The Theoretical Foundations of Quantum Mechanics

Cover......Page 1
Title Page......Page 5
Preface......Page 7
Acknowledgments......Page 9
Contents......Page 11
1 Synopsis......Page 15
2.1 What Is a Classical Entity?......Page 19
2.2 The Entity in Quantum Mechanics......Page 22
2.3 Describing an Indeterminate Quantum Entity......Page 24
2.4 The Copenhagen Quantum Postulate......Page 26
2.5 Five Pillars of Quantum Mechanics......Page 28
2.7 State Space V(F)......Page 29
2.8 Operators O(F)......Page 30
2.9 The Schrödinger Equation for State ψ(t,F)......Page 31
2.10 Indeterminate Quantum Paths......Page 32
2.11 The Process of Measurement......Page 34
2.12 Summary: Quantum Entity......Page 36
3 Quantum Mechanics: Empirical and Trans-empirical......Page 39
3.1 Real Versus Exist......Page 40
3.2 Empirical, Trans-empirical, and Indeterminate......Page 41
3.3 Quantum Mechanics and the Trans-empirical......Page 43
3.4 Quantum Degree of Freedom F Is Trans-empirical......Page 45
3.5 The Quantum State ψ: Transition......Page 47
3.6 Trans-empirical Domain and Laws of Physics......Page 49
3.7 Quantum Superposition: Trans-empirical Paths......Page 50
3.8 Trans-empirical Interpretation of Two-Slit Experiment......Page 56
3.9 The Trans-empirical Quantum Principle......Page 57
3.10 Does the Quantum State ψ(t,F) “Exist”?......Page 58
3.11 Summary......Page 60
4 Degree of Freedom F; State Space V......Page 63
4.2 State Space and Experiment......Page 64
4.3 Quantum Degree of Freedom F......Page 66
4.4 Binary Degree of Freedom and State Space......Page 67
4.5 Degree of Freedom F_{(2N+1)}: State Space V_{(2N+1)}......Page 71
4.6 Continuous Degree of Freedom......Page 73
4.7 Basis States for State Space......Page 76
4.8 Unitary Transformation: Momentum Basis......Page 77
4.9 State Space V......Page 80
4.10 Hilbert Space......Page 82
4.11 Summary......Page 83
5 Operators......Page 85
5.1 Operators: Trans-empirical to Empirical......Page 86
5.2 Hermitian Operators......Page 87
5.3 Eigenstates: Projection Operators......Page 89
5.4 Operators and Quantum Numbers......Page 92
5.5 Periodic Degree of Freedom......Page 94
5.6 Position and Momentum Operators x̂ and p̂......Page 96
5.7 Heisenberg Commutation Equation......Page 99
5.8 Expectation Value of Operators......Page 101
5.9 The Schrödinger Equation......Page 102
5.10 Heisenberg Operator Formulation......Page 103
5.11 Summary......Page 104
6 Density Matrix: Entangled States......Page 107
6.1 Tensor Product......Page 108
6.2 The Outer Product......Page 109
6.3 Partial Trace for Outer Products......Page 111
6.4 Density Matrix ρ......Page 112
6.5 The Schmidt Decomposition......Page 114
6.6 Reduced Density Matrix......Page 116
6.7 Separable Quantum Systems......Page 118
6.8 Entangled Quantum States......Page 119
6.9 A Pair of Entangled Spins......Page 121
6.10 Quantum Entropys......Page 122
6.11 Pure and Mixed Density Matrix......Page 125
6.12 Summary......Page 127
7 Quantum Indeterminacy......Page 129
7.1 The EPR Paradox......Page 130
7.2 The Bell-CHSH Operator......Page 133
7.3 Classical Probability: Objective Reality......Page 135
7.4 The Bell Inequality......Page 137
7.5 The Bell Inequality Non-violation......Page 139
7.6 Bell Inequality Violation: Entangled States......Page 142
7.7 The Bell–Kochen–Specker Inequality......Page 145
7.8 Commuting and Non-commuting Operators......Page 149
7.9 Quantum Probability......Page 150
7.10 A Metaphor......Page 155
7.11 Summary......Page 156
8 Quantum Superposition......Page 159
8.1 Superposing State Vectors......Page 160
8.2 Probability and Probability Amplitudes......Page 162
8.3 Empirical and Trans-Empirical Paths......Page 164
8.4 Successive Slits......Page 167
8.5 The Mach–Zehnder Interferometer......Page 168
8.6 Determinate Empirical Paths: No Interference......Page 170
8.7 Indeterminate Trans-Empirical Paths: Interference......Page 171
8.8 Quantum Eraser......Page 173
8.9 Erasing Interference......Page 174
8.10 Restoring Interference......Page 176
8.11 Partial Quantum Eraser......Page 179
8.12 Summary......Page 183
9 Quantum Theory of Measurement......Page 185
9.1 Measurement: Trans-Empirical to Empirical......Page 187
9.2 Position Projection Operator......Page 188
9.3 Repeated Observations in Quantum Mechanics......Page 190
9.4 Expectation Value of Projection Operators......Page 191
9.5 The Experimental Device......Page 195
9.6 The Process of Measurement......Page 197
9.7 Mixed Density Matrix ρ_M......Page 199
9.8 Reduced Density Matrix ρ_R......Page 201
9.9 Preparation of a Quantum State......Page 206
9.10 The Heisenberg Uncertainty Principle......Page 209
9.11 Theories of Quantum Measurement......Page 215
9.12 Summary......Page 216
10.1 The Experiment......Page 219
10.2 Classical and Quantum Predictions......Page 221
10.3 The Stern–Gerlach Hamiltonian......Page 222
10.4 Electron's Time Evolution......Page 225
10.5 Entanglement of Spin and Device......Page 228
10.6 Summary of Spin Measurement......Page 229
10.7 Irreversibility and Collapse of State Vector......Page 231
10.8 Interpretation of Spin Measurement......Page 232
10.9 Summary......Page 233
11.1 Probability Amplitude and Time Evolution......Page 235
11.2 Evolution Kernel......Page 237
11.3 Superposition of Trans-Empirical Paths......Page 240
11.4 The Dirac–Feynman Formula......Page 241
11.5 The Lagrangian......Page 243
11.6 The Feynman Path Integral......Page 245
11.7 Path Integral for Evolution Kernel......Page 248
11.8 Composition Rule for Probability Amplitudes......Page 253
11.9 Trans-Empirical Paths and Path Integral......Page 255
11.10 State Vector and Trans-Empirical Paths......Page 257
11.11 Path Integral Quantization: Action......Page 258
11.12 Hamiltonian from Lagrangian......Page 259
11.13 Summary......Page 262
12 Conclusions......Page 265
12.1 Three Formulations of Quantum Mechanics......Page 266
12.2 Interpretations of Quantum Mechanics......Page 267
Glossary of Terms......Page 271
List of Symbols......Page 273
References......Page 277
Index......Page 279