ESSENTIAL QUANTUM MECHANICS

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Essential QuantumMechanics

GARY E. BOWMANDepartment of Physics and Astronomy

Northern Arizona University

1

3Great Clarendon Street, Oxford OX2 6DP

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1 3 5 7 9 10 8 6 4 2

Contents

Preface ix

1 Introduction: Three Worlds 11.1 Worlds 1 and 2 11.2 World 3 31.3 Problems 4

2 The Quantum Postulates 52.1 Postulate 1: The Quantum State 62.2 Postulate 2: Observables, Operators, and Eigenstates 82.3 Postulate 3: Quantum Superpositions 10

2.3.1 Discrete Eigenvalues 112.3.2 Continuous Eigenvalues 12

2.4 Closing Comments 152.5 Problems 16

3 What Is a Quantum State? 193.1 Probabilities, Averages, and Uncertainties 19

3.1.1 Probabilities 193.1.2 Averages 223.1.3 Uncertainties 24

3.2 The Statistical Interpretation 263.3 Bohr, Einstein, and Hidden Variables 28

3.3.1 Background 283.3.2 Fundamental Issues 303.3.3 Einstein Revisited 32

3.4 Problems 33

4 The Structure of Quantum States 364.1 Mathematical Preliminaries 36

4.1.1 Vector Spaces 364.1.2 Function Spaces 39

4.2 Diracs Bra-ket Notation 414.2.1 Bras and Kets 414.2.2 Labeling States 42

4.3 The Scalar Product 434.3.1 Quantum Scalar Products 434.3.2 Discussion 45

vi Contents

4.4 Representations 474.4.1 Basics 474.4.2 Superpositions and Representations 484.4.3 Representational Freedom 50

4.5 Problems 52

5 Operators 535.1 Introductory Comments 545.2 Hermitian Operators 56

5.2.1 Adjoint Operators 565.2.2 Hermitian Operators: Definition and Properties 575.2.3 Wavefunctions and Hermitian Operators 59

5.3 Projection and Identity Operators 615.3.1 Projection Operators 615.3.2 The Identity Operator 62

5.4 Unitary Operators 625.5 Problems 64

6 Matrix Mechanics 686.1 Elementary Matrix Operations 68

6.1.1 Vectors and Scalar Products 686.1.2 Matrices and Matrix Multiplication 696.1.3 Vector Transformations 70

6.2 States as Vectors 716.3 Operators as Matrices 72

6.3.1 An Operator in Its Eigenbasis 726.3.2 Matrix Elements and Alternative Bases 736.3.3 Change of Basis 756.3.4 Adjoint, Hermitian, and Unitary Operators 75

6.4 Eigenvalue Equations 776.5 Problems 78

7 Commutators and Uncertainty Relations 827.1 The Commutator 83

7.1.1 Definition and Characteristics 837.1.2 Commutators in Matrix Mechanics 85

7.2 The Uncertainty Relations 867.2.1 Uncertainty Products 867.2.2 General Form of the Uncertainty Relations 877.2.3 Interpretations 887.2.4 Reflections 91

7.3 Problems 93

8 Angular Momentum 958.1 Angular Momentum in Classical Mechanics 958.2 Basics of Quantum Angular Momentum 97

8.2.1 Operators and Commutation Relations 97

Contents vii

8.2.2 Eigenstates and Eigenvalues 998.2.3 Raising and Lowering Operators 100

8.3 Physical Interpretation 1018.3.1 Measurements 1018.3.2 Relating L2 and Lz 104

8.4 Orbital and Spin Angular Momentum 1068.4.1 Orbital Angular Momentum 1068.4.2 Spin Angular Momentum 107

8.5 Review 1078.6 Problems 108

9 The Time-Independent Schrodinger Equation 1119.1 An Eigenvalue Equation for Energy 1129.2 Using the Schrodinger Equation 114

9.2.1 Conditions on Wavefunctions 1149.2.2 An Example: the Infinite Potential Well 115

9.3 Interpretation 1179.3.1 Energy Eigenstates in Position Space 1179.3.2 Overall and Relative Phases 118

9.4 Potential Barriers and Tunneling 1209.4.1 The Step Potential 1209.4.2 The Step Potential and Scattering 1229.4.3 Tunneling 124

9.5 Whats Wrong with This Picture? 1259.6 Problems 126

10 Why Is the State Complex? 12810.1 Complex Numbers 129

10.1.1 Basics 12910.1.2 Polar Form 13010.1.3 Argand Diagrams and the Role of the Phase 131

10.2 The Phase in Quantum Mechanics 13310.2.1 Phases and the Description of States 13310.2.2 Phase Changes and Probabilities 13510.2.3 Unitary Operators Revisited 13610.2.4 Unitary Operators, Phases, and Probabilities 13710.2.5 Example: A Spin 12 System 139

10.3 Wavefunctions 14110.4 Reflections 14210.5 Problems 143

11 Time Evolution 14511.1 The Time-Dependent Schrodinger Equation 14511.2 How Time Evolution Works 146

11.2.1 Time Evolving a Quantum State 14611.2.2 Unitarity and Phases Revisited 148

viii Contents

11.3 Expectation Values 14911.3.1 Time Derivatives 14911.3.2 Constants of the Motion 150

11.4 Energy-Time Uncertainty Relations 15111.4.1 Conceptual Basis 15111.4.2 Spin 12 : An Example 153

11.5 Problems 154

12 Wavefunctions 15712.1 What is a Wavefunction? 158

12.1.1 Eigenstates and Coefficients 15812.1.2 Representations and Operators 159

12.2 Changing Representations 16112.2.1 Change of Basis Revisited 16112.2.2 From x to p and Back Again 16112.2.3 Gaussians and Beyond 163

12.3 Phases and Time Evolution 16512.3.1 Free Particle Evolution 16512.3.2 Wavepackets 167

12.4 Bra-ket Notation 16812.4.1 Quantum States 16812.4.2 Eigenstates and Transformations 170

12.5 Epilogue 17112.6 Problems 172

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

B Quantum Measurement 183

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

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

D.3.1 Symmetry Transformations 197D.3.2 Symmetries of Physical Law 197D.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 didnt understand quantum mechanics. To thishe promptly replied: Youll spend the rest of your life trying to understandquantum mechanics! Despite countless books that the subject has spawnedsince it first assumed a coherent form in the 1920s, quantum mechanicsremains notoriously, even legendarily, difficult. Some may believe studentsshould be told that physics really isnt that hard, presumably so as not tointimidate them. I disagree: what can be more demoralizing than strugglingmightily with a subject, only to be told that its really not that difficult?

Let me say it outright, then: quantum mechanics is hard. In writingthis book, I have not found any magic bullet by which I can renderthe subject easily digestible. I have, however, tried to write a book that isneither a popularization nor a standard text; a book that takes a modernapproach, rather than one grounded in pedagogical precedent; a book thatfocuses 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 firstquantum course. But my hope is that the book will also serve biologists,philosophers, engineers, and other thoughtful peoplepeople who are fasci-nated by quantum physics, but find the popularizations too simplistic, andthe textbooks too advanced and comprehensiveby providing a footholdon real quantum mechanics, as used by working scientists.

Popularizations of quantum mechanics are intended not to expoundthe subject as used by working scientists, but rather to discuss quantumweirdness, such as Bells theorem and the measurement problem, in termspalatable to interested non-scientists. As such, the mathematical level ofsuch books ranges from very low to essentially nonexistent.

In contrast, the comprehensive texts used in advanced courses oftenmake daunting conceptual and mathematical demands on the reader.Preparation for such courses typically consists of a modern physics course,but these tend to be rather conceptual. Modern physics texts generallytake a semi-historical approach, discussing topics such a