Essential Quantum Mechanics

Gary Bowman9780199228928, 9781435631144, 0199228922

Quantum Mechanics- central not only to physics but also chemistry, materials science and other fields- is notoriously abstract and difficult. Essential Quantum Mechanics is a uniquely concise and explanatory book that fills the gap between introductory and advanced courses, between popularizations and technical treatises. By focusing on the fundamental structure, concepts and methods of quantum mechanics this introductory yet sophisticated work emphasizes both physical and matehmeatical understanding. A modern perspective is adopted throughout-the goal in part, being to gain entry into the world of “real” quantum mechanics, as used by practicing scientists. With over 60 original problems, Essential Quantum Mechanics is suitable as either a text or a reference. It will be invaluable to physics students as well as chemists, electrical engineers, philosophers, and others whose work is impacted by quantum mechanics, or who simply wish to better understand this fascinating subject.

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Table of contents :
Contents……Page 6
Preface……Page 10
1.1 Worlds 1 and 2……Page 14
1.2 World 3……Page 16
1.3 Problems……Page 17
2 The Quantum Postulates……Page 18
2.1 Postulate 1: The Quantum State……Page 19
2.2 Postulate 2: Observables, Operators, and Eigenstates……Page 21
2.3 Postulate 3: Quantum Superpositions……Page 23
2.3.1 Discrete Eigenvalues……Page 24
2.3.2 Continuous Eigenvalues……Page 25
2.4 Closing Comments……Page 28
2.5 Problems……Page 29
3.1.1 Probabilities……Page 32
3.1.2 Averages……Page 35
3.1.3 Uncertainties……Page 37
3.2 The Statistical Interpretation……Page 39
3.3.1 Background……Page 41
3.3.2 Fundamental Issues……Page 43
3.3.3 Einstein Revisited……Page 45
3.4 Problems……Page 46
4.1.1 Vector Spaces……Page 49
4.1.2 Function Spaces……Page 52
4.2.1 Bras and Kets……Page 54
4.2.2 Labeling States……Page 55
4.3.1 Quantum Scalar Products……Page 56
4.3.2 Discussion……Page 58
4.4.1 Basics……Page 60
4.4.2 Superpositions and Representations……Page 61
4.4.3 Representational Freedom……Page 63
4.5 Problems……Page 65
5 Operators……Page 66
5.1 Introductory Comments……Page 67
5.2.1 Adjoint Operators……Page 69
5.2.2 Hermitian Operators: Definition and Properties……Page 70
5.2.3 Wavefunctions and Hermitian Operators……Page 72
5.3.1 Projection Operators……Page 74
5.4 Unitary Operators……Page 75
5.5 Problems……Page 77
6.1.1 Vectors and Scalar Products……Page 81
6.1.2 Matrices and Matrix Multiplication……Page 82
6.1.3 Vector Transformations……Page 83
6.2 States as Vectors……Page 84
6.3.1 An Operator in Its Eigenbasis……Page 85
6.3.2 Matrix Elements and Alternative Bases……Page 86
6.3.4 Adjoint, Hermitian, and Unitary Operators……Page 88
6.4 Eigenvalue Equations……Page 90
6.5 Problems……Page 91
7 Commutators and Uncertainty Relations……Page 95
7.1.1 Definition and Characteristics……Page 96
7.1.2 Commutators in Matrix Mechanics……Page 98
7.2.1 Uncertainty Products……Page 99
7.2.2 General Form of the Uncertainty Relations……Page 100
7.2.3 Interpretations……Page 101
7.2.4 Reflections……Page 104
7.3 Problems……Page 106
8.1 Angular Momentum in Classical Mechanics……Page 108
8.2.1 Operators and Commutation Relations……Page 110
8.2.2 Eigenstates and Eigenvalues……Page 112
8.2.3 Raising and Lowering Operators……Page 113
8.3.1 Measurements……Page 114
8.3.2 Relating L[sup(2)] and L[sub(z)]……Page 117
8.4.1 Orbital Angular Momentum……Page 119
8.5 Review……Page 120
8.6 Problems……Page 121
9 The Time-Independent Schrödinger Equation……Page 124
9.1 An Eigenvalue Equation for Energy……Page 125
9.2.1 Conditions on Wavefunctions……Page 127
9.2.2 An Example: the Infinite Potential Well……Page 128
9.3.1 Energy Eigenstates in Position Space……Page 130
9.3.2 Overall and Relative Phases……Page 131
9.4.1 The Step Potential……Page 133
9.4.2 The Step Potential and Scattering……Page 135
9.4.3 Tunneling……Page 137
9.5 What’s Wrong with This Picture?……Page 138
9.6 Problems……Page 139
10 Why Is the State Complex?……Page 141
10.1.1 Basics……Page 142
10.1.2 Polar Form……Page 143
10.1.3 Argand Diagrams and the Role of the Phase……Page 144
10.2.1 Phases and the Description of States……Page 146
10.2.2 Phase Changes and Probabilities……Page 148
10.2.3 Unitary Operators Revisited……Page 149
10.2.4 Unitary Operators, Phases, and Probabilities……Page 150
10.2.5 Example: A Spin ½ System……Page 152
10.3 Wavefunctions……Page 154
10.4 Reflections……Page 155
10.5 Problems……Page 156
11.1 The Time-Dependent Schrödinger Equation……Page 158
11.2.1 Time Evolving a Quantum State……Page 159
11.2.2 Unitarity and Phases Revisited……Page 161
11.3.1 Time Derivatives……Page 162
11.3.2 Constants of the Motion……Page 163
11.4.1 Conceptual Basis……Page 164
11.4.2 Spin ½: An Example……Page 166
11.5 Problems……Page 167
12 Wavefunctions……Page 170
12.1.1 Eigenstates and Coefficients……Page 171
12.1.2 Representations and Operators……Page 172
12.2.2 From x to p and Back Again……Page 174
12.2.3 Gaussians and Beyond……Page 176
12.3.1 Free Particle Evolution……Page 178
12.3.2 Wavepackets……Page 180
12.4.1 Quantum States……Page 181
12.4.2 Eigenstates and Transformations……Page 183
12.5 Epilogue……Page 184
12.6 Problems……Page 185
A.1 Complex Numbers and Functions……Page 188
A.2 Differentiation……Page 189
A.3 Integration……Page 191
A.4 Differential Equations……Page 193
B: Quantum Measurement……Page 196
C.1 Energy Eigenstates and Eigenvalues……Page 199
C.2 The Number Operator and its Cousins……Page 201
C.3 Photons as Oscillators……Page 202
D.1 Unitary Operators……Page 205
D.2 Finite Transformations and Generators……Page 208
D.3.2 Symmetries of Physical Law……Page 210
D.3.3 System Symmetries……Page 212
Bibliography……Page 214
D……Page 218
M……Page 219
S……Page 220
Z……Page 221