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ESSENTIAL QUANTUM MECHANICS

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Essential Quantum Mechanics

GARY E. BOWMAN Department of Physics and Astronomy

Northern Arizona University

1

3 Great Clarendon Street, Oxford OX2 6DP

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Published in the United States by Oxford University Press Inc., New York

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First published 2008

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British Library Cataloguing in Publication Data Data available

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Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain

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ISBN 978–0–19–922892–8 (Hbk.) ISBN 978–0–19–922893–5 (Pbk.)

1 3 5 7 9 10 8 6 4 2

Contents

Preface ix

1 Introduction: Three Worlds 1 1.1 Worlds 1 and 2 1 1.2 World 3 3 1.3 Problems 4

2 The Quantum Postulates 5 2.1 Postulate 1: The Quantum State 6 2.2 Postulate 2: Observables, Operators, and Eigenstates 8 2.3 Postulate 3: Quantum Superpositions 10

2.3.1 Discrete Eigenvalues 11 2.3.2 Continuous Eigenvalues 12

2.4 Closing Comments 15 2.5 Problems 16

3 What Is a Quantum State? 19 3.1 Probabilities, Averages, and Uncertainties 19

3.1.1 Probabilities 19 3.1.2 Averages 22 3.1.3 Uncertainties 24

3.2 The Statistical Interpretation 26 3.3 Bohr, Einstein, and Hidden Variables 28

3.3.1 Background 28 3.3.2 Fundamental Issues 30 3.3.3 Einstein Revisited 32

3.4 Problems 33

4 The Structure of Quantum States 36 4.1 Mathematical Preliminaries 36

4.1.1 Vector Spaces 36 4.1.2 Function Spaces 39

4.2 Dirac’s Bra-ket Notation 41 4.2.1 Bras and Kets 41 4.2.2 Labeling States 42

4.3 The Scalar Product 43 4.3.1 Quantum Scalar Products 43 4.3.2 Discussion 45

vi Contents

4.4 Representations 47 4.4.1 Basics 47 4.4.2 Superpositions and Representations 48 4.4.3 Representational Freedom 50

4.5 Problems 52

5 Operators 53 5.1 Introductory Comments 54 5.2 Hermitian Operators 56

5.2.1 Adjoint Operators 56 5.2.2 Hermitian Operators: Definition and Properties 57 5.2.3 Wavefunctions and Hermitian Operators 59

5.3 Projection and Identity Operators 61 5.3.1 Projection Operators 61 5.3.2 The Identity Operator 62

5.4 Unitary Operators 62 5.5 Problems 64

6 Matrix Mechanics 68 6.1 Elementary Matrix Operations 68

6.1.1 Vectors and Scalar Products 68 6.1.2 Matrices and Matrix Multiplication 69 6.1.3 Vector Transformations 70

6.2 States as Vectors 71 6.3 Operators as Matrices 72

6.3.1 An Operator in Its Eigenbasis 72 6.3.2 Matrix Elements and Alternative Bases 73 6.3.3 Change of Basis 75 6.3.4 Adjoint, Hermitian, and Unitary Operators 75

6.4 Eigenvalue Equations 77 6.5 Problems 78

7 Commutators and Uncertainty Relations 82 7.1 The Commutator 83

7.1.1 Definition and Characteristics 83 7.1.2 Commutators in Matrix Mechanics 85

7.2 The Uncertainty Relations 86 7.2.1 Uncertainty Products 86 7.2.2 General Form of the Uncertainty Relations 87 7.2.3 Interpretations 88 7.2.4 Reflections 91

7.3 Problems 93

8 Angular Momentum 95 8.1 Angular Momentum in Classical Mechanics 95 8.2 Basics of Quantum Angular Momentum 97

8.2.1 Operators and Commutation Relations 97

Contents vii

8.2.2 Eigenstates and Eigenvalues 99 8.2.3 Raising and Lowering Operators 100

8.3 Physical Interpretation 101 8.3.1 Measurements 101 8.3.2 Relating L2 and Lz 104

8.4 Orbital and Spin Angular Momentum 106 8.4.1 Orbital Angular Momentum 106 8.4.2 Spin Angular Momentum 107

8.5 Review 107 8.6 Problems 108

9 The Time-Independent Schrödinger Equation 111 9.1 An Eigenvalue Equation for Energy 112 9.2 Using the Schrödinger Equation 114

9.2.1 Conditions on Wavefunctions 114 9.2.2 An Example: the Infinite Potential Well 115

9.3 Interpretation 117 9.3.1 Energy Eigenstates in Position Space 117 9.3.2 Overall and Relative Phases 118

9.4 Potential Barriers and Tunneling 120 9.4.1 The Step Potential 120 9.4.2 The Step Potential and Scattering 122 9.4.3 Tunneling 124

9.5 What’s Wrong with This Picture? 125 9.6 Problems 126

10 Why Is the State Complex? 128 10.1 Complex Numbers 129

10.1.1 Basics 129 10.1.2 Polar Form 130 10.1.3 Argand Diagrams and the Role of the Phase 131

10.2 The Phase in Quantum Mechanics 133 10.2.1 Phases and the Description of States 133 10.2.2 Phase Changes and Probabilities 135 10.2.3 Unitary Operators Revisited 136 10.2.4 Unitary Operators, Phases, and Probabilities 137 10.2.5 Example: A Spin 12 System 139

10.3 Wavefunctions 141 10.4 Reflections 142 10.5 Problems 143

11 Time Evolution 145 11.1 The Time-Dependent Schrödinger Equation 145 11.2 How Time Evolution Works 146

11.2.1 Time Evolving a Quantum State 146 11.2.2 Unitarity and Phases Revisited 148

viii Contents

11.3 Expectation Values 149 11.3.1 Time Derivatives 149 11.3.2 Constants of the Motion 150

11.4 Energy-Time Uncertainty Relations 151 11.4.1 Conceptual Basis 151 11.4.2 Spin 12 : An Example 153

11.5 Problems 154

12 Wavefunctions 157 12.1 What is a Wavefunction? 158

12.1.1 Eigenstates and Coefficients 158 12.1.2 Representations and Operators 159

12.2 Changing Representations 161 12.2.1 Change of Basis Revisited 161 12.2.2 From x to p and Back Again 161 12.2.3 Gaussians and Beyond 163

12.3 Phases and Time Evolution 165 12.3.1 Free Particle Evolution 165 12.3.2 Wavepackets 167

12.4 Bra-ket Notation 168 12.4.1 Quantum States 168 12.4.2 Eigenstates and Transformations 170

12.5 Epilogue 171 12.6 Problems 172

A Mathematical Concepts 175 A.1 Complex Numbers and Functions 175 A.2 Differentiation 176 A.3 Integration 178 A.4 Differential Equations 180

B Quantum Measurement 183

C The Harmonic Oscillator 186 C.1 Energy Eigenstates and Eigenvalues 186 C.2 The Number Operator and its Cousins 188 C.3 Photons as Oscillators 189

D Unitary Transformations 192 D.1 Unitary Operators 192 D.2 Finite Transformations and Generators 195 D.3 Continuous Symmetries 197

D.3.1 Symmetry Transformations 197 D.3.2 Symmetries of Physical Law 197 D.3.3 System Symmetries 199

Bibliography 201

Index 205

Preface

While still a relatively new graduate student, I once remarked to my advi- sor, Jim Cushing, that I still didn’t understand quantum mechanics. To this he promptly replied: “You’ll spend the rest of your life trying to understand quantum mechanics!” Despite countless books that the subject has spawned since it first assumed a coherent form in the 1920s, quantum mechanics remains notoriously, even legendarily, difficult. Some may believe students should be told that physics really isn’t that hard, presumably so as not to intimidate them. I disagree: what can be more demoralizing than struggling mightily with a subject, only to be told that it’s really not that difficult?

Let me say it outright, then: quantum mechanics is hard. In writing this book, I have not found any “magic bullet” by which I can render the subject easily digestible. I have, however, tried to write a book that is neither a popularization nor a “standard” text; a book that takes a modern approach, rather than one grounded in pedagogical precedent; a book that focuses on elucidating the structure and meaning of quantum mechanics, leaving comprehensive treatments to the standard texts.

Above all, I have tried to write with the student in mind. The pri- mary target audience is undergraduates about to take, or taking, their first quantum course. But my hope is that the book will also serve biologists, philosophers, engineers, and other thoughtful people—people who are fasci- nated by quantum physics, but find the popularizations too simplistic, and the textbooks too advanced and comprehensive—by providing a foothold on “real” quantum mechanics, as used by working scientists.

Popularizations of quantum mechanics are intended not to expound the subject as used by working scientists, but rather to discuss “quantum weirdness,” such as Bell’s theorem and