as pure states, and are

1-4

Discrete Quantum Mechanics

completely defined by the vector ∣ϕ〉, called the state vector, obeying 〈ϕ∣ϕ〉 = 1.2 Quantum states that are not pure states are referred to as mixed states and are superpositions of pure states. Mixed state density operators can be expressed as

ρ=

∑ pi ϕi

ϕi ,

i

where the ∣ϕi 〉s are an ensemble of state vectors and the pis are real non-negative values satisfying ∑i pi = 1. One such ensemble representation of ρ has the functions ∣ϕi 〉 identified with its d eigenvectors and the parameters pi equal to corresponding real eigenvalues. This demonstrates that the trace of ρ is also the sum of its eigenvalues. For a pure state with vector components vi, the diagonal elements of its density matrix are ∣vi∣2 . The trace of the density matrix is thus ∑i ∣vi∣2 and is equal to 1 by virtue of the normalization condition for state vectors. The trace of the density matrix for a mixed state is the sum of pure state traces, weighted by probabilities 2 summing to one, and is thus also equal to 1 as postulated. Evaluating Tr(ρms ), on the other hand, can be used as a test that distinguishes a pure state density matrix from a mixed state one. Assuming an ensemble of pure states, ρj , the state vectors of which are not necessarily orthogonal to one another, we construct

ρms =

∑ pj ρj . j

Noting that Tr( ρ j2 ) = Tr( ρj ) = 1 and Tr( ρj ρk ) ⩽ 1, compare term-by-term the 2 expressions for (∑j pj )2 and Tr( ρms ) = Tr(∑j pj ρj ∑k pk ρk ) to see in a straightforward manner that the latter sum is less than or equal to the former, equality holding only when the mixed-state sum consists of a single term, the case representing a pure 2 state. We conclude that in the case of a pure state, Tr( ρps ) = 1, while for non-trivial 2 mixed states, Tr(ρms ) < 1.

1.2 Time evolution Classical systems change with time according to Newton’s second law, which prescribes changes in velocity in reaction to applied forces. Whether we are observing these changes or not, we ascribe objective reality to the predicted trajectories. Quantum systems evolve with time in a particularly simple way as long as we do not monitor their progress: their state functions and density operators change under the influence of a unitary operator. Such an operator, Uˆ , in the context of finite-dimensional Hilbert spaces satisfies (by definition) the relation ˆ ˆ † = Uˆ †Uˆ = I , where I is the identity operator, the matrix representation of which UU is a diagonal matrix with all 1s along the principal diagonal. 2 The state vector corresponds to the normalized ‘wave function’ or ‘state function’ of many presentations of quantum mechanics.

1-5

Discrete Quantum Mechanics

Postulate The change of an isolated quantum system over time is described by a unitary transformation Uˆ . If the state of the system at time t1 is described by the density operator ρ(t1), at a different time t2 the system is described by † ρ(t2 ) = Uˆ ρ(t1)Uˆ .

(1.1)

The operator Uˆ depends only on the time interval t2 − t1. If the system is in a pure state, isolated from other quantum systems, the information in this postulate can be recast in the following way. Representing the state vector at t1 as ∣ϕ〉, equation (1.1) becomes † ρ(t2 ) = Uˆ ϕ ϕ Uˆ = Uˆ ϕ (Uˆ ϕ

)† ,

showing that the state at t2 is also pure, with state vector Uˆ ∣ϕ〉. Pure states evolve with time via a unitary transformation:

ϕ(t2 ) = Uˆ ϕ(t1) .

(1.2)

ˆ ˆ † = Uˆ †Uˆ ) it has a spectral decomposition, i.e. there is a basis Since Uˆ is normal (UU in which it can be represented by a diagonal matrix. Furthermore, none of the diagonal elements in this representation are zero (see appendix A). From this, Hˆ = −i ln(Uˆ ) can be defined3, and † † −1 Hˆ = i ln(Uˆ ) = i ln(Uˆ ) = −i ln(Uˆ ) = Hˆ ,

i.e. Hˆ is Hermitian. In general, therefore, any unitary operator can be represented as

Uˆ = exp(iHˆ ) , where Hˆ is Hermitian. If Uˆ is the time evolution operator, the operator Hˆ is called the Hamiltonian4, and time evolution can be stated as

ϕ(t2 ) = exp( −iHˆ (t2 − t1)) ϕ(t1) . For cases where the Hamiltonian is independent of time, this state function can be found as the solution to the equation

i

d∣ϕ〉 = Hˆ ∣ϕ〉, dt

(1.3)

the time-dependent Schrödinger equation. 3 If a basis exists for which an operator Aˆ is diagonal with diagonal elements ai, then the operator defined to be a function of that operator f (Aˆ ) has in the same basis a diagonal representation for which the ith diagonal element is f (ai ). 4 We adopt a convention of physical units in which the reduced Planck constant, ℏ , is set equal to 1.

1-6

Discrete Quantum Mechanics

1.3 Quantum measurement The notion of measurement does not appear as a fundamental principle in describing classical physical systems. It is assumed, usually tacitly, that with sufficient care one can always measure the properties of a classical system while keeping the effects of these measurements on the system below an acceptable level. We assume, for instance, that the Moon not only exists when we are not watching it, but that it is moving in the same way it would had we chosen to continuously monitor its progress. Perhaps the most profound difference between quantum and classical descriptions of nature is the quantum tenet that systems behave differently than they would otherwise while being observed. This reveals itself in a postulate about quantum measurement which proposes that behavior whenever a quantum system is measured is radically different from unitary evolution.

Postulate The result of a measurement on a quantum system is fully determined by a set of ˆ i that operate on the system’s state space, and which measurement operators M ˆ i†M ˆ i = 1. The index i is in one-to-one correspondence satisfy the relation ∑ M i

with the possible outcomes of the measurement. If the state is described by the density operator ρ immediately prior to the measurement, the probability that the outcome corresponding to i will occur is

(

ˆ iρM ˆ i† pi = Tr M

)

(1.4)

and the state of the system immediately after measurement will be

ρ′ =

ˆ iρM ˆ i† M

(

ˆ iρM ˆ i† Tr M

)

.

(1.5)

The term quantum measurement as described in this postulate refers to the inspection of a quantum system via a physical process related to a particular variable that can have one of a finite set of values. As a result of the measurement, the measurement device will manifest one of the possible values and the quantum system will usually be altered, having lost much of the information about its pre-measurement state. The theory predicts only probabilities of the possible outcomes, but gives a full description of the post-measurement state once the outcome variable is known. If the environment of the system is known (i.e. the Hamiltonian), the evolution of the system’s state is fully predictable up until the moment it is interrupted by a subsequent measurement. If immediately before measurement the system is in a pure state ρ = ∣ϕ〉〈ϕ∣, the probability of outcome i can be expressed as

(

)

(

)

ˆ iρM ˆ i† = Tr M ˆi ϕ ϕ M ˆ i† = ϕ M ˆ i†M ˆi ϕ , pi = Tr M

1-7

(1.6)

Discrete Quantum Mechanics

by making use of the cyclic property within the trace operator. The density operator after measurement is

ρ′ =

†

( Mˆ ∣ϕ〉)( Mˆ ∣ϕ〉) =

ˆ i∣ϕ〉〈ϕ∣M ˆ i† M

(

ˆ iρM ˆ i† Tr M

i

)

i

pi

,

a pure state with state vector

ˆ i∣ϕ〉 M . pi

∣ϕ′〉 =

(1.7)

The most common sub-category of quantum measurements is that of projective measurements, in which the measurement operators are projections into subspaces of the appropriate Hilbert space:

ˆi = M

∑ ej

ej ,

j

where the ∣ej 〉s are members of an orthonormal set. The number of terms in the j sum can be as few as 1 or as many as the dimension of the Hilbert space, but this latter extreme is the trivial case of a measurement operator equal to the identity operator. A projective measurement is often depicted as a generalization of the Stern–Gerlach experiment. In this historically significant process, a particle with total spin S (with a proportional magnetic dipole moment) is passed through an inhomogeneous magnetic field and emerges in one of 2S + 1 possible directions, each direction corresponding to one possible value of the particle’s spin component along the field direction. A simple example of the generalization for a three-dimensional Hilbert space is depicted in figure 1.1 as a device into which a particle is sent to be measured for the property Qˆ with three distinct outcomes. The particle will emerge from one of the ports on the right side of the device, depending upon the outcome of the measurement (q0, q1 or q2), and immediately upon its emergence it will exhibit the particular quantum state ( ∣0〉, ∣1〉 or ∣2〉) associated with the measured value.

Figure 1.1. Projective measurement device in three dimensions.

1-8

Discrete Quantum Mechanics

Figure 1.2. Projective measurement device with degeneracy.

Often, a quantum property with a degenerate spectrum is to be measured, i.e. there is more than one distinct state of the system being measured that exhibits the same value of the property. This is the case, for instance, when the energy of an electron is examined, assuming the electron’s environment has no magnetic field. While one can depict the electron as being in either of the distinct states ‘spin up’ or ‘spin down’, the energy of the electron is the same for the two cases. An example exhibiting degeneracy is depicted in figure 1.2. Here, the particle being examined has four degrees of freedom, existing in a four-dimensional Hilbert space. The property examined, R, has only three distinct values (r0, r1 and r3), the second of which is exhibited by all states that are linear combinations of the basis vectors ∣1〉 and ∣2〉. When the measurement is complete, if we know the value of the property is r0 (r3), we know the immediate post-measurement state of the system will be ρ0 = ∣0〉〈0∣ (ρ3 = ∣3〉〈3∣) exactly. If, however, we measure the value r1, the postmeasurement state carries some information about the state before measurement, in particular it retains its orientation in the ∣1〉, ∣2〉 subspace. In the language of the quantum measurement postulate, this case has measurement operators Mˆ1 = ∣0〉〈0∣, Mˆ 2 = ∣1〉〈1∣ + ∣2〉〈2∣ and Mˆ 3 = ∣3〉〈3∣, and depicting the pre-measurement quantum state as ρ, its post-measurement state is ∣0〉〈0∣ if r0 is the measurement outcome; ∣3〉〈3∣ if r3 is the outcome; and

( ∣1〉〈1∣ + ∣2〉〈2∣)ρ( ∣1〉〈1∣ + ∣2〉〈2∣) 〈1∣ρ∣1〉 + 〈2∣ρ∣2〉 if r2 is the outcome. A measurement category more general than that of projective measurements is allowed by the quantum measurement postulate: positive operator-valued measures (POVM). It is often the case that only the probabilities associated with the various measurement outcomes are of interest, and not the post-measurement state of the measured system. This is obviously the case when the quantum measurement under consideration is the final step in a process. In such cases the important operators 1-9

Discrete Quantum Mechanics

† associated with the measurement are the POVM elements Eˆi ≡ Mˆ i Mˆ i , since by using the cyclic property of the trace we can identify

(

)

(

)

ˆ iρM ˆ i† = Tr M ˆ i†M ˆ iρ = Tr(Eˆiρ) . pi = Tr M The Eˆi are easily seen to be positive operators and to satisfy ∑i Eˆi = 1. In fact, it is straightforward to demonstrate that any set of operators satisfying these two properties corresponds to a set of measurement operators obeying the conditions of the postulate. Projective measurements themselves therefore lead to Eˆi s that are POVM elements. More interesting are those measurements that possess a POVM measure but are not projective measurements, and which cannot be represented as generalizations of the Stern–Gerlach experiment as illustrated in figures 1.1 and 1.2. One example is that of a measurement to distinguish as fully as possible whether a system is in one of two non-orthogonal pure states ∣ϕ〉 and ∣ψ 〉. It is impossible to make this distinction with certainty, since if there is a measurement that reads ‘true’ if the state is prepared as ∣ϕ〉, there is a non-zero probability of a false positive when the same measurement is made on ∣ψ 〉 due to the fact that 〈ϕ∣ψ 〉 ≠ 0. One can, however, devise a scheme using a POVM that allows one to say with certainty in some cases which of the states was prepared, and to say (confidently) ‘I don’t know’ in the remaining cases. Such a POVM has three elements:

Eˆ1 = a ϕ⊥ ϕ⊥ , Eˆ 2 = b ψ⊥ ψ⊥ and

Eˆ 3 = 1 − Eˆ1 − Eˆ 2. The states ∣ϕ⊥〉 and ∣ψ⊥〉 are chosen such that 〈ϕ∣ϕ⊥〉 = 0 and 〈ψ ∣ψ⊥〉 = 0. Real constants a and b must be non-negative to ensure that Eˆ1 and Eˆ2 are positive operators, and they are further restricted by the requirement that Eˆ3 be positive. When this quantum measurement is performed, a result in the first channel (that of Eˆ1) assures that the pre-measurement state could not have been ∣ϕ〉 since 〈ϕ∣ϕ⊥〉 = 0. The prepared state must, therefore, have been ∣ψ 〉. Similarly, a result in the second channel (that of Eˆ2 ) implies that the prepared state must have been ∣ϕ〉. Only if the result corresponding to element Eˆ3 appears is the result inconclusive. The optimal case obtains when a and b are chosen so as to minimize p3 = Tr(Eˆ3 ρ ). Assuming there is equal likelihood that the state is prepared in either of the states, the best result comes when a = b = 1/(1 + ∣〈ϕ∣ψ 〉∣), resulting in the probability of an uncertain result of

p3 = ∣〈ϕ ψ ∣.

(1.8)

Projective measurements, when combined with a unitary operation, can be shown to be equivalent to the most general operations allowed by the quantum measurement

1-10

Discrete Quantum Mechanics

postulate. By combining the system to be measured using a POVM with another system in a known state, and performing a particular unitary operation on the combined systems, one obtains a larger quantum system. A projective measurement on this combined system can be found that performs the desired POVM measure on the original system. Unitary operations of the desired nature can be carried out by allowing the systems (original system plus auxiliary system) to evolve in the presence of a selected Hamiltonian. The unitary operation serves to entangle the two systems. This concept will be covered in section 3.2, where we give an example of this particular kind of transformation. The universality of projective measurements justifies a characterization of quantum measurements by way of observables and associated observable operators. An observable operator is defined as

Oˆ =

∑ fi Mˆˆ i ,

(1.9)

i

where the real variables fi are the possible (real) values the observable can exhibit and the Mˆˆi are corresponding measurement operators, which in this case are projection operators. These measurement operators are Hermitian and thus the observable operator Oˆ is also Hermitian. In cases where the Mˆˆi projects into a onedimensional subspace,

ˆi = i i , M the unit vector ∣i 〉 is an eigenvector of Oˆ with eigenvalue fi. For observables with degenerate spectra, there are projectors into higher-dimensional subspaces

ˆj = j M 1

j1 + j2

j2 + ⋯.

Every vector in this subspace is an eigenvector of Oˆ with eigenvalue fj. The postulate requires that

I=

†

∑Mˆ i Mˆ i = ∑Mˆ i , i

(1.10)

i

the last equality following since each of the Mˆ i s are projection operators. The collection of subspaces into which the Mˆ i s project thus spans the Hilbert space. Note that if zero is an allowed value of the observable, there are more terms in equation (1.10) than in equation (1.9). Uncertainty principle Two physical properties are called incompatible if their measurement operators Oˆ 1 and Oˆ 2 do not commute, i.e. [Oˆ 1, Oˆ 2 ] ≡ Oˆ 1Oˆ 2 − Oˆ 2Oˆ 1 ≠ 0. Following a standard procedure, we show here that two such properties cannot be perfectly known at the same time. We begin with the straightforward observation that

2Oˆ 1Oˆ 2 = [Oˆ 1, Oˆ 2 ] + (Oˆ 1Oˆ 2 + Oˆ 2Oˆ 1) .

1-11

Discrete Quantum Mechanics

Note that the first term on the right is anti-Hermitian (i.e. its Hermitian conjugate is its negative) and the second is Hermitian. Taking the expectation value5 of the left and right sides of this expression relative to an arbitrary quantum state ∣ψ 〉, we get

2 Oˆ 1Oˆ 2 = ⎡⎣ Oˆ 1, Oˆ 2⎤⎦ +

( Oˆ Oˆ 1

2

)

+ Oˆ 2Oˆ 1 ,

in which the first term on the right is pure imaginary and the second pure real since the expectation value of a Hermitian operator is real (see appendix A). Taking the absolute square of both sides, it follows that

4

Oˆ 1Oˆ 2

2

=

⎡ Oˆ , Oˆ ⎤ ⎣ 1 2⎦

2

+

( Oˆ Oˆ 1

2

2

)

+ Oˆ 2Oˆ 1

⩾

⎡ Oˆ , Oˆ ⎤ ⎣ 1 2⎦

2

.

From the Cauchy–Schwarz inequality (see appendix A),

Oˆ 1Oˆ 2

2

2 ⩽ Oˆ1

2 Oˆ 2 ,

leading to 2 4 Oˆ1

2 Oˆ 2 ⩾

⎡ Oˆ , Oˆ ⎤ ⎣ 1 2⎦

2

.

The ultimate step in this procedure is to replace the operators Oˆj by related Hermitian operators Oˆj − 〈Oˆj 〉, producing

1 Δ Oˆ 1 Δ Oˆ 2 ⩾ 2

( ) ( )

⎡ Oˆ , Oˆ ⎤ ⎣ 1 2⎦

,

(1.11)

where Δ(Oˆj ) = 〈Oˆ 1 − 〈Oˆj 〉〉2 is the standard deviation of the observable associated with Oˆj , based on an ensemble of measurements of identically prepared systems. This result says that, in general, incompatible properties cannot be known simultaneously to arbitrarily fine precision. The most familiar example of this result is the momentum–position uncertainty relation, based upon the commutation relation for one component of the position variable of a particle and the corresponding component of its momentum, [x , px ] = i , producing

ΔxΔ( px ) ⩾

1 . 2

1.4 Composite systems An elementary classical system would be an entity with no internal degrees of freedom, or ones that can be ignored. In systems made up of two or more such elementary systems, each component continues to move as predicted by Newton’s second law, and the center of mass of the system as a whole reacts similarly to the vector sum of all the forces applied to its component systems. Elementary quantum systems change as described in the two preceeding postulates. Rules for composite 5 The expectation value of an operator Aˆ denoted 〈Aˆ 〉 is equivalent to 〈φ Aˆ φ〉 where φ〉 is a state vector. This construct is discussed in section 2.2, and its properties discussed in appendix A.

1-12

Discrete Quantum Mechanics

quantum systems result from the remaining postulate and are relatively straightforward if all the component systems are pure states. We will see examples of the complications encountered when component systems are in mixed states. Postulate The state space of a quantum system made up of multiple systems is the tensor product of the state spaces of the individual systems. If each of the individual systems is in a pure state described by the density function ρi = ∣vi〉〈vi∣, then the density function of the composite system is

ρ1 ⊗ ρ2 ⊗ ρ3 ⋯ = ( v1 ⊗ v2 ⊗ v3 ⋯)( v1 ⊗ v2 ⊗ v3 ⋯) , also a pure state. Of the postulates we have presented, this is the only one that makes a statement about pure states that is not paralleled by a statement about mixed states. The situation regarding mixed states as components of a composite state is a bit messy. Consider two quantum systems, labeled A and B, that combine to a composite state AB. In the simplest case—the one explicitly addressed in the postulate—both component states are pure and thus can be expressed in terms of their state vectors: ∣A〉 and ∣B〉. The composite state is also a pure state, with a state vector ∣AB〉 ≡ ∣A〉 ⊗ ∣B〉. The density function ρAB of a composite state, be it pure or mixed, uniquely specifies component density functions ρA and ρB if we require (quite reasonably) that the probabilities predicted regarding measurements on either of the component states are the same whether we compute them with the component state density function or with the composite state density function. Surprisingly, however, when the component states are mixed, they do not uniquely specify a composite state. When ρAB is pure, the component states are pure if and only if the state vector (and also the density function) of the composite state can be factored into two terms, one involving only component state A and the other involving only B. When this factoring is not possible, the systems A and B are said to be entangled. Independent and thus unentangled component states can become entangled through unitary evolution under the influence of a Hamiltonian that describes interaction between the states. When the composite state is mixed, it will always lead to mixed component states. Mixed states typically come about as a result of entanglement. We show the challenge in describing composite states with this rather straightforward example. Assume A and B are both in mixed states, both resulting from entanglement, and that we wish to describe the composite state AB made up of A and B. If, on the one hand, A and B are entangled with each other but isolated from other quantum systems, they could be the result of a pure but unfactorable composite state. On the other hand, A could be entangled with a third system C, and B with a fourth system D, yet both have the same mixed state density functions as in the former case. The composite system AB is entangled with systems C and D and AB’s density function must be mixed to show this dependence. Two quite different density functions are seen to represent component states with mathematically identical density functions. 1-13

Discrete Quantum Mechanics

The mixed states for individual components of entangled systems do not carry information about how they came about (was A entangled with B, or another state?) and thus while the density matrix for A can successfully predict the probabilities associated with measurements on A, it is not sufficient to inform us how A behaves when combined with other quantum systems. By using the evolution postulate, we can utilize unitary evolution to express quantum theory entirely in terms of state vectors, avoiding difficulties such as that just discussed. Nevertheless, the convenience of density functions in many applications is so great that their introduction is well worth the effort.

1.5 The genie The above postulates drive a theory intended to replace the useful but flawed Newtonian theory of motion. To be put to use, they must be supplemented by explicit expressions for the Hamiltonian operator appropriate to particular situations. Since quantum predictions are probabilities, we will have to rely upon experiments that examine large numbers of replicate systems so as to produce ensemble averages for comparison. In more recent applications, however, we are attempting to exploit single quantum systems for the great resources in information storage and transmission promised in the state space postulate: even the simplest single quantum system can store an infinite amount of information in its limitless Hilbert space. But there is a hitch, found in the measurement postulate. Quantum theory appears as a genie, who hands us a tiny volume with the promise that it contains the complete text of Finnegan’s Wake6 and, as a bonus, all the other written works of human history. The book is delivered with an impish warning, however: do not try to read the book, because the moment you open the cover, its contents will be reduced to a single character that forever replaces the original contents. The challenge of quantum information theory is to outwit the genie. Can we find ways to extract more information from a quantum system than we get by a single act of observation? Well, yes, if we can find a way to make an exact copy of a quantum system about which we know nothing initially. We could make such a copy, then a copy of the copy, and so on, until we have enough copies to make multiple measurements of multiple properties and recover, to whatever accuracy we desire, the pre-measurement density function of the unknown state. We recover Finnegan’s Wake, character by character, until the full text is there for the reading. This copying strategy remained a hope until the early 1980s, when it was definitively proven that making an exact copy of a quantum state is not possible— the ‘no-cloning theorem’. The proof is disarmingly simple. Starting with a pure system we wish to copy, ∣ψ 〉, we acquire another pure state of the same dimensionality in a state we call ∣0〉 and form a composite system of the two, ∣ψ 〉A ∣0〉B , using subscripts to distinguish the two component systems. The postulates describe two ways in which systems change: via a unitary operation as told in the time evolution postulate, and via the act of measurement described in the quantum measurement 6

James Joyce’s famously long novel, published in 1939.

1-14

Discrete Quantum Mechanics

postulate. It is easy to dismiss the quantum measurement postulate as promising in our search for a way to clone ψ. We know that measurement changes a state unless we measure relative to an observable for which the state is an eigenvalue. Since we do not know the state ∣ψ 〉, we cannot choose a measurement without danger of destroying the state. Measurement on our state ∣0〉 has the possibility of casting it into the state ∣ψ 〉, but only if we know the observable to measure, which we do not. Suppose there is a unitary operation that acts on our composite state in such a way as to produce the desired effect of leaving us with the original state and an exact copy:

Uˆ ( ψ

0 B) = ψ

A

A

ψ

B.

The cloning operator we seek cannot be unique to the state ψ, since we do not know that state, so it must work for other states also:

Uˆ ( ψ ′

A

0 B) = ψ′

A

ψ′ B.

The inner product of the left-hand sides of the two previous equations is

ψ

A

† 0 B Uˆ Uˆ ψ ′

A

0

= ψ ψ′

B

A

00

B

= ψ ψ ′ A.

This equals the inner product of the right-hand sides of the same two equations:

ψ

A

† ψ B Uˆ Uˆ ψ ′

A

ψ′

B

=

(

ψ ψ′

A

)2 ,

with this equality implying that the inner product of the two unknown states must be either 1 or 0. We can thus only find a unitary operator that can clone a single state, or a single state and a state or states orthogonal to it. Neither possibility realizes a cloning operator for unknown states, thus creating exact clones of unknown quantum states cannot be accomplished. Because mixed states have been shown previously to be a part of a pure composite state, the proof shown above for pure states can be considered universal. The no-cloning theorem is a disappointment in our quest to foil the genie that is quantum theory, but it is not a game-stopper. We will see in future chapters some examples from the huge body of work carried out in recent decades to tease out the power implicit in quantum states, including efforts to produce near-perfect copies of unknown states that will allow us to access much of the information therein.

Exercises 1. From the definition of the Hermitian conjugate of an operator, † y Aˆ x = Aˆ y x ,

show that if ai,j are the elements of a matrix representation of Aˆ , the † corresponding matrix elements of Aˆ in the same basis are aj ,i . 2. Prove the result in equation (1.8) using straightforward minimization relative to the parameters a and b. 1-15

Discrete Quantum Mechanics

Bibliography [Planck1900] Planck M 1900 Zur Teorie des Gesetzes der Energieverteilung in Normalspektrum Verhandlung der Deutscher Physikalischen Gesellschaft 2 237 [Heisenberg1925] Heisenberg W 1925 Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen Z. Phys. 33 879–93 [Schrödinger1926] Schrödinger E 1926 Quantisierung als Eigenwert Problem Ann. Phys. 79 361–7 [vonNeumann1932] von Neumann J 1932 Mathematische Grundlagen der Quantenmechanik (Berlin: Springer) (English translation 1955 Mathematical Foundation of Quantum Mechanics (Princeton, NJ: Princeton University Press))

1-16

IOP Concise Physics

Discrete Quantum Mechanics H Thomas Williams

Chapter 2 Two-state systems

The Universe is a messy place. It has lots of stuff in it, and every bit of stuff seems to be influenced by every other bit. To have any hope at all of understanding the workings of the Universe so as to be able to make predictions about its future, we rely upon fictions like ‘isolated systems’ and ‘simple systems’. We have been quite successful, admittedly by our own assessment, at learning the workings of our environment through the development first of classical mechanics and then, during the last century, of quantum mechanics. Both of these world views have relied on the notions of isolated and simple systems as a first step in working to understand more complex and realistic systems. The idea of an isolated system of two elementary objects proved valuable to Kepler, Newton and others in considering the Earth– Moon system and the Sun–planet system. The construct of the solar system as an object explainable via a series of two-body problems allowed quite accurate predictions of the motions of objects within the Sunʼs gravitational influence. Things became more challenging as we took on the investigation of atomic systems. What seemed at the time to be the two-body analog of the Earth–Sun system—the proton–electron interaction of the neutral hydrogen atom—did not obey Keplerʼs laws of planetary motion. Even with the assumption that the proton (by virtue of its mass) was a static center of eletromagnetic forces, we were challenged by an electron with an infinite number of degrees of freedom, yet nevertheless fewer than those of the planets. It seemed that the planets could move within any elliptical orbit, given an influence to move them there, while bound electrons are limited to discrete orbits. Also, while the planets seemed content to remain in their orbits over vast spans of time, electrons were subject to many influences that compelled them to visit huge numbers of the orbits allowed them. Thus, even this ‘simple’ atomic two-body problem forced us to create over a period of a few decades a new and mysterious way of envisioning its behavior: quantum mechanics. Yet more vexing was the challenge of dealing with atoms with multiple electrons, which, unlike the planets, are often influenced more strongly by one another than by the nucleus.

doi:10.1088/978-1-6817-4125-3ch2

2-1

ª Morgan & Claypool Publishers 2015

Discrete Quantum Mechanics

Our model of the workings of an isolated hydrogen atom is extraordinarily accurate, validating both the quantum view and the assumption that such an atom in many circumstances is sufficiently isolated from external forces. Regardless, for reasons of pedagogy and of pragmatism, we seek to identify and examine quantum systems that are even simpler. The simplest non-trivial quantum system is the twodimensional system, which can manifest itself in one of only two configurations relative to any observable. Even though this, too, is a fiction1, it is a useful and often extremely accurate one. In this chapter we examine several physically important systems that can be approximated as two-state systems.

2.1 Schrödinger’s cat Having declared isolated two-state systems to be fictions, we begin by allowing quantum mechanics’ most famous fictional being to serve as the opening act: Schrödinger’s cat. Recall that this unfortunate creature is placed in a sealed compartment, away from view, along with a device that will eventually release a fatal poison into the air of the compartment. The cat, being Schrödinger’s, is to be given a fully quantum description. To enable a quantitative description of Schrödinger’s cat, I will introduce a physical property, that of ‘animation’. The eigenstates of this property and their corresponding normalized state vectors and density operators are: • ‘alive’, described by state function ∣♡〉 and density operator ∣♡〉〈♡∣; and • ‘dead’, described by state function ∣♠〉 and density operator ∣♠〉〈♠∣. Since no other degrees of freedom of the cat (location, speed of movement, contentment, etc) will be of interest in this exercise, the cat will be described as a two-state system. The eigenstates of animation (as of all other observables) will be orthogonal, thus 〈♡∣♠〉 = 〈♠∣♡〉 = 0. Upon entering the compartment, the cat, in full health, is described by a pure state with density function ρSC (t = 0) = ∣♡〉〈♡∣. The device that triggers the release of poison is usually described as a single radioactive nucleus, the decay of which causes the poison to fill the compartment. This mechanism is a combination of a quantum system—the radioactive nucleus with its decay product—and a profoundly classical one—the thermal processes that cause the irreversible spread of the poison throughout the air within the compartment. We will talk about the consequences of this hybrid mechanism a bit later. A simple description of the radioactive nucleus, sufficient for the purposes of this example, is that if it is an undecayed state at the time the cat enters the compartment, after time interval t it will have decayed with probability 1 − exp( −λt ), λ denoting the decay constant of the nuclear decay involved. Assuming it takes no time at all for the poison release to kill the cat (this is fiction, remember) the cat after the same interval is described as being dead with the same probability, and alive with the complementary probability exp( −λt ). Only if we open the compartment do we observe the cat’s situation in classical terms—alive or dead. 1

We have ignored degrees of fredom such as momentum, for instance.

2-2

Discrete Quantum Mechanics

The state-space postulate suggests two distinct ways we might describe the cat just before we open the compartment door to observe its fate. One is to assume it is in a pure quantum superposition with state vector

⎛ λ ⎞ ∣ψ (t )〉 = exp⎜ − t⎟∣♡〉 + exp(iφ) 1 − exp( −λt ) ∣♠〉 ⎝ 2 ⎠ and the corresponding density operator ρSCp (t ) = ∣ψ (t )〉〈ψ (t )∣. The real parameter φ is the relative complex phase between the two components of the superposition. Understanding what we will see upon opening the compartment door requires use of the quantum measurement postulate. Our examination is a measurement of the animation property. This is a projective measurement, with two measurement operators,

M0 = ♡ ♡

and

1

M1 = ♠ ♠ , 1

that satisfy the requirement ∑ j = 0M j†Mj = ∑ j = 0Mj = I , the latter equality holding since the states ∣♡〉 and ∣♠〉 constitute an orthonormal basis for the Hilbert space of the cat. We evaluate the probability of finding (upon looking) a live cat via the calculation,

(

tr M0 ρSCp (t )M0†

)

which reduces in a straightforward manner to

⎛⎛ ⎞ ⎛ λ ⎞ 〈♡∣ρSCp (t )∣♡〉 = 〈♡∣⎜⎜ exp⎜ − t⎟∣♡〉 + exp(iφ) 1 − exp( −λt ) ∣♠〉⎟ ⎝ 2 ⎠ ⎠ ⎝⎝ ⎛ ⎛ λ ⎞ ⎞⎞ ⎜exp⎜ − t⎟〈♡∣ + exp( −iφ) 1 − exp( −λt ) 〈♠∣⎟⎟⎟∣♡〉 ⎠⎠ ⎝ ⎝ 2 ⎠ = exp( −λt ). The final simplification requires only the orthonormality of the basis vectors, and the result is the one expected from the narrative above. The second straightforward way to describe the cat’s state after time interval t is via the mixed state density function

ρSCm = exp( −λt ) ♡ ♡ + (1 − exp( −λt )) ♠ ♠ . Again using the measurement postulate, we calculate the probability of finding a live cat in this case as

tr(M0ρSCm (t )M0† ) = exp( −λt )tr(M0 ♡ ♡ M0†) + (1 − exp( −λt )tr(M0 ♠ ♠ M0†)) = exp( −λt ). Just as before, the ultimate result comes from the orthonormality of the basis kets, and the result (as before) is the anticipated one.

2-3

Discrete Quantum Mechanics

The two proposed descriptions of Schrödinger’s cat just before the compartment is opened and examined are quite different. The former (ρSCp) is a quantum superposition. It describes a cat that exhibits the properties of ‘alive’ and ‘dead’ simultaneously—a situation that is not a part of our everyday experience. We do not see such cats, a situation enforced by the measurement postulate that has the act of observation, i.e. of opening the compartment, replacing the superposition with a state of either live cat (ρ = ∣♡〉〈♡∣) or dead cat (ρ = ∣♠〉〈♠∣). On the other hand, the latter description as a mixed state (ρSCm ) represents a classical superposition of live cat and dead cat. It describes a cat, even before observation, that is either alive or dead. The two probabilities in the density function, p0 = exp( −λt ) and p1 = 1 − p0, describe our ignorance of which state the cat is in. We know only the likelihood of the two possibilities. We would like to have a measurement that would help us decide which of these two descriptions is an accurate depiction of nature. The only measurement we have mentioned so far cannot do that. Certainly we could run a large number of experiments identical to what we have described, given enough cats and no intervention by the humane society. Opening the door to the enclosure after the same interval in each case would allow us to establish the probability of finding the cat alive, and confirm that it is exp( −λt ). Unfortunately, as we have seen, both the pure state and mixed state descriptions of the cat at time t predict this result, so either is possible. To determine which is the better description, we need to measure another observable. It must be an observable that is incompatible with the observable ‘animation’. This means the observable operators for the two observables do not commute. As a result, the eigenvectors of one operator cannot be eigenvectors of the other (see appendix A). In introducing such an incompatible observable, we place further strain on the credibility of the story by introducing ‘ambiguity’, with orthogonal eigenvectors ∣a↑〉 = 1 (∣♡〉 + ∣♠〉) and ∣a↓〉 = 1 (∣♡〉 − ∣♠〉).2 2 2 Assuming we have found a way of observing the cat by measuring the ambiguity variable, we could replicate the process of: (1) confining the cat; (2) waiting for time t; and (3) repeating the measurement enough times to determine to arbitrary accuracy the probability of observing the cat to be in the state ρ = ∣a↑〉〈a↑∣. Let us therefore examine what the two alternative density functions predict this probability to be. The measurement operator corresponding to this outcome is Ma↑ = ∣a↑〉〈a↑∣, a projective measure. The probability of measuring the cat thus, when the pure state description is assumed, is

(

)

tr Ma↑ ρSCp (t )Ma†↑ = 〈a↑∣(∣ψ (t )〉〈ψ (t )∣)∣a↑〉 = ∣〈a↑∣ψ (t )〉∣2 ⎞ 2 ⎛ ⎛ λ ⎞ 1 ⎜ ⎟ = (〈♡∣+〈♠∣)⎜ exp − t ∣♡〉 + exp(iφ) 1 − exp( −λt ) ∣♠〉⎟ . ⎝ 2 ⎠ ⎠ ⎝ 2 ⎞ ⎛ λ ⎞ 1⎛ (2.1) = ⎜1 + 2 cos( φ)exp⎜ − t⎟ 1 − exp( −λt ) ⎟. ⎝ ⎠ ⎠ 2⎝ 2 2 The author will avoid trying to describe how a cat in either of these states might appear, or what an apparatus to measure this property might consist of, but there might be clues in one or more zombie movies.

2-4

Discrete Quantum Mechanics

When the mixed state description of the cat is used, the same probability evaluates as

(

)

(

tr Ma↑ ρSCm (t )Ma†↑ = exp( −λt )tr Ma↑∣♡〉〈♡∣Ma†↑

(

)

(

+ 1 − exp( −λt )tr Ma↑∣♠〉〈♠∣Ma†↑

)) = 12 .

(2.2)

If we choose to measure ambiguity, it seems we have a good chance to experimentally distinguish between the pure and mixed state descriptions of the cat after time t, since the expressions equation (2.1) and equation (2.2) are not the same. 1 Should our measured value of the probability be in the neighborhood of 2 , however, we would not easily be able to distinguish the mixed state option from the pure state option with cos(φ ) ≈ 0. This case can be resolved by using yet a third variable: ‘pseudoambiguity’, with eigenvectors ∣p↑ 〉 = 1 (∣♡〉 + i∣♠〉) and ∣p↓ 〉 = 1 (∣♡〉 − i∣♠〉). The reader 2 2 should be able, by following the steps of the previous paragraphs, to show that the predicted probabilities of finding the cat in the ∣p↑ 〉 state are: for the pure state

⎞ ⎛ λ ⎞ 1⎛ ⎜1 + 2 sin( φ)exp⎜ − t⎟ 1 − exp( −λt ) ⎟ ⎝ ⎠ ⎠ 2⎝ 2 and for the mixed state

1 . 2 In just that circumstance where our measurement of ambiguity left the issue unresolved (cos(φ ) ≈ 0), measurement of pseudoambiguity can be used to distinguish the pure state option, since sin(φ ) ≈ ±1.3 We have demonstrated that by making appropriate measurements we can determine whether Schrödinger’s cat, while in the compartment, behaves classically or quantum mechanically. We should note, however, that regardless of this possibility, our fictional cat will fail to behave as one would expect from the time evolution postulate. The evolution in time of a quantum system when not observed is described by a unitary operation and is thus intrinsically reversible. Every change in a system, as mediated by the unitary operator U, can be undone by the unitary operator U †. The Schrödinger’s cat scenario has two macroscopic stages that are irreversible: the spread of the poisonous fumes throughout the sealed compartment, and the effect of the poison on the cat. Short of treating each of the molecules of the contents of the compartment as a quantum system at the molecular level, we have no chance of a reversible description of its evolution. This would take us far beyond the comfort of describing it as a two-state system. In subsequent sections we will describe two-state systems that are much better approximations to natural systems, and which will be seen to undergo reversible changes as they evolve unobserved.

Note that a judicious choice of t relative to the decay constant λ can maximize the time-dependent term that distinguishes the two cases.

3

2-5

Discrete Quantum Mechanics

2.2 Expectation value; energy operator In this section we consider two issues that apply not only to two-state systems, but to those of all dimensions. Nonetheless, this seems a natural place to introduce the concepts. In probability theory, the expected value of a property represents the average value of the property after a large number of measurements. For example, if an event can occur N different ways, with absolute probabilities pi, i = 1,…, N , and when the ith situation occurs the property takes on the value ai, the expected value is N defined to be E (a ) ≡ ∑i = 1aipi . For example, the roll of a single fair die produces each of the six possible outcomes with equal probabilities of 1/6. If a game is played in which for each roll the person rolling will receive i dollars if i is an even number, and must pay i dollars if it is odd, the average gain of the roller after many rolls is the expected value

E = ( −1)

1 1 1 1 1 1 1 + (2) + ( −3) + (4) + ( −5) + (6) = , 6 2 6 6 6 6 6

one half-dollar. The quantum analog of the expected value is the expectation value. In the case of ˆ relative to that a pure quantum state ∣ψ 〉, the expectation value of an operator O, state, is

Oˆ

ψ

≡ ψ Oˆ ψ .

As is obvious from the definition, the expectation value depends not only upon the ˆ but also upon the state of the system being considered. Quite often, operator O, however, the state is clear from the context and in such cases explicit reference to it is omitted. An equivalent expression, written in terms of the density function ρ of a pure state, is

Oˆ

ρ

= Tr(ρOˆ ) = Tr(Oˆ ρ) .

A mixed quantum state can be interpreted as a probabilistically weighted sum of pure state density functions, ρm = ∑j pj ρj , so the expectation value relative to a mixed state is expressible as a linear combination of pure state expectation values with the same weights:

Oˆ

ρm

=

∑ pj

Oˆ

j

ρj

.

The interpretation we have given this construct is made clearer when we consider the expectation value of an observable operator relative to a pure state: d −1

Oˆ =

d −1

∑ fi Mi i=0

=

∑ fi

Mi ,

(2.3)

i=0

where the Mi are a collection of projective measurement operators associated with a system property, and the fi are the corresponding measurement outcome values. 2-6

Discrete Quantum Mechanics

Projective measurement operators are also Hermitian Mi†Mi = Mi2 = Mi . Therefore, equation (2.3) is equivalent to

and

thus

obey

d −1

Oˆ =

∑ fi

Mi†Mi ,

i=0

whose ith summand is measurement outcome fi times the probability that fi is observed, exactly paralleling the definition of expected value. The calculation of the expectation value of a property’s observable operator is the average value we would find upon making many replicate measurements of the property relative to the same pure state. This is quite often the value one observes in macroscopic measurements, making the expectation value the value anticipated classically. Relationships among quantum expectation values mirror the results of a classical Newtonian analysis, thus highlighting the importance of the concept introduced here. Look in particular at the Hamiltonian operator H and consider its expectation value relative to an arbitrary time-dependent pure state ∣ψ (t )〉:

H ≡ ψ (t ) H ψ (t ) = ψ (0) exp(iHt )H exp( −iHt ) ψ (0) = ψ (0) H ψ (0) , the final step enabled by the fact that an operator commutes with any function of itself. Assuming (as we do throughout) that H has no explicit time dependence, we conclude that 〈H 〉, relative to any quantum pure state, is constant in time. Note, too, that 〈H 〉 has units of energy4. The classical analog of H is seen to be constant for all isolated systems, and to carry energy units, justifying the identification of the Hamiltonian operator as the operator for the total energy observable.

2.3 Spin one-half Quantum systems described as having spin one-half are ubiquitous in the subatomic world. They range from the most familiar, e.g. electrons, protons and neutrons, to twelve of the sixteen5 members of the standard model of particle physics. Many composite nuclei, atoms and molecules also exhibit a total angular momentum corresponding to spin one-half. Often these systems can be isolated in such a way that the only degree of freedom important for their behavior is that of the total spin variable. It is this kind of situation that we consider in this section. We will discuss the behavior of a generic massive particle of spin one-half, with a magnetic diople moment μ⃗ that is parallel to the total spin vector μ ⃗ = μs ,⃗ so that we can speak of magnetic fields (B ⃗ ) as a means to examine and manipulate the particle. The state space for a spin one-half particle is a two-dimensional Hilbert space. Density functions describing a single particle can be represented by 2 × 2 positive matrices with trace one. Positive matrices are also Hermitian (see appendix A),

4 5

Our choice ℏ = 1 renders the units of energy the inverse of those of time. Seventeen, should we grant the Higgs boson membership in the model.

2-7

Discrete Quantum Mechanics

thus the general form of a density function for a spin one-half, two-state quantum system is ⎛a b ⎞ ⎟, ρ=⎜ ⎝ b¯ 1 − a ⎠ where 0 ⩽ a ⩽ 1 and a(1 − a ) − ∣b∣2 ⩾ 0. Pure states can be represented by a two-component complex (column) state vector

ϕ =

( dc ),

with the normalization condition 〈ϕ∣ϕ〉 = 1 requiring ∣c∣2 + ∣d ∣2 = 1. The density function corresponding to this pure state is

⎛ c2 cd¯ ⎞ ⎟. ρps = ϕ ϕ = ⎜ ¯ 1 − c 2⎠ ⎝ cd The trace-one property of the density function for a pure state is equivalent to normalization of the state vector. This density function is characterized by one real parameter (∣c∣2 ) and one complex parameter (cd¯ ), and describes the system as fully as possible. It is clear, thus, that the state vector—characterized by two complex parameters (c and d)—is over-specified, i.e. it carries a redundant variable. Without loss of generality, we can choose the first component c of the state vector to be real, as is often done. In fact, a common characterization of the state vector for a two-state system is

⎛ ϑ ⎞ ⎜ cos ⎟ 2 ⎟. ∣ϕ〉 = ⎜ ⎜⎜ e iφ sin ϑ ⎟⎟ ⎝ 2⎠ This parameterization maps every distinct spin one-half state vector onto a point on the unit sphere for which ϑ is the polar angle (measured downward from the positive z axis) and φ is the azimuthal angle measured along the equator relative to the x axis. This sphere is referred to as the Bloch sphere. It is a useful visualization of the state vector for a two-state system, but does not conveniently extend to states of higher spin, or to composite states of multiple spin one-half particles. We will make only limited use of this construct for these reasons. To construct a predictive quantum description of a particular two-state system, we will eventually have to construct its Hamiltonian operator in analogy with the corresponding classical situation. We begin, therefore, with a description of the classical dynamics of a spinning positively charged particle, with spin angular momentum s ⃗ and parallel magnetic dipole moment μ ⃗ = μs ,⃗ influenced only by a magnetic field B.⃗ Classical electrodynamics prescribes a potential energy for the particle of U = −μ ⃗ · B ⃗ and a torque on the particle of τ ⃗ = μ ⃗ × B.⃗ The force on the particle is found through

(

)

F ⃗ = −∇⃗U = μ∇⃗ s ⃗ · B ⃗ .

2-8

Discrete Quantum Mechanics

Spatial dependence of the potential energy lies entirely in the magnetic field term. We will be interested in a relatively simple case: a magnetic field with only a z component, and this varying only along the z direction, thus ∂B z ⃗ F ⃗ = μ∇⃗(szBz ) = μsz z . ∂z z If the the magnetic field is non-zero, but does not vary along any direction, then F ⃗ = 0. The equation describing rotational dynamics relates the torque to the rate of change of angular momentum,

τ ⃗ = μs ⃗ × B ⃗ =

ds ⃗ , dt

indicating that the instantaneous change in angular momentum is always normal to the angular momentum itself as well as to the field direction. As a result, the torque causes the angular momentum vector s ⃗ to precess about the field direction B ⃗ at a rate proportional to the magnetic moment and field strength, and the sine of the angle between s ⃗ and B.⃗ Continuing to think classically, we predict the behavior of a particle that, while traveling along the x direction, enters a region of magnetic field in the upward (+z ) direction and increasing in that direction, and finally emerges again into a field-free region. While in the field region it experiences a vertically directed force proportional to its spin alignment with the z axis, i.e. to sz. We would predict, therefore, that a collection of particles with their spin axes randomly aligned would emerge from the field region symmetrically fanned out with velocities in the x–z plane. The particles, while in the field region, would precess about the z direction, emerging with the same sz value they had upon entry, but with sx and sy likely changed. Nature, however, defies these classical expectations. The experiment described here was first performed by Otto Stern and Walther Gerlach in 1922 in Frankfurt, Germany, and was honored by the Nobel Prize in Physics in 1943.6 Stern and Gerlach used an oven to vaporize silver, sending randomly aligned atoms, each of spin one-half, through an inhomogeneous magnetic field, as described above. The classically anticipated fan of atoms emerging from the field region was not observed: rather, only two clusters of emerging atoms were seen. One cluster corresponded to atoms that behaved within the magnetic region as if their spins were maximally aligned along the +z direction, and the other to atoms maximally aligned along the −z direction. Randomized input leading to highly organized output: how could this be? 2.3.1 Spin orientation measurements Returning to the ‘projective measurement devices’ introduced in chapter 1, we will illustrate related measurements that will further motivate our development of the quantum mathematics of spin one-half systems. Note that the experiment of Stern and 6 The prize cited only Stern, possibly due to Gerlach’s active work for the government of Nazi Germany at the time of the award.

2-9

Discrete Quantum Mechanics

Gerlach is, in fact, a measurement: that of the z component of spin angular momentum, a projective measurement with equally spaced measurement values that can be quantified as (d − 1)/2, (d − 3)/2,…, −(d − 1)/2 for a d-dimensional quantum system. Also note that with straightforward changes to the geometry of the entering beam and magnet we can similarly measure the x and y components of the particle’s spin. We will introduce a diagrammatic representation of Stern–Gerlach devices, into which particles are sent and from which they emerge in two (or more, in the case of d > 2) beams with distinct values of a component of angular momentum. These devices are not necessarily measurement devices, since we can merge the emerging beams without examining or altering them and thereby reconstitute the original incoming beam. Should, however, we examine one or more of the output beams, or block one or more of them, the device can be described utilizing the measurement postulate. Figure 2.1 represents a beam of particles sent into a device to measure sz, with those that emerge from the spin-down port blocked from further interaction. This turns the device into a selector for a desired property, in this case spin up. The beam thus prepared is immediately sent into an identical device and we observe that all particles entering the second device emerge, likewise, from its spin-up port. Remeasurement of any property will always yield the same result as long as there are no intervening operations on the particles. (Calculations verifying the results claimed in this and successive figures will be presented in the following section.) Figure 2.2 shows a similar situation but for the fact that the prepared spin-up beam enters a second device that measures sy.7 The classical expectation would be

Figure 2.1. Remeasurement of sz.

Figure 2.2. Sequential measurement of sz, sy. 7 This second device could be designed identically to the first, but physically rotated 90 degrees about the axis of the incoming particle (x axis).

2-10

Discrete Quantum Mechanics

Figure 2.3. Sequential measurement of sz, sy, sz.

that the prepared beam, consisting entirely of particles maximally aligned along the z axis, will exhibit a zero value when measured for sy. Instead, after the second measurement, half the beam exhibits maximal alignment along the +y direction, and the other half along the −y direction. Should we combine the beams from the two ports of the final apparatus without observation, thus treating the combined beams as a pure state of a quantum superposition, the expectation value of sy for the combined beams will be zero, consistent with the classical expectation. Should we now select only those that show +y alignment in the second device and do a final measurement of sz, as shown in figure 2.3, we encounter another challenge to our classical intuition. We begin with a beam of particles and in the first step discard all those not aligned along the +z axis, then, using only those that remain, in the second step we discard those not aligned in the +y direction, so that only particles exhibiting both these properies should remain. Instead, we find a resulting beam that contains half its constituents aligned along −z ! It seems that the particles examined for their y alignment have forgotten that they were all previously aligned along +z, and have split 50−50 between +z and −z. The intermediate measurement of sy = + 12 has created a new future for the particles, which has erased any remnant of its previous spin direction. Note that these results, as counterintuitive as they may be, would be the same if we interchanged sz and sy measurements. We would obtain similar results for other permutations of the three principal axes. These results imply that the three quantum measurement operators sˆx, sˆy and sˆz are mutually incompatible in the following sense. Measurement of any one will provide one of two results, with relative probabilities depending on the state before measurement, but the post-measurement state of each particle will hold no information about its premeasurement state: we cannot know the spin projection eigenvalue of a particle for more than one direction at a time. As ˆ e.g.—do not have a set of mentioned earlier, two incompatible operators—Qˆ and R, identical eigenvectors. This implies directly that the two operators do not commute: ˆ ˆ − RQ ˆ ˆ ≠ 0. We will see that no two of the three spin operators [Qˆ , Rˆ ] ≡ QR commute, thus no measurements of two different corresponding properties are compatible. The failure of classical physics is resolved, as we will show, by the application of quantum physics. Working from the postulates, we can predict the outcome of these and like experiments perfectly. Quantum mechanics provides the best answers we 2-11

Discrete Quantum Mechanics

know to ‘what will happen?’ questions, but as always gives no answer to ‘how can this happen?’ questions, which we judiciously avoid. 2.3.2 Quantum mathematics of spin one-half We begin the process of establishing the quantum language for discussing these processes by making an arbitrary choice of basis vectors for our representations of ⎛ ⎞ Hilbert space elements for a spin one-half particle8. We select the vector ∣z↑〉 = ⎜ 1 ⎟ to ⎝0⎠ ⎛0⎞ represent spin alignment along the positive z axis and ∣z↓〉 = ⎜ ⎟ to represent spin ⎝1⎠ alignment opposite the positive z axis. We construct the operator corresponding to a measurement of spin direction along the z axis to be

sˆz =

1 1 ⎛1 0 ⎞⎟ , σz = ⎜ 2 2 ⎝ 0 −1⎠ 1

1

with eigenvectors ∣z↑〉 and ∣z↓〉 and corresponding eigenvalues 2 and − 2 . We further introduce a pair of companion matrices corresponding to measurement of spin alignment along the x and y directions, respectively

sˆx =

1 1 ⎛ 0 1 ⎞⎟ and σx = ⎜ 2 2 ⎝1 0 ⎠

sˆy =

1 1 ⎛0 i ⎞ σy = ⎜ − ⎟ . 2 2⎝i 0 ⎠

The three matrices σx, σy and σz are known as the Pauli spin matrices9. They obey commutation relations implied by those of the quantum angular momentum 1 operators sˆj = 2 σj :

[σx, σy ] = 2iσz and likewise for the cyclic permutation of the indices into ( y, z, x ) and (z, x , y ). We further claim that the operator measuring spin alignment along any direction identified by unit vector n ⃗ is

sˆn =

1 1 n ⃗ · σ ⃗ = (nxσx + n yσy + nzσz ). 2 2

It is a straightforward exercise to show that sˆn has eigenvectors

∣n ↑〉 =

⎛ 1 + nz ⎞ 1 ⎜ ⎟ and 2(1 + nz ) ⎝ nx + in y ⎠

8

∣n ↓〉 =

1 2(1 + nz )

⎛ nx − in y ⎞ ⎜ ⎟ ⎝− 1 − nz ⎠

(2.4)

While arbitrary, the choices made here are nearly universal for such applications. Named for Wolfgang Pauli, awarded the 1945 Nobel Prize in Physics for the discovery of what we now call the Pauli exclusion principle. 9

2-12

Discrete Quantum Mechanics

1

1

with eigenvalues + 2 and − 2 respectively. Determination of spin orientation in any direction is a projective measurement, so the two measurement operators for spin orientation relative to the n ⃗ direction are

M↑ = n ↑ n ↑

and

M↓ = n ↓ n ↓ .

We will use these elements to examine quantitatively the processes of figures 2.1, 2.2 and 2.3. Each of the illustrated processes begins with a beam of particles in an arbitrary state passing through a device that measures the z-component of the spin, and the selection after measurement of those measured to be spin up, i.e. ∣z↑〉. Except for the case where the incoming beam is in the pure state ∣z↓〉〈z↓∣, there will be particles emerging from the spin-up port. We will for obvious reasons ignore this perverse possibility. Assuming the original beam to be in the (otherwise) most general possible state ρin = ∑j pj ∣ϕj 〉〈ϕj ∣, following the results from section 1.3 we predict that particles will emerge from the spin-up port with probability

⎛ p = tr Mz↑ ρin Mz†↑ = tr⎜⎜ z↑ z↑ ∑ pj ϕj ⎝ j

(

)

⎞ ϕj z↑ z↑ ⎟⎟ = ⎠

∑

z↑ ϕj

2

j

and in the state

Mz↑ ρin Mz†↑ p

= ∣z↑〉〈z↑∣ .

(2.5)

The emerging state is a pure state with state vector ∣z↑〉. Note that in this case, as in all such cases, the denominator in equation (2.5) serves to produce a normalized resulting state. The final remeasuring step in figure 2.1 will produce a probability of producing spin up ( ∣z↑〉〈z↑∣) of

(

)

p = tr Mz↑ ρMz†↑ = tr(( z↑ z↑ )( z↑ z↑ )( z↑ z↑ )) = 1, requiring that the probability of spin down be 0, as desired. The preceding gives an example of an important general rule, applicable to systems with any number of degrees of freedom: If a pure state characterized by state vector ∣ψ 〉 is measured relative to an observable operator that has a nondegenerate eigenstate ∣ϕ〉, the probability of it emerging from the measurement device with pure state vector ∣ϕ〉 is p = ∣〈ψ ∣ϕ〉∣2 . Using this rule, it is straightforward to analyze the remaining two figures. Figure 2.2 consists of two measurements in quick succession. As discussed earlier, the leftmost measurement serves to produce a beam of particles in the pure state ∣z↑〉 to enter the

2-13

Discrete Quantum Mechanics

second device. The probability of each entering particle passing out through the spinup port of the second device is ∣〈z↑∣y↑〉∣2 . Vector forms for the two pure state vectors are found by evaluation of the expression for ∣n↑〉 (equation (2.4)): for n ⃗ = (0, 0, 1) we find ⎛ ⎞ 1 ⎛1⎞ ⎜ ⎟ , thus ∣z↑〉 = ⎜ 1 ⎟ and for n ⃗ = (0, 1, 0) we find ∣y↑〉 = 2 ⎝i⎠ ⎝0⎠

p=

z↑ y↑

2

= (1 0 )

1 ⎛⎜1⎞⎟ 2 ⎝i⎠

2

=

1 . 2

The lower, spin-down port probability must thus be 1 − p = 12 , in accord with the earlier description. If we take only those particles that emerge from the ∣y↑〉 port of the second device in figure 2.2 and send them through a third device for measuring sˆz , we obtain the situation depicted in figure 2.3. The probability that a particle from the ∣y↑〉 port of the second device passes through the third device and emerges from the rightmost 1 ∣z↑〉 port is ∣〈y↑∣z↑〉∣2 , which can be evaluated as in the previous example to be 2 , in agreement with experimental outcomes. 2.3.3 Spin precession As explained earlier in this chapter, we expect a spin one-half particle in the presence of a uniform magnetic field to precess about the field direction based on classical arguments. Let us now examine the quantum behavior in a similar circumstance. We construct the quantum Hamiltonian operator from the classical expression for energy: for a spin one-half particle, magnetic moment of magnitude μ, in a uniform field of magnitude B pointing in the positive z direction, E = −μ ⃗ · B.⃗ From this,

⎛ μB ⎞ ⎟σ , H = −⎜ ⎝ 2 ⎠ z and following from equation (1.3) we construct the unitary time evolution operator as

⎛ ⎛ μBt ⎞ ⎞ ⎟σz⎟ . U = exp( −iHt ) = exp ⎜ i ⎜ ⎝ ⎝ 2 ⎠ ⎠ The key to simplifying the exponential with operator argument is noting that the operator obeys σz2 = I , thus all even powers of σz equal the 2 × 2 identity, and all odd powers equal σz itself. Expanding the exponential in a Taylor series, and separating into even terms and odd terms, produces the Euler formula: exp(ix ) = cos(x ) + i sin(x ). Combining,

⎛⎛ μBt ⎞ ⎞ ⎛⎛ μBt ⎞ ⎞ ⎛ ⎛ μBt ⎞ ⎞ ⎛ μBt ⎞ ⎛ μBt ⎞ ⎟σz⎟ = cos⎜⎜ ⎟σz⎟ + i sin⎜⎜ ⎟σz⎟ = I cos ⎜ ⎟ + iσz sin⎜ ⎟. U = exp⎜i ⎜ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝⎝ 2 ⎠ ⎠ ⎝⎝ 2 ⎠ ⎠ ⎝⎝ 2 ⎠ ⎠

2-14

Discrete Quantum Mechanics

Let use this to see what happens over time to a pure state vector ∣ψ (t )〉 that at time t = 0 is an eigenstate of the sˆx operator with eigenvalue + 12 and that is in a constant z-directed magnetic field. From equation (2.4) with n ⃗ = (1, 0, 0) we obtain 1 ⎛1⎞ ⎜ ⎟ , thus ∣ψ (0)〉 = 2 ⎝1⎠ ⎛ ⎛ μBt ⎞ ⎛ μBt ⎞⎞ 1 ⎛1⎞ ⎜ ⎟ ⎟ + iσz sin⎜ ⎟⎟ ∣ψ (t )〉 = U ∣ψ (0)〉 = ⎜ I cos⎜ ⎝ 2 ⎠ ⎝ 2 ⎠⎠ 2 ⎝1⎠ ⎝

⎛ ⎛ t⎞ ⎞ ⎜ exp ⎝⎜ iπ ⎠⎟ ⎟ 1 ⎜ τ ⎟ , = ⎜ ⎛ t ⎞⎟⎟ 2 ⎜ exp i − π ⎜ ⎟ ⎝ ⎝ τ ⎠⎠ using the definition τ ≡

2π . μB

(2.6)

It is a matter of straightforward evaluation to trace the

time evolution of this state: starting with ∣x↑〉 at t = 0, it becomes (within a complex phase10) ∣y↓〉 at t = τ /4, ∣x↓〉 at t = τ /2 and ∣y↑〉 at t = 3τ /4, returning to ∣x↑〉 at t = τ. This indeed describes a precession with period τ in a complex plane containing the eigenvectors of sˆx and sˆy. 2.3.4 Heisenberg picture The preceding discussion leads us towards another concept worthy of brief mention: the ‘Heisenberg picture’. The version of quantum theory described by the postulates of chapter 1 is called the ‘Schrödinger picture’, in which observable operators carry no time dependence, and state functions and density functions change with time, as described in the time evolution postulate. A completely equivalent version, called the Heisenberg picture, has the state vectors carrying no time dependence and the observable operators evolving with time. In examining the measurement postulate we saw that for pure states, probabilities associated with pure states take the form (from equation (1.5)) 〈ϕ∣M †M ∣ϕ〉, where M is a measurement operator. Since density operators for mixed states can be expressed as linear combinations of pure states, it follows that the most general probability is expressable as a linear combination of terms of the form 〈ϕ∣M †M ∣ψ 〉. It is of interest now to examine how such expressions evolve with time. Let Oˆ be an operator in the Schrödinger picture, and consider the expression 〈ϕ∣Oˆ ∣ψ 〉 at a fixed time t = 0. At later times the state vectors evolve into ∣ϕ(t )〉 = U (t )∣ϕ〉 and ∣ψ (t )〉 = U (t )∣ψ 〉, so the expression becomes

ˆ (t ) ψ . ϕ U (t )†OU The Heisenberg picture follows from the obvious observation that this can be seen as ˆ (t ) with respect the matrix element of the time-dependent operator Oˆ (t ) = U (t )†OU 10 Note that an overall phase exp(iφ ) multiplying a state vector has no relevance, in that it disappears on creation of the corresponding density operator.

2-15

Discrete Quantum Mechanics

to static state vectors ∣ϕ〉 and ∣ψ 〉, and that results of measurements can always be expressed in terms of such matrix elements. We will follow the logic of section 2.3.3 to look at the time evolution of the spin one-half operators. We begin by considering, as we did there, the Hamiltonian associated with a uniform magnetic field along the positive z direction and its associated time evolution operator

⎛ μBt ⎞ ⎛ μBt ⎞ ⎟ + iσz sin⎜ ⎟. U = I cos ⎜ ⎝ 2 ⎠ ⎝ 2 ⎠ The evolution of the sˆx operator in the Heisenberg picture is as follows:

⎛ μBt ⎞ ⎛ μBt ⎞⎞ ⎛ μBt ⎞ ⎛ μBt ⎞⎞ ⎛ 1 1⎛ ⎟ − iσz sin⎜ ⎟⎟σx⎜I cos⎜ ⎟ + iσz sin⎜ ⎟⎟ U † σxU = ⎜I cos⎜ ⎝ 2 ⎠ ⎝ 2 ⎠⎠ ⎝ 2 ⎠ ⎝ 2 ⎠⎠ ⎝ 2 2⎝ ⎞ ⎛ μBt ⎞ ⎛ μBt ⎞ ⎛ μBt ⎞ ⎛ μBt ⎞ 1⎛ ⎟σx − i cos⎜ ⎟sin⎜ ⎟( σzσx − σxσz ) + sin2⎜ ⎟σzσxσz⎟ = ⎜cos2⎜ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎠ 2⎝ =

1 (cos(μBt )σx + sin(μBt )σy ). 2

In this picture, the operator precesses with the same period that we saw the state vector precess—τ = 2π /μB —beginning as sˆx, becoming sˆy after a quarter-period, −sˆx after one-half period, −sˆy after three-quarter period and returning to sˆx after time τ.

2.4 Ammonia molecule An important molecular system—the NH3 molecule—can be isolated in such a way as to have it behave as a two-state system. Its gross structure is that of an equilateral triangle of hydrogen atoms with a nitrogen situated along the axis of symmetry of the triangle, some distance beyond its plane (figure 2.4). The particular way electrons are shared among the atomic constituents dictates this pyramidal structure. In general, this molecule has degrees of freedom related to its translation motion as a whole, to rotations about its center of mass, vibrations of the atoms relative to their equilibrium positions, and another type of motion that we will focus on in this section. We assume isolation of the molecule so that its translation and vibrational

Figure 2.4. Two tetrahedral ammonia molecule structures: filled circles represent H and open circles, N.

2-16

Discrete Quantum Mechanics

motions can be ignored, and further assume a small constant rotation of the molecule about its symmetry axis11. This rotation is important in that it establishes an up/down distinction relative to the location of the N relative to the three H: the nitrogen atom can occupy symmetric locations above and below the plane of the hydrogens. Using very low energy stimuli, we can limit the changes of the molecule solely to the transfer of the N from its up state (∣up〉) to its down state (∣down〉) and back. Ammonia, in this state of isolation, behaves as a quantum two-state system. A two-by-two Hamiltonian matrix describes the low energy dynamics of the molecule. The energy of the nitrogen atom in its up-state (〈up∣H ∣up〉) and its downstate (〈down∣H∣down〉) should be equal, based on the symmetry of the two configurations. We will call this value ϵ. We have noted that there is a probability, albeit small, that the molecule in one of these states can transform into the other, and symmetry again suggests that we can assign equal rates to the up–down and down– up transitions. We accomplish this by setting 〈up∣H∣down〉 = 〈down∣H∣up〉 = a and with foresight we choose a to be a real constant. With these choices, the Hamiltonian matrix in the ∣up〉, ∣down〉 basis is

H = ϵI + aσx, where we utilize the Pauli matrix σx for its form and familiarity, but not in relationship to spin. The unitary time-evolution matrix can be found using methods like those in the previous two sections:

U (t ) = exp( −iϵt )(I cos(at ) − iσx sin(at )). Using this we can track the time behavior of a molecule initially in the ∣up〉 state:

U up = exp( − iϵt )(cos(at ) up − i sin(at ) down ). Recalling that a multiplicative phase of a state vector carries no physical information, we find that the state representing a nitrogen atom above the plane becomes, in time π t = 2a , one with the nitrogen atom below the plane. After the same additional amount of time, it returns to its ‘above’ position. If, however, we start with a molecule in the superposition state 1 (∣up∣〉 + ∣down〉), we see it evolve as ∣+〉 = 2

U + = exp( −iϵt )(cos(at ) + − i sin(at ) + ) = exp( − i(ϵ + a )t ) + , showing it to be a stationary state, with only a trivial time-dependent phase carrying no physical significance. It is simple to show that this state is an eigenvector of H (the energy operator) with eigenvalue E+ = ϵ + a. Likewise, the state ∣−〉 = 1 (∣up∣〉 − ∣down〉) is also stationary, with no non-trivial time dependence, 2

and it is a second eigenvector of H with the lower eigenvalue E− = ϵ − a. The behavior of this quantum system is a tight analog to that of the classical problem of two coupled pendula. Picture two identical children’s swings supported 11 Most quantum variables exhibit zero-point energy, which causes their lowest-energy configuration to be other than zero.

2-17

Discrete Quantum Mechanics

by a single horizontal bar. Each swing will move with the same frequency if it swings alone, reflecting the equal energies associated with the quantum states ∣up〉 and ∣down〉. However, if one swing is set into motion with the other at rest, eventually (due to the slight flex in the bar as it moves) the other swing will start moving with an amplitude that increases until it takes all the energy of the system, leaving the first swing hanging stationary. Over time, the process will operate in reverse and the two swings return to their original states of motion and rest. If we set both swings into identical synchronized motion, they continue together, swinging at a slightly higher frequency than that of either swing moving alone; if they are put into identical but opposing motion, they will continue in this motion, moving at a slightly lower frequency than either swing alone. These two types of joint motion are called the normal modes of the swing system, reflecting the stationary states in the quantum behavior of the ammonia molecule. This section began by referring to the ammonia molecule as ‘important’. The particular application that inspired that claim was the ammonia maser. This device was developed in the years 1952–1954 by Nickloay Basov, Aleksandr Prokarev and Charles Townes, and they received the Nobel Prize in Physics for it in 1964. The maser—a device for amplifying microwave radiation—was the precursor to the amplification of light in the visible spectrum by way of the now ubiquitous laser. Here we will give a brief qualitative explanation of the operation of the ammonia maser, making reference to processes discussed earlier in other contexts. To begin, let us relax the idealized picture we have given for ammonia’s molecular structure. The two states ∣up〉 and ∣down〉, earlier said to have identical energies, in fact have a small but significant energy difference on the order of 10−5 eV, as a result of the zero-point rotational motion of the molecule that distinguishes the two states. Furthermore, due to the electronic structure of the molecule, there is an electric dipole moment p ⃗ that points along the molecular axis of symmetry, away from the location of the nitrogen atom. In the presence of an electric field ,⃗ , classical dynamics predicts a potential energy U = −p ⃗ · ,⃗ and a torque on the molecule of τ = p ⃗ × ,⃗ . This torque would suggest that the dipole would move to align with the field direction, but the rotation of the molecule about its axis prohibits this and should lead to a precession of the molecular axis about the field axis. But precisely as we found when we considered a spinning magnetic dipole in a uniform magnetic field, the actual behavior is otherwise. Our molecule, once placed in a uniform electric field, will act as if its (electric dipole) axis is aligned either in the direction of ,⃗ , or in the opposite direction. Furthermore, if the field has a gradient along its direction, the electric dipole will experience a force along the field axis and proportional to the dipole moment component along that axis. Obviously, we can construct an electric field device that serves as a projective measurement tool for the molecule’s electric dipole moment component in any direction. This device will allow us to take a beam of randomly oriented molecules and convert it into two beams, one of ∣up〉 molecules, and the other of ∣down〉 molecules. The first stage of our stylized maser consists of a beam of ammonia molecules, sent through a projective measurement in order to select out those in the state of

2-18

Discrete Quantum Mechanics

higher energy. These ‘excited’ molecules are then sent into a resonance cavity that supports electromagnetic radiation of 2.4 GHz, this being the frequency of microwave photons emitted should the excited molecule transform to its opposite, lowerenergy state. We need only ‘seed’ this cavity with photons of the appropriate frequency to cause the molecules to de-excite via a process known as stimulated emission of radiation. We extract radiation from the cavity consisting of many more photons than we used to seed it, due to the additional ones from molecular deexcitation. The process, then, is aptly referred to as Microwave Amplification by Stimulated Emission of Radiation.

2.5 Photons We conclude this chapter with a brief mention of a two-state system that could receive much greater attention. Since our goal here is only to introduce the formalism of two-state systems, and the theoretical considerations used to describe photon systems are like those for spin one-half systems, here we only draw attention to the parallels. Let us focus on the properties of a collimated beam of monochromatic photons, as might be emitted by a gas laser. Each photon in the beam has one unconstrained degree of freedom: polarization. A photon can be plane polarized in any direction in the plane perpendicular to the beam direction. Horizontal polarization and vertical polarization are two orthogonal states for a so-called plane-polarized photon in a beam moving in a horizontal direction, and these are often chosen to be polarization basis states. Any other two orthogonal directions in the same plane can also be thought of as bases for a plane-polarized photon’s description12. Complex superpositions of basis states describing plane-polarized photons can form bases for circular-polarized or elliptical-polarized photons. In these cases, the electric and magnetic field directions spiral forward rather than oscillating in a single plane. Simple devices are available for the creation and manipulation of photon states. Birefringent crystals can be cut so as to serve as projective measurement devices for plane polarization: an incoming beam of randomly polarized photons emerges as two beams, plane-polarized in orthogonal directions. Quarter-wave plates are slabs of birefringent crystal cut at such a thickness so as to make plane-polarized light entering the slab emerge as circularly polarized, and vice versa. Devices such as these serve the same roles for photons as Stern–Gerlach magnets do for spin one-half systems with magnetic moments. Due to their ease of manipulation, photons have taken on many roles in quantum information devices based upon two-state system analyses.

12 The polarization direction for a plane-polarized photon, along with the direction of its propagation, defines the plane in which the electric field component of its electromagnetic field oscillates.

2-19

IOP Concise Physics

Discrete Quantum Mechanics H Thomas Williams

Chapter 3 Entanglement

We have demonstrated that the simplest of quantum examples—an isolated two-state system—provides a multitude of behaviors outside our experience, outside our understanding. A mythical cat that can be both dead and alive at the same moment, as proxy for a real electron that can be spinning clockwise and counterclockwise simultaneously. A system like the proton that chooses to direct its angular momentum vector instantly towards or directly counter to the direction in which we point, using a magnetic field rather than a forefinger. Properties of a system that can never be known simultaneously, like the y and z components of the spin vector. Perhaps the closest we come to grasping properties such as these is when we throw up our hands, say it just cannot be that way, and then simply ‘shut up and calculate’1. Yet we need only take another tiny step into the quantum forest to encounter behavior stranger yet—the strangest quantum scientists have so far encountered: entanglement. Entanglement of two quantum systems gives rise to what Albert Einstein derisively referred to as ‘spooky action at a distance’. A 1935 paper by Albert Einstein, Nathan Rosen and Boris Podolsky titled ‘Can quantum-mechanical description of physical reality be considered complete?’ [Einstein1935] first brought prominent attention to this attribute of the then new theory. This paper discussed a thought experiment involving two quantum systems with correlated (entangled) positions and momenta. After the two systems were allowed to move a great distance apart, one particle was measured for its momentum, implying the momentum of the other via the correlation. At the same time, the other particle was measured to determine its position, providing simultaneous knowledge of both the position and the momentum of the particle. Quantum mechanics strictly forbids 1

This phrase, describing Niels Bohr’s Copenhagen interpretation of quantum mechanics, is likely due to David Mermin.

doi:10.1088/978-1-6817-4125-3ch3

3-1

ª Morgan & Claypool Publishers 2015

Discrete Quantum Mechanics

this via the Heisenberg uncertainty principle2. According to the paper, this so-called ‘EPR paradox’ (named after Einstein, Podolsky and Rosen) points to an incompleteness in the basic principles of quantum mechanics. It took over a half-century before this challenge to the theory was resolved experimentally—in favor of quantum mechanics. Entanglement has been proven to be real. We will explore here some intriguing consequences of this phenomenon.

3.1 Entangled qubit pairs Information theory, beginning in the middle of the twentieth century, has explored the notion of information as a quantifiable, physical quantity. Information storage, transmission, manipulation and encryption have become subject to quantitative study with profound practical consequences. The basic unit of information has come to be known as the bit, short for the term ‘binary digit’. It is the abstract notion of a quantity that can take on one of only two values, most often 0 and 1. Physically, bits can be manifested in many ways: a hole in a punched card, or the absence of a hole; a positively charged capacitor, or one negatively charged; a wire at a potential of +5 volts, or 0 volts; etc. Quantum information theory examines the same properties of information, but with a purely quantum-mechanical basic unit—the qubit, from the term ‘quantum bit’. A qubit can be defined as an abstraction, that of a variable that can take on two values, like 0 and 1 as well as any complex-valued superposition thereof. Physical representations of a qubit include the spin states of a spin one-half particle and the polarization states of a photon. The term qubit is used to describe both the abstraction and one of its physical manifestations, with context making it clear which is intended. Herein we will most often use qubit to refer to a physical two-dimensional quantum system. The intriguing difference between a bit and a qubit is that the latter can take on an infinite number of configurations—e.g. all the points on the Bloch sphere—as long as we do not examine it. The moment we try to look, i.e. to make a measurement of the state of a qubit, it exhibits one of only two values, behaving for all the world like a classical bit. One of the most challenging aspects of quantum information theory is that of trying to extract some of the copious storage capacity of groups of qubits, which infinitely exceeds that of an equivalent number of classical bits. We want to characterize a quantum system in a pure two-dimensional state. A single qubit in a pure state can be characterized in terms of normalized basis states ∣0〉 and ∣1〉, and has the general form

ϕ = a 0 + a′ 1 , with a a real constant and a′ a complex one, obeying a 2 + ∣a′∣2 = 1 to assure normalization. A pair of qubits, for instance with the first in state ∣0〉A and the second

2

In the version of quantum mechanics that admits position and momentum as continuous variables, their observable operators do not commute. As a result, one cannot simultaneously know the two variables to arbitrarily high precision.

3-2

Discrete Quantum Mechanics

in state ∣1〉B , can be jointly represented, according to the composite systems postulate, as

0

A

⊗ 1

B

≡ 0

A

1

B

≡ 01 .

It is quite common to refer to the distinguishable members of a pair of qubits as ‘A’ and ‘B’ and equally common to speak of them as ‘Alice’s qubit’ and ‘Bob’s qubit’, so as to easily discuss situations in which the two qubits are operated upon individually. Most often, the tensor product symbol ⊗ is omitted. In many circumstances we talk about the product of two basis states with a single ket (e.g. ∣01〉), always assuming that the first of the two integers within refers to the state of Alice’s qubit, and the second to Bob’s. A pair of qubits in states (a∣0〉 + a′∣1〉)A and (b∣0〉 + b′∣1〉)B have a joint representation following from distributive multiplication as follows:

(a 0 + a′ 1 )A ⊗ (b 0 + b′ 1 )B = ab 00 + ab′ 01 + a′b 10 + a′b′ 11 . A standard basis set of state vectors for the four-dimensional Hilbert space of two qubits is {∣00〉, ∣01〉, ∣10〉, ∣11〉}, and thus the most general pure state for the pair can be characterized as

ψ = a 00 + b 01 + c 10 + d 11 ,

(3.1)

where due to the arbitrary nature of the overall phase of a state vector we can pick a to be real and positive. Normalization requires a 2 + ∣b∣2 + ∣c∣2 + ∣d ∣2 = 1. All such states are pure, but only those which obey ad = bc can be factored into separate pure states for each individual qubit. Others, to some degree at least, are defined to be entangled. Our examination of entangled states will be simplified by making use of the Schmidt decomposition, which in our context can be stated as follows: If ∣ψ 〉 is a pure state of two qubits, there is a choice of basis for Alice and one for Bob, for which

ψ = a′ 00 + d ′ 11

(3.2)

and both a′ and d ′ are real and non-negative, obeying a′2 + d ′2 = 1. This state is entangled as long as neither a′ nor d ′ is zero. Alice and Bob are free to choose different bases for their qubits, and unless we specify otherwise we assume that when Alice or Bob makes a measurement, each makes it relative to her or his own chosen basis3. The Schmidt decomposition expresses the fact that should we have a state like that in equation (3.1), Alice and Bob can use unitary operations to independently transform their bases into new ones for which the same state can be expressed as in equation (3.2). 3 As we shall see, action on a qubit by a properly chosen unitary operator before measurement with respect to a fixed basis is equivalent to measurement with respect to a different basis.

3-3

Discrete Quantum Mechanics

If we possess a single unentangled qubit, we can attempt to learn something about its state using measurements. If we know that it is prepared in one of the two eigenstates of a particular measurement operator, using a single measurement we can learn its state exactly, possibly leaving that state intact while doing so. Without such specific information, we can still often extract considerable information should we have many replicate copies of the qubit, by making multiple measurements. This is also true in the case where our qubit (or multi-qubit system) is in a non-entangled composite state. The situation is more complicated, and thus more interesting, if our qubit is entangled with one or more others. Entanglement is considered a resource, enabling us to do things much more efficiently than can be done via classical means—things like computation and communication—and other things that are impossible classically, like teleportation or totally secure encryption. As a resource, we would like to quantify entanglement as much as possible. Compare two states in the form of equation (3.2), one with a′ = .9999, d ′ = 0.01 and the other with a′ = d ′ = 1 . 2

The former behaves with high probability like the pure unentangled state ∣00〉, and the latter is entangled to the greatest degree possible. Clearly there is a hierarchy of entangled states, but no single measure of the entanglement property has been found that predicts for different applications which states will be the most useful. We will herein be content with understanding which states are unentangled (i.e. those with factorizable state functions) and which states are partially and fully entangled. Unitary operations by Alice on her half of a shared entangled pair, or by Bob on his half, are referred to as local operations. Such manipulations cannot change the degree of entanglement of the pair, since they are equivalent to basis changes for Alice or Bob and the resulting state can be re-expressed in its original form via the Schmidt decomposition. Within this chapter we will make a change of notation, so as to refer to the typical single-qubit unitary operations in the way most often used in discussions of entanglement and quantum information theory: we introduce the symbols Xˆ , Yˆ , Zˆ to be identified with the Pauli operators σx, σy, σz, respectively. A fully entangled state is one that is equivalent, via local operations, to the state

∣η00〉 ≡

1 (∣00〉 + ∣11〉), 2

and this and the other three fully entangled states

∣η01〉 ≡

1 (∣01〉 + ∣10〉) 2

(3.3)

∣η10〉 ≡

1 (∣01〉 − ∣10〉) 2

(3.4)

∣η11〉 ≡

1 (∣00〉 − ∣11〉) 2

(3.5)

3-4

Discrete Quantum Mechanics

are known collectively as Bell states4, forming an orthonormal basis for two-qubit state functions. The Bell states can be transformed one into another by purely local operations. Let us examine the effect on ∣η00〉 when Alice applies her operator (ZˆA) while Bob does nothing (IB):

1 ˆ ZˆA ⊗ IB∣η00〉 = ZA ⊗ IB(∣00〉 + ∣11〉) 2 1 = (Zˆ ∣0〉)A∣0〉B + (Zˆ ∣1〉)A∣1〉B . 2

(

(3.6)

)

Using a matrix representation for Zˆ and vector representations for ∣0〉 and ∣1〉, we find ⎛ ⎞⎛ ⎞ ⎛ ⎞ Zˆ 0 = ⎜ 1 0 ⎟ ⎜1 ⎟ = ⎜ 1 ⎟ = 0 ⎝ 0 −1⎠ ⎝ 0 ⎠ ⎝ 0 ⎠ and

⎛ ⎞⎛ ⎞ ⎛ ⎞ Zˆ 1 = ⎜ 1 0 ⎟ ⎜ 0 ⎟ = −⎜ 0 ⎟ = − 1 . ⎝ 0 −1⎠ ⎝1 ⎠ ⎝ 1⎠ Inserting these results into equation (3.6), we see

1 (3.7) ZˆA ⊗ IB∣η00〉 = ( ∣0〉A∣0〉B − ∣1〉A∣1〉B ) = ∣η11〉. 2 Similar calculations yield ∣η01〉 = XˆA∣η00〉 and ∣η10〉 = XˆAZˆA∣η00〉. All these transformations are accomplished solely by Alice applying unitary transformations on her qubit. All other transformations among the Bell states can be accomplished by Alice, as can be shown using Alice’s Zˆ and Xˆ operators.

3.2 Quantum gates We now begin to talk about operations on one or more qubits that will manipulate them in useful ways, and to do so we introduce the notion of quantum gates, with diagrams that usefully illustrate complex multi-step operations. All quantum gates are represented by unitary operations that act on one or more qubits. The simplest gates are those that transform one qubit and are represented by 2 × 2 unitary matrices. The action of these gates is reversible, as will also be the case for multiqubit gates. Since the Hermitian conjugate of a unitary operator is also unitary, and † since Uˆ is the inverse of unitary operator Uˆ , for every gate there is another that ‘undoes’ its transformation. We are already familiar with three common one-qubit gates—Xˆ , Yˆ and Zˆ —and we introduce a fourth, the Hadamard gate 1 ⎛⎜1 1 ⎞⎟ , Hˆ ≡ 2 ⎝1 −1⎠

4

After physicist John S Bell, a pioneer in understanding the full power of entangled states.

3-5

Discrete Quantum Mechanics

Figure 3.1. Single qubit gate.

Figure 3.2. CNOT gate.

which we will find invaluable in the applications to be discussed. In figure 3.1 we represent the evolution of a single qubit that begins in the state ∣0〉, is transformed by the operation of a gate operator Uˆ , and is then in the state ∣ψ 〉 = Uˆ ∣0〉. The horizontal line tracks the qubit through the process, with the time axis running from left to right. A two-qubit gate operates on two qubits in a way that cannot be factored into a sequence of single-qubit operations. It can be represented by a 4 × 4 matrix, but we will not utilize such representations in our discussion of quantum gates. The most important of such operations is the controlled-NOT gate (CNOT). The importance of this gate comes from its universality, i.e. any unitary operation on two or more qubits can be decomposed into a sequence of CNOT gates and single-qubit gates [DiVencenzo1995]. The name we use for this particular gate is derived from the classical NOT gate that transforms a 0 bit into a 1 and vice versa, and its quantum ˆ which is often called the quantum NOT gate. In a CNOT gate the input analog X, qubits are referred to as the control qubit and the target qubit. Its operation is straightforward: when the control qubit enters the gate in the state ∣0〉, the target qubit is left unchanged; when the control qubit is in the state ∣1〉, an Xˆ (NOT) transformation is applied to the target qubit. In either case, the control qubit is unchanged. In figure 3.2 we symbolically represent a CNOT gate operating on the initially unentangled state ∣ϕ〉∣ψ 〉. When the two initial state vectors are written as ∣ϕ〉 = a∣0〉 + b∣1〉, with neither a nor b equal to zero, and ∣ψ 〉 = c∣0〉 + d ∣1〉, with neither c nor d equal to zero, the process of figure 3.2 can be broken down as

CNOT ϕ ψ = a 0 (c 0 + d 1 ) + b 1 (c 1 + d 0 ) = ac 00 + ad 01 + bd 10 + bc 11 . From our earlier observation, we find this state to be entangled unless c 2 = d 2, i.e. unless the target qubit is changed at most by an overall phase under the action of Xˆ . We demonstrate the entangling capability of the CNOT gate by showing how to create the Bell basis states from a starting point of ∣00〉. Consider the two-qubit circuit in figure 3.3, in which both qubits begin in the state ∣0〉. The control qubit is acted upon by a Hadamard gate that transforms it into the state 1 (∣0〉 + ∣1〉). This 2 and the target qubit are then sent into a CNOT gate, producing the entangled 3-6

Discrete Quantum Mechanics

Figure 3.3. Bell state creation.

ˆ ˆ state ∣η00〉. We noted at the end of the previous section how one-qubit gates Xˆ , XZ ˆ applied to the target qubit, will transform ∣η 〉 into ∣η 〉,∣η 〉 and ∣η 〉, and Z, 00 10 01 11 respectively, so the circuit of figure 3.3 can produce each of the Bell states, depending upon the choice of Uˆ .

3.3 Utilizing entanglement 3.3.1 Superdense coding Consider communication between two parties distant from one another in the following sense: one party translates a message into binary code and transmits the bits of the coded message to the other party, who decodes and reads the message. Classically, the amount of information sent with each bit transmitted is, well…one bit. Is there a way to send messages more efficiently by sending qubits rather than bits? On the face of it, this should not be possible since despite the prodigious amount of information stored in each qubit, it collapses into behaving just like a classical bit once we look at it. Using entanglement as a resource, however, we will show a simple way to double the amount of information transmitted per qubit. Alice and Bob, anticipating the desire to send a binary-coded message some time in the future, create an entangled pair of qubits in Bell state ∣η00〉, each taking possession of one of the pair before they separate. Alice wants to send to Bob a twodigit binary string, i.e. two bits, as part of a message she plans to transmit. As we saw in the entanglement exercise in the previous section, Alice can transform the entangled pair into any of the other three Bell states using operations on her qubit alone. Thus, by a pre-arranged agreement with Bob, she will convey to him Bell state ∣η00〉 to represent the string ‘00’, ∣η01〉 to represent the string ‘01’, ∣η10〉 to represent the string ‘10’ and ∣η11〉 to represent the string ‘11’. To send message ‘00’, therefore, she simply sends her half of the pair to Bob, who upon receipt has the entangled pair ∣η00〉; to send message ‘01’, she applies Xˆ to her qubit before sending it to Bob, who ˆ ˆ upon receipt now has the entangled pair ∣η01〉; to send message ‘10’, she applies XZ to her qubit before sending it to Bob, who upon receipt now has the entangled pair ∣η10〉; and to send message ‘11’, she applies Xˆ to her qubit before sending it to Bob, who upon receipt now has the entangled pair ∣η11〉. Since the four Bell states constitute an orthonormal basis for a two-qubit system, Bob can measure in that basis and know with certainty which message Alice has sent. Another way for Bob to accomplish the same task would be for him to transform the state he receives into one of the four states ∣00〉, ∣01〉, ∣10〉, ∣11〉 and then measure 3-7

Discrete Quantum Mechanics

with respect to this basis, i.e. using measurement operators M00 = ∣00〉〈00∣, M01 = ∣01〉〈01∣, M10 = ∣10〉〈10∣ and M11 = ∣11〉〈11∣. The transformations needed are straightforward to deduce. Recall that in our discussion of figure 3.3 we saw that ∣η00〉 = CNOT Hˆ control∣00〉. The process is reversible, and both gates are their own inverses, so if Bob wishes to go from ∣η00〉 back to ∣00〉 he needs to apply the two operations in the opposite order: Hˆ control CNOT. It is a simple exercise to show that the same two-gate sequence will change ∣η01〉 into ∣01〉; ∣η10〉 into ∣11〉; and ∣η11〉 into ∣10〉. 3.3.2 Teleportation The no-cloning theorem prohibits making perfect copies of qubits, but does not prevent us from moving a qubit without damage from one place to another, making use of a noiseless quantum (communication) channel. Imagine that Alice is in possession of a qubit ∣ϕ〉 = a∣0〉 + b∣1〉. She does not know its form and it is critical to a larger quantum process that she is unable to conduct. She knows that Bob can complete the process only if he has this particular qubit. How can she move ∣ϕ〉, intact, to Bob, who is presumably quite far away? If there is a noiseless quantum channel connecting Alice and Bob, Alice need only place this qubit in the channel and send it on its way. There may be technological challenges to doing this, however. For instance, perhaps the qubit is encoded in a spin one-half atomic state, and it is unlikely that an atom could make the trip without interacting with other objects that might change its state—the channel for atoms is ‘noisy’. On the other hand, if the information were carried by a photon, the chances of noiseless transmission would be much greater, and so Alice undertakes to move the information from one qubit (the atom) to another (a photon). This she accomplishes by using the quantum circuit pictured in figure 3.4. Describing the two qubits as ∣ϕ〉 = a∣0〉 + b∣1〉 and ∣ψ 〉 = c∣0〉 + d ∣1〉, the first CNOT gate transforms the product state ∣ϕ〉∣ψ 〉 into

a 0 (c 0 + d 1 ) + b 1 (d 0 + c 1 ) = (ac 0 + bd 1 ) 0 + (ad 0 + bc 1 ) 1 . That state is changed by the second CNOT gate into

(ac 0 + bd 1 ) 0 + (bc 0 + ad 1 ) 1 = 0 (ac 0 + bc 1 ) + 1 (bd 0 + ad 1 ) , which the last gate takes to

0 (ac 0 + bc 1 ) + 1 (ad 0 + bd 1 ) = (c 0 + d 1 )(a 0 + b 1 ) in the form of the original pure state with its constituents swapped. Now Alice can safely send the qubit to Bob via a noiseless quantum photon channel.

Figure 3.4. Qubit swapping.

3-8

Discrete Quantum Mechanics

Figure 3.5. Quantum teleportation.

A yet more interesting challenge arises if Alice and Bob share no noiseless quantum channel, but can communicate via a noiseless classical channel through which they can pass classical bits. Alice can still safely pass ∣ϕ〉 to Bob using a process called quantum teleportation5, as long as they have been prudent enough to have previously shared halves of an entangled pair of qubits. To accomplish the task, Alice first entangles the qubit ∣ϕ〉 with her half of the entangled pair using a CNOT gate with ∣ϕ〉 as the control qubit, then she sends her first qubit (that remains in state ∣ϕ〉) through a Hadamard gate, and after that she measures her two qubits. Using a classical channel, perhaps her cell phone, she tells Bob the result of her measurement. With this information, Bob chooses a unitary gate to apply to his half of the shared pair, which transforms it into the exact form of Alice’s original bit ∣ϕ〉. This process is depicted in figure 3.5, in which the vertical arrow represents Alice’s classical communication with Bob. To follow the logic of this process quantitatively, assume that the state shared by Alice and Bob is the Bell basis state ∣η00〉. The kets in order from top-to-bottom in figure 3.5 become the kets from left to right in the following expressions. The starting three-qubit state is 1 (∣0〉∣0〉 + ∣1〉∣1〉). (a∣0〉 + b∣1〉) 2 The CNOT gate acts on Alice’s two qubits (the two leftmost in these expressions) to produce 1 1 (∣1〉∣0〉 + ∣0〉∣1〉) (∣0〉∣0〉 + ∣1〉∣1〉) + b∣1〉 a∣0〉 2 2 and the subsequent action of the Hadamard gate transforms this into 1 1 1 1 (∣1〉∣0〉 + ∣0〉∣1〉) (∣0〉 − ∣1〉) (∣0〉∣0〉 + ∣1〉∣1〉) + b (∣0〉 + ∣1〉) 2 2 2 2 1 1 1 = ∣0〉∣0〉(a∣0〉 + b∣1〉)B + ∣0〉∣1〉(a∣1〉 + b∣0〉)B + ∣1〉∣0〉(a∣0〉 − b∣1〉)B 2 2 2 1 + ∣1〉∣1〉(a∣1〉 − b∣0〉)B , 2 The term teleportation has been borrowed from science fiction, where it typically refers to the transfer of things (often beings) between distant locations without the inconvenience of traveling through the intervening space. 5

3-9

Discrete Quantum Mechanics

where in the final, re-factored form, we explicitly label Bob’s qubit for clarity. Alice’s measurement produces one of four outcomes for her two qubits. • ∣0〉∣0〉, in which case Bob’s qubit becomes (a∣0〉 + b∣1〉), a perfect replica of the qubit Alice wished to transmit. The operation Bob applies in figure 3.5 is Uˆ = I. • ∣0〉∣1〉, in which case Bob’s qubit becomes (a∣1〉 + b∣0〉). If Bob chooses Uˆ = Xˆ , he transforms it into the desired state. • ∣1〉∣0〉, in which case Bob’s qubit becomes (a∣0〉 − b∣1〉), which Bob transforms ˆ into the desired state by choosing Uˆ = Z. • ∣1〉∣1〉, in which case Bob’s qubit becomes (a∣1〉 − b∣0〉), which Bob transforms ˆ ˆ. into the desired state by choosing Uˆ = ZX Bob, knowing Alice’s result from her phone call, chooses the correct Uˆ to transform his qubit into ∣ϕ〉, completing the task. There are two final points worth noting here: (1) the qubit is not transferred from Alice to Bob superluminally, since the required classical communication between them can transmit no faster than light speed, and (2) the no-cloning theorem is not violated since by the time Bob has the state ∣ϕ〉, Alice is left with only one of her four basis states. 3.3.3 Bell’s inequality This chapter began with mention of the 1935 paper by Einstein, Podolski and Rosen (EPR) that questioned the completeness of quantum mechanics. This paper compared the predictions regarding a particular thought experiment that would follow from quantum theory’s postulates to those of a theory that exhibited local realism. According to the special theory of relativity, no action taken by Alice can influence a distant Bob until there has been sufficient time for this action to have caused a signal that, traveling at the speed of light, reaches Bob: theories that exhibit this property are called local theories. In a theory exhibiting realism, objects have a complete set of properties that exist independently of their being observed. EPR posited that any reasonable theory would be local and exhibit realism, and by pointing out a prediction of quantum mechanics that did not, they argued that quantum theory was missing something. Alice and Bob, with a shared Bell state ∣η00〉, can illustrate this objection of EPR. Assume they are at some distance from one another. At some moment, before either has measured her/his qubit, Bob’s has the property that when measured in his basis will reveal the states ∣0〉 and ∣1〉 with equal probability. Should Alice measure her qubit first, and obtain for example ∣0〉, Bob’s qubit is immediately changed into one that can only reveal ∣0〉 in the same measurement. This would seem to be a matter of Alice’s measurement influencing Bob’s qubit instantaneously, in violation of the principle of locality. Otherwise, Bob’s particle must have carried with it all along a property predetermining what it would do in his measurement (realism), contradicting the quantum mechanical description of the entangled state. One must choose the quantum mechanical description, or local realism: one cannot have both.

3-10

Discrete Quantum Mechanics

Alice cannot transmit physical objects to Bob instantaneously by making measurements. Neither can she send information to him, because the results of a string of measurements on her half of a collection of ∣η00〉 pairs will produce a random string of zeros and ones that Bob can read, but that carries no message. What quantum theory allows to travel instantaneously is merely the influence of one measurement upon the results of another. The challenge of EPR to quantum mechanics seemed, then, a philosophical difference of world views rather than a practical matter open to experiment. That changed, however, due to a paper of singular importance published in 1964 by John Bell [Bell1964]. Bell described an experiment for which quantum mechanics gave specific outcome predictions, and contrasted these outcomes with ones expected from a large class of classical theories, i.e. so-called hidden variable theories, that exhibited local realism, finding situations where the two theoretical constructs predicted different results. This paper challenged experimental physicists to take on the very delicate measurements that might resolve the paradox presented in EPR. Bell’s paper was followed shortly by another [Clauser1969] that proposed a different test, using photons, that was more amenable to practical experiments. These papers precipitated decades of experimental tests of what is now called Bell’s inequality, a relationship that follows merely from the assumption of local realism. These tests have universally supported the quantum mechanical view of nature, but successive results have increased the confidence level of the outcome, and more recently they have tested more and more esoteric loopholes that might allow hidden-variable theories to prevail. As the year 2015 comes to a close, based upon the results of such experiments, the overwhelming consensus favors the quantum mechanical view of nature. Alice and Bob again serve as useful allies in producing a proof of a straightforward version of Bell’s inequality. We assume, as usual, that they get together at the start of our procedure and each take possession of one member of a pair of spin onehalf particles that have been prepared to have complementary properties relative to future measurements of spin in a particular direction. Ultimately, each will measure their half of the pair for its spin orientation in one of three directions in the x–z plane: θ1, θ2, θ3, each angle measured from the positive z axis. Anticipating some of the results of measurements that Alice and Bob will record once they have traveled apart, we foreshadow the point (to be proven below) that the two parts of the pair are anti-correlated: if Alice and Bob make measurements on a pair relative to the same direction, whenever Alice measures spin up, Bob will measure spin down, and vice versa. With that in mind, we give a description of the pair consistent with local realism. Each particle carries an instruction set that will predict the outcome regardless of which of the three measurements the particle encounters. For example, a particle that will reveal itself to be spin up for each potential measurement will be said to be carrying the instruction set [↑↑↑]. In this notation, the first arrow predicts the outcome for a measurement along θ1, the second for θ2 and the third for θ3. The anti-correlation property implies that the instruction set carried by Alice’s particle (e.g. [↓↑↑]) is complementary to the one carried by Bob ( [↑↓↓]). We make no assumption about the population of pairs with the eight possible instruction set pairs, but assign to each a statistical description: f (↑↑↑) will represent the fraction of

3-11

Discrete Quantum Mechanics

pairs created with Alice’s particle carrying the instruction set of the argument, with f (↑↑↓), f (↑↓↑), etc describing the fraction created with Alice’s other possible sets (with Bob’s, of course, being the corresponding complement). Once in possession of an anti-correlated pair of particles, Alice and Bob say goodbye and travel far from one another, to have no further communication until each makes a measurement on her or his particle to determine spin projection in one of the three directions. Each chooses which measurement to make randomly, makes the measurement, and records the result. They come together again, share another correlated pair, and repeat the process. We examine the results of their measurements after many iterations of this process. As foreshadowed, on each occasion when Alice and Bob measure relative to the same angle, regardless of which of the three angles this is, their results are opposite—if Alice records ↑, Bob records ↓, and vice versa. This is not the case, however, when they measure relative to different angles. Consider the subset of measurements when Alice measures spin relative to angle θ1, and Bob measures it relative to θ2. We denote the probability that in this case both record ‘spin up’ as C12, and realize that this can happen only in cases where the instruction set for Alice’s particle reads either [↑↓↑] or [↑↓↓]. Therefore, in the case of many such measurements we expect

C12 = f (↑↓↑) + f (↑↓↓). A similar analysis of data for cases when Alice measures relative to θ3 and Bob measures relative to θ2 —and denoting the fraction of such cases when both get spin up as C32—will yield

C32 = f (↑↓↑) + f (↓↓↑). Considering C13, the fraction of cases when Alice and Bob measure relative to θ1 and θ3, respectively, and both measure spin up, produces

C13 = f (↑↑↓) + f (↑↓↓). Combining these results, we get

C32 + C13 − C12 = f (↓↓↑) + f (↑↑↓). Since all the fractions f are non-negative, we conclude

C32 + C13 − C12 ⩾ 0,

(3.8)

a version of Bell’s inequality that expresses a requirement for correlations under the assumption of local realism. We wish to compare the prediction of Bell’s inequality with a quantum explanation of the same situation. In order to explain the anti-correlation observation for cases when Alice and Bob measure their halves of the same pair relative to the same angle, the two particles must be entangled. We will describe the entangled pairs that Alice and Bob share as

∣Ψ〉 =

1 2

( ∣1↑ 〉∣1↓ 〉 − ∣1↓ 〉∣1↑ 〉), 3-12

Discrete Quantum Mechanics

often referred to as the singlet state of the two-particle system6. We define the states ∣1↑ 〉 and ∣1↓ 〉 to be the eigenstates of the operator for measurement along direction θ1. It is significant that this state has the analogous form relative to measurement along any other direction. That is, it can be expressed in terms of the basis ∣2↑〉 and ∣2↓〉 for measurement along θ2 as:

∣Ψ〉 =

1 2

( ∣1↑ 〉∣1↓ 〉 − ∣1↓ 〉∣1↑ 〉) =

1 2

( ∣2↑〉∣2↓〉 − ∣2↓〉∣2↑〉),

(3.9)

regardless of the angles θ1 and θ2, as can be proven straightforwardly by utilizing equations (2.4). This is a reflection of the fact that states of zero angular momentum are spherically symmetric. Using the quantum formalism, we will calculate the three correlations that constitute equation (3.8). We noted in section 2.3.2 that a particle in state ∣ψ 〉 will transform into ∣ϕ〉 as a result of a single measurement with probability ∣〈ψ ∣ϕ〉∣2 (θ − θ ) and we can use equations (2.4) to show that ∣〈1↑ ∣2↑〉∣2 = cos2( 2 2 1 ) and ∣〈1↑ ∣2↓〉∣2 = θ −θ

sin2( 2 2 1 ). To calculate C12, we evaluate the probability of the singlet state producing the state ∣↑〉∣2↑〉 upon measurements by Alice and Bob:

C12 =

〈Ψ∣AB (∣1↑ 〉A ∣2↑〉B )

2

2

=

1 2

=

1 〈1↑ ∣1↑ 〉A 〈1↓ ∣2↑〉B 2

( 〈1↑ ∣A 〈1↓ ∣B − 〈1↓ ∣A 〈1↑ ∣B )(∣1↑ 〉A∣2↑〉B ) 2

⎛ θ − θ1 ⎞ 1 ⎟, = sin2⎜ 2 ⎝ 2 ⎠ 2

(3.10)

where we have added Alice and Bob subscripts for clarity. Similarly, we evaluate

C13 =

⎛ θ − θ1 ⎞ 1 ⎟ sin2⎜ 3 ⎝ 2 ⎠ 2

C32 =

⎛ θ − θ3 ⎞ 1 ⎟. sin2⎜ 2 ⎝ 2 ⎠ 2

and

Thus for the quantum predictions for this set of experiments and those of local realism to be in agreement,

⎛ θ − θ3 ⎞ ⎛ θ3 − θ1 ⎞ ⎛ θ2 − θ1 ⎞ ⎟ + sin2⎜ ⎟ − sin2⎜ ⎟ ⩾ 0 sin2⎜ 2 ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠

6 This state is redundant with the Bell state ∣ η10〉, but is also an eigenstate of total angular momentum for the system with eigenvalue zero. Spectroscopic examination of zero angular momentum states produces a single isolated spectral line, in contrast with multiplets of lines associated with higher angular momentum states, thus the name ‘singlet’.

3-13

Discrete Quantum Mechanics

for all angles θ1, θ2 and θ3. We check this inequality for the case Θ ≡ θ2 − θ3 = θ3 − θ1 = (θ2 − θ1)/2, yielding

2 sin2

Θ Θ Θ − sin2 Θ = 4 sin4 − 2 sin2 ⩾ 0. 2 2 2

This is violated for values 0 ⩽ Θ ⩽ π /2, and this contradiction justifies the conclusion of Bell’s paper: no physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.

3.4 Large-scale quantum algorithms The previous few sections served to illustrate the advantages of quantum-information techniques over classical ones, but none of them in the form shown provide any practical advances. The promise of the field is that the technical difficulties in storing and manipulating quantum states can be overcome so as to allow quantum computation, for instance, to manipulate data sets as large as those handled by the most powerful current digital computers. Even ignoring these implementation difficulties, it has proven challenging to conceive of quantum algorithms that outperform classical ones for practical problems. The 1994 algorithm of Peter Shor for finding the prime factors of integers [Shor1994, Shor1997] and the 1996 algorithm of Grover for searching an unsorted database [Grover1996] are the prominent exceptions. There are a greater number of quantum algorithms that have been developed to exhibit the advantages of quantum over classical computation, but solve problems with no apparent practical applications. These should serve as useful steps towards other successful and practical quantum computation schemes. In this section we will look at one such example: Simon’s problem. In 1994, Daniel Simon [Simon1994] presented a quantum algorithm for solving the following problem. Given a function F that maps a binary string x of N bits into another such string, with the property that there is an N bit string s (not all zeros) such that F (x ) = F (y ) if and only if y = x or y = x ⊕ s, find s. Before explaining in detail the nature of this problem, we note the feature that makes it of interest: the number of times the function F must be called in the best classical algorithm for solving this problem grows exponentially with N, the number of bits upon which the function F acts, while Simon’s quantum algorithm requires only a linearly increasing number of calls to F to extract a result—a quite considerable improvement. Simon’s problem postulates a function (F) that operates like a ‘black box’, i.e. it operates on a set of inputs, producing a set of outputs, but the user has no information about the construction of the function or its computational structure. In this case, the challenge is to learn one feature of the black box while calling the function as few times as possible. Inputs to this black box are N bits and it delivers N bits as output. We use x to designate one of the 2N possible states of the N-bit input, and F (x ) to designate the corresponding N-bit output. For each of the 2N −1 distinct outputs of F 3-14

Discrete Quantum Mechanics

there are exactly two distinct inputs that yield that output. The two inputs are related to one another by y = x ⊕ s, where the ⊕ operator represents bitwise addition modulo 2. We represent x, y and s each as an ordered sequence of binary digits. The bitwise sum of two such strings is fully characterized by the table below.

Bitwise addition modulo 2 nth digit of x 0 0 1 1

nth digit of s

nth digit of x ⊕ s

0 1 0 1

0 1 1 0

Note that this operation is somewhat like binary addition, but with no ‘carry’, thus it operates on one digit of its two arguments at a time. In terms of the language of logic gates, it performs an ‘exclusive or’ or ‘XOR’ operation on the nth digit of x and the nth digit of s. Since s ≠ 0,7 x = x ⊕ s maps every possible x onto a different and unique N-bit string. From the table, it is clear that x ⊕ x = 0, thus if y = x ⊕ s it follows that x ⊕ y = s. The set of 2N possible input strings can therefore be separated into 2N −1 disjoint pairs—x and x ⊕ s. F maps the two elements of each pair onto a single output string of N elements. 3.4.1 Classical algorithm for Simon’s problem Using classical computation, there is no approach more efficient that feeding unique x values, in any order, into the black box F one at a time until we find two (xk , xl ) that produce the same output. When that happens, we can deduce s from s = xk ⊕ xl . We start by picking, at random, input value x1 and use the black box to determine F (x1). Subsequently, we select other input values, each time sending one into the black box, and see if our new F (x ) matches any of the previous black box results. We trace the steps in the process, defining the probability of success (i.e. finding for the first time a match of F (x ) with a previously calculated F output) in the kth selection as αk and the probability of failure after k selections as βk . The steps are as follows. • We pick a second input, x2, from the 2N − 1 remaining input values and determine F (x2 ). The probability of a match (F (x2 ) = F (x1)) is α2 = N1 and 2

N

−1

the probability of failure is β2 = 2N − 2 . 2 −1 • If there was no success in the previous pick, we now pick a third x3, from the 2N − 2 remaining input values and determine F (x3). The probability of a 7

By s = 0 we mean s is a string of N zeros.

3-15

Discrete Quantum Mechanics

match with either of the previous black box outputs (F (x3) = F (x1) or F (x3) = F (x2 )) is α3 =

previous picks is β3 =

(2N − 2) 2 . (2N − 1) 2N − 2 N N (2 − 2)(2 − 4)

The probability of failure in this and

. (2N − 1)(2N − 2)

• Continuing likewise, we find our first match in the kth pick with probability

(2 (2

N

αk =

N

)( − 1)( 2

)( − 2)( 2

) ( − 3) ⋯ ( 2

) + 2 − k)

− 2 2N − 4 2N − 6 ⋯ 2N + 4 − 2k N

N

N

k−1 k−1 ⩽ N . 2 +1−k 2 −1 N

Using the upper bound for αk we just established, we can calculate an upper bound for the probability of finding a match on or before the kth pick as k

j−1

∑ 2N − 1 j=2

=

k (k − 1)

(

)

2 2N − 1

.

A standard for judging the effectiveness of a computer algorithm is the computational complexity, defined as the trend in the time required for it to produce a result as the size of the data set increases. We consider this time to be proportional to the number of calls to the black box function F. For large values of N and k, we can see from this bound that finding a match, and thus determining s, can be achieved with a probability P after k or fewer calls to the black box when P ⩽

k2 2N

and thus

N /2

k ⩾ P 2 . Clearly, when using classical computation, the number of calls to the function F required to produce a solution to Simon’s problem with the desired certainty increases exponentially with N, expressed as a computational complexity N for this classical algorithm of O(2 2 ).

3.4.2 Quantum algorithm for Simon’s problem The reversible quantum computation (figure 3.6) for attacking this problem is carried out on 2N qubits, the first N of which constitute the input register and the remaining N of which constitute the output register. Each qubit is initialized in the

Figure 3.6. N = 3 Simon’s problem algorithm.

3-16

Discrete Quantum Mechanics

basis state ∣0〉, which we use to represent the binary digit 0. A general basis state for the complete input (or output) register will be denoted by ∣x〉in ∣y〉out , where x and y are strings of N 0s and 1s that can be referred to by their interpretation as binary numbers between 0 and 2N − 1. The first step in Simon’s algorithm is to apply a Hadamard transformation to each qubit in the input register. This transforms each qubit into the form 1 (∣0〉 + ∣1〉), a superposition of the two basis states with equal weighting. It is 2 easily seen that the product of N such superpositions can be expressed as an N-qubit state that is a superposition of all possible basis states of the input register, also with equal weighting:

1 2N 2

2N − 1

∑ ∣x〉 . x=0

This step prepares us to send into the computer implementation of the function F—the 2N qubit unitary gate UˆF —every possible input x at the same time. The possibility of using superposition states so as to operate on multiple inputs simultaneously is referred to as quantum parallelism, in that it accomplishes what is done using multiple processors in classical parallel computers. The operator UˆF is applied to all 2N qubits and it is designed to leave the input state ∣x〉 intact in the input register, and insert the state ∣F (x )〉 into the output register, producing

1 2N 2

2N − 1

∑ ∣x〉in ∣F ( x)〉out .

(3.11)

x=0

We next make a measurement of the output register in its natural basis. This selects one of the output states of F, which we designate as F (x0 ). This output state will correspond to precisely two distinct values of x, that we denote as x0 and x0 ⊕ s, so the equation (3.12) becomes

1 (∣x0〉 + ∣x0 ⊕ s〉)in ∣F ( x0)〉out . 2 The remaining challenge is to extract some useful information about s from this yetto-be-examined input register. A final step of applying Hadamard gates to each qubit in the input register will prove useful in this regard. Examine the effect of the Hadamard gates on the state ∣x0〉. Since ∣x0〉 is a basis state of the N-qubit register, its constituent one-qubit states are either ∣0〉 or ∣1〉, which a Hadamard transformation changes into 1 (∣0〉 + ∣1〉) or 1 (∣0〉 − ∣1〉), 2 2 respectively. The product of n such superpositions again produces a linear combination of all possible n-qubit states, but this time weighted by a phase term 2n− 1 ⊗N Hˆ x0 = ∑ ( −1)φy y .

y=0

3-17

(3.12)

Discrete Quantum Mechanics

The phase ( −1)φy for the yth term in the sum on the right-hand side of equation (3.12) picks up a multiplicative ( −1) from one of the N digits in the binary string x if and only if • that particular digit of x is a 1, so Hˆ ∣1〉 = 1 (∣0〉 − ∣1〉) is contributed to the 2 product of qubits on the left of equation (3.12); and • the term y in the sum has a 1 in the same digit, since it comes from the second term in the expression for Hˆ ∣1〉. Another ( −1) is contributed to the phase, therefore, whenever corresponding digits in x and y are both equal to 1. We will use the notation

φy = x · y to capture this notion, where we mean by x · y the sum, modulo 2, of the products of corresponding digits of x and y.8 We thus rewrite equation (3.12) as 2N − 1 ⊗N Hˆ x0 =

∑ ( −1)x · y y 0

y=0

and the input register, as a result of the applied Hadamard gates, becomes

Hˆ

⊗N

1 1 (∣x0〉 + ∣x0 + s〉) = N +1 2 2 2 =

1

2N − 1

∑ ( ( −1)x · y + ( −1)(x ⊕s )· y )∣y〉 0

0

y=0 2N − 1

∑ (1 + ( −1)s · y )( −1)x ·y ∣y〉. 0

N +1 2 2 y=0

(3.13)

The final form of this expression shows that the sum is limited to those values of y for which s · y = 0. The final step in the algorithm, measuring the input register in its natural basis, will give as a result only values of y that satisfy this condition, thus revealing information about the parameter s. Each time the algorithm is repeated, a random value of y satisfying s · y = 0 is revealed from among the 2N −1 possibilities. We need to find N − 1 linearly independent values of y (necessarily nonzero) to completely determine s. The first nonzero y to be picked, y1, has a pattern of 1s, implying that if one sums the bits of s found in that same pattern of locations, the sum is even. Half of the 2N possible s values have that property. The probability of obtaining information about s from a y pick is thus

2N −1 − 1 = 1 − 21−N . 2N −1

8

For example, if x = 100111 and y = 101101, then x · y = (1 + 0 + 0 + 1 + 0 + 1) mod 2 = 1.

3-18

Discrete Quantum Mechanics

To be useful, the next pick, y2, must be different from y1 and y = 0, and the probability of it being so is

2N − 1 − 2 = 1 − 22−N . 2N − 1 This has a different pattern of 1s from that of y1, implying a different sum of s values that must be even, again halving the number of possible s values, taking it down to 2N −2 . Each subsequent pick, yk for example, we wish to be linearly independent of y1, …, yk−1,9 which span a space with 2k−1 elements, so the probability of such a y being picked will be

( 2N −1 − 2k−1)

2N −1 = 1 − 2k−N .

Each such step halves the number of possible s values. The probability of exactly N − 1 calls of F leading to that many halvings of the set of possible s values and thus exactly determining s is given by

PN −1 = ( 1 − 21−N )( 1 − 22−N ) ⋯ ( 1 − 2−2)( 1 − 2−1). We can find an upper bound for this product utilizing (1 − a )(1 − b ) = 1 − a − b + ab ⩾ 1 − a − b for every term but the rightmost:

(

PN −1 ⩾ 1 − 21−N − 22−N − ⋯ −2−2 ⎛ 1 ⩾ ⎜⎜ 1 − 4 ⎝

∞

1 ⎞1

∑ 2n ⎟⎟ 2 n=0

⎠

=

) 12

1 . 4

This probability can be made much closer to 1 by allowing a handful more picks. The straightforward, albeit inefficient process of repeating the entire algorithm j times would increase the probability of success to

⎛ 3 ⎞j P⩾1−⎜ ⎟, ⎝ 4⎠ a result that approaches 1 rapidly as j increases. To achieve success with a probability arbitrarily close to one, the necessary number of trials j can be determined independently of the size N of the data set. For this reason, the computation complexity of finding a collection of N − 1 distinct y, each orthogonal to s, is O(N ). The final step in solving Simon’s problem is to extract s from N − 1 values of y obeying y · s = 0. This can be done efficiently via the classical process of Gaussian elimination, a process with a computational complexity of O(N 3). Assuming Gaussian elimination as the final step, the overall computational complexity of

9

This subspace includes y = 0.

3-19

Discrete Quantum Mechanics

Simon’s algorithm is O(N 3). Nonetheless, this is far more efficient than the classical solution’s computational complexity of O(2N ). As stated earlier, Simon’s problem is a ‘toy’ problem with an interesting solution. Its importance lies in the other algorithms it has inspired, particularly that of Shor. Peter Shor’s 1994 algorithm solved a problem that sounds similar to Simon’s problem, but is significantly more difficult. Given a function F that maps a binary string x of N bits into another such string, with the property F (x ) = F (y ) if and only if x and y differ by an integer multiple of the period s, find the positive integer parameter s of the function. The only difference between this and Simon’s problem is that the function’s parameter s relates x and y through ordinary addition rather than bitwise, modulo 2 addition, but this makes its solution much more complicated. Since Shor’s algorithm makes the finding of the prime factors of integers much faster than any known classical algorithm, its consequences are profound. Should we build (when we build) a quantum computer on which we can implement Shor’s algorithm, we would immediately render Rivest–Shamir–Adleman (RSA)-encrypted data [1, 2] insecure. Since this encryption scheme is currently used widely for both military and commercial applications, the effect would be enormous. Other important large-scale algorithms for quantum computers and methods for other quantum information applications have been developed and refined in the last two decades. An excellent explanation of many of the most important of these can be found in the book Quantum Computer Science by N David Mermin [Mermin2007].

Q4

Exercises 1. Show that states of the form of equation (3.1) can be factored into two pure states only if ad = bc. 2. Following steps similar to those that led to equation (3.7), show that ∣η01〉 = XˆA∣η00〉 and ∣η10〉 = XˆAZˆA∣η00〉. 3. In discussion of figure 3.3 we saw that ∣η00〉 = CNOT Hˆ control∣00〉. If Bob wishes to go from ∣η00〉 back to ∣00〉 he only needs to apply the two operations in the opposite order: Hˆ control CNOT. Show how the same two-gate sequence will change ∣η01〉 into ∣01〉; ∣η10〉 into ∣11〉; and ∣η11〉 into ∣10〉. 4. Prove equation (3.9) utilizing equations (2.4). 5. Show that the product of N qubits, each in a superposition state of the form 1 (∣0〉 + ∣1〉), can be expressed as an N-qubit state that is a superposition of all 2 possible basis states of the input register, also with equal weighting:

1 2N 2

2N − 1

∑ ∣x〉 . x=0

3-20

Q5

Q6

Discrete Quantum Mechanics

Bibliography [Einstein1935] Einstein A 1935 Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47 777–80 [DiVencenzo1995] DiVencenzo D P 1995 Two-bit gates are universal for quantum computation Phys. Rev. A 51 1015–22 [Bell1964] Bell J S 1964 On the Einstein Podolsky Rosen paradox Physics 1 195–200 [Clauser1969] Clauser J, Horne M, Shimony A and Holt R 1969 Proposed experiment to test local hidden-variable theories Phys. Rev. Lett. 23 880 [Shor1994] Shor P W 1994 Algorithms for quantum computation: discrete logarithm and factoring Proc. 35th Annu. Symp. on Foundations of Computer Science Santa Fe, NM [Shor1997] Shor P W 1997 Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer’ SIAM J. Comput. 26 1484–509 [Grover1996] Grover L K 1996 A fast quantum mechanical algorithm for database search Proc. 28th Annu. ACM Symp. on the Theory of Computing p 212 [Simon1994] Simon D R 1994 On the power of quantum computation Proc. 35th Annu. Symp. on the Foundations of Computer Science (Los Alamitos, CA: IEEE) pp 116–93 [1] [Mathworld] MathWorld—A Wolfram Web Resource http://mathworld.wolfram.com/ RSAEncryption.html [2] [Coutinho1999] Coutinho S C 1999 The Mathematics of Ciphers: Number Theory and RSA Cryptography (Wellesley, MA: A K Peters) [Mermin2007] Mermin D N 2007 Quantum Computer Science (New York: Cambridge University Press)

3-21

IOP Concise Physics

Discrete Quantum Mechanics H Thomas Williams

Chapter 4 Quantum angular momentum

We may face in whatever direction we wish, and yet the physical laws that affect us are the same. This universal indifference to orientation has, as a consequence, an associated conserved quantity that goes by the name of angular momentum. In classical physics, the vector angular momentum of a particle with momentum p ⃗ relative to a point is L⃗ ≡ r ⃗ × p ⃗ , where r ⃗ connects the point to the location of the particle. For a particle moving under the influence of a radial force, and thus derivable from a spherically symmetric potential energy function, the angular momentum about the force center is a conserved quantity1. Since the Coulomb potential energy and that of Newtonian gravitation are both spherically symmetric, angular momentum is a ubiquitous and important quantity in classical systems. In the continuous infinite-dimensional Hilbert space version of quantum mechanics that follows from Schrödinger’s differential equation, a clear analog of classical angular momentum can be introduced by way of three differential operators, Lˆ x , Lˆ y and Lˆ z , which correspond to components of orbital angular momentum. It has been shown theoretically and confirmed by experiment that these Hermitian operators correspond to observables with integer eigenvalues2. In addition to the angular momentum associated with moving particles, composite structures such as atoms are seen to have nonzero angular momentum with respect to their centers of mass as a result of the motion of constituent particles, e.g. atomic electrons. Quantum theory allows, and experiment again confirms, that point particles with no internal structure—electrons, for instance—can nonetheless exhibit angular momentum about axes through their centers. This is referred to as intrinsic angular momentum, and operators related to this property have eigenvalues that can take on

1 The connection between symmetries in forces and conservation laws was first established by German mathematician Emmy Noether [Noether1918]. 2 The physical units of ℏ are those of angular momentum, and in our numerical unit system in which ℏ = 1, the possible values of an orbital angular momentum component are 0, ±1, ±2, ….

doi:10.1088/978-1-6817-4125-3ch4

4-1

ª Morgan & Claypool Publishers 2015

Discrete Quantum Mechanics

both integer and half-integer values. The terms spin and intrinsic spin are synonyms for intrinsic angular momentum. Our subsequent discussion will consider both orbital and intrinsic angular momentum.

4.1 Operators for orbital angular momentum In the Schrödinger continuous theory, expressed in configuration space, the state functions are differentiable functions of the spatial coordinates x, y and z, and operators are functions of the spatial coordinates and their derivatives. Position operators are simply the functions xˆ = x , yˆ = y and zˆ = z. Momentum operators are pˆx = −i ∂∂x , pˆy = −i ∂∂y and pˆz = −i ∂∂z . Quantum angular momentum operators are formed from operator analogs of the components of the classical angular momentum operator L⃗ ≡ r ⃗ × p ⃗ :

⎛ ∂ ∂ ⎞ ˆ ˆz − zp Lˆ x = yp − z ⎟; ˆ ˆy = −i ⎜y ∂y ⎠ ⎝ ∂z ⎛ ∂ ∂ ⎞ ˆ ˆz = −i ⎜z Lˆ y = zp − x ⎟; ˆ ˆx − xp ⎝ ∂x ∂z ⎠ ⎛ ∂ ∂ ⎞ ˆ ˆy − yp ˆ ˆx = −i ⎜x Lˆ z = xp − y ⎟. ∂x ⎠ ⎝ ∂y

(4.1)

A fourth operator of note represents the square of the total orbital angular momentum: 2 2 2 2 Lˆ ≡ Lˆ x + Lˆ y + Lˆ z .

(4.2)

We examine now commutators among these operators, beginning with [Lˆ x, Lˆ y ]. Look first at ⎛ ∂ ∂ ⎞ ∂ ⎞⎛ ∂ Lˆ xLˆ y = −⎜ y −x ⎟ − z ⎟⎜z ∂z ⎠ ∂y ⎠⎝ ∂x ⎝ ∂z ⎛ ∂ ∂ ∂ ∂ ⎞ ∂ ∂ ∂ ∂ +z x ⎟ −z z −y x = −⎜ y z ∂y ∂z ⎠ ∂y ∂x ∂z ∂z ⎝ ∂z ∂x ⎛ ∂ ∂2 ⎞ ∂2 ∂2 ∂2 + xz − xy 2 − z 2 + yz = −⎜ y ⎟; ∂y∂z ⎠ ∂x∂y ∂z ∂x∂z ⎝ ∂x and next at

⎛ ∂ ∂ ⎞ ∂ ⎞⎛ ∂ Lˆ yLˆ x = −⎜ z −z ⎟ − x ⎟⎜y ⎝ ∂x ∂y ⎠ ∂z ⎠⎝ ∂z ⎛ ∂ ∂ ∂ ∂ ⎞ ∂ ∂ ∂ ∂ +x z ⎟ −x y −z z = −⎜ z y ∂z ∂y ⎠ ∂z ∂z ∂x ∂y ⎝ ∂x ∂z 2 2 ⎛ ∂2 ∂2 ⎞ ∂ ∂ ∂ + xz − xy 2 + x − z2 = −⎜ yz ⎟. ∂y∂z ⎠ ∂y ∂z ∂x∂y ⎝ ∂x∂z

4-2

Discrete Quantum Mechanics

Subtraction leads to

⎛ ⎞ ⎡ Lˆ , Lˆ ⎤ = Lˆ Lˆ − Lˆ Lˆ = −⎜ y ∂ − x ∂ ⎟ = iLˆ . x y y x z ⎣ x y⎦ ∂y ⎠ ⎝ ∂x

(4.3)

Similar straightforward calculations lead to

⎛ ⎞ ⎡ Lˆ , Lˆ ⎤ = Lˆ Lˆ − Lˆ Lˆ = −⎜ z ∂ − y ∂ ⎟ = iLˆ y z z y x ⎣ y z⎦ ∂z ⎠ ⎝ ∂y

(4.4)

⎡ Lˆ , Lˆ ⎤ = Lˆ Lˆ − Lˆ Lˆ = −⎛⎜ x ∂ − z ∂ ⎞⎟ = iLˆ . z x x z y ⎣ z x⎦ ⎝ ∂z ∂x ⎠

(4.5)

and

We evaluate the remaining commutators of interest without resorting to explicit 2 expansion of the operators in terms of partial derivatives. First, since [Lˆ x, Lˆ x ] = 0,

⎡Lˆ , Lˆ 2⎤ = ⎡Lˆ , Lˆ 2⎤ + ⎡Lˆ , Lˆ 2⎤ . ⎣ x ⎦ ⎣ x y⎦ ⎣ x z ⎦ Using equation (4.3), we see that

⎡ ˆ ˆ 2⎤ ˆ ˆ2 ˆ2ˆ ˆ ˆ ˆ ˆ ˆ2ˆ ⎣Lx, L y ⎦ = LxL y − L y Lx = L yLx + iLz L y − L y Lx

(

)

2 = Lˆ y Lˆ yLˆ x + iLˆ z + iLˆ zLˆ y − Lˆ y Lˆ x = i Lˆ yLˆ z + Lˆ zLˆ y ,

(

)

(

)

and similarly with the help of equation (4.5)

⎡ ˆ ˆ 2⎤ ˆ ˆ2 ˆ2ˆ ˆ ˆ ˆ ˆ ⎣Lx, L z ⎦ = LxL z − L z Lx = −i L yLz + LzL y ,

(

)

thus upon addition,

⎡Lˆ , Lˆ 2⎤ = 0. ⎣ x ⎦

(4.6)

⎡Lˆ , Lˆ 2⎤ = ⎡Lˆ , Lˆ 2⎤ = 0. ⎣ y ⎦ ⎣ z ⎦

(4.7)

Similar manipulations lead to

4.2 Operators for spin one-half We use the spin one-half system as the base constituent of the theory of intrinsic angular momentum. This system was introduced and described in some detail in chapter 2. We repeat here some of the key notions there introduced. The operators that are components of the angular momentum vector for spin one-half are sˆx , sˆy and sˆz . 4-3

Discrete Quantum Mechanics

Central to our considerations in this chapter are the commutation relations among these operators:

⎡⎣sˆj , sˆk ⎤⎦ = isˆl ,

(4.8)

where j , k , l represents any cyclic permutation of the ordered set of indices {x , y, z}. Since no two of these operators commute, an eigenstate of one cannot be simultaneously an eigenstate of another. The eigenstates of sˆz will be our basis states unless we explicitly state otherwise:

sˆz∣z↑〉 =

1 ∣z↑〉, 2

sˆz∣z↓〉 = −

1 ∣z↓〉. 2

2 The spin operator analogous to Lˆ is important for general cases, but is rather trivial in the case of spin one-half:

3 1 ⎛ 0 1 ⎞⎟2 1 ⎛⎜ 0 − i ⎞⎟2 1 ⎜⎛1 0 ⎟⎞2 2 Sˆ ≡ sˆx2 + sˆy2 + sˆz2 = ⎜ + + = I. ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 4 0 −1 4 4 1 0 4 i 0

(4.9)

2 Since it is proportional to the identity operator, Sˆ commutes with each of the angular momentum component operators. As a consequence, we can label our basis 2 states using the eigenvalues of both Sˆ and sˆz . Comparing the commutation relations of equations (4.8) and (4.9) to their counterparts in the case of orbital angular momentum from equations (4.3)–(4.7), we find a striking parallelism. This is by no means mere coincidence, but is in fact the basis of the theory of quantum angular momentum. In the following section, we will begin with these commutation relations alone and deduce the general theory.

4.3 Generalized angular momentum theory Assume the existence of three Hermitian operators—Jˆx , Jˆy , Jˆz —and a fourth defined as 2 2 2 2 Jˆ ≡ Jˆx + Jˆy + Jˆz .

(4.10)

2 It is straightforward to prove that Jˆ is also Hermitian, given the Hermiticity of its constituent operators. Furthermore assume that the three component operators obey the commutation relations

⎡Jˆ , Jˆ ⎤ = iJˆ , l ⎣ j k⎦

(4.11)

where j , k , l represents any cyclic permutation of the indices {x , y, z}. As we saw at the end of section 4.1, the commutation relations among the component operators is sufficient to prove that

⎡Jˆ , Jˆ 2⎤ = 0 ⎣ j ⎦ for j = x , y, z. 4-4

(4.12)

Discrete Quantum Mechanics

We establish a set of normalized basis states ∣κ , m〉 that are simultaneously 2 eigenstates of both Jˆ and Jˆz: 2 Jˆ κ , m = κ κ , m ,

Jˆz κ , m = m κ , m .

The calculations to come are enabled through the introduction of a raising operator Jˆ+ ≡ Jˆx + iJˆy and a lowering operator Jˆ− ≡ Jˆx − iJˆy. From the commutation relations among the operators already introduced, it is straightforward to show that

⎡Jˆ 2, Jˆ ⎤ = 0 ±⎦ ⎣

[Jˆz, Jˆ±] = ±Jˆ± ,

(4.13)

and 2 2 2 Jˆ = Jˆ− Jˆ+ + Jˆz + Jˆz = Jˆ+ Jˆ− + Jˆz − Jˆz .

(4.14)

The action of Jˆ+ on an eigenstate of Jˆz produces another of its eigenstates. Using equation (4.13), this is easily demonstrated:

Jˆz Jˆ+ κ , m = (Jˆ+Jˆz + Jˆ+) κ , m = (m + 1)Jˆ+ κ , m . The Jˆz eigenstate Jˆ+∣κ , m〉 has eigenvalue one greater than the original, thus justifying the descriptor raising operator for Jˆ+. In nearly identical fashion we can show that J−, the lowering operator, acts on 2 ∣κ , m〉, producing an eigenstate of Jˆz with eigenvalue (m − 1). Since Jˆ commutes with both the raising and lowering operators, we deduce that these operators change the m value while leaving κ unchanged. 2 We can use the Jˆ± operators to generate members of a family of eigenstates of Jˆ and Jˆz with m values differing by steps of one. We should, however, find out if that progression continues without limit or if there are maximum and minimum allowed values of m. To examine this, we take the expectation value of both sides of equation (4.10) relative to ∣κ , m〉, 2 2 2 2 Jˆ = Jˆx + Jˆy + Jˆz ,

leading to 2 2 κ = Jˆx + Jˆy + m 2 .

Since the expectation value of the square of a Hermitian operator is non-negative (see appendix A), we conclude

κ ⩾ m2 . Since m is bounded from above and below, there must be a maximum value such that Jˆ+∣κ , m max〉 = 0, and a minimum value such that Jˆ−∣κ , m min〉 = 0. It follows that

(

)

2 2 2 Jˆ κ , m max = κ κ , m max = Jˆ− Jˆ+ + Jˆz + Jˆz κ , m max = (m max + m max ) κ , m max ,

4-5

Discrete Quantum Mechanics

thus κ = m max(m max + 1). Similar reasoning shows that κ = m min(m min − 1), and these two results combine to yield m max = −m min . We henceforth identify j ≡ m max and adopt the notation ∣ j , m〉 ≡ ∣κ , m〉, pointing out that κ = j ( j + 1). We have seen that m changes in steps of ±1, from which we deduce that j is restricted to integer or half-integer non-negative values. In the most general case, therefore, there is a family of states ∣j , m〉 that are 2 simultaneously eigenstates of the Jˆ operator, 2 Jˆ j , m = j ( j + 1) j , m ,

with j = 0, 12 , 1, 32 , …; and of the Jˆz operator,

Jˆz j , m = m j , m , with m = −j , −j + 1, …, j − 1, j . These are called states of total angular momen2 tum j and angular momentum projection m. Since the two operators Jˆ and Jˆz are Hermitian, states with different eigenvalues are mutually orthogonal:

j ; m j ′ ; m′ = δj , j′δm,m′. There are many important physical systems that exhibit zero total angular momentum. The mathematics of the angular momentum piece of such systems represents the ‘trivial case’, j = m = 0 with state vector ∣0; 0〉. The first non-trivial angular momentum example is that of intrinsic angular momentum one-half, which we have found already to be rich in both its complexity and consequences. From 2 equation (4.9) we found the eigenvalue of Jˆ in this case to be j ( j + 1) = 34 , thus 2 1 j = . The two simultaneous eigenstates of Jˆ and Jˆz discussed in section 4.2, in the 2

generalized notation can be written as ∣ 12 ; 12 〉 (equivalent to ∣z↑〉) and ∣ 12 ; − 12 〉 (equivalent to ∣z↓〉). Other states exhibiting intrinsic angular momentum can exist with both integer and half-integer j values. In section 4.1 we explored the properties of systems with angular momentum resulting from the motion of constituent particles, so-called orbital angular momentum. Such systems can only exhibit integer-valued j, including j = 0, which represents extended entities exhibiting spherical spatial symmetry. Their eigenstates are most often written as ∣l ; m〉, with l = 0, 1, 2, … and m = −l , −l + 1, …, l − 1, l .

4.4 Angular momentum addition It is frequently useful to express composite systems in terms of angular momentum eigenstates, when the constituent states are so described. For example, an atomic electron may be described by giving its spin angular momentum state and its orbital angular momentum state, or alternatively as a state with a particular total angular momentum and projection. In collision experiments one often envisions an intermediate stage of merged projectile and target particles (or one particle from each of two colliding beams) in terms of total angular momentum properties which affect the angular distribution of final state particles. 4-6

Discrete Quantum Mechanics

From the composite state postulate of section 1.4, we know that state functions for composite states can be formed by taking tensor products of the separate constituent state functions. The tensor product of two quantum mechanical state vectors does not describe a composite state with specific angular momentum properties in every case, however. To accomplish this, we must typically create linear combinations of composite states of the form ∣ j , m〉. Consider a system of two particles with state functions ∣ j1; m1〉 and ∣ j2; m2〉. These particles can be elementary or themselves composite, and can be states of intrinsic or orbital angular momentum, or both. There are 2j1 + 1 states (with various m1 values) of the first kind and likewise 2j2 + 1 states of the second kind, thus the dimensionality of the product state is (2j1 + 1)(2j2 + 1). The operator for the z component of angular momentum for the combined state is

Jˆz = Jˆ1z + Jˆ2z , where Jˆ1z and Jˆ2z are the corresponding operators for particles one and two respectively. Operators for the x and y components of angular momentum for the composite state are formed similarly, and 2 2 2 2 Jˆ = Jˆz + Jˆy + Jˆz .

The (2j1 + 1)(2j2 + 1) product states ∣ j1; m1〉∣ j2; m2〉 form an orthonormal basis for states of the combined system, each element of which is a simultaneous eigenstate of 2 2 Jˆ1 , Jˆ2 , Jˆ1z and Jˆ2z. Likewise, there is a collection of states (the coupled states) ∣ j , m〉, 2 2 2 consisting of simultaneous eigenstates of Jˆ1 , Jˆ2 , Jˆ and Jˆz , which also constitute an orthonormal basis for the same space. There is therefore a unitary transformation with matrix elements

j , m j1, m1; j2 , m2

(4.15)

that connects these two bases3. We will examine the general properties of this transformation and show how to calculate its matrix elements in a simple example. We begin by introducing raising and lowering operators for the combined states: Jˆ+ ≡ Jˆ1+ + Jˆ2+ and Jˆ− ≡ Jˆ1− + Jˆ2−, where Jˆk ± represents the raising (lowering) operator for the kth particle. These definitions follow from the recognition that the angular momentum component operators for the combined states are simply the sum of the corresponding operators for the individual states. If we apply the raising operator Jˆ+ to any one of the product states multiple times, we eventually reach a zero result. In the penultimate step, the state produced is ∣ j1, j1 〉∣ j2, j2 〉. This state, with Jˆz eigenvalue j1 + j2 , and which vanishes when acted upon by Jˆ+, is also the coupled state ∣ j1 + j2, j1 + j2 〉. From this state, acting successively with Jˆ− we can generate the 2( j1 + j2 ) + 1 states ∣ j1 + j2 , m〉, ending with

j1 + j2 , −j1 − j2 = j1, −j1 j2, −j2 . 3

In this compressed notation, ∣ j1, m1; j2 , m2〉 ≡ ∣ j1, m1〉∣ j2, m2〉.

4-7

Discrete Quantum Mechanics

2 From these observations it follows that the largest eigenvalue of Jˆ present among the coupled states is jmax ( jmax + 1), where jmax = j1 + j2. We begin now a search for jmin by examining the state

Jˆ− jmax, jmax = Jˆ− j1, j1 j2, j2 = j1, j1 − 1 j2, j2 + j1, j1 j2, j2 − 1 . The lowering operator leaves the value of j intact, thus this (after normalization) is the state ∣ jmax, jmax − 1〉. A second state with m = j1 + j2 − 1 and orthogonal to ∣ jmax, jmax − 1〉 must be associated with a j-value below jmax and no smaller than j1 + j2 − 1, thus j = j1 + j2 − 1:

jmax − 1, jmax − 1 =

1 2

(

j1, j1 − 1

j2, j2

− j1, j1

j2, j2 − 1

).

Applying Jˆ− successively to this state will generate the 2( j1 + j2 − 1) + 1 states in the family of coupled states with total angular momentum j1 + j2 − 1. If j1 and j2 are sufficiently large, we can continue likewise, noting that there are three product states with summed projection quantum numbers of jmax − 2 that form a three-dimensional subspace. Two orthogonal coupled state vectors within 2 this subspace are proportional to Jˆ − ∣ jmax, jmax 〉 and Jˆ−∣ jmax − 1, jmax − 1〉; the third within the subspace, orthogonal to these two, must be ∣ jmax − 2; jmax − 2〉. As before, successive applications of Jˆ− to this third state will generate the 2( j1 + j2 − 2) + 1 members of the j = j1 + j2 − 2 family. This process reveals that the total angular momentum quantum numbers represented among the coupled state basis are moving downward in sequence from the maximum value: j = j1 + j2 , j1 + j2 − 1, j1 + j2 − 2, …. We can now find the minimum allowed j-value by finding how deep in this sequence we need go to account for a number of states equal to the total number of states in the product basis: j1 + j2

∑ (2j + 1) = (2j1 + 1)(2j2

+ 1) .

j = jmin k

1

The sum on the left is evaluated using ∑1 j = 2 k (k + 1), and from this we learn that 2 jmin = ( j1 − j2 )2 and thus jmin = ∣ j2 − j1 ∣. The limits we have derived for the total angular momentum quantum number j, jmax = ( j1 + j2 ) and jmin = ∣ j1 − j2 ∣, reflect the triangularity limits on the ordinary sum of Euclidean vectors of length r1 and r2: the maximum length of the resultant is r1 + r2 , occurring when the vectors are oriented parallel to one another, and the minimum length is ∣r1 − r2∣, when the vectors are anti-parallel. We have already used the straightforward result that Jˆz∣ j1; m1〉∣ j2; m2〉 = (m1 + m2 )∣ j1; m1〉∣ j2; m2〉. This, too, is reflective of a property of the Euclidean vector sum r1⃗ + r2⃗ : each Cartesian component of the sum is equal to the numeric sum of the corresponding components of the individual vectors, e.g. (r1⃗ + r2⃗ )z = (r1⃗ )z + (r2⃗ )z . This similarity between classical and quantum vector summation is obviously not perfect, as reflected by the

4-8

Discrete Quantum Mechanics

quantized values of the quantum sum resultant, and the fact that the squared length 2 of quantum angular momentum j, i.e. the eigenvalue of Jˆ , is j ( j + 1). Matrix elements of the unitary operator that connects the coupled and product basis states, equation (4.15), are important components of many calculations of atomic, nuclear and elementary particle properties, and thus have been carefully studied and thoroughly documented. These matrix elements go by a variety of names, e.g. Clebsch–Gordan coefficients, C-coefficients and vector coupling coefficients. Notations abound, as do phase conventions (resulting from the phase ambiguity of quantum state vectors). Closely related are the Wigner coefficients and 3-j symbols, which are within an overall normalization and phase of the Clebsch– Gordan coefficients. We will not argue or mediate regarding the merits of the various choices, but rather briefly show how these matrix elements can be determined in the simplest of non-trivial cases. Take two spin one-half particles with state vectors ∣ 12 , m1〉1 and ∣ 12 , m2〉2 , considered as a single composite state. The product basis states of this fourdimensional entity are ∣ 12 , + 12 〉1 ∣ 12 , + 12 〉2 , ∣ 12 , + 12 〉1 ∣ 12 , − 12 〉2 , ∣ 12 , − 12 〉1 ∣ 12 , + 12 〉2 , and 1

1

1

1

∣ 2 , − 2 〉1 ∣ 2 , − 2 〉2 . From our earlier work on the general case, we see that the coupled representation has j = 1 and j = 0 and thus the coupled representation states are ∣1, 1〉, ∣1, 0〉, ∣1, −1〉 and ∣0, 0〉. We can also extract from the earlier discussion the relations

∣1, 1〉 =

1 , 2

1

+2

1 ⎛ ∣1, 0〉 = ⎜ 2⎝ ∣1, −1〉 =

1 , 2

1 , 2

1

1 , 2

1

+2

1

−2

1 ⎛ ∣0, 0〉 = ⎜ 2⎝

1

1

−2

+ 2

1 , 2

1

−2

1 , 2

1

1

+2

2

1

−2

1

+2

2

1 , 2

1

1 , 2

1

1 , 2

1

+2

2

1 , 2

1

−2

1 , 2

− 2

1

−2

1

1 , 2

1

+2

. 2

From these equations we can simply read off values for the set of transformation coefficients for the coupling of two spin one-half particles: 1

1

1

1, 1 2 , 2 ; 2 ,

1 2

= 1, −1 2 , − 2 ; 2 , − 2

1

1

1

1

= 1, 0 2 , − 2 ; 2 ,

1

1

1

1

= − 1, 0 2 , − 2 ; 2 ,

0, 0 2 , 2 ; 2 , − 2

1

1

1

1, 0 2 , 2 ; 2 , − 2

1

1

1

1

1

1

1 2 1 2

1

=1

=

1 2

=

1 2

.

The general properties of these coefficients, as well as their specific values, can be found in a variety of standard sources, including [Edmunds1957], [Rose1957] and [Biedenharn1981].

4-9

Discrete Quantum Mechanics

4.5 Interaction operators Angular momentum is conserved, we know, due to the angular homogeneity of the Universe, but try to tell that to an electron as it works its way through that magnetic field-littered environment within an atom: it will not believe you. A proton or neutron in the nucleus would be similarly sceptical. The viewpoint of a longsuffering elementary particle, which sees the rest of the Universe as existing only to shove it around, reflects our own tendency to focus on a single particle and approximate the influence of other particles by way of terms in a potential energy. Such an approach, to be accurate, requires us to consider interaction operators that have angular and spin dependence. We first consider interaction terms with non-uniform angular dependence, 2 examining explicit representations of the eigenstates of Lˆ and Lˆ z operators. If we describe three-dimensional space using standard spherical components—radial distance r, polar angle relative to the z axis of θ and azimuthal angle relative to the x axis of ϕ—we can transform the commuting angular momentum operators of equations (4.1) and (4.2) into

∂ Lˆ z = −i ∂ϕ

∂ ⎞ 1 ∂2 1 ∂ ⎛ 2 ⎜sin θ ⎟ − . Lˆ = − ∂θ ⎠ sin2 θ ∂ϕ 2 sin θ ∂θ ⎝

and

The eigenvalue equation Lˆ z Φ(ϕ ) = mΦ(ϕ ) has solutions of the form Φ = exp(imϕ ) with the restriction to integer values of m coming from the boundary condition 2 Φ(ϕ ) = Φ(ϕ + 2π ). The second eigenvalue equation Lˆ P (θ )Φ(ϕ ) = l (l + 1)P (θ )Φ(ϕ ) is more challenging, but can be solved, showing P(θ ) to be proportional to the associated Legendre polynomials [Weisstein2002a], with indices l and m, where l is a non-negative integer obeying l ⩾ ∣m∣. The inner product of two such functions is accomplished by way of integration over the unit sphere, so when properly normalized

l ′ , m′ l , m =

∫0

π

sin θ dθ

∫0

2π

dϕ Yl*′m′Ylm = δl ′l δm′m,

in which Ylm(θ , ϕ ) = NPlm(θ )Φ(ϕ ), with Plm the associated Legendre function and N the required normalization constant. The orthonormal functions Ylm are known as the spherical harmonics [Weisstein2002b] and form an orthonormal basis of functions defined on the unit sphere. By way of example, we visit an atomic electron whose orbit is perturbed by a neighboring electron, or perhaps a passing photon. The largest interaction term with angular variation is often of the form v ⃗ · r ⃗ , in which r ⃗ is the position vector of the electron and v ⃗ is a vector independent of the electron’s coordinates. We are at liberty to choose the direction of v ⃗ to be the z direction, in which case

v ⃗ · r ⃗ = vr cos θ =

4-10

3 vrY10. 4π

Discrete Quantum Mechanics

Energy shifts and the rates of transition caused by this interaction, to lowest order of approximation, are calculated by way of the matrix element of this interaction term between angular momentum states. The angular part of such a matrix element is

l ′ , m′ Y10 l , m =

∫0

π

sin θ dθ

∫0

2π

dϕ Yl*′m′Y10Ylm.

(4.16)

Note that the last two terms of the integrand here are mathematically equivalent to the product state ∣1, 0〉∣l , m〉 expressed in the representation of spherical harmonics. This product can be expressed as a linear combination of coupled states using the coupling coefficients of equation (4.15):

1, 0 l , m =

∑

L; M 1; 0, l ; m L, M ,

L, M

producing the identity

Y10Ylm =

∑

L; M 1; 0, l ; m YLM .

L, M

Substituting this into the integral of equation (4.16) and using the orthonormality of the spherical harmonics produces the simplified result

⎛ ⎞ dϕYl*′m′⎜⎜ ∑ L; M 1; 0, l ; m YLM ⎟⎟ ⎝ L, M ⎠ = l ′ , m′ 1; 0; l , m . (4.17)

l ′ , m′ Y10 l , m =

∫0

π

sin θ dθ

∫0

2π

The angular integral needed for this calculation is equivalent to the Clebsch–Gordan coupling coefficient, illustrating one reason these coefficients have received so much attention. Based on the properties we have already explained, we know this result vanishes unless m′ = m and l ′ and l differ at most by 1, producing so-called ‘selection rules’ that explain the absence of certain otherwise acceptable transitions in composite systems. Interaction operators are often more complicated than simple functions of angular variables, however. We have already seen that orbital angular momentum operators not only have θ and ϕ dependence, but also contain derivative operators with respect to these variables. Intrinsic angular momentum operators are square matrices of dimension 2j + 1. In order to enable calculations of the matrix elements of generalized angular and spin dependent operators, work in the mid-twentieth century yielded a formalism for characterizing such operators in terms of angular momentum variables, the so-called quantum ‘irreducible tensors’ [Danos1990, Fano1959]. To understand this topic, it is worth first examining Cartesian tensors, which in three-dimensional Euclidean space are collections of spatially dependent variables that transform in specified ways when the coordinate axes are rotated. A scalar,

4-11

Discrete Quantum Mechanics

a single variable that is unchanged by rotations of coordinates, is a tensor of rank zero. A vector, or tensor of rank one, has three components that change under rotation, as specified by a rotation matrix: 3

v j′ =

∑ Rjkvk , k=1

where elements of the matrix R are functions of the angle of rotation and rotation axis. A tensor of rank two has two indices, each of which runs through three values. These are most often represented by 3 × 3 matrices. As an example of a tensor of rank 2, take the nine values represented by the outer (dyadic) product of two position vector components: Tjk ≡ rj′rk . When the coordinate axes are rotated, each vector is separately rotated using the rotation matrix R, so the rank two tensor components are altered as 3

T jk′ =

∑

RjmR knTmn.

m, n = 1

This tensor is reducible in the following sense: • there is a linear combination of tensor elements—r ′⃗ · r ⃗ = ∑j Tjj , the matrix trace—that is invariant under rotation, transforming like a tensor of rank zero; • there are three additional independent linear combinations— (r ′⃗ × r ⃗ )i = Tjk − Tkj , with i , j , k a cyclic permutation of the three coordinate indices— that transform like a vector, thus a tensor of rank one; • five remaining tensor components have linear combinations— Tjk + Tkj − 1 tr(T )δj ,k —that transform as elements of a traceless symmetric matrix, a 3 tensor of rank 2 that is not further reducible. The search for quantum irreducible tensors begins with the rotation operator for quantum state vectors. The work of section 2.3.3, culminating in equation (2.6) implies that a spin one-half state function is rotated through angle θ about the z axis by the operator exp( −isˆzθ ). This result, generalized, produces the operator for rotation of a spin one-half state through angle θ about an axis parallel to unit vector n ⃗ :

Rˆs = exp( −isˆ · n ⃗θ ) . A meta-generalization can show that for a state function of any angular momentum J, the operator that affects a rotation through angle θ about axis n ⃗ is

Rˆ J = exp( −iJˆ · n ⃗θ ) . (These generalizations can be taken on as exercises by readers with ample supplies of scrap paper and spare time.) The generalized operator RˆJ is a function of the 2 angular momentum component operators, and thus commutes with the Jˆ operator,

4-12

Discrete Quantum Mechanics

implying that when it is applied to an angular momentum state it produces another with the same j value:

exp( −iJˆ · nˆ θ ) j , m =

∑ D mj ′m

j , m′ ,

m′

where the Dmj ′m represent a matrix expression (with indices m′, m) for the rotation operation. Components of D depend upon the angle of rotation and two variables that define the rotation axis, which are most often characterized by the Euler angles that denote the rotation. Closed form expressions for the Dmj ′m as a finite sum, in terms of Euler angles, were first developed by Wigner [Wigner1931] and can be found in many standard quantum mechanics sources [Rose1957]. L Quantum irreducible tensor operators ; M are defined by the requirement that under rotation they behave in the same way as quantum states with the same quantum numbers. Since quantum matrix elements are scalars, and are thus unchanged by a rotation of coordinates,

ϕ′ Oˆ ′ ψ ′ = ϕ Oˆ ψ , where the primed entities have their coordinates expressed in a rotated coordinate system. If the state functions’ coordinates are rotated by the operator UR, this implies that

ϕ UR†Oˆ ′UR ψ = ϕ Oˆ ψ , from which it follows that

ˆ R†. Oˆ ′ = UROU The defining equation for irreducible tensor operators is therefore L exp( −iLˆ · nˆ θ ); M exp(iLˆ · nˆ θ ) =

∑ DML ′M ; ML ′. M′

These operators, unlike state functions, are restricted to integer values of L and M, since they are defined in the context of quantum matrix elements, for which the bra and ket state functions both have integer, or both have half-integer, angular momentum states. As we shall soon see, the normalization constant that is not specified by this definition, and which is independent of M, is of but minor concern due to the way these operators are used. The definition above produces useful identifications of operators as irreducible tensors, albeit with formidable amounts of algebra. Scalars, left unchanged by any rotation of coordinate axes, are identified with ; 00. Quantum operators that represent vector observables, like position, momentum and angular momentum, have components that are related to tensors of rank one:

vz = ; 10;

−

1 vx + ivy = ; 11; 2

(

)

4-13

and

1 vx − ivy = ; 1−1. 2

(

)

Discrete Quantum Mechanics

Scalar products of vector operators, e.g. Jˆ1 · Jˆ2 , can be shown to be two tensor operators of rank one coupled using Clebsch–Gordan coefficients to a tensor of rank zero. Vector cross products, like rˆ × pˆ , are two tensors of rank one coupled, again using Clebsch–Gordan coefficients, to a tensor of rank one. These, and a handful of other such identifications, allow the characterization of the vast majority of common quantum coupling terms in terms of irreducible tensor operators. Indeed, the collection of irreducible tensors form a basis for all possible interaction operators, so every conceivable coupling term can be so characterized. The power of irreducible tensorial sets lies in the Wigner–Eckart theorem [Eckart1930, Wigner1931]: L 〈 j′; m′∣; M

j ; m = j ; m j1 ; m1, j2 ; m2

j′ ; L j .

(4.18)

The term on the far right is called the reduced matrix element, and as its notation suggests it is independent of the projection quantum numbers m′, M and m. It contains the constants that relate the operators being used to the corresponding irreducible tensor operators, as well as other parts of the matrix element being calculated, such as the integral over radial state functions. This theorem shows that all the regularities related to angular symmetry are expressed by the Clebsch– Gordan coefficient that multiplies the reduced matrix element. As mentioned earlier in our example of an interaction operator represented by a spherical harmonic, the triangularity properties of the Clebsch–Gordan coefficient indicate which of these matrix elements will vanish identically. The values of non-zero Clebsch–Gordan coefficients allow the evaluation of the relative strengths of transitions enabled by the interactions. A great many physical predictions follow from knowledge of Clebsch–Gordan coefficients. Many web resources provide look-up tables or calculators for these coupling coefficients, such as the Wolfram Alpha site4. The formalism of angular momentum coupling of states, and operator characterizations in terms of irreducible tensor operators, enables all these predictions using results from finite-dimensional quantum mechanics.

4.6 Isospin Symmetry begets regularity. Particles acting under the influence of a Hamiltonian that is spherically symmetric exhibit conserved angular momentum. In addition, states with common j values and differing m values have identical energies and reaction rates. Examination of the properties of atomic electrons shows this property to a reasonable approximation. Within the atom, however, magnetic fields are ubiquitous due to the motion of electrons and protons, and the intrinsic magnetic properties of protons and neutrons. As we have seen, magnetic fields break the spherical symmetry, and this causes energy splitting among otherwise degenerate states, and differing transition and reaction rates among states with identical

4

http://wolframalpha.com/input/?i=Clebsch-Gordan+calculator.

4-14

Discrete Quantum Mechanics

j values. Nonetheless, we find it useful to begin with angular momentum eigenstates and approximate the small effects that come from symmetry-breaking effects. The isospin formalism arises from a quite similar scenario. Perhaps due to the order of discovery, we have never thought of the electron as two particles, despite our current understanding that a spin-up electron behaves differently in some circumstances from a spin-down one. The 1897 discovery of the negatively charged electron and the realization that it was a constituent of all atoms, in the context of the electrical neutrality of most atoms and molecules, led to the realization that there must be a positively charged atomic particle that also carried the majority of the atom’s mass. The term ‘proton’ entered into common use in the 1920s. James Chadwick in 1932 confirmed the existence of a neutral particle within the nucleus—the neutron—differing by only a fraction of a per cent in mass from the proton. Initially, neutrons seemed to play only the role of packing peanuts, spacers keeping protons from such close proximity that their Coulomb repulsion would overwhelm their strong force attraction, thus preventing bound nuclei. With such an obvious distinction—charge—and such different roles, protons and neutrons were thought to be unrelated, with their near-equal masses seen as a coincidence. As nuclear properties came under intense scrutiny in the 1930s, the proton and neutron began to appear less like members of different species and more like twins. The term ‘nucleon’ came into use, with the proton and neutron being the two cases of this category, analogous with ‘spin-up electron’ and ‘spin-down electron’ being cases of the category ‘electron’. A formalism was proposed to mirror that of electron spin. The nucleon was defined to be a two-state system with eigenstates ‘isospin up’ (the proton) and ‘isospin down’ (the neutron)5. The strong nuclear force was described as invariant under rotations in isospin space, a symmetry that is broken by the electromagnetic force. As properties of complex nuclei were discovered, isospin symmetry revealed itself in many ways, and when the symmetry was not apparent, this was explained straightforwardly through the effects of Coulomb repulsion among protons. The state function of nucleons had now expanded to include, in addition to radial, angular and intrinsic angular momentum components, an isospin degree of freedom: 1

ΨN = ψ (r )∣L; M 〉 2 ; m

1 spin 2

; mI

. isospin

By 1935 an exchange model of the strong nuclear force had been proposed by Hideki Yukawa, who suggested that the strong nucleon–nucleon force was the consequence of their exchange of a new category of massive, strongly interacting particles—mesons. In the absence of particle accelerators to search for these newcomers, cosmic ray particles were studied, leading to the 1947 discovery of the pi meson. These two pieces of work, opening the door to a teeming menagerie of 5

The term isospin describes a property that has no physical similarity to intrinsic angular momentum (‘spin’), but was chosen because of the mathematical similarity of the two formalisms. The term originated as a contraction of ‘isotopic spin’, a name later converted to ‘isobaric spin’, considered to be less misleading.

4-15

Discrete Quantum Mechanics

strongly interacting particles (hadrons), were recognized by the Nobel Prize in Physics in 1949 (Yukawa) and 1950 (Cecil Powell for the discovery of the pion). Hadrons in two categories—baryons, which were closely related to the nucleon, and mesons—were found in abundance and their properties were studied. All, with the exception of the proton, were unstable. Their mean lifetimes varied from 880 s (nearly forever) for the neutron, to 2.610−8 s for the charged pions (the longest lived of the mesons), to a paltry 10−24 s or so for many baryon resonance states. These particles frequently appear as families with nearly equal masses (rest energies) but different charges. • Nucleon: two charge states (proton charge +e and neutron charge 0); average mass 939.0 MeV, mass spread 1.3 MeV. • Δ(1232): four charge states (+2e , +e , 0, −e); average mass 1232 MeV, mass spread 4 MeV. • Pion: three charge states ( ±e, 0); average mass 138.1 MeV, mass spread 4.6 MeV. • η meson: one charge state (0); mass 547.9 MeV. • ρ(770) meson: three charge states ( ±e, 0); average mass 775.3 MeV, mass spread ≈2 mev.6 This evidence led to widespread use of the full formalism of isospin, reflecting exactly the formalism already developed for intrinsic angular momentum. This ˆ centered on a vector isospin operator I ⃗ , with Hermitian components Iˆ1, Iˆ2 and Iˆ3,7 2 along with Iˆ = I12 + I22 + I32 , which obey the same commutation relations as 2 Jˆx, Jˆy, Jˆz and Jˆ . From section 4.3 we have learned that the whole formalism associated with angular momentum follows from these commutation relations, thus the formalism of isospin is the same. The spectrum of isospin states 1 3 ∣I ; Iz〉 has I = 0, 2 , 1, 2 , … with Iz = −I , −I + 1, −I + 2, … I. This reveals the

nucleon to be a particle with I = 12 , the pion and ρ(770) meson to have I = 1, the 3

η meson to have I = 0 and the Δ(1232) to have I = 2 . Coupling of isospin states proceeds as coupling of angular momentums states does, with identical coupling coefficients. Operators in isospin space can be classified in terms of irreducible tensors, and matrix elements expressed in terms of vector coupling coefficients. The fact that the Hamiltonian operator describing the strong force was invariant under ˆ rotations in isospin space (exp(−iθI ⃗ · n ⃗ ) with n ⃗ a real three-dimensional unit vector) allowed adoption of the whole machinery developed for dealing with angular momentum entities to the purpose of computation of isospin properties8.

6

Based upon estimates from 2015 data in http://pdg.lbl.gov. Indices 1, 2 and 3 are preferred to x, y and z so as to avoid any suggestion that isospin space has any relationship to configuration space. 8 There is yet another variable, the weak isospin, that is conserved in weak interactions and has four operators 2 paralleling Jˆx, Jˆy, Jˆz and Jˆ , and thus adopts the identical formalism for another use. 7

4-16

Discrete Quantum Mechanics

The mathematical symmetry underlying angular momentum theory and isospin theory9 is that of the group10 SU (2), the special unitary group in two dimensions. This group is represented by a set of four complex 2 × 2 matrices that are unitary and have unit determinant. In the case of intrinsic angular momentum, these matrices are the unit I and the rotation operators along the coordinate axes exp( −iθJˆx ), exp( −iθJˆ2 ), exp(−iθJˆz ).

4.7 And… If our Universe were only of a single spatial dimension, we would be spared of all this.

Exercises 1. Following the steps that were used in deriving equation (4.3), derive equations (4.4) and (4.5). 2. Use the fact that Jˆx, Jˆy and Jˆz are Hermitian, prove that 2 2 2 2 Jˆ ≡ Jˆx + Jˆy + Jˆz

(4.19)

is also Hermitian. 2 3. Use the commutation relations among the operators Jˆx, Jˆy, Jˆz and Jˆ to prove equations (4.13) and (4.14). 4. Show that J− acts on ∣κ , m〉, producing an eigenstate of Jˆz with eigenvalue (m − 1). 2 5. Prove, by evaluating Jˆ ∣κ , m min〉, that κ = m min(m min − 1).

Bibliography [Noether1918] Nöther E 1918 Invariante Variationsproblema, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen 235–57 Nöther’s original paper on this topic and an English translation by M A Tavel can be found at ‘Noether, Amalie Emmy’ CWP http://cwp.library. ucla.edu/ [Rose1957] Rose M E 1957 Elementary Theory of Angular Momentum (New York: Wiley) (Reprinted by Dover in 1995) [Edmunds1957] Edmonds A R 1957 Angular Momentum in Quantum Mechanics (Princeton, NJ: Princeton University Press) [Biedenharn1981] Biedenharn L C and Louck J D 1981 Angular Momentum in Quantum Physics (Reading, MA: Addison-Wesley) [Weisstein2002a] Weisstein E W 2002 CRC Concise Encyclopedia of Mathematics 2nd edn (Weisstein E W Associated Legendre Polynomial MathWorld—A Wolfram Web Resource http://mathworld.wolfram.com/AssociatedLegendrePolynomial.html) (Boca Raton, FL: CRC) [Weisstein2002b] Weisstein E W 2002 CRC Concise Encyclopedia of Mathematics 2nd edn (Weisstein E W Spherical harmonic MathWorld—A Wolfram Web Resource http://mathworld.wolfram.com/SphericalHarmonic.html) (Boca Raton, FL: CRC) 9

And also weak isospin theory. A brief introduction to the notions of group theory can be found in appendix A.

10

4-17

Discrete Quantum Mechanics

[Fano1959] Fano U and Racah G 1959 Irreducible Tensorial Sets (New York: Academic) [Danos1990] Danos M and Gillet V 1990 Angular Momentum Calculus in Quantum Mechanics (Singapore: World Scientific) [Wigner1931] Wigner E 1931 The application of group theory to the quantum dynamics of monatomic systems Gruppenteorie (Braunschweig: Friedrich Vieweg und Sohn) [Eckart1930] Eckart C 1930 The application of group theory to the quantum dynamics of monatomic systems Rev. Mod. Phys. 2 305

4-18

IOP Concise Physics

Discrete Quantum Mechanics H Thomas Williams

Chapter 5 Quantum many-body problem

Thermodynamics, dating from the early nineteenth century, examines relationships among variables that describe matter in bulk, such as temperature, pressure and entropy. Beginning in the second half of the nineteenth century, there were successful attempts (notably due to Maxwell, Boltzmann and Gibbs) to explain the equilibrium thermodynamics of macroscopic bodies based upon the microscopic interactions among molecules obeying Newton’s laws of motion, so establishing the field of statistical mechanics. As soon as quantum mechanics found a firm mathematical footing in the late 1920s, attempts were made to create useful statistical mechanical models based upon the quantum behavior of molecular constituents of matter. Significant among such early models was the Ising–Heisenberg model of magnetism, in which N microscopic systems—each possessing a quantum-mechanical spin—were arranged in a d-dimensional static array, subject to nearest-neighbor spin–spin interactions. Even in one dimension, solutions to such models proved challenging, but over time great progress has been made [Sunderland2004]. In this chapter we examine a simple version of the Ising–Heisenberg model: the Heisenberg XXX spin chain. This and related models have their most practical applications in the thermodynamic limit, where the number of array elements approaches a countable infinity. Nonetheless, there are things that can be learned by examining small systems. Our primary goal is to introduce techniques that have proven useful in this and many other quantum mechanical contexts. Our strategy will be first to introduce the Hamiltonian operator for a system of N spins, and then to seek its eigenstates and eigenvalues (energies), making use of the symmetries of the Hamiltonian to simplify the search.

5.1 The general case: Heisenberg XYZ spin chain The Bohr–van Leeuwen theorem [Bohr1972, vanLeeuwen1921] demonstrates that consistent application of statistical mechanics using Newtonian mechanics cannot

doi:10.1088/978-1-6817-4125-3ch5

5-1

ª Morgan & Claypool Publishers 2015

Discrete Quantum Mechanics

explain magnetic effects in matter. The Heisenberg spin chain model, also know as the Ising–Heisenberg model, was introduced to utilize quantum mechanics and statistical principles to explain mechanisms such as paramagnetism, diamagnetism and ferromagnetism. The chain is a one-dimensional array of N sites, each occupied by a particle exhibiting only a spin one-half degree of freedom, and with the only interactions being between the spins of adjacent particles. Finite length chains will be considered, with periodic boundary conditions identifying site N + 1 with site 1. The general form of the Hamiltonian operator for the Heisenberg spin chain is N

(

H = −∑ kx j sˆx

j+1

sˆx + ky j sˆy

j+1

sˆy + kz j sˆz

j+1

)

sˆz + E ′ ,

(5.1)

j=1

in which jsˆx represents the x-component (for example) of the spin one-half operator on the jth site, and where E ′ is a constant that can be used to adjust the energy of the lowest eigenstate of H to be zero. When the three constants kx, ky and kz are distinct, this is referred to as the XYZ spin chain.

5.2 Ising model The simplest special case of the Heisenberg XYZ spin chain is one in which the constants kx = ky = 0 and kz are the same for every site in the chain: N

(

H = −kz∑ j sˆz

j+1

)

sˆz + kz

j=1

N . 4

(5.2)

This problem has a straightforward solution that yields important thermodynamic properties for such a system. It is a special case of the one-dimensional Ising model [Ising1925, Lenz1920] that uses the Hamiltonian N

(

HI = −4K ∑ j sˆz

j+1

N

)

sˆz − 2h∑ j sˆz .

j=1

(5.3)

j=1

We will examine the Ising model since it, too, is easily solved and results that follow from the Hamiltonian of equation (5.2) are easily recovered. The Ising model Hamiltonian HI describes nearest-neighbor spin–spin interactions between spin one-half entities at N locations along the chain, in addition to a second term describing the interaction of each spin with an external magnetic field represented by the parameter h. We assume the system of N spins to be in contact with an environment at absolute temperature T. Labeling the eigenstates of HI as ∣c〉, with eigenvalues Ec, the probability of finding the system in any eigenstate is given by the Boltzmann distribution

Pc =

⎛ E ⎞ exp⎜ − c ⎟ ⎝ kBT ⎠ Z

5-2

,

Discrete Quantum Mechanics

where kB is the Boltzmann constant and Z, the normalization constant, is known as the partition function: Z ≡ ∑ exp( −βEc ), (5.4) c

defining the inverse temperature variable β = k 1T . From the partition function we B can derive many important thermodynamic (macroscopic) variables, for instance: • the average energy of the system

〈E 〉 =

1

1∂

∑EcPc = Z ∑Ec exp( −βEc ) = − Z ∂ c

β

c

Z=−

∂ ln(Z ); ∂β

• the variance of the energy

〈(ΔE )2 〉 ≡ 〈(E − 〈E 〉)2 〉 =

∂2 ln(Z ); ∂β 2

• the heat capacity

∂ 〈E 〉 = kBβ 2〈(ΔE )2 〉; ∂T

Cv ≡ • and the entropy

S = −kB∑Pc ln(Pc ) = kB(ln Z + β E ). c

For this reason, we will establish the eigenvectors of HI and calculate therefrom a closed form expression for the partition function. We will work from a basis in which the state of the kth particle in the chain, ∣k 〉, is ⎛ ⎞ ⎛ ⎞ represented by ⎜ 1 ⎟ for spin up along the z axis, and ⎜ 0 ⎟ for spin down along the ⎝ 1⎠ ⎝ 0⎠ z axis. A basis state of the N element chain is ∣c〉 = ∣1〉∣2〉⋯∣N 〉, for which there are 2N distinct orthonormal possibilities. In this basis, every term in HI is diagonal, so every N-spin basis state is an eigenstate. In preparation for calculating the partition function, we make a series of observations regarding the Ising Hamiltonian. We can recast HI as a sum over nearest-neighbor pairs (1,2; 2,3; 3,4; etc) in the form N

HI =

∑(

j

−4K sˆz

(

j+1

N

sˆz − h j sˆz +

j+1

j=1

sˆz

)) = ∑H . j

j=1

Note that the individual terms (Hj) commute with one another, therefore

⎞ ⎛ exp( −βHI ) = exp⎜⎜ −β∑Hj ⎟⎟ = ⎠ ⎝ j

N

∏ exp(−βHj ).

(5.5)

j=1

(This property of exponentiated commuting operators is proven in appendix A.) 5-3

Discrete Quantum Mechanics

Because they are diagonal in the basis over which we will sum, we treat the operators of the form jsˆz that are found in Hj as classical variables jsz with two values, 12 and − 12 , and the piece of the sum over configurations involving the jth particle as a sum over these two values. We can thus cast the operator exp(−βHj ) = exp(−β (− 4K ( jsz j + 1sz ) − h( jsz + j + 1sz )), in terms of its four possible values, as a 2 × 2 matrix Qˆ with matrix elements 1 1 • Q11 = exp( β (K + h )) for jsz = 2 and j + 1sz = 2 ; • Q12 = exp( −βK ) for jsz =

1 2

and

1

j+1

sz = − 2 ;

1

• Q21 = exp( −βK ) for jsz = − 2 and

1

j+1

sz = 2 ; and

• Q22 = exp( β (K − h ) for jsz = − 12 and

1

j+1

sz = − 2 .

Using this characterization, it is straightforward to evaluate 1 2

∑

exp( −βH1)exp( −βH2 )

2

sz =− 12

as • exp(2β (K + h )) + exp(−2βK ) for 1sz = • exp(βh ) + exp( −βh ) for 1sz =

1 2

1 2

1

and 3sz = 2 , 1

and 3sz = − 2 , 1

1

• exp(βh ) + exp( −βh ) for 1sz = − 2 and 3sz = 2 , and 1

1

• exp(2β (K − h )) + exp(−2βK ) for 1sz = − 2 and 3sz = − 2 , 2 2 2 2 which are seen to be the elements (Qˆ )11; (Qˆ )12 ; (Qˆ )21; and (Qˆ )22 , respectively of the square of the 2 × 2 matrix Qˆ introduced above. Using this result iteratively, we deduce that 1 2

1 2

∑ ∑

1 2

2

sz =− 12 3sz =− 12

n

∑ ∏ exp( −βHj )

⋯ n

sz =− 12 j = 1

is n • (Qˆ )11 for 1sz = 12 and n + 1sz = 12 , n • (Qˆ )12 for 1sz = 12 and n + 1sz = − 12 , n • (Qˆ )21 for 1sz = − 12 and n + 1sz = 12 , and n • (Qˆ )22 for 1sz = − 1 and n + 1sz = − 1 .

2

2

For an N-particle chain with periodic boundary conditions, which identifies particle N + 1 with particle 1, we use this result with n = N and complete the final sum (over 1 sz ) to arrive at 1 2

Z=

∑ exp( −βHI ) = c

1 2

1 2

∑ ∑

⋯

sz =− 12 3sz =− 12

2

N

5-4

n

∑ ∏ exp( −βHj ) = Tr( Qˆ N ).

sz =− 12 j = 1

Discrete Quantum Mechanics

Qˆ is a real, symmetric matrix and can be brought into diagonal form using a unitary ˆ , thus by using the cyclic property of matrix U through the operation Qˆ diag = U †QU the trace we see that

(

)

N Z = Tr Qˆ diag = q+N + q−N ,

(5.6)

where q± are the eigenvalues of Qˆ , specifically

(

q± = exp(βK ) cosh(βh) ±

)

sinh2(βh) + exp( −4βK ) .

Note that q+ > q−, so that in the thermodynamic limit of large N, we can use Z = q+N and from this calculate the macroscopic variables listed above, among others. Such results can be found discussed in detail elsewhere [Baxter2007].

5.3 Heisenberg XXX spin chain Our focus now shifts to the more challenging XXX Heisenberg spin chain that uses the Hamiltonian of equation (5.1) with kx = ky = kz ≡ K . This is the model that Hans Bethe [Bethe1931] solved for exact eigenvalues and eigenstates of the Hamiltonian for arbitrary N in a way that avoids matrix diagonalization. His method utilizes an assumption about the form of the eigenstates now known as the Bethe ansatz. This methodology has since been extended and utilized in countless applications of quantum statistical mechanics models. As in our preceding discussion of the Ising model, we will use as a basis for each spin one-half particle eigenstates of sˆz , spin along the z axis. To facilitate the discussion to follow, however, we introduce the notations ∣+〉 representing spin up, ∣−〉 representing spin down and N-particle states of the form ∣+−−−−++⋯〉 for the complete chain. The brute force approach to this problem would be to create the Hamiltonian matrix with matrix elements 〈c′∣Hxxx∣c〉, ∣c〉 and ∣c′〉 representing two of the 2N basis state configurations of the chain, and then diagonalize the matrix to find its eigenvectors and eigenvalues. From that, the partition function could be found, at least in principle, and thermodynamic properties derived1. Direct diagonalization of a 2N × 2N matrix, particularly to find the analytic dependence upon the parameter J in the Hamiltonian, is a prohibitive challenge for all but quite small values of N. A way to simplify this challenge somewhat is to identify and utilize symmetries of the Hamiltonian. The Heisenberg XXX spin chain Hamiltonian is N

(

HXXX = −K ∑ jsx

j+1

sx + jsy

j=1

j+1

sy + jsz

j+1

)

sz +

N K, 4

(5.7)

where (as we will see) the constant has been chosen to set the lowest eigenvalue at zero. If the parameter K is greater than zero, the favored lower energy states are 1

The strategy used for the Ising model problem cannot work here, primarily because the configurations c summed over in the definiton of Z of equation (5.4) are not basis states, but are rather the eigenstates of the non-diagonal Hamiltonian.

5-5

Discrete Quantum Mechanics

those with the greatest number of aligned spins. This choice models paramagnetic or ferromagnetic behavior, and in the discussion to follow we assume K > 0. We can produce an alternative and often useful expression for HXXX through the introduction of raising and lowering operators on the jth particle as jσ+ ≡ jsx + i jsy and jσ − ≡ jsx − i jsy (where i represents the imaginary unit) with properties j

σ+ +

j

= 0,

j

σ+ −

j

j

= + j,

σ− +

j

= − j,

and

j+1

j+1

j

σ− −

j

= 0.

Straightforward algebra leads to, j

σ+

j+1

σ − + jσ −

(

j+1

σ+ = 2 jsx

sx + jsy

sy

)

enabling re-expression of the Hamiltonian as N ⎛j σ HXXX = −K ∑⎜⎜ + j = 1⎝

j+1

σ − + jσ − 2

j+1

σ+

+ jsz

⎞ N sz⎟⎟ + K . 4 ⎠

j+1

(5.8)

To enable yet another version of HXXX , which we will utilize in upcoming work, note the effect of j

S⃗ ·

j

j+1

S ⃗ ≡ jsx

j+1

sx + jsy

j+1

sy + jsz

j+1

sz =

σ+

j+1

σ − + jσ − 2

j+1

σ+

+ jsz

j+1

sz

on adjacent spins in the j and j + 1 locations, j

j

S⃗ ·

S⃗ ·

S ⃗∣++〉 =

1 ∣++〉, 4

⃗ S ∣+−〉 =

1 1 ∣−+〉 − ∣+−〉, 2 4

j+1

j+1

j

S⃗ ·

j+1

⃗ S ∣−−〉 = j

S⃗ ·

1 ∣−−〉 4

j+1

⃗ S ∣−+〉 =

1 1 ∣+−〉 − ∣−+〉. 2 4

This enables the definition of a permutation operator that swaps the spin states of adjacent particles,

⎛j Pj , j +1 ≡ 2⎜ S ⃗ · ⎝

j+1

S⃗ +

1 ⎞ I ⎟, 4 ⎠

(5.9)

which in turn enables us to express the Hamiltonian for the XXX chain as N

HXXX = −

K N ∑Pj, j+1 + K . 2 j=1 2

(5.10)

From the expression for HXXX of equation (5.7) it is clear that we should find the lowest energy when all spins are aligned, as for instance ∣+++⋯+〉. It is easiest to

5-6

Discrete Quantum Mechanics

1

evaluate HXXX∣+++⋯+〉 using equation (5.8), utilizing σ+∣+〉 = 0 and sz∣+〉 = 2 ∣+〉 to arrive at N

1 N + K = 0. 4 4 j=1

HXXX∣+++⋯+〉 = −K ∑

There are three symmetries of HXXX that we will use to simplify its diagonalization. From the form of equation (5.7) we see that HXXX is quadratic in sz operators. It follows that a reflection through the x–y plane, which produces a simultaneous sign change on all spin z components, has no effect on the Hamiltonian. The operator, Rˆxy , which evokes this reflection thus commutes with HXXX . If, therefore, HXXX∣ψ 〉 = E ∣ψ 〉, it follows that

RˆxyHXXX ψ = HXXXRˆxy ψ = ERˆxy ψ , showing that Rˆxy∣ψ 〉, like ∣ψ 〉, is an eigenstate of HXXX with the same eigenvalue E. 2 Rˆxy , as a reflection operator, satisfies Rˆ xy = I , from which it is an easy exercise to show that its eigenvalues are r = ±1. A second symmetry relates to a translation operator Tˆ , the effect of which is to move the spin in the jth site to the j + 1th site, for 1 ⩽ j ⩽ N . From equation (5.7), the Hamiltonian is seen to be a sum of identical interactions between every pair of neighboring spins, so HXXX is unchanged by the action of Tˆ . Stated otherwise, the operators Tˆ and HXXX commute. Furthermore, since Rˆxy involves the inversion of the spin of every site in the chain, the translation of all sites caused by Tˆ has no effect on the inversion, meaning that Tˆ also commutes with Rˆxy . An operator that accomplishes this translation can be written in terms of a product of permutation operators:

Tˆ = P1,2P2,3…PN −1,N , as can be demonstrated convincingly by example. Since the operators Tˆ and HXXX commute, there is a complete set of states that are simultaneous eigenstates of the two operators (see appendix A). Define t to be an eigenvalue of Tˆ and note that applying Tˆ N times produces one complete N rotation of the ring, and thus Tˆ = I . If ∣t〉 is an eigenstate of Tˆ with eigenvalue t, N then Tˆ ∣t〉 = t N ∣t〉 = ∣t〉, and therefore the N eigenvalues of Tˆ are the N complex roots of 1:

⎛ n⎞ tn = exp⎜ 2π i ⎟ n = 1, 2, … N . ⎝ N⎠

(5.11)

Finally, define the operator corresponding to the z component of spin for the whole chain N

Sˆz =

∑ jsz . j=1

5-7

Discrete Quantum Mechanics

It is clear from the sum in the definition of Sˆz that this operator commutes with the translation operator Tˆ . We want also to show that it commutes with HXXX by noting the following: • jsz commutes with every operator in HXXX (in the form exhibited in equation (5.8)) except for jσ+ and jσ−; • [ jsz , jσ+ ] = 2 jσ+ and [ jsz , jσ −] = −2 jσ −, as follows from the commutation relations for the spin component operators; • for each term in the Hamiltonian of the form jσ+ j + 1σ − there are exactly two terms in Sˆz that give non-zero contributions to the commutator [Sˆz, HXXX ]: jsz , which contributes 2 jσ+ j + 1σ − and j + 1sz , which contributes −2 jσ+ j + 1σ −—these two terms cancel exactly; • each term in the Hamiltonian of the form jσ − j + 1σ+ likewise has exactly two terms in Sˆz that give non-zero contributions to the commutator and similarly these two terms cancel exactly; • since such cancellations apply to every term in the commutator that is not trivially zero, it follows that [Sˆz, HXXX ] = 0. Eigenstates of Sˆz are linear combinations of basis states with a fixed number of ∣−〉, for example for such an eigenstate ∣s〉 of length N with n spin-down sites N (and N − n spin ups), Sˆz∣s〉 = 12 (N − 2n )∣s〉, so eigenvalues range from s = 2 to N

s = − 2 in steps of −1, s serving as a measure of the excess of spin-up sites over spin-down sites. As we have shown, the three operators HXXX, Tˆ and Sˆz mutually commute. As proven in appendix A, this implies the existence of a complete set of eigenstates that are mutually eigenstates of each operator, and these states can be labeled by eigenvalues E, t and s, which are called the quantum numbers of the state. Our goal is to seek these mutual eigenstates and their quantum numbers. We have noted that the energy is minimized when nearest-neighbor spin alignment is maximized, thus when all spins are in the same state, either spin up or spin down. It is straightforward to see from equation (5.10) that either of these states has energy zero, and Tˆ eigenvalue 1, since translation leaves the state invariant. The allN N up state is an eigenstate of Sˆz with eigenvalue 2 , thus ∣E = 0, t = 1, s = 2 〉, and its N reflection—the all-down state—is likewise an eigenstate of Sˆz with eigenvalue − , 2

N

thus ∣E = 0, t = 1, s = − 2 〉. N . 2−1

Eigenstates of ˆ T can be constructed as linear combinations of these basis states. They can be fully labeled by their translation operator eigenvalue t, as follows. Define the state There are N basis states with exactly one spin-down site, s =

s=

N − 1, t = 2

5-8

N

∑ aj ( t ) ∣ j 〉 , j=1

Discrete Quantum Mechanics

where ∣ j 〉 denotes the state with all spins up except for the jth, which is down. Noting that Tˆ ∣ j 〉 = ∣ j +1〉, we see that

N − 1, t = Tˆ s = 2

N

N

∑aj(t )∣ j + 1〉 = t∑aj(t )∣ j〉, j=1

j=1

and from the orthogonality of the ∣ j 〉 states with differing js we conclude

taj (t ) = aj −1(t ) and therefore

aj (t ) = t −ja1(t ). Normalized eigenstates of the translation operator for total spin z-component equal N to 2 − 1 are thus

s=

⎛ N m⎞ − 1, t = exp⎜ 2π i ⎟ ⎝ 2 N⎠

=

N ⎛ 1 jm ⎞ ∑ exp⎝⎜ −2π i N ⎠⎟∣ j〉 N j=1

(5.12)

for m = 1, 2, …, N . These states have the form of the discrete Fourier transform of the states ∣ j 〉 and can be thought of as complex waves consisting of single spin-down states. These wave-like excitations are referred to as magnons. The reflection operator Rˆxy commutes with the translation operator Tˆ , thus

N ˆ ˆxy s = N − 1, t = tRˆxy s = N − 1, t . RˆxyTˆ s = − 1, t = TR 2 2 2 Since the effect of the reflection on an eigenstate of Sˆz simply changes the sign of s, this leads to

⎞ ⎞ N N Tˆ s = − + 1, t ⎟ = t s = − + 1, t ⎟ , ⎠ ⎠ 2 2 revealing a second set of N simultaneous eigenstates of Tˆ and Sˆz that are effectively magnons of single spin-up states:

s=−

⎛ N m⎞ + 1, t = exp⎜ 2π i ⎟ ⎝ 2 N⎠

=

N ⎛ 1 jm ⎞ ∑ exp⎝⎜ −2π i N ⎠⎟∣ j¯ 〉, N j=1

in which ∣ ¯j 〉 represents a state of all spins down but for one spin up at location j, i.e. ∣ j¯ 〉 = Rˆxy∣ j 〉. We anticipate the magnon states to be eigenstates of the Hamiltonian, since HXXX and Tˆ commute, thus we seek their energies by applying the Hamiltonian operator directly to them in the form of equation (5.12). The action of the Hamiltonian is exclusively on the terms ∣ j 〉 and using equation (5.10) we note that

HXXX∣ j 〉 =

K (2∣ j 〉 − ∣ j − 1〉 − ∣ j + 1〉), 2

5-9

Discrete Quantum Mechanics

which leads us to

HXXX s =

⎛ N m⎞ − 1, t = exp⎜ 2π i ⎟ ⎝ 2 N⎠

=

N ⎛ K 1 mj ⎞ exp⎜−2π i ⎟(2∣ j 〉 − ∣ j − 1〉) − ∣ j + 1〉 ∑ ⎝ 2 N j=1 N⎠

=

N ⎛ ⎛ ⎛ ⎛ mj ⎞ K 1 m( j + 1)⎞ m( j − 1) ⎞⎞ ⎟ − exp⎜−2π i ⎟⎟∣j 〉 ⎜−2 exp⎜2π i ⎟ − exp⎜−2π i ∑ ⎝ ⎝ ⎝ ⎠⎠ 2 N j = 1⎝ N⎠ N ⎠ N

N ⎛ ⎛ ⎛ m⎞ 1 mj ⎞ exp⎜ −2π i ⎟∣k〉 = ⎜1 − cos⎜ 2π ⎟K ∑ ⎝ N⎠ ⎝ ⎝ N⎠ N j=1

⎛ ⎛ m⎞ ⎛ N m⎞ = ⎜ 1 − cos⎜ 2π ⎟K E , s = − 1, t = exp⎜ 2π i ⎟ . ⎝ ⎝ N⎠ ⎝ N⎠ 2

(5.13)

The one-magnon states of one spin up are thus shown to have energies m E = (1 − cos(2π N )K that range from E = 0 for m = N and t = 1, to E = 2K for N

m = 2 and t = −1. For other m values, the energies are between 0 and 2K and are twofold degenerate, with E (m ) = E (N − m ). For large values of N this constitutes a band of energies between the extremes. Applying this analysis to the case of one magnon of a single spin down, the same set of energies will be found. Continuing on this agenda, we should look successively at states of two and more spin downs, taking advantage of the commutation of HXXX and Sˆz , which casts the Hamiltonian into block-diagonal form, with each block of fixed s being diagonalized separately. The difficulty of this strategy is easily seen. The number of basis states of n an N chain with exactly n spin-down sites is given by the binomial coefficient ( ), so N the number of states with n = 2 is N (N − 1)/2. Thus for chains with N ⩾ 4 the number of n = 2 states exceeds N, the number of eigenstates of Tˆ , so such states force us into non-trivial matrix diagonalization. Even in the N = 4 case, taking maximum advantage of the symmetry relative to Tˆ , this case resolves into two 2 × 2 matrices that must be diagonalized. Admittedly, this is quite simple, but the complications grow rapidly as N grows. For N = 10, for instance, there are 45 basis states with two spin downs, and even utilizing symmetries there is much work to be done. The largest block in the block diagonal form of HXXX for N = 10 is that with five spins down and five up, and this has 252 basis states, presenting a formidable challenge to the diagonalization strategy. As mentioned at the start of this section, the eigenvalues and eigenvectors of HXXX have been determined exactly for arbitrarily large N by utilizing an ingenious surmise about the form of the eigenstates, known as the Bethe ansatz. For a taste of this method, consider the block of eigenstates of HXXX, N ⩾ 4, with two spin-down N sites – ∣E , s = 2 − 1, t〉. The N (N − 1)/2 basis states that contribute to this case we 5-10

Discrete Quantum Mechanics

denoted by ∣n1, n2〉, 1 ⩽ n1 < n2 ⩽ N, in which n1 and n2 designate the sites of the spin-down states, thus

E, s =

N − 2, t = 2

N

∑

a n1,n2(t )∣n1, n2〉.

(5.14)

1 ⩽ n1< n 2

The Bethe ansatz posits a form for the coefficients in this case that is modeled on the form (equation (5.3)) for translation-invariant states for one spin-down:

a n1,n2(t ) = exp(k1n1 + k2n2 )) + exp(k1n2 + k2n1 + θ ). The parameters k1, k2 and θ are determined by the requirement that N ∣E , s = 2 − 2, t〉 in the form of equation (5.14) is an eigenstate of HXXX , and enforcement of periodicity under transformations ni → ni + N , i = 1, 2. These conditions produce the required N (N − 1)/2 eigenvectors and eigenvalues for two spin downs. By a like process, conditions that must be satisfied by the full set of eigenstates for all values of s can be found. An excellent presentation of the details of using the Bethe ansatz for this Hamiltonian can be found in the 1997 paper of Karbach and Müller [Karbach1997]. Therein they present the full resolution for cases N = 4, 5 and 6, and show aspects of the N = 32 and N = 128 solution. While pointing out that the agenda for finding all eigenstates grows prohibitively tedious as N increases, they explain how useful generalizations regarding the thermodynamic limit (N → ∞) are enabled by the ansatz. From this work, one can see that many solutions following from the form of the Bethe ansatz (equation (5.14)) represent multiple magnons that scatter off one another as they move about the ring, with θ representing the phase shift caused by their interactions. Other solutions, however, lead to complex values for ki and can be interpreted as magnon bound states. In the following section we briefly present the eigenstates and eigenvalue of HXXX for the N = 4 case, resolving the two-magnon states using first direct diagonalization, and then the Bethe ansatz. 5.3.1 N = 4 case: diagonalization We here fully resolve the eigenvalue problem for the N = 4 case of HXXX . For clarity, we will describe every basis vector for the chain using the symbol ‘+’ for spin up and ‘−’ for spin down. From the previous discussion, we can easily extract the eigenstates and eigenvalues of HXXX when N = 4 and there are 0, 1, 3 and 4 spins down, as displayed in table 5.1. The six remaining eigenstates of HXXX are states with two up spins and two down spins. We will begin by looking for eigenstates of Tˆ composed of such states:

s = 0, t = a1 −−++ + a2 +−−+ + a3 ++−− + a 4 −++− + b1 −+−+ + b2 +−+− ,

(5.15)

noting that the four terms with aj coefficients transform among themselves under the action of Tˆ , as do the two with bj coefficients. For each of the four eigenvalues of Tˆ (t = i, −1, −i, 1) we will evaluate Tˆ ∣s = 0, t〉, first by applying the translation

5-11

Discrete Quantum Mechanics

operator directly to the basis states, and then by using the eigenvalue equation, equating the two to evaluate the coefficients in equation (5.15). For example, for t = i , this gives Tˆ s = 0, t = i = a1 +−−+ + a2 ++−− + a3 −++−

+ a 4 −−++ + b1 +−+− + b2 −+−+ = i(a1 −−++ + a2 +−−+ + a3 ++−− + a 4 −++− + b1 −+−+ + b2 +−+− ) . Since the basis states are orthogonal, this implies

a1 = ia2 ,

a2 = ia3,

a3 = ia 4,

a 4 = ia1,

b1 = ib2 and b2 = ib1

and thus

s = 0, t = i = a1( +−−+ − i ++−− − −++− + i −−++ ), with a1 =

1 2

(5.16)

for normalization. The same process will yield

s = 0, t = −i = a1( +−−+ + i ++−− − −++− − i −−++ ),

(5.17)

s = 0, t = 1 = a1( +−−+ + ++−− + −++− + i −−++ ) + b1( +−+− + −+−+ )

(5.18)

and

s = 0, t = −1 = a1( +−−+ − ++−− + −++− − −−++ ) + b1( +−+− − −+−+ ).

(5.19) The states ∣s = 0, t = i〉 and ∣s = 0, t = −i〉 are non-degenerate eigenstates of Tˆ and are therefore also eigenstates of HXXX . To find the energies of these states we use the eigenvalue equation for energy. Utilizing the form of HXXX in equation (5.10), it is straightforward to show HXXX s = 0, t = i = K s = 0, t = i and HXXX s = 0, t = −i = K s = 0, t = −i . The form of equations (5.18) and (5.19), with the normalization requirement ∣a1∣2 + ∣b1∣2 = 1, indicates that there is in each case a two-dimensional subspace of basis vectors within which each vector is an eigenstate of Tˆ . There are at least two vectors in the subspace that are also eigenstates of HXXX . To find those eigenstates with corresponding energy eigenvalues, we examine the effect of the Hamiltonian on the degenerate spaces. For the case t = 1,

HXXX s = 0, t = 1 = HXXX(a1( +−−+ + ++−− + −++− + −−++ ) + b1( +−+− + −+−+ )) = 2K (b1 − a1)( +−−+ + ++−− + −++− + −−++ ) + K (a1 − b1)( +−+− + −+−+ ) = E (a1( +−−+ + ++−− + −++− + −−++ ) + b1( +−+− + −+−+ )).

5-12

Discrete Quantum Mechanics

This can be satisfied if: (i) a1 = b1 and E = 0; and (ii) 2a1 = −b1 and E = 3K . For the final case, t = −1,

HXXX s = 0, t = −1 = HXXX(a1( +−−+ − ++−− + −++− − −−++ ) + b1( +−+− − −+−+ )) = 2K (b1 − a1)( +−−+ + ++−− + −++− + −−++ ) + K (a1 − b1)( +−+− + −+−+ ) = E (a1( +−−+ + ++−− + −++− + −−++ ) + b1( +−+− + −+−+ )). This can be satisfied if: (i) b1 = 0 and E = K ; and (ii) a1 = 0 and E = 2K . The full spectrum of eigenstates and eigenvalues has now been resolved, enabling the completion of the entries in table 5.1, shown in table 5.2.

Table 5.1. Eigenstates for Heisenberg XXX chain: N = 4, s ≠ 0 .

E

s

t

Eigenstate

0 K

2 1

1 i

∣++++〉 1 (i∣−+++〉 − ∣+−++〉 − i∣++−+〉 + ∣+++−〉 ) 2

2K

1

−1

K

1

−i

0

1

1

K

−1

i

2K

−1

−1

K

−1

−i

0

−1

1

0

−2

1

1 ( −∣−+++〉 + ∣+−++〉 − ∣++−+〉 + ∣+++−〉 ) 2 1 ( − i∣−+++〉 − ∣+−++〉 + i∣++−+〉 + ∣+++−〉 ) 2 1 ( ∣−+++〉 + ∣+−++〉 + ∣++−+〉 + ∣+++−〉 ) 2 1 (i∣+−−−〉 − ∣−+−−〉 − i∣−−+−〉 + ∣−−−+〉 ) 2 1 ( −∣+−−−〉 + ∣−+−−〉 − ∣−−+−〉 + ∣−−−+〉 ) 2 1 ( − i∣+−−−〉 − ∣−+−−〉 + i∣−−+−〉 + ∣−−−+〉 ) 2 1 ( ∣+−−−〉 + ∣−+−−〉 + ∣−−+−〉 + ∣−−−+〉 ) 2

∣−−−−〉

Table 5.2. Eigenstates for Heisenberg XXX chain: N = 4, s = 0 .

E

s

t

Eigenstate

K

0

i

K

0

−i

0

0

1

3K

0

1

K

0

−1

2K

0

−1

1 ( ∣+−−+〉 − i∣++−−〉 − ∣−++−〉 + i∣−−++〉 ) 2 1 ( ∣+−−+〉 + i∣++−−〉 − ∣−++−〉 − i∣−−++〉 ) 2 1 ( ∣+−−+〉 + ∣++−−〉 + ∣−++−〉 + ∣−−++〉 + ∣+−+−〉 + ∣−+−+〉 ) 6 1 ( ∣+−−+〉 + ∣++−−〉 + ∣−++−〉 + ∣−−++〉 − 2∣+−+−〉 − 2∣−+−+〉 ) 2 3 1 ( ∣+−−+〉 − ∣++−−〉 + ∣−++−〉 − ∣−−++〉 ) 2 1 ( ∣+−+−〉 − ∣−+−+〉 ) 2

5-13

Discrete Quantum Mechanics

5.3.2 N = 4 case: Bethe ansatz For this small chain, the process of finding the s = 0 states using the Bethe ansatz is only a bit less complicated than what was done in the immediately preceding work, but the advantage of this technique becomes considerably more apparent for larger chains, approaching the point where it is the only viable resolution. We briefly exhibit how to utilize this method as an introduction to its more general use. Our starting point, as before, is the general expression for s = 0 states,

s = 0, t = a1 −−++ + a2 +−−+ + a3 ++−− + a 4 −++− + b1 −+−+ + b2 +−+− ,

(5.20)

with the added assumption, the Bethe ansatz, that each of the aj and bj coefficients are of the form

exp(k1n1 + k2n2 ) + exp(k1n2 + k2n1 + θ ),

(5.21)

with integers n1 < n2 giving the respective locations of the spin-down particles within the chain. The periodic boundary condition requirement, that equation (5.21) is unchanged when both location indices (n1, n2 ) are increased by 4, produces

exp( i4(k1 + k2 )) = 1. Explicit evaluation of the energy eigenvalue equation HXXX∣s = 0, t〉 = E ∣s = 0, t〉, utilizing the orthogonality of the basis states, yields six conditions on the coefficients in equation (5.20):

−

K (2ak + b1 + b2 ) = (E − 2K )ak 2

for 1 ⩽ k ⩽ 4

(5.22)

and

−

K (a1 + a2 + a3 + a 4) = (E − 2K )bk 2

for k = 1, 2.

(5.23)

Expressing the ak and bk coefficients using the Bethe ansatz yields, for example,

a1 = exp( i(k1 + 2k2 )) + exp( i(2k1 + k2 + θ )) b1 = exp( i(k1 + 3k2 )) + exp( i(3k1 + k2 + θ )) ,

and (5.24)

along with a2 = ηa1, a3 = η2 and b2 = ηb1, utilizing the definition η ≡ exp(i(k1 + k2 )). From the periodic boundary condition η 4 = 1, so we examine separately the possibilities η = 1, −1, i and −i. Picking η = 1, we see from the Bethe ansatz that a1 = a2 = a3 and b1 = b2 . From K−E equations (5.22) we extract b1 = K a1 and from equations (5.23) a1 = 2K2K− E b1, both of which can be satisfied only if E = 0 or 3K . When E = 0, equations (5.22) can be used to show that a 4 = a1 and that all ak and bk coefficients are equal. This solution

5-14

Discrete Quantum Mechanics

is that of the third eigenstate in table 5.2. When E = 3K , equations (5.22) give a1 = a2 = a3 = a 4 = −b1/2 = −b2 /2 and thus correspond to the fourth eigenstate in table 5.2. When η = −1, equations (5.22) and (5.23) are compatible only when • b1 = b2 = 0 and a1 = −a2 = a3 = −a 4 , producing the fifth eigenstate in the s = 0 table, or • a1 = a2 = a3 = a 4 = 0 and b1 = −b2 , producing the last eigenstate in the same table. For η = i , equations (5.23) along with b2 = ib1 imply b1 = b2 = 0; and ∑k ak = 0 with a2 = ia1 = −ia3, implying a 4 = ia1. Equation (5.22) with k = 1 implies E = K . Together, these conditions produce the first eigenstate in the s = 0 table. A nearidentical process for η = −i gives b1 = b2 = 0 and a1 = ia2 = −a3 = −ia 4 , and E = K, filling in the second and final state of the table.

Exercises 2 1. Show that if Rˆ = I, the eigenvalues of Rˆ are ±1. 2. Apply the analysis that led to equation (5.13) to one-magnon states of one spin down to show that the same energies are found. 3. Derive equations (5.17)–(5.19).

Bibliography [Sunderland2004] Sunderland B 2004 Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems (Hackensack, NJ: World Scientific) [Bohr1972] Bohr N 1911 PhD Thesis Copenhagen University (in Rosenfeld L and Nielsen J R 1972 Niels Bohr Collected Works 1. Early Works (1905–1911) (Amsterdam: Elsevier) pp 163, 165–393 [vanLeeuwen1921] van Leeuwen H J 1921 Problémes de la théorie électronique du magnétisme J. Phys. Radium 2 361–77 [Lenz1920] Lenz W 1920 Beiträge zum Verständnis der magnetischen Eigenschaften in fester Körpern Phys. Z. 21 613 [Ising1925] Ising E 1925 Beitrag zur Theorie des Ferromagnetismus Z. Phys. 31 253–8 [Baxter2007] Baxter R J 1982 Exactly Solved Models in Statistical Mechanics (London: Academic) [Bethe1931] Bethe H A 1931 Zur Theorie der Metalle 1 Z. Phys. 71 205–26 [Karbach1997] Karbach M and Müller J 1997 Introduction to the Bethe Ansatz I Comput. Phys. 11 36–43

5-15

IOP Concise Physics

Discrete Quantum Mechanics H Thomas Williams

Chapter 6 Infinity, and beyond

This volume began with reference to Heisenberg and Schrödinger who, in 1925 and 1926, created the first systematic theories that encompassed the paradigm-shifting quantum discoveries of the previous quarter-century. Beginning with Planck’s quantum interpretation of the blackbody radiation spectrum in 1900, this period included the work of Einstein on the photo-electric effect, Bohr on atomic structure, Millikan on electronic charge and behavior, Compton on photon–electron scattering, deBroglie on the wave nature of electrons and Davisson and Thompson on diffraction of electrons by crystals. The consolidation of these and many other pieces of experimental and interpretational work is credited in large part to Heisenberg, Schrödinger, Born and Dirac. All of the contributions mentioned above were recognized by the award of a Nobel Prize in Physics: the first to Planck (awarded in 1918) and the latest to Born (awarded in 1954). This amazing period resulted in ten Nobel Prizes going to twelve physicists, representing the progression of quantum mechanics from birth until its coming of age. Of age, perhaps: with the theories of Heisenberg and Schrödinger, the subject by 1926 was capable of bearing fruit. In the minds of some, however, it was not yet capable of consistently responsible behavior. John von Neumann, in 1932, set himself to the task of bringing quantum theory into adulthood, capable of mature mathematical conduct [vonNeumann1932]. His self-stated goal was ‘to present the new quantum mechanics in a unified representation which, so far as it is possible and useful, is mathematically rigorous’. By introducing a Hilbert space formulation of the topic, von Neumann provided a unified form of quantum mechanics based upon postulates, blazing the trail followed since by myriad physicists, including the formulation used in this book. Von Neumann’s work proved the equivalence of Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics. This had been addressed previously, with somewhat limited success. In the same year that he published his wave mechanics, Schrödinger attempted to show the equivalence of the two approaches [Schroedinger1926–2].

doi:10.1088/978-1-6817-4125-3ch2

6-1

ª Morgan & Claypool Publishers 2015

Discrete Quantum Mechanics

Unable to establish the mathematical equivalence of the two approaches, in this paper he was able to show that his wave mechanics was contained within Heisenberg’s matrix approach, but not the converse. In 1930 the first edition of Dirac’s The Principles of Quantum Mechanics [Dirac1930] appeared, in which, at least to Dirac’s satisfaction, the equivalence was fully established. To von Neumann, the Dirac approach was flawed due to his use of ‘improper functions’. In what follows, we examine a series of problems that involve observables with continuous values, and their resolution via Dirac’s improper functions.

6.1 Schrödinger equation The first step in many quantum calculations, and often the first step in teaching elementary quantum mechanics, is the determination of a Hamiltonian operator appropriate to the problem at hand. Paralleling the formalism of classical Hamiltonian mechanics, the Hamiltonian is expressed in terms of a position and a momentum variable for each degree of freedom. The transition from classical to quantum mechanics requires the introduction of Hermitian operators for position and momentum and their insertion into the analogous classical Hamiltonian. In the Schrödinger approach (in configuration space), the position operator (e.g. qˆ ) is the act of multiplication by a position variable (q): the corresponding operator for momentum along the q direction is the differential operator pˆ = −i ∂∂q . For each degree of freedom, corresponding position and momentum operators obey the canonical commutation relation

[qˆ , pˆ ] = i.

(6.1)

Two operators, each corresponding to a different degree of freedom, commute. The order in which non-commuting operators appear in multiplicative terms in the Hamiltonian matters in the quantum case in a way that it does not in the classical case. In the examples we use in this chapter, this issue will not prove problematic. The classical Hamiltonian function for N particles of various masses (mn), moving under the influence of forces associated with a position-dependent potential energy function V, is N

H=

1 mn

3

∑ 2 ∑ npj2 +V (1q1…Nq 3), n=1

j=1

where nq j and n pj are the values of the position and momentum, respectively, of the jth Cartesian component of particle n, and the potential energy V can, but does not necessarily, depend upon every position coordinate.

6.2 Schrödinger equation in one dimension 6.2.1 Free particle From this brief introduction, it would appear that the simplest problem to solve in the Schrödinger approach would be that of a single particle of mass m with 6-2

Discrete Quantum Mechanics

momentum p, moving along a line (coordinate q) in the absence of forces. The classical Hamiltonian can be cast as

H=

1 2 1 2 p + V ( q) = p , 2m 2m

which leads unambiguously to the quantum Hamiltonian

1 ∂2 Hˆ = − . 2m ∂q 2 The state function ∣ψ 〉 = ψ (q, t ) obeys the time-dependent Schrödinger equation (see chapter 1)

i

1 ∂2 ∂ ψ (q , t ) = Hˆ ψ (q , t ) = − ψ (q , t ). 2m ∂q 2 ∂t

Postulating a separable solution, it is straightforward to arrive at:

ψ (q , t ) = A exp(ikq )exp( −iωt ),

(6.2) 2

in which k and ω are real constants that must obey ω = 2km and A is an arbitrary constant that is to be used to normalize the state function. The constant k, which can assume any real value, is the eigenvalue of the momentum operator and ω, which ˆ can take on any non-negative real value is the eigenvalue of H. Standard normalization for a state function in finite dimensional Hilbert space, e.g. one with components ϕn(t ), n = 1, …, d , requires d

∑

2

ϕn(t ) = 1.

n=1

The analogous requirement for the state function of equation (6.2) would be ∞

∫−∞

∞

ψ ( q , t ) 2 dq = A 2

∫−∞ dq = 1.

(6.3)

The integral in this equation is clearly infinite, so there is no non-zero value of A that will satisfy the equation. The common technique to work around this problem is to normalize to infinity instead of 1 through the introduction of the so-called ‘Dirac delta function’, δ (x ). Dirac’s ‘improper’ function, δ (x ), has the general nature of a sharply peaked function. It is defined by two properties: 1. δ (x ) = 0 for x ≠ 0; and 2.

∫a

b

δ(x )dx =

{

1 for a < 0 < b 0 otherwise.

6-3

Discrete Quantum Mechanics

These properties lead to the expression

∫a

b

⎧ f (c ) for a < c < b f (x )δ(x − c )dx = ⎨ ⎩ 0 otherwise,

as long as f (x) is continuous in the neighborhood of x = c. Hereafter we will refer to this entity as the ‘Dirac delta’, avoiding the incorrect descriptor ‘function’. The Dirac delta arises in the normalization of state functions for a free particle in one spatial dimension in the statement of orthonormality: ∞

∞

∫−∞ ψ ′†(q, t )ψ (q, t )dq = ∫−∞ exp(−ik′q + iω′t )) exp(ikq − iωt ))dq = δ(k′ − k ). (6.4) This integral can be easily shown to vanish in the case k′ ≠ k, and it produces an infinity of ‘just the right kind’ when k′ = k. Consider next the operator for the position observable for this problem, q, ˆ with its eigenvalue equation

qˆψq(q , t ) = q0ψq(q , t ), where q0 takes on any value along the real line. Since the operation consists of multiplication by a real number it is easily shown to be Hermitian. In the Dirac formulation, the eigenstate must be a ‘function’ that vanishes when q ≠ q0, it must be non-zero precisely at q = q0, and it is expressed as ψq(q, t ) = δ (q − q0 )exp( −iωt ). It is the mathematical pathology of this proposed eigenfunction, which fails to satisfy the mathematical properties of a function, that caused von Neumann to comment as follows: ‘For example, the method [of Dirac] adheres to the fiction that every selfadjoint operator can be put in diagonal form. In the case of operators for which this is not actually the case, this requires the introduction of ‘improper’ functions with self-contradictory properties. The introduction of such a mathematical ‘fiction’ is frequently required in Dirac’s approach….’ 6.2.2 Infinitely deep square well The most common first exercise in teaching the Schrödinger approach to quantum mechanics is finding the eigenvalues and eigenstates for the one-dimensional infinite pˆ 2

square well. The Hamiltonian for this problem consists of a kinetic energy term 2m and a potential energy term that describes a particle that moves free of forces in a single, finite-width region of the spatial axis and encounters infinite forces (directed towards the force-free region) at either end of this region: V(q) is written as zero in the force-free region (0 < q < L ) and infinite elsewhere. In solving the Schrödinger equation, the infinite potential regions are accounted for by way of boundary

6-4

Discrete Quantum Mechanics

conditions on the state function that require it to vanish in the regions of infinite potential. This enables straightforward evaluation of the state function with finite domain along the q axis, allowing a finite normalization, leading to

ψ (q , t ) =

⎛ nπq ⎞ 2 ⎟ exp( − iEt ) sin⎜ ⎝ L ⎠ L

0 ⩽ q ⩽ L,

2

with E = n 2 π 2 and n = 1, 2, …. The time dependence comes from the separation 2mL of variables, as it did in the free particle case. The normalization discomfort, present for the free particle, is thus avoided in the square well situation. Nonetheless, the problem of finding eigenstates of the position operator is precisely as it was for the free particle: they must be expressed using the Dirac delta, damaging the mathematical purity of the approach. We next examine a final one-dimensional problem in which the energy spectrum can be developed using familiar techniques from discretely indexed Hilbert spaces, and a tempting methodology for the development of position eigenstates. 6.2.3 Harmonic oscillator in one dimension The linear motion of a mass (m) attached to a simple spring—one which provides a restoring force proportional to its stretch or compression (F = −kq )—is known as simple harmonic motion and is a standard example of the use of Newtonian mechanics. Such motion exhibits a conserved total mechanical energy of

E=

p2 1 + kq 2 , 2m 2

where p is the linear momentum of the mass, q is the displacement of the mass from its position when the spring is relaxed and k is the spring constant. The motion of the mass has the simple oscillating form x(t ) = A cos(ωt + φ ), in which the amplitude A of the oscillation and the phase angle φ are functions of the position and momentum of the mass at time t = 0. The angular velocity of the oscillation is ω = k /m . The analogous quantum problem has a Hamiltonian operator mimicking the classical energy expression, but with momentum and displacement replaced by quantum observable operators. Taking advantage of notational prescience, we choose to replace the spring constant by its equivalent expression in terms of ω, thus

1 2 mω 2 2 Hˆ = pˆ + qˆ . 2m 2

(6.5)

The momentum and position operators obey the commutation relation of equation (6.1), leading to a factored form of the Hamiltonian

mω 2 ⎛ i ⎞⎛ i ⎞ 1 ⎜qˆ − Hˆ = pˆ ⎟⎜qˆ + pˆ ⎟ + mω. 2 ⎝ mω ⎠⎝ mω ⎠ 2

6-5

Discrete Quantum Mechanics

With the operator definitions

aˆ + ≡

mω ⎛ i ⎞ ⎜qˆ − pˆ ⎟ 2 ⎝ mω ⎠

and

aˆ − ≡

mω ⎛ i ⎞ ⎜qˆ + pˆ ⎟ mω ⎠ 2 ⎝

(6.6)

we can express the Hamiltonian more simply, as

⎛ 1⎞ Hˆ = ω⎜Nˆ + ⎟ , ⎝ 2⎠

(6.7)

where Nˆ ≡ aˆ+aˆ−. Note that the aˆ+, aˆ− operators are not Hermitian, but rather aˆ− = (aˆ+ )† . It can easily be demonstrated that [aˆ−, aˆ+ ] = 1. The operator Nˆ is Hermitian, as is the Hamiltonian, so both have real eigenvalues. If we denote the eigenvalues and normalized eigenvectors of Nˆ as n (real) and ∣n〉, the corresponding eigenvalues and eigenvectors of Hˆ are ω(n + 12 ) and ∣n〉. We begin exploration of the eigenvalue spectrum of ∣H 〉 by considering

Nˆ (aˆ − n ) = aˆ +aˆ −aˆ − n = (aˆ −aˆ + − 1)aˆ − n = aˆ −(Nˆ − 1) n = (n − 1)(aˆ − n ), revealing aˆ−∣n〉 to be an eigenvector of Nˆ with eigenvalue n − 1, so aˆ−∣n〉 = cn∣n − 1〉. The constant cn can be evaluated simply by noting

(cn n − 1 , cn n − 1 ) = cn 2 = n aˆ +aˆ − n = n . Making the simplest choice of the arbitrary phase of cn, we write aˆ−∣n〉 = n ∣n − 1〉. This property reveals why the operator aˆ− is referred to as a lowering operator. Following the steps of the prior paragraph, beginning with examination of the expression Nˆ (aˆ+∣n〉), we can conclude that aˆ+, the raising operator, has the property aˆ+∣n〉 = n + 1 ∣n + 1〉. From equation (6.5) we see that Hˆ can be written as the sum of two terms, each of which is the square of a Hermitian operator. Expectation values of the square of a Hermitian operator are non-negative (see appendix A), so equation (6.7) implies ⎛ 1⎞ 〈n∣Hˆ ∣n〉 = ω⎜n + ⎟ ⩾ 0, ⎝ 2⎠ revealing that the eigenvalue spectrum for Nˆ has a lower limit given by n ⩾ − 12 . Thus there is a lowest state ∣nLS 〉 that obeys aˆ−∣nLS 〉 = 0. Examining the expression

Nˆ nLS = aˆ +aˆ − nLS = 0, we conclude that the lowest eigenstate of Nˆ has eigenvalue n = 0 and the lowest ˆ the ground state energy of the one-dimensional harmonic oscillator, eigenvalue of H, 1 is 2 ω. Using the raising operator successively, we generate all the energy eigenvalues:

⎛ 1⎞ En = ω⎜n + ⎟ for n = 0, 1, 2, …. ⎝ 2⎠

6-6

Discrete Quantum Mechanics

Explicit configuration space representations of the eigenstates can be found by solving the differential equation that results from Hˆ ∣n〉 = En∣n〉, resulting in

⎛ mω 2⎞ 1 ⎛⎜ mω ⎞⎟1 4 exp ⎜ − x ⎟/n( mω x ), n ⎝ 2 ⎠ 2 n! ⎝ π ⎠

∣n〉 =

(6.8)

where /n signifies the nth Hermite polynomial [WeissteinHP]. Despite the infinite domain of these states, the normalization integral converges and we can establish that

〈n∣n〉 =

1 ⎛⎜ mω ⎞⎟1 2 2n n ! ⎝ π ⎠

∞

∫−∞

exp ( −mωx 2 )/ n2( mω x )dx = 1.

There is no upper limit on n, so there is a countable infinity of energy eigenstates for this Hamiltonian. Not so, however, for the eigenvalue spectrum for the position operator, q, ˆ as we show next. From the definitions of the raising and lowering operators, equation (6.6), we can write the position operator as follows:

1 (aˆ + + aˆ −). 2mω

qˆ =

(6.9)

The eigenvalue equation for this operator can be expressed as ∞

∞

(aˆ + + aˆ −) ∑ αn n = q ∑ αn n , n=0

(6.10)

n=0

in which the expansion coefficients (into which we have absorbed the constant multiplier on the right of equation (6.9)) depend on the eigenvalue q and are to be determined from this equation. Applying the raising and lowering operators directly to the states ∣n〉 changes equation (6.10) straightforwardly into ∞

∑ αn(

n+1 n+1 +

n n−1 −q n

) = 0,

n=0

leading to ∞

∑(

n αn − 1 +

n + 1 αn+1 − qαn) n = 0.

n=0

Since the eigenstates are mutually orthogonal, we arrive at a recursion relation for the αn coefficients that reads

n αn − 1 +

n + 1 αn+1 − qαn = 0

6-7

Discrete Quantum Mechanics

for n = 1, 2, 3, … and

α1 = qα0. The αns are polynomials in the eigenvector q, and with some diligence they can be evaluated as n − 1 or n

αn =

∑ j = 0 even

j

⎛ −1 ⎞ 2 ⎜ ⎟ ⎝ 2 ⎠

n!

() j 2

! (n − j )!

q n −j α 0 .

It is significant to note that this expression allows us to find an eigenstate for every position eigenvalue −∞ ⩽ q ⩽ ∞ . Thus we have found one observable, position, that can take on a continuous infinity of values, and another, energy, that has a countable infinity of values, a situation that is mathematically awkward at the very least. Convergence of the αns is most easily examined in the case x = 0, for which

⎛ − 1 ⎞ 2 n! ⎟ αn = ⎜ α. ⎝ 2 ⎠ n ! 0 2 n

()

We should seek to evaluate the coefficient α0 via normalization of the eigenstate ∑n αn∣n〉, which in the case of x = 0 leads to ∞

∑ ∣αn∣2 n=0

∞

=

⎛ 1 ⎞n

∑ ⎜⎝ 2 ⎟⎠

n=0

n! ⎡ ⎣

() n 2

2 !⎤⎦

∣α0∣2 = 1.

The asymptotic form of terms in the sum can be examined using Sterling’s approximation for the factorials, showing that these terms slowly approach a constant value, implying a divergent sum. Normalization can be achieved only with a vanishing value of α0, implying that αn = 0 for all n and thus a vanishing state function. This conundrum can be understood when we realize that the eigenstate we are attempting to build is a Dirac delta, carrying all the pathology noted by von Neumann.

6.3 Schrödinger equation in three dimensions The one-dimensional examples just described serve as a gateway to understanding the behavior of single particles moving in our more familiar space of three dimensions. They have the virtue of relatively straightforward solutions, provide insights into quantum behavior in more realistic scenarios, and in a few circumstances serve as reasonable approximations to actual physical circumstances. Extension of the Schrödinger protocol to three dimensions is mathematically straightforward, but leads us deeper into the morass of challenges for the Schrödinger formulation of quantum theory. The (non-relativistic) three-dimensional Hamiltonian operator for a particle of mass m moving under the influence of a spatially dependent potential energy

6-8

Discrete Quantum Mechanics

function is an obvious generalization of the one-dimensional case, in which there are three momentum operators: pˆx = −i ∂∂x , pˆy = −i ∂∂y and pˆz = −i ∂∂z . The potential energy can depend on each of the spatial coordinate components (V (x , y, z )). The time-dependent state function obeys

1 ⎛ ∂2 ∂2 ⎞ ∂2 Hˆ ψ (x , y , z , t ) = − ⎜ 2 + 2 + 2 ⎟ψ (x , y , z , t ) + V (x , y , z )ψ (x , y , z , t ) 2m ⎝ ∂x ∂z ⎠ ∂y ∂ = i ψ (x , y , z , t ) ∂t and has a solution in which the spatial and temporal parts of the state function are separated:

ψ (x , y , z , t ) = ψ (x , y , z )exp( −iEt ). The spatial part of the state function obeys the time-independent Schrödinger equation

1 ⎛ ∂2 ∂2 ⎞ ∂2 Hˆ ψ (x , y , z ) = − ⎜ 2 + 2 + 2 ⎟ψ (x , y , z ) + V (x , y , z )ψ (x , y , z ) 2 m ⎝ ∂x ∂z ⎠ ∂y (6.11) = Eψ (x , y , z ). The separation constant E has units of energy and represents the energy of a stationary state solution, i.e. a state whose modulus does not change with time. 6.3.1 Free particle When the potential vanishes throughout three-space, the solution of the threedimensional time-independent Schrödinger equation,

1 ⎛ ∂2 ∂2 ⎞ ∂2 Hˆ ψ (x , y , z ) = − ⎜ 2 + 2 + 2 ⎟ψ (x , y , z ) = Eψ (x , y , z ), 2m ⎝ ∂x ∂z ⎠ ∂y postulating a solution separable in the three spatial coordinates, is a simple extension of the one-dimensional case:

ψ (x , y , z ) = A exp(ikxx )exp(ikyy )exp(ikzz ) = A exp(ik ⃗ · r ⃗ ). This state function is simultaneously an eigenstate of the Hamiltonian and pˆx , pˆy and pˆz . The time-dependent state function for this case is ψ (x , y , z, t ) = A exp(ik ⃗ · r ⃗ − iEt ), with the condition k 2 ≡ k x2 + k y2 + k z2 = 2mE . Normalization of this state function encounters the same challenge that we saw in one dimension, since

∫

ψ ( x , y , z , t ) 2 d x dy d z = A 2

6-9

∫ d x dy d z = 1

Discrete Quantum Mechanics

implies A = 0. The Dirac ‘trick’ thus must be applied here threefold via the statement of the orthonormality of state functions Ψ(x , y, z ) = A exp((ik ⃗ · r ⃗ ) and Ψ′(x , y, z ) = A′ exp((ik′⃗ · r ⃗ ) as

∫ ψ †(x, y, z)ψ ′(x, y, z)dx dy dx = δ(kx − kx′)δ(ky − ky′)δ(kz − k′z). Akin to the one-dimensional case, section 6.2.1, the eigenstates of the three position operators are themselves represented by Dirac deltas: δ (x − x0 ), δ (y − y0 ) and δ (z − z0 ). 6.3.2 Infinitely deep well If the potential vanishes within a three-dimensional box (rectangular parallelepiped), and is infinite elsewhere in three-space, the solution to the Schrödinger equation is again easily constructed from solutions in the one-dimensional case. The time-independent state function vanishes outside the box, and inside it satisfies

1 ⎛ ∂2 ∂2 ⎞ ∂2 Hˆ ψ (x , y , z ) = − ⎜ 2 + 2 + 2 ⎟ψ (x , y , z ) = Eψ (x , y , z ). 2m ⎝ ∂x ∂z ⎠ ∂y A separable solution that obeys this equation with boundary conditions that require it to be zero at the box’s edges (x = 0, Lx; y = 0, Ly; z = 0, Lz ) normalized to one, is

ψ (x , y , z , t ) =

⎛ n πx ⎞ ⎛ n yπy ⎞ ⎛ nzπz ⎞ 8 ⎟ sin⎜ sin⎜ x ⎟ sin⎜ ⎟exp( −iEt ), ⎝ Lx ⎠ ⎝ L y ⎠ ⎝ Lz ⎠ LxL yLz

where the energy and momentum variables are related by E = (

n x2 L x2

+

n y2 L y2

+

n z2 π 2 ) L z2 2m

and the three quantum numbers nx , ny , nz can each independently take nonnegative integer values. Eigenstates for the position operator xˆ (and likewise yˆ , zˆ ) are proportional to δ (x − x0 ) (δ (y − y0 ), δ (z − z0 )). 6.3.3 Harmonic oscillator The time-independent state function for the three-dimensional harmonic oscillator satisfies

1 2 mω 2 2 pˆx + pˆy2 + pˆz2 ψ (x , y , z ) + (x + y 2 + z 2 )ψ (x, y, z ) = Eψ (x, y, z ). 2m 2

(

)

Since the left side of this equation factors straightforwardly into three additive terms depending separately upon the variables x, y and z, a separable solution is easy to find, resulting in

ψ (x , y , z ) = nx ∣n y〉∣nz〉, in which the term ∣nx〉 can be represented as an x-dependent expression of the form of equation (6.8) with integer-valued nx, and ∣ny〉 and ∣nz〉 are exactly parallel y- and

6-10

Discrete Quantum Mechanics

z-dependent terms with independent positive integer-valued ny and nz values. The energy of this state is

(

E = ω nx + n y + nz +

3 2

).

Position eigenstates require that the Dirac delta has the same form as those for the infinitely deep well of the previous paragraph. 6.3.4 Spherically symmetric potentials Nature frequently presents us with radial forces, which can be described using potential energy functions that are a function of r (the distance from the force center) alone. The most efficient way to deal with such potentials is to re-express the Schrödinger equation (equation (6.11)) in terms of r and two angular variables: a polar angle θ and an azimuthal angle ϕ. These angles are uniquely defined by their relationships to the three Cartesian coordinates: x = r sin θ cos ϕ, y = r sin θ sin ϕ and z = r cos θ. With a bit of persistence, the Laplacian operator of equation (6.11) can be re-expressed in spherical coordinates 1 ∂2 ⎞ 1 ⎛∂ ⎛ 2∂ ⎞ 1 ∂ ⎛ ∂ ⎞ ∂2 ∂2 ∂2 ⎜r ⎟ + ⎜sin θ ⎟ + = + + ⎜ ⎟. r 2 ⎝ ∂r ⎝ ∂r ⎠ sin θ ∂θ ⎝ ∂θ ⎠ sin2 θ ∂ϕ 2 ⎠ ∂z 2 ∂y 2 ∂x 2 Inserting this into equation (6.11) along with a potential V (r ), and postulating a separable solution ψ (x , y, z ) = R(r )Y (θ , ϕ ), allows us to generate distinct equations for the angular variables,

−

1 ∂ 2Y 1 ∂ ⎛ ∂Y ⎞ ⎜sin θ ⎟ − = λY , sin θ ∂θ ⎝ ∂θ ⎠ sin2 θ ∂ϕ 2

(6.12)

and for the radial variable,

−

⎞ 1 ∂ ⎛ 2 ∂R ⎞ ⎛ λ ⎜r ⎟ + ⎜ + V ( r ) − E ⎟R = 0, 2 2 ⎝ ⎠ ⎝ ⎠ 2mr ∂r 2mr ∂r

(6.13)

in which λ is a separation constant. Referring back to section 4.5 in chapter 4, we recognize the operator applied to Y on the left side of equation (6.12) as the one that 2 represents the square of total angular momentum Lˆ , and from there see that the separation constant takes the form λ = l (l + 1), with l = 0, 1, 2, …. The angular parts of separable state functions for r-dependent potentials are the spherical harmonics. These functions were shown in chapter 4 to be normalized to one when integrated over the unit sphere. The radial equation takes on its simplest form when we introduce the function u(r ) = rR(r ), which allows equation (6.13) to be transformed into

1 ∂ 2u ⎛ l (l + 1) ⎞ ⎟u = Eu , + ⎜ V ( r) + 2 ⎝ 2m ∂r r2 ⎠ taking the form of the one-dimensional Schrödinger equation with the potential l (l + 1) energy function supplemented with the term , related to the ‘fictitious’ r2 centrifugal force. 6-11

Discrete Quantum Mechanics

There are three exercises with spherically symmetric potentials that are common in textbook presentations of Schrödinger techniques: the infinite spherical well, the spherical harmonic oscillator and the Coulomb potential. Each can be solved analytically, but each involves some tedium. Here we will only briefly mention these cases, leaving readers who are interested in the detailed solutions to consult existing familiar references like Messiah’s excellent two-volume treatise on quantum theory [Messiah1961]. The three-dimensional analog of the infinite square well is a spherical region of radius r0 about the origin, in which V (r ) = 0, and outside of which the potential is infinite. This forces a boundary condition on the radial part of the state function of R(r0 ) = 0. Non-trivial solutions are only possible when the slope of the state function at r0 is non-zero. In this case, the solutions that are non-singular at the origin, as is required, are constant multiples of the spherical Bessel functions of the first kind

R(r ) = Ajl ( 2mE r ) , which are restricted by the boundary condition to satisfy 2mEr02 = a l2,n, where al,n denotes the nth zero of jl (r) beyond r = 0. For each angular momentum quantum number l there are an infinite number of zeros, thus there is a doubly infinite spectrum of energy eigenvalues

El ,n =

al2,n 2mr02

and associated state functions

⎛ r⎞ Rl ,n( r ) = Ajl ⎜al ,n ⎟ . ⎝ r0 ⎠ A finite value for A can be found for each pair l ,n that normalizes the radial part of the state function, i.e.

∫0

r0

Rl ,n(r ) 2 r 2dr = 1.

(6.14)

Issues we have seen previously regarding the eigenstates of the position operator (and other operators associated with continuous observables) will, however, still require the use of the Dirac delta for this case. 1 For a spherically symmetric harmonic oscillator potential, V (r ) = 2 mω 2r 2, the radial state function obeys

−

⎞ 1 ∂ ⎛ 2 ∂R ⎞ ⎛ l (l + 1) mω 2 2 ⎜r ⎟ + ⎜ + r − E ⎟R = 0 2 2 ⎠ 2mr ∂r ⎝ ∂r ⎠ ⎝ 2mr 2

and yields solutions of the form

⎛ mω 2⎞ l + 12 R n,l ( r ) = Ar l exp⎜ − r ⎟L (mωr 2 ). ⎝ 2 ⎠ n

6-12

Discrete Quantum Mechanics

Here n = 0, 1, 2, … is the radial quantum number; A is the normalization constant l+ 1

that depends upon n and l; L n 2 is a generalized Laguerre polynomial [WeissteinLP]; and the energy eigenvalue associated with Rn,l is En,l = (2n + l + 32 )ω. This radial function is strongly convergent, allowing a well-defined finite value for A for all allowed values of n,l via ∞

∫0

R n,l 2 r 2dr = 1.

Again in this case, eigenstates of the position operators (and other operators associated with continuous observables) require the use of the Dirac delta. The most challenging and most important standard result is the determination of the eigenvalues and eigenstates of the three-dimensional Schrödinger equation for 2

the Coulomb potential, V (r ) = − er :

−

⎞ 1 ∂ ⎛ 2 ∂R ⎞ ⎛ l (l + 1) e2 ⎜r ⎟ + ⎜ E − − ⎟R = 0. ⎠ 2mr 2 ∂r ⎝ ∂r ⎠ ⎝ 2mr 2 r

Its eigenstates separate into two distinct pieces: discrete states with E < 0 (bound states) enumerated by a single radial quantum number n = 1, 2, …, ∞ , En varying as − 12 ; and a continuum of unbound states with energies assuming every value from n 0 to ∞. The bound-state state functions are strongly convergent and can be normalized to one. The continuum wave functions behave asymptotically as free particles and require Dirac delta normalization. Eigenstates of observables, such as position, that have a continuous spectrum of eigenvalues require (as always) the Dirac delta.

6.4 Defending the delta From a pragmatic perspective, the Dirac delta requires no defense: it simply works. The examples of the Schrödinger wave equation approach cited above (and many others requiring the Dirac delta) can produce results that yield stunningly accurate predictions of physical properties. An oft-cited example is the value of the anomalous magnetic momentum of the electron, calculated using quantum electrodynamics—a relativistic quantum field theory bristling with Dirac deltas—which agrees with experiment to an accuracy of more than one part in a billion. The pathology of the Dirac delta that so exercised John von Neumann involves its status as a function, an entity normally assumed to be well-defined at every point in its domain. At the only point where δ (x ) does not vanish by definition, i.e. at x = 0, it is undefined. The meaning of the Dirac delta is clear only when it is part of an integrand. In modern terminology, the Dirac delta is not a function, but rather a distribution [Gelfand1964], i.e. a functional that maps test functions1 onto the set of real numbers.

1

A test function is an infinitely differentiable function that vanishes outside of a bounded domain.

6-13

Discrete Quantum Mechanics

Dirac himself recognized that the ‘δ function’ (his terminology [Dirac1930]) was mathematically problematical, and thus he referred to it as an ‘improper function’. He furthermore noted that it was dispensable, since quantum mechanical theory could be written in such a way that the delta appeared only within integrals that yield well-defined values. He several times described the delta as simply a convenience, allowing one to focus on the physics without becoming buried in unnecessary mathematical formalism. The disagreement between Dirac and von Neumann in the early 1930s was thus one of philosophy rather than results. Both thought of quantum mechanics as a powerful tool for understanding nature, particularly at the sub-atomic scale. Both understood that their separate approaches to wave mechanics arrived at identical results. Dirac held that ‘...the mathematics is only a tool and one must learn to hold the physical ideas in one’s mind without reference to the mathematical form’ and ‘...use of improper functions is thus not really connected with any essential lack of rigor in the theory. It is, rather, a convenient notation, enabling us to express in a concise form certain fundamental formulas which we could, if necessary, rewrite in a rigorous form, but only in a cumbersome way in which the parallelism with the case of discrete eigenvalues would be obscured’ [Dirac1930]. By contrast, von Neumann [vonNeumann1932] addresses Dirac’s ‘improper functions’ in his preface, stating ‘There would be no objection here if these concepts, which cannot be incorporated into the present day framework of analysis, were intrinsically necessary for the physical theory [...] But this is by no means the case. It should rather be pointed out that the quantum mechanical ‘transformation theory’ can be established in a manner wihch is just as clear and unified, but which is also without mathematical objections.’ To a great degree, subsequent developments in quantum theory have proceeded without regard to this dispute, using the Dirac delta (and similar ‘improper functions’) whenever useful, and von Neumann’s elegant Hilbert space formulation when it works best.

Exercises 1. Show that the raising and lowering operators introduced in section 6.2.3 obey (i) aˆ− = (aˆ+ )† and (ii) [aˆ−, aˆ+ ] = 1. 6-14

Discrete Quantum Mechanics

2. Following the steps that were used to show

aˆ − n =

n n−1

for eigenstates of the one-dimensional quantum harmonic oscillator eigenstates, justify the claim that

aˆ+ n =

n+1 n+1 .

3. Prove that the integral in equation (6.4) satisfies the two defining conditions of the Dirac delta.

Bibliography [vonNeumann1932] von Neumann J 1932 Mathematische Grundlagen der Quantenmechanik (Berlin: Springer) English translation 1955 Mathematical Foundation of Quantum Mechanics (Princeton, NJ: Princeton University Press) [Schroedinger1926–2] Schrödinger E 1926 Ann. Phys. 79 734–56 [WeissteinHP] Weisstein E W Hermite polynomial MathWorld—A Wolfram Web Resource http://mathworld.wolfram.com/HermitePolynomial.html [Messiah1961] Messiah A (transl. G M Temmer) 1961 Quantum Mechanics (North Holland: Amsterdam) [WeissteinLP] Weisstein E W Associated Laguerre polynomial MathWorld—A Wolfram Web Resource http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html [Dirac1930] Dirac P A M 1930 The Principles of Quantum Mechanics (Oxford: Oxford University Press) (Subsequent editions in 1935, 1947 and 1958) [Gelfand1964] Gel’fand I M and Shilov G E 1964 Generalized Functions vol. 1 (New York: Academic)

6-15

IOP Concise Physics

Discrete Quantum Mechanics H Thomas Williams

Appendix A Relevant results from linear algebra

Linear algebra is the study of finite-dimensional vector spaces and linear maps within and between such spaces. As presented in some detail in chapter 1, quantum theory can be expressed within a complex vector space in d dimensions consisting of a collection of d-dimensional complex vectors with operations of vector addition (with commutativity, additive inverses and an additive identity) and scalar multiplication of a vector (distributive, associative and with a multiplicative identity). A linear map from a vector space = to another, > is an operation that, when applied to any element v within = , produces a vector w within >, M (v ) = w , such that for vectors v and v′ within = , M (v + v′) = M (v ) + M (v′) (additivity) and for any complex constant a, M (av ) = aM (v ) (homogeneity). A linear map within a vector space = associates every vector v within = with a vector w in the same space, obeying like properties of additivity and homogeneity. We refer to a linear map ˆ = v′. within a vector space as an operator, e.g. Oˆ , writing Ov Use of linear algebra in this volume relies on a subset of the topic, applying to a restricted set of vector spaces and operators. Vector spaces of quantum state vectors are Hilbert (or inner product) spaces, i.e. vector spaces with the additional requirement of an inner product that maps two vectors into a complex constant. For any pair of vectors v and v′ in a finite-dimensional Hilbert space there is defined an inner product (v′, v ) such that: (i) (v, v ) ⩾ 0 with 0 occurring only if v is the zero vector (additive identity element); (ii) if v″ is in the same space, then (v + v′, v″) = (v, v″) + (v′, v″); (iii) if a is a complex constant, (v′, av ) = a(v′, v ); and (iv) (v′, v ) = (v, v′). Dirac notation is used extensively herein, in which the notation ∣v〉, referred to as a ket, is used to represent any element of a Hilbert space. The inner product of two Hilbert space vectors, e.g. (v′, v ), is represented in Dirac notation as the bracket 〈v′∣v〉. The norm of a vector v, v , is equivalent to 〈v∣v〉 . Two Hilbert space vectors v and v′ are said to be orthogonal if 〈v′∣v〉 = 〈v∣v′〉 = 0.

doi:10.1088/978-1-6817-4125-3ch7

A-1

ª Morgan & Claypool Publishers 2015

Discrete Quantum Mechanics

A set of vectors {∣vi〉; i = 1, …, n} in a complex vector space is said to be linearly independent if a weighted sum of the vectors n

∑ai vi

,

i=1

with complex coefficients ai, vanishes if and only if ai = 0 for all i. A basis of a complex vector space is defined to be a subset of vectors in the space that is linearly independent, and such that every vector in the space can be written as a linear combination of the basis vectors with complex coefficients. The coefficients appropriate to the expression of a vector ∣v〉 are called the components of v relative to the basis, and linear independence of the basis implies that each vector’s components are uniquely determined. A set of unit vectors {∣ei〉; i = 1, …, d } in a d-dimensional vector space, such that 〈ei∣ej 〉 = δi, j , constitute an orthonormal basis for the vector space. Any d-dimensional Hilbert space can be represented as d-tuples of complex numbers ( d ) by picking a basis of the space and identifying each element of the space with its coordinate vector relative to that basis. Every operator Oˆ on the space can be represented by a d × d complex matrix with matrix elements Oij, such that the mapping of v into w obeys d

wi =

∑Oijvj , j

in which wi is the ith complex component of ∣w〉, vj is the jth complex component of ∣v〉, and the sum runs over the d values of the index j1. In such a representation, the inner product obeys d

vw =

∑vjwj ,

(A.1)

j

which implies d

ww =

∑ wj 2 ⩾ 0.

(A.2)

j

A.1 Properties of linear operators Assume / to be a Hilbert space. For every operator Oˆ on / there is an associated † operator, the Hermitian conjugate (or adjoint) of Oˆ , denoted Oˆ and defined by

ˆ ) = (Oˆ †h′ , h) (h′, Oh

1 The d values of j typically run from 0 to j − 1, or 1 to d, depending upon context. The summation notation used here and later in the appendix represents whichever is appropriate.

A-2

Discrete Quantum Mechanics

for every pair of vectors h and h′ in / . When Oˆ is represented by a d × d matrix, this definition becomes d

⎛

⎞

d

⎛

d

⎞

d

∑hj′⎜⎜∑Ojihi ⎟⎟ = ∑⎜⎜∑ (O )ij h′j ⎟⎟hi , †

j

⎝

⎠

i

⎝

i

⎠

j

leading to d

d

∑Ojihj′hi =

∑ (O )ij hj′hi . †

i, j

i, j

Since h and h′ are arbitrary vectors within / , it follows that

Oji = (O †)ij †

for every i , j , i.e. the matrix Oˆ is the conjugate transpose of the matrix Oˆ . A.1.1 Unitary operators A linear operator Uˆ on a Hilbert space is called unitary if † ˆ ˆ † = I. Uˆ Uˆ = UU

Unitary operators preserve the inner product, that is for two elements v and w of a Hilbert space † ˆ ˆ Uw ˆ = v Uˆ Uw = vw . Uv

A special case of this result is that unitary operators preserve the norm of a Hilbert space element. Assume that {∣ei〉; i = 1, …, d } and {∣ fi 〉; i = 1, …, d } are distinct orthonormal bases of = , a d-dimensional complex vector space. Each vector element of the first basis can be written as a linear combination of the vectors in the second basis, as d

ei =

∑ aki

fk ,

k=1

where the aki are complex coefficients. Note then that

⎛ d ei ej = ⎜⎜ ∑ aki fk , ⎝ k=1

⎞ a f ∑ lj l ⎟⎟ = ⎠ l=1 d

d

d

d

∑∑ aki alj fk fl =

∑ akiakj

k=1 l=1

k=1

and thus the matrix A with elements aij obeys A†A = I , i.e. A is a unitary matrix. Any vector in = can be expressed in any basis, for example using the two orthonormal bases just discussed, d

v =

d

∑νj ej

=

j=1

∑ν j′ f j j=1

A-3

,

Discrete Quantum Mechanics

where the νj and ν ′j are the complex vector components in each basis. From this follows d

⎛

d

∑νj⎜⎜∑ akj j=1

⎝ k=1

⎞ fk ⎟⎟ = ⎠

d

⎛

d

⎞

∑ ⎜⎜∑νjakj⎟⎟ fk k = 1⎝ j = 1

⎠

d

=

∑ν ′j f j

,

j=1

from which we conclude (using orthogonality of basis elements) d

ν k′ =

∑νjakj . j=1

Denoting the explicit vector representation of ∣v〉 in the ej basis as ν⃗ (with components νj ) and in the fj basis as ν′⃗ (with components ν ′j ), the previous expression can be written

ν′⃗ = A ν ⃗ and we conclude that transforming vectors in one orthonormal basis into another is accomplished by multiplication of the vector by a unitary matrix. A.1.2 Hermitian operators Since every observable property in quantum mechanics corresponds to a Hermitian operator, this category of operator is central to our subject. Hermitian operators are also referred to as self-adjoint operators. Any Hermitian operator Rˆ must obey † Rˆ = Rˆ

and have matrix elements that obey

Rij = Rji . Cauchy–Schwarz inequality Two elements of a Hilbert space have an inner product that is less than or equal to the product of their norms Let ∣v〉 and ∣w〉 be two arbitrary vectors in a d-dimensional Hilbert space that has a basis of mutually orthogonal unit vectors ∣ek〉, k = 1, …, d . If either of the vectors is the zero vector the inequality is trivially satisfied, so we assume neither to be zero. We can expand each vector in terms of the basis: N

v =

∑ αk ek k=1

and N

w =

∑ βk ek k=1

A-4

,

Discrete Quantum Mechanics

where the αk s and βk s are complex components of the vectors. We use the components to express the inner product of the two vectors, N

wv =

∑ βk αk , k=1

and also the squared norm of each N

vv =

∑ αk 2 , k=1

N

ww =

∑ βk 2 . k=1

We wish to prove the Cauchy–Schwarz inequality in this context, i.e. 2

∑βk αk

∑ αk 2 ∑ βk 2 .

⩽

k

k

(A.3)

k

The proof follows in a straightforward way from examination of the non-negative sum N

N

∑∑ ( j=1 k=1

αj βk − αkβj

)

2

N

N

(

2

2

= ∑ ∑ αj βk + αk 2 βj

2

− αj βk αkβj − αj βk αkβj

j=1 k=1 N

N

= 2 ∑ αj j=0

2

∑

βj

2

)

N

− 2 ∑ αj βj ⩾ 0.

j=0

(A.4)

j=0

Comparison of the final expression of equation (A.4) with equation (A.3) confirms the Cauchy–Schwarz inequality. □ The eigenvalue equation for any operator Oˆ is

Oˆ v = o v , where o is a complex constant, an eigenvalue of Oˆ , and ∣v〉 is the corresponding eigenvector.

For a Hermitian operator (Rˆ ) all eigenvalues (r) are real If ∣h〉 is a non-zero eigenvector of Rˆ corresponding to eigenvalue r, then Rˆ h = r h .

A-5

Discrete Quantum Mechanics

It follows that

ˆ = Rh ˆ h =r hh , r h h = h rh = h Rh and since 〈h∣h〉 ≠ 0, r = r as claimed2. □

Eigenvectors of a Hermitian operator Rˆ corresponding to distinct eigenvalues are orthogonal Assume r to be an eigenvalue of Rˆ with corresponding eigenvector ∣h〉, and r′ ≠ r to also be an eigenvalue of Rˆ with corresponding eigenvector ∣h′〉. The Hermiticity of Rˆ implies ˆ ′h h′ Rˆ h = Rh and therefore

r h′ h = r ′ h ′ h = r ′ h ′ h . Since r ≠ r′, necessarily 〈h∣h′〉 = 0. □

The expectation value of a Hermitian operator, with respect to any vector, is real The expectation value of an operator Oˆ relative to a vector ∣h〉, denoted 〈Oˆ 〉, is defined by Oˆ ≡ h Oˆ h . Assume Rˆ is a Hermitian operator and ∣h〉 is an arbitrary normalized vector. Then

ˆ h = Rˆ , Rˆ = h Rˆ h = Rh thus 〈Rˆ 〉 is real. □

The expectation value of the square of a Hermitian operator, with respect to any vector, is non-negative Assume Rˆ is a Hermitian operator and ∣h〉 is an arbitrary vector. Then 2 Rˆ

2 ˆ Rh ˆ ⩾ 0, = h Rˆ h = Rh h

the final inequality following from the definition of the inner product. □ A.1.3 Positive operators Qˆ is a positive operator in a complex vector space = if, for every vector v in = , v Qˆ v ⩾ 0. 2 ˆ ′〉 can also be written 〈v∣Oˆ ∣v′〉. We will henceforth favor the latter In Dirac notation, the inner product 〈v∣Ov expression.

A-6

Discrete Quantum Mechanics

If an operator Qˆ on a complex vector space is positive, it is also Hermitian We first break a positive operator Qˆ into two pieces as follows: † † Qˆ + Qˆ Qˆ − Qˆ +i ≡ Qˆ1 + iQˆ2. Qˆ = 2 2i

† Qˆ + Qˆ 2

†

Qˆ − Qˆ and Qˆ2 = 2i are Hermitian, easily seen by taking the Hermitian conjugate of each. Since Qˆ is positive,

Both Qˆ1 =

(v, Qˆ v) + i(v, Qˆ v) ⩾ 0. 1

2

(A.5)

For any vector v, the first term in this expression is real since Qˆ1 is Hermitian. The second term is pure imaginary because Qˆ2 is also Hermitian, so (v, Qˆ2v ) = 0 for all ∣v〉. ˆ ) = 0 for all vectors ∣v〉 in = , then Oˆ is the null operator If (v, Ov ˆ ), using v = w + w′, v = w − w′, Create four expressions of the form (v, Ov v = w + iw′ and v = w − iw′, in which w and w′ are arbitrary distinct and non-zero vectors in = , and combine them as follows:

w + w′ Oˆ w + w′ − w − w′ Oˆ w − w′ − i w + iw′ Oˆ w + iw′ +i w − iw′ Oˆ w − iw′ .

(A.6)

Assuming 〈v∣Oˆ ∣v〉 = 0 for every vector v in = , each term in this expression is zero, and expanding each term using the properties of inner products and summing produces w Oˆ w′ = 0. An operator for which the expectation value relative to each vector v vanishes must, thus, be the null operator. □ We conclude that Qˆ , a general positive operator, can be written as Qˆ = Qˆ1 = and is thus Hermitian. □

† Qˆ + Qˆ , 2

While all positive operators are Hermitian, the converse is not true, as is seen by noting that the eigenvalues of Hermitian operators must be real but can be negative. For Rˆ with negative eigenvalue λ corresponding to eigenvector ∣r〉, then 〈r∣Rˆ∣r〉 < 0. Such an Rˆ is not a positive operator. A.1.4 Projection operators, subspaces > is a subspace of vector space = if the elements of > are a subset of those of = and > is itself a vector space. A projection operator Pˆ from = onto > is an ˆ in > , and which obeys Pˆ 2 = Pˆ . The operator that for every vector v in = yields Pv kernel or nullspace of projection operator Pˆ is the set of vectors v′ in = such that A-7

Discrete Quantum Mechanics

ˆ , thus the subspace ˆ ′ = 0: the range of Pˆ is the collection of vectors of the form Pv Pv ˆ > . An orthogonal projection P from a Hilbert space / into itself is one for which its range and kernel are orthogonal subspaces, i.e. every vector in the range of Pˆ is orthogonal to every vector in the kernel of Pˆ . For the remainder of this section, Pˆ will represent an orthogonal projection. Every vector h in Hilbert space / can be trivially separated into two components:

ˆ + Pˆ ⊥h, h = Ph ⊥ where Pˆ ≡ I − Pˆ is called the orthogonal complement of Pˆ . The first component, ˆ , is in the range of Pˆ , since Ph

ˆ ) = Pˆ 2h = Ph ˆ . Pˆ (Ph ⊥ The second component, Pˆ h is in the kernel of Pˆ , since

(

)

⊥ 2 Pˆ (Pˆ h) = Pˆ (I − Pˆ )h = Pˆ − Pˆ h = 0.

Since every vector in the range is orthogonal to every one in the kernel, it follows that

ˆ ∣(I − Pˆ )h′〉 = 〈(I − Pˆ )h∣Ph ˆ ′〉 = 0, 〈Ph and from these two equalities we easily see that

ˆ ∣h′〉 = 〈h∣Ph ˆ ′〉 . 〈Ph Pˆ , a general projection operator, is equal to its adjoint, thus projection operators are Hermitian. In the language of vector and matrix components, an operator in a d-dimensional Hilbert space / that projects into a d ′ < d dimensional subspace can be written as d′

Pˆ =

∑ ej

ej ,

j=1

where the set {∣ej 〉} is an orthonormal basis for the subspace into which Pˆ projects and 〈ej ∣ = ∣ej 〉†. It is straightforward to show that this expression for Pˆ satisfies † 2 Pˆ = Pˆ and Pˆ = Pˆ . A.1.5 Normal operators; spectral decomposition In general, an operator does not commute with its Hermitian conjugate. Those that do (Nˆ ), obeying ˆ ˆ † = Nˆ †Nˆ , NN are called normal operators. Clearly, every Hermitian operator is a normal operator, but many normal operators are not Hermitian, e.g. those representable by diagonal matrices with complex entries on the diagonal. In anticipation of the spectral theorem, which addresses the eigenvalues and eigenvectors of normal operators, we first prove the following. A-8

Discrete Quantum Mechanics

Every operator on a finite-dimensional vector space has at least one eigenvalue Assume Oˆ to be an operator in a complex vector space = of finite dimension d > 0, and ∣v〉 to be a nonzero vector in = . The set of vectors formed using a sequence of j powers of Oˆ , {Oˆ ∣v〉: 0 ⩽ j ⩽ d }, must be linearly dependent since it has d + 1 elements, i.e. there is a collection of complex constants aj, not all of which are zero, such that d

∑ ajOˆ j v

= 0.

(A.7)

j=0

Define j = J to be the largest index for which aj is nonzero, and consider the polynomial J

∑ aj z j , j=0

in which z is a complex constant. The fundamental theorem of algebra [Axler1997] states that every polynomial over the complex numbers has at least one zero. A simple inductive argument leads from this to the conclusion that J

J

∑ ajz j = c ∏ (z − λj ) j=0

j=0

j for complex numbers {λj } and c. It is easy to see that this also holds with Oˆ ∣v〉 replacing z, and λj ∣v〉 replacing λj , and thus from equation (A.7)

J

d j

∑ ajOˆ v =

∏

j=0

j=0

(Oˆ − λ I ) v j

= 0,

in which I represents the identity operator. From this it is clear that there is at least one nonzero vector (call it ∣v′〉) such that (Oˆ − λj′I )∣v′〉 = 0 for some j ′ such that 0 ⩽ j ′ ⩽ J . Thus λj′ is an eigenvalue of Oˆ , proving our assertion. □

If ∣v〉 is an eigenstate of a normal matrix Nˆ with eigenvalue λ, it is also an † eigenstate of Nˆ Since

(Nˆ − λI ) v = 0, it follows that

0 = v (Nˆ − λI )†(Nˆ − λI ) v = v (Nˆ − λI )(Nˆ − λI )† v † = v (Nˆ − λI )(Nˆ − λ¯I )∣v〉 =

(Nˆ † − λ¯I ) v

2

,

† demonstrating that ∣v〉 is also an eigenstate of Nˆ with eigenvalue λ¯ . □

A-9

Discrete Quantum Mechanics

We now present proof of a version of the spectral theorem. Every normal operator Nˆ on a d-dimensional complex vector space = has d orthogonal eigenvectors that form a basis for = . Nˆ must have at least one eigenvalue λ. Define Pˆ to be the projection operator onto ⊥ the subspace of = that contains all eigenvectors of Nˆ with eigenvalue λ, and Pˆ to be the orthogonal complement of Pˆ . Consider ⊥ ⊥ ˆ ˆ ˆ + Pˆ ⊥NP ˆ ˆ ⊥ + PNP ˆ ˆ ˆ ⊥ + Pˆ ⊥NP. ˆ ˆ Nˆ = (Pˆ + Pˆ )Nˆ (Pˆ + Pˆ ) = PNP

(A.8)

ˆ ˆ∣v〉 is in = for all ∣v〉 and thus Observing that = is invariant under Nˆ , we see that NP ⊥ ˆ ˆ ) vanishes. the final term in equation (A.8) (Pˆ NP † ⊥ † Likewise, = is invariant under Nˆ , so Pˆ Nˆ Pˆ is zero, as is its Hermitian conjugate ˆ ˆ ˆ ⊥, the penultimate term in equation (A.8). PNP ⊥ † Consider next the expression Pˆ Nˆ Pˆ∣v〉 for any vector ∣v〉 in = . ∣v′〉 = Pˆ∣v〉 is either † zero or an eigenvector of Nˆ with eigenvalue λ; ∣v′〉 = Nˆ ∣v′〉 is either zero or an ⊥ † ⊥ eigenstate of Nˆ ; and Pˆ ∣v′〉, in either case, vanishes. Thus the operator Pˆ Nˆ Pˆ is ˆ ˆ ˆ ⊥, the penultimate term in equation (A.8). zero, as is its Hermitian conjugate PNP Considering now the second term beyond the equal sign in equation (A.8), we ⊥ ˆ ˆ⊥ seek to prove that Pˆ NP is normal by examining ⊥ † ⊥ ⊥ ˆ ˆ⊥ ⊥ ˆ ˆ⊥ ˆ⊥ ˆ † ˆ⊥ ⊥ † ⊥ ˆ ˆ⊥ ⊥ ˆ ˆ⊥ ˆ † ˆ⊥ − Pˆ NP − Pˆ NP Pˆ Nˆ Pˆ Pˆ NP P N P = Pˆ Nˆ Pˆ NP N P .

(A.9)

⊥ ˆ ˆ⊥ ˆ ˆ ⊥, since the subspace Pˆ ⊥∣v〉, ∣v〉 in =, is invariant under Nˆ Note that Pˆ NP = NP ⊥ † ⊥ ⊥ † as is the Hermitian conjugate of this expression Pˆ Nˆ Pˆ = Pˆ Nˆ . This allows us to transform equation (A.9) into

(

)

⊥ † ˆ ˆ⊥ ⊥ ˆ ˆ † ˆ⊥ ⊥ † ˆ ˆ † Pˆ ⊥, − Pˆ NN Pˆ Nˆ NP P = Pˆ Nˆ Nˆ − NN

⊥ ˆ ˆ⊥ which vanishes because Nˆ is normal, thus proving that Pˆ NP is normal. We have proven, starting from equation (A.8), that

ˆ ˆ ˆ + Pˆ ⊥NP ˆ ˆ ⊥, Nˆ = PNP

(A.10)

which, in the language of matrix operators, shows that Nˆ is of block-diagonal form, with one block operating in the space of eigenvectors of Nˆ with eigenvalue λ. The ⊥ ˆ ˆ⊥ final operator in equation (A.10), Pˆ NP , operates in a space smaller than that of Nˆ , but since it is normal it too has at least one eigenvalue. Applying the same logic that we used above for Nˆ , we can ‘peel off’ another block from Nˆ with a different eigenvalue, iterating this process until Nˆ is completely block-diagonalized, with each block corresponding to eigenstates with a distinct eigenvalue. Such a matrix, fully diagonalized, has as eigenvectors a complete set of orthogonal vectors that serve as an orthonormal basis of = made up of eigenstates of Nˆ . □ A-10

Discrete Quantum Mechanics

Identifying state vectors ∣ej 〉 as elements of an orthonormal basis for a d-dimensional Hilbert space / , any vector ∣v〉 in / can be represented as d

v =

∑αj ej

,

j

where each αj is a complex constant. It follows that

ek v =

∑αj

ek ej = αk .

j

The resulting expression for ∣v〉 is, d

v =

( ej v ),

∑ ej j

in which the parentheses set off the inner product, which is a complex constant. Using associativity, we can also write d

v =

∑(

) )

ej ej v

j

⎞ ⎛ d = ⎜⎜ ∑ ej ek ⎟⎟ v , ⎠ ⎝ j

(

true for any vector in / . Thus d

I=

∑ ej

ej ,

(A.11)

j

valid for any orthonormal basis. This is a complicated albeit frequently useful alternative expression of the identity operator.

A.2 Simultaneous eigenvectors A complete set of eigenvalues of Aˆ can be found that also consists of eigenvalues of Bˆ , when the two operators are Hermitian and commute with one another. Assume Aˆ and Bˆ to be Hermitian operators in Hilbert space / and that [Aˆ , Bˆ ] = 0. Any Hermitian operator is also normal, so both Aˆ and Bˆ have sets of eigenvalues each of which constitutes a basis for / . Consider ∣v〉 to be an eigenvector of Aˆ with eigenvalue λ and expand it in terms of normalized eigenstates of Bˆ, ∣bj 〉 (with corresponding eigenvalues bj): v =

∑αj bj

.

(A.12)

j

It follows that

(Aˆ − λI ) v = 0 =

∑αj(Aˆ − λI ) bj j

A-11

.

(A.13)

Discrete Quantum Mechanics

Our goal now is to break the sum on the right into terms that are linearly independent and thus must separately equal zero. Towards this end, consider the effect of applying Bˆ to individual terms in the sum:

Bˆ(Aˆ − λI ) bj = (Aˆ − λI )Bˆ bj = bj (Aˆ − λI ) bj .

(A.14)

This demonstrates that (Aˆ − λI )∣bj 〉 is an eigenstate of Bˆ with eigenvalue bj. If there is only one such (non-degenerate) eigenstate, then (Aˆ − λI )∣bj 〉 is in the direction of ∣bj 〉. If ∣bj 〉 is one of two or more (degenerate) eigenstates with the same eigenvalue, then (Aˆ − λI )∣bj 〉 is in the subspace of ∣bj 〉s with eigenvalues bj. Since non-degenerate eigenstates of Bˆ are orthogonal to one another, and orthogonal to subspaces spanned by degenerate eigenstates, equation (A.14) implies that either • (Aˆ − λI )∣bj 〉 = 0 for non-degenerate Bˆ eigenstates ∣bj 〉 and thus they are also eigenstates of Aˆ ; or • (Aˆ − λI )∑j′αj′∣bj′〉 = 0, where the sum is over a multi-dimensional subspace corresponding to a single degenerate eigenvalue of Bˆ . There are as many such sums (which are distinct eigenstates of Aˆ ) as the dimensionality of the subspace, thus each such sum is a simultaneous eigenstate of Aˆ and Bˆ . It is clear, then, that while each basis of Aˆ is not necessarily made up of eigenstates of Bˆ when there is degeneracy, we can always find an orthonormal basis consisting of simultaneous eigenstates of Aˆ and Bˆ . □ Repeated use of this argument can show that for any number of mutually commuting Hermitian operators in / , there is an orthonormal basis for the space for which each basis state is an eigenstate of all the operators.

A.3 Operator functions We have seen already that scalar multiples of linear operators and sums of linear operators are themselves linear operators and thus can be worked with in ways we have described. In addition, powers and products of linear operators are also linear operators. These facts allow one to generate a linear operator corresponding to any function of an operator that can be given a Taylor series expansion, as long as appropriate convergence criteria are satisfied3. This was done in the case of the exponential function in chapter 2, for example. For normal operators, since they have a spectral decomposition, there is another way to express functions with operator arguments. Express the spectral decomposition

∞ Assuming the Taylor series takes the form f (z ) = ∑ j =0aj z j and the linear operator is Tˆ , the radius of convergence of the Taylor series must exceed the operator norm of Tˆ , i.e. the maximum of the moduli of the eigenvalues of Tˆ if Tˆ is normal. 3

A-12

Discrete Quantum Mechanics

of the d-dimensional normal operator Nˆ in terms of its eigenvalues nj and eigenvectors ∣nj 〉 as d

Nˆ =

∑nj nj

nj .

j

Define the operator function f (Nˆ ), mapping Nˆ into a normal operator of the same dimension, by d

f (Nˆ ) =

∑ f (n j ) n j

nj .

(A.15)

j

This definition is consistent with the process above, as can be seen by considering Nˆ as a d × d matrix. Using the eigenvectors of Nˆ as basis, it becomes a diagonal matrix with its d eigenvalues along the diagonal. The nth power of this diagonal matrix is itself diagonal, with its diagonal elements equal to the nth power of the eigenvalues of Nˆ . Any function of the matrix representation of Nˆ that can be represented by a Taylor series expansion can be seen to be represented in this basis by a diagonal matrix with diagonal elements equal to the function with the corresponding eigenvalue as argument. As an example of equation (A.15), consider the function exp(Aˆ + Bˆ ), where Aˆ and Bˆ are normal matrices. As we have seen, if Aˆ and Bˆ commute, there is an orthonormal set of basis elements ∣ej 〉 that are simultaneously eigenvectors of the two operators. The sum Aˆ + Bˆ can thus be expressed via spectral decomposition as d

∑(aj + bj ) ej

ej ,

j

where aj and bj are eigenvalues respectively of Aˆ and Bˆ corresponding to eigenvector ∣ej 〉. We express the exponential of the sum of operators as d

d

exp(Aˆ + Bˆ ) = ∑ exp(aj + bj ) ej ej = j

∑ exp(aj ) exp(bj ) ej

ej

j

= exp(Aˆ ) exp(Bˆ ).

(A.16)

This relationship, reflective of that for the exponential of the sum of two complex numbers, fails to hold when [Aˆ , Bˆ ] ≠ 0. This can be seen by comparing the lowest order terms in the Taylor series expansions of the two sides of equation (A.16), keeping Aˆ operators to the left of Bˆ operators in multiplicative terms whenever possible,

(Aˆ + Bˆ )2 +⋯ exp(Aˆ + Bˆ ) = 1 + Aˆ + Bˆ + 2 2 ˆ ˆ + Bˆ 2 + [Bˆ , Aˆ ] Aˆ + 2AB = 1 + Aˆ + Bˆ + +⋯ 2 A-13

Discrete Quantum Mechanics

and 2 2 ⎛ ⎞⎛ ⎞ Aˆ Bˆ ˆ ˆ ˆ ˆ ⎜ ⎟ ⎜ exp(A) exp(B ) = ⎜ 1 + A + + ⋯⎟⎜1 + B + + ⋯⎟⎟ 2 2 ⎝ ⎠⎝ ⎠ 2 ˆ ˆ + Bˆ 2 Aˆ + 2AB ˆ ˆ =1 + A + B + + ⋯. 2 The dependence on [Aˆ , Bˆ ] seen in the first expression becomes more complicated in higher order terms, while the second expression keeps all Aˆ dependence to the left of that of Bˆ , thus avoiding commutator dependence.

A.4 Trace Assuming Rˆ to be a Hermitian operator in a d-dimensional Hilbert space / , we define the trace of Rˆ by d

Tr(Rˆ ) ≡

∑

ej Rˆ ej ,

(A.17)

j

where the vectors ej are elements of an orthonormal basis for / .

The trace is independent of the choice of orthonormal basis As we have seen in section A.1.1, the elements ∣ej 〉 of one orthonormal basis are related to the elements ∣ f j 〉 of another through the matrix elements of a unitary ˆ operator Uˆ : d

ej =

∑Uˆjk fk

.

k

Inserting this into equation (A.17), we get d

d

d

Tr(Rˆ ) = ∑ ∑ Uˆjk fk Rˆ ∑ Uˆjl fl j

k

d

d

l d

† = ∑ ∑ ∑(Uˆ )kj Uˆjl

k

l

d

fk Rˆ fl

j

=

d

∑ ∑(Uˆ †Uˆ )kl k

fk Rˆ fl

l

d

= ∑ fk Rˆ fk , k

as claimed. □ The trace as it relates to any matrix representation of operator Rˆ becomes Tr(Rˆ ) = ∑Rjj , j

the sum of its diagonal elements.

A-14

Discrete Quantum Mechanics

Since the trace has been seen to be independent of the choice of orthonormal basis, d

Tr(Rˆ ) =

∑

d

f j Rˆ f j

=

j

∑Rjj , j

regardless of the orthonormal basis chosen to represent Rˆ . □ Choosing as the basis set ∣rj 〉, the eigenvectors of the matrix Rˆ , d

Tr(Rˆ ) =

∑

d

rj Rˆ rj =

d

∑

j

rk λj rk =

j

∑λj , j

where λj is the eigenvalue corresponding to eigenvector ∣rj 〉. The trace of Rˆ , therefore, is also the sum of its eigenvalues (including repeated values). The trace of the sum of operators is the sum of traces, i.e. d

d

Tr(Aˆ + Bˆ ) = ∑ ej (Aˆ + Bˆ ) ej =

∑

j

d

ej Aˆ ej +

∑

j

ej Bˆ ej

j

= Tr(Aˆ ) + Tr(Bˆ ) ,

(A.18)

where the vectors ej are elements of an orthonormal basis for / . Also, for any complex constant a d

Tr(aBˆ ) =

∑

d

ej aBˆ ej = a∑ ej Bˆ ej = a Tr(Bˆ ) .

j

(A.19)

j

Another useful property of the trace can be simply shown using the identity operator as exhibited in equation (A.11): d

ˆ ˆ) ≡ Tr(AB

∑ j

d ⎛ d ⎞ ˆ ˆ ej = ∑ ej Aˆ ⎜∑ ek ek ⎟Bˆ ej ej AB ⎜ ⎟ ⎝ k ⎠ j d

d

(

)( e

(

)( e Aˆ e )

= ∑ ∑ ej Aˆ ek j

k

d

d

= ∑ ∑ ek Bˆ ej j

k

Bˆ ej

j

k

k

⎞ ⎛ d d = ∑〈ek∣Bˆ⎜⎜∑ ej ej ⎟⎟Aˆ ∣ek〉 ⎠ ⎝ j k d

ˆ ˆ ek = Tr(BA ˆ ˆ ). = ∑ ek BA k

A-15

)

Discrete Quantum Mechanics

This result generalizes simply to the cyclic property of the trace, which states that the trace of a product of operators is invariant under the operation of removing one or more operators from one end of the product and moving them (without rearrangeˆ ˆ ˆ ) = Tr(BCA ˆ ˆ ˆ ) = Tr(CAB ˆ ˆ ˆ ). ment) to the other end, e.g. Tr(ABC

A.5 Vector space of operators The linear operators that act upon state vectors in a d-dimensional Hilbert space / themselves form a Hilbert space, since • Oˆ = Oˆ 1 + Oˆ 2 = Oˆ 2 + Oˆ 1 is a linear operator in / if Oˆ 1 and Oˆ 2 are; • the zero operator in /, 0, obeys 0 + Oˆ = Oˆ + 0 = Oˆ for every linear operator Oˆ in / ; • for any complex constant a and linear operator Oˆ in /, aOˆ is also an operator in / ; and • the Hilbert–Schmidt inner product

(Oˆ 1, Oˆ 2 )HS ≡ Tr

(Oˆ Oˆ ) † 1

2

maps any two operators Oˆ 1 and Oˆ 2 in / into a complex constant. (Oˆ 1, Oˆ 2 )HS is an inner product because: • (Oˆ , Oˆ )HS ⩾ 0, since

( )

d

(

)

† † ˆ Tr Oˆ Oˆ = ∑ ej , Oˆ Oe j =

j

d

d

d

∑∑ j

† ej Oˆ ek

d

ek Oˆ ej = ∑∑ ek Oˆ ej

k

j

2

⩾ 0;

k

• (Oˆ 1 + Oˆ 2, Oˆ )HS = (Oˆ 1, Oˆ )HS + (Oˆ 2, Oˆ )HS, easily proven from the property of the trace shown in equation (A.18); • if a is a complex constant, (aOˆ 1, Oˆ 2 )HS = a (Oˆ 1, Oˆ 2 )HS, as a result of the property of the trace from equation (A.19); and • (Oˆ 1, Oˆ 2 )HS = (Oˆ 2, Oˆ 1)HS . By way of example, consider the collection of linear operators on two-dimensional complex vectors, which we identify with the collection of 2 × 2 matrices with complex entries. This collection forms a four-dimensional Hilbert space having natural basis

⎛ ⎞ Oˆ 1 = ⎜ 1 0 ⎟ , ⎝ 0 0⎠

⎛ ⎞ Oˆ 2 = ⎜ 0 1 ⎟ , ⎝ 0 0⎠

⎛ ⎞ Oˆ 3 = ⎜ 0 0 ⎟ , ⎝ 1 0⎠

⎛ ⎞ Oˆ4 = ⎜ 0 0 ⎟ . ⎝ 0 1⎠

It is easy to verify that this natural basis is actually an orthonormal basis relative to the Hilbert–Schmidt inner product. In the next section, we point out that the Pauli spin matrices, together with the identity matrix, form another basis for this space (which is orthogonal).

A-16

Discrete Quantum Mechanics

A.6 Spin one-half component matrices and the SU(2) group In chapter 2 we introduced three matrices corresponding to components of a Hermitian observable operator for the measurement of spin component along coordinate axes for a particle with spin one-half:

sˆx =

1 ⎛⎜ 0 1 ⎞⎟ 2 ⎝1 0 ⎠

sˆy =

1 ⎛⎜ 0 −i ⎞⎟ 2⎝i 0 ⎠

sˆz =

1 ⎛⎜1 0 ⎞⎟ . 2 ⎝ 0 −1⎠

(A.20)

These are each one-half the corresponding member of the Pauli spin matrices. In addition to being Hermitian, it is easy to show that they are also unitary, and have trace zero. They are mutually orthogonal operators using the Hilbert–Schmidt inner product, for example

( sˆ , sˆ ) x

y

HS

( )

= Tr sˆx†sˆy =

1 ⎛⎛⎜ 0 1 ⎞⎟ ⎛⎜ 0 −i ⎞⎟⎞ 1 ⎛⎜ i 0 ⎞⎟ Tr⎜ ⎟ = Tr = 0. 4 ⎝⎝1 0 ⎠ ⎝ i 0 ⎠⎠ 4 ⎝ 0 −i ⎠

Similarly, (sˆx, sˆz )HS = (sˆy, sˆz )HS = 0. Furthermore, the fact that each of the sˆj s have trace zero means that the Hilbert–Schmidt inner product of I with each is also zero. Thus, the four-operator set {sˆx, sˆy, sˆz , I } forms an orthogonal basis for the vector space of all 2 × 2 complex matrices. In this basis each element has norm 1 . 2 The three operators of equation A.20 are the generators for the symmetry group SU(2), the special unitary group of degree two. A group . , for our purposes, is defined to be a collection of operators Gˆ 1, Gˆ 2, … along with a binary multiplication operation (denoted e.g. by Gˆ 1Gˆ 2 ) with the properties: • if Gˆ 1 and Gˆ 2 are elements of . , then Gˆ 1Gˆ 2 is also an element of . ; • for all Gˆ 1, Gˆ 2, Gˆ 3 in ., (Gˆ 1Gˆ 2 )Gˆ 3 = Gˆ 1(Gˆ 2Gˆ 3) ≡ Gˆ 1Gˆ 2Gˆ 3; • there is an element I in . such that for every Gˆ 1 in ., IGˆ 1 = Gˆ 1I = Gˆ 1; • for every Gˆ 1 in . there is another, Gˆ 2 , in . such that Gˆ 1Gˆ 2 = Gˆ 2Gˆ 1 = I . The special unitary group of degree 2, SU(2), is the group of 2 × 2 unitary matrices with determinant 1. It is a Lie group in that its elements are differentiable with respect to their parameters. Each element in the group can be represented in the form

(

)

Gˆ = exp −i(sˆxθx + sˆyθy + sˆzθz ) , where the θk s are the three real parameters of the group and the spin one-half operators sk are the group’s generators. The group elements represent rotations of a spin one-half quantum state from one orientation to another in the Bloch sphere. The three parameters encode the axis of rotation and the angle of rotation. The group structure is completely determined by the algebra of its generators, and for SU(2) this is fully expressed by

⎡⎣sˆi , sˆj ⎤⎦ = isˆk ,

A-17

Discrete Quantum Mechanics

where i , j , k represent any cyclic permutation of the indices x , y, z . In dimensions d > 2, one can find three complex operators analagous to the sj operators that satisfy the same commutation relations. Using them as generators, one finds a subgroup of all the trace-one unitary operators in d dimensions and these are thus d-dimensional representations of SU(2). These representations are the basis of angular momentum and isospin algebras for states of spin greater than one-half.

Bibliography [Axler1997] Axler S 1997 Linear Algebra Done Right (New York: Springer)

A-18

IOP Concise Physics

Discrete Quantum Mechanics H Thomas Williams

Appendix B Directory of definitions and notation

B.1 Notation a

a¯ ∣·〉

〈·∣

〈a ∣ b〉

〈a ∣Oˆ ∣ b〉 ∣ a〉〈b ∣ † Oˆ

I 〈Oˆ 〉ϕ ρ σx, σy, σz

Xˆ , Yˆ , Zˆ

Lowercase roman letters refer to complex constants, except for a few exceptions that are mentioned when introduced. The complex conjugate of complex number a. A ket, representative of any vector in a Hilbert space except for the zero (null) vector which is denoted as 0. A bra, an element of a Hilbert space that is dual to the Hilbert space of the kets. A bra can also be thought of as a functional that maps kets into complex numbers, as 〈a ∣ ( ∣ b〉 ) ≡ 〈a ∣ b〉, the final bracket evaluating as a complex number. A bracket, representing the complex number that is the inner product of ∣ a〉 with ∣ b〉. The inner product definition is contained within the description of the Hilbert space of the kets. The bracket is not symmetric relative to its arguments, but rather 〈a ∣ b〉 = 〈b ∣ a〉. A bracket construction representing the inner product of vector ∣ a〉 with vector Oˆ ∣ b〉. Operator that maps Hilbert space vectors onto vectors in the same Hilbert space, as ∣ a〉〈b ∣ ( ∣ c〉 ) ≡ ( 〈b ∣ c〉 ) ∣ a〉. The Hermitian conjugate of the operator Oˆ , defined by the fact that for arbitrary Hilbert space vectors ∣ x〉 and ∣ y〉, the inner product of ∣ x〉 with Oˆ ∣ y〉 is equal to the † inner product of Oˆ ∣ x〉 with ∣ y〉. Identity operator, satisfying I ∣ x〉 = ∣ x〉 for every ∣ x〉. Expectation value of operator Oˆ with respect to quantum state ϕ, equal to 〈ϕ ∣ Oˆ ∣ ϕ〉. When it is obvious from the context, the quantum state subscript is often omitted. Density operator, often used with subscripts. ⎞ ⎛ ⎛ ⎞ ⎛ ⎞ Pauli spin operators with matrix representations ⎜ 0 1 ⎟, ⎜ 0 − i ⎟, ⎜ 1 0 ⎟. ⎝ 1 0 ⎠ ⎝ i 0 ⎠ ⎝ 0 − 1⎠ Proxies for the Pauli spin matrices used when discussing quantum information theory: Xˆ ≡ σx , Yˆ ≡ σy , Zˆ = σz .

doi:10.1088/978-1-6817-4125-3ch8

B-1

ª Morgan & Claypool Publishers 2015

Discrete Quantum Mechanics

(Continued.)

sˆx , sˆy, sˆz

Operators for the spin one-half observables for spin projection in the x, y, z directions: 1 1 1 sˆx = 2 σx , sˆy = 2 σy , sˆz = 2 σz .

ℏ=1

Throughout this volume, physical units are used in which the ubiquitous quantum constant ℏ is set to one. This makes all energies have units of inverse time. Elements of an orthonormal basis with integer subscripts j, satisfying 〈ej ∣ ek〉 = δj ,k .

∣ ej 〉

B.2 Definitions Terms that are not specific to a discussion localized within the text. basis, orthonormal basis

A collection of d vectors, ∣ vj 〉, is a basis for a d-dimensional Hilbert space if every vector ∣ w〉 in the space can be represented as a linear d combination of the basis vectors: ∣ w〉 = ∑ j aj ∣ vj 〉. The basis is orthonormal if and only if 〈vj ∣ vk〉 = δj ,k .

bit

The basic unit of classical information. It is the abstract notion of a quantity that can take on one of only two values, e.g. 0 and 1. A positive, trace-one operator that provides the most general expression for a quantum state. If an operator Oˆ obeys the equation Oˆ ∣v〉 = a∣v〉, the constant a is called an eigenvalue and the vector ∣ v〉 is called an eigenvector of Oˆ . Two or more pure quantum states are said to be entangled if the joint state function cannot be expressed as a product of individual quantum state vectors. † ˆ An operator Oˆ that obeys Oˆ = O.

density operator eigenvalue, eigenvector entanglement

Hermitian operator linearly independent normal operator observable operator

operator orthogonal vectors positive operator projection operator

pure state

A collection of vectors ∣ vj 〉 is linearly independent if and only if no linear combination ∑j aj ∣vj 〉 vanishes unless each coefficient aj vanishes. † ˆ ˆ †. An operator Oˆ that obeys Oˆ Oˆ = OO A Hermitian operator, the eigenvalues of which are the possible outcomes of a measurement of a particular observable, and the corresponding eigenvectors are the states that emerge from the measurement if that eigenvalue is the result of the measurement. A linear map within a vector space. If 〈v ∣ w〉 = 0, the vectors ∣ v〉 and ∣ w〉 are said to be orthogonal. An operator Oˆ for which 〈v ∣ Oˆ ∣ v〉 ≥ 0 for every vector ∣ v〉. k An operator that can be expressed in the form Pˆ = ∑ j ∣ ej 〉〈ej ∣ in which the vectors ∣ ej 〉 are elements of an orthonormal basis. Such 2 operators are idempotent, i.e. Pˆ = Pˆ .

A quantum state that can be completely defined by a state vector. General quantum states are always superpositions of pure states, called mixed states, and their corresponding density operators can be expressed as

ρ=

∑ pi

ϕi

i

in which the ∣ ϕi 〉 are state vectors.

B-2

ϕi

Discrete Quantum Mechanics

(Continued.)

quantum numbers qubit

raising operator (lowering operator) state space state vector trace

unitary operator

Numbers, often integer of half-integer, that serve to distinguish eigenvalues of observable operators. An abstraction, representing a variable that can take on two values, like 0 and 1, as well as any complex-valued superposition thereof. The term is also used to refer to a physical manifestation of such a variable, such as the polarization state of a photon or the spin state of an electron. Operators that transform an eigenvector of an observable operator into another eigenvector with the next-highest (next-lowest) eigenvalue. The Hilbert space associated with a particular quantum system. The Hilbert space vector that represents all that can be known about a particular configuration of a quantum system. A property of an operator that is independent of the operator’s representation. It is equal to the sum of the eigenvalues of the operator. It also equals the sum of the diagonal elements of any matrix representation of the operator. † ˆ ˆ† = I . An operator Oˆ that obeys Oˆ Oˆ = OO

B-3