Time in Quantum Mechanics

Lecture Notes in Physics: Monographs 72......Page 1
Time in Quantum Mechanics......Page 3
Preface......Page 5
List of Contributors......Page 11
Contents......Page 9
1.1 Role of Time in the Early Days of Quantum Theory......Page 13
1.1.1 Born’s Collision Papers......Page 14
1.1.2 Pauli’s First Encyclopedia Article......Page 15
1.1.3 Uncertainty Relations......Page 16
1.2.1 Pauli’s “Theorem”......Page 17
1.2.2 Von Neumann......Page 18
1.3.1 Bohm’s Book on Quantum Theory......Page 19
1.3.3 Are There Quantum Jumps?......Page 21
1.3.5 Quantum Clocks and Their Limitations......Page 22
1.4.2 Aharonov-Bohm, 1961......Page 23
1.4.3 Allcock, 1963......Page 24
1.4.4 Aharonov-Bergmann-Lebowitz, 1964......Page 26
1.5.1 Kijowski’s Time-of-Arrival Distribution......Page 27
1.5.3 Times of Occurrence in Quantum Mechanics and POVMs......Page 28
1.6.1 Theories That Include Events in Quantum Mechanics......Page 29
1.6.2 The Resurrection of Quantum Jumps......Page 30
1.6.3 Tunneling Times......Page 31
1.7 Discussion......Page 35
References......Page 36
2.1 Introduction......Page 41
2.2.2 Basic Abstract and Parameterized Operators......Page 43
2.2.4 Eigenstates of H^......Page 47
2.2.5 Relation Between Scattering Amplitudes and Basic Operators......Page 48
2.2.6 The Diagonal S_d Matrix......Page 50
2.2.7 Complex Momentum......Page 51
2.3 A Measure of the Collision Duration: The Dwell Time......Page 52
2.4 Importance of the Phases. Time Delays......Page 56
2.4.1 The Hartman Effect......Page 58
2.4.2 The Lifetime and Delay Time Matrices......Page 60
2.4.3 Breit–Wigner Resonances......Page 62
2.4.4 Negative Delays......Page 63
2.5 Time Dependence of Survival Probability: Exponential Decay and Deviations......Page 65
2.5.1 Predicted Time Behaviour......Page 66
2.5.2 Short Time Behaviour......Page 67
2.5.3 Large Time Behaviour......Page 69
2.6 Other Characteristic Times of Wave Propagation......Page 71
2.6.1 Role of the Traversal Time for a Source with a Sharp Onset......Page 74
Appendix: Properties of w-Functions......Page 76
References......Page 77
3.2 The Threefold Role of Time in Quantum Theory......Page 81
3.2.2 Intrinsic Time......Page 82
3.2.3 Observable Time......Page 83
3.3 Relation Between External Time and Energy Spread......Page 84
3.3.1 Aharonov–Bohm Energy Measurement Model......Page 85
3.3.2 Relation Between Preparation Time and Energy......Page 89
3.4.1 Mandelstam–Tamm Relation......Page 90
3.4.2 Lifetime of a Property......Page 91
3.4.3 Bohr–Wigner Uncertainty Relation......Page 92
3.4.4 Further Relations Involving Intrinsic Time......Page 94
3.5 Quantum Clock......Page 96
3.6 Relations Based on Time Observables......Page 98
3.6.1 Event Time Observables......Page 99
3.6.2 Einstein’s Photon Box......Page 104
3.6.3 Temporal Interference and Time Indeterminacy......Page 105
3.7 Conclusion......Page 108
References......Page 109
4.1 Introduction......Page 111
4.2 Jump Time......Page 114
4.3.1 Comparison with Tunneling Time......Page 115
4.3.2 Time-Energy Uncertainty Principle......Page 116
4.3.3 Reconciling Continuous Measurement with the QZE......Page 117
4.3.4 Experiments on the Quadratic Regime of Decay......Page 119
4.4.1 Fleming’s Bound and the Ersak Equation......Page 121
4.4.2 Implications of the Bound in Measurements......Page 122
4.5.2 Special States for Decay......Page 124
4.6 Discussion......Page 127
Acknowledgments......Page 129
References......Page 130
5.1 Introduction......Page 133
5.2.1 Brief Introduction......Page 134
5.2.2 Derivation of a General 1D Arrival-Time Distribution......Page 137
5.2.3 Transmission and Reflection Times......Page 139
5.3.1 Can a Freely-Evolving Quantum Particle Turn Around?......Page 142
5.3.3 The Systematic Projector Approach of Brouard, Sala and Muga......Page 144
5.3.4 Approaches Based on the Probability Current......Page 147
5.3.5 The Quantile Approach......Page 148
5.4 Spin-Dependent Arrival-Time Distributions for Nonrelativistic Electrons......Page 149
5.5 Protective Measurements and Bohm Trajectories......Page 152
5.6 Concluding Comments......Page 158
Appendix......Page 160
References......Page 162
6.1 Introduction......Page 165
6.2 Decoherent Histories Approach to Quantum Theory......Page 169
6.3 Space-Time Coarse Grainings......Page 172
6.4 Decoherence of Space-Time Coarse-Grained Histories in the Quantum Brownian Motion Model......Page 174
6.4.1 The Crossing Time Problem in Classical Brownian Motion......Page 175
6.4.2 The Crossing Time Problem in Quantum Brownian Motion......Page 178
6.4.3 Properties of the Solution......Page 180
6.5 A Detector Model......Page 181
6.6 A Comparison of the Decoherent Histories Result with the Detector Result......Page 184
6.7 Timeless Questions in Quantum Theory......Page 187
6.8 Discussion......Page 189
References......Page 191
7.1 Introduction......Page 195
7.2 Path Decomposition. The Clocked Schrödinger Equation. Coarse Graining......Page 196
7.3 Meters and Measurements. Uncertainty Relation......Page 200
7.4 Averages. Complex Times. Weak Measurements......Page 202
7.5 Examples: Free Motion and Tunneling......Page 204
7.6 Semiclassical Limit. How Long Does It Take for a Particle to Tunnel?......Page 207
7.7 Larmor Clock as a Realistic Meter......Page 208
7.8 Traversal Time Analysis......Page 214
7.9 Traversal Time and the ‘Superluminal’ Tunneling......Page 217
7.10 Relativistic Traversal Time......Page 222
7.11 Concluding Remarks......Page 225
References......Page 226
8.1 Introduction......Page 229
8.2 What Is a Clock?......Page 231
8.3.1 Mathematical Description of a Quantum Clock......Page 232
8.3.2 The Quantum Clock Analysis of Peres......Page 234
8.3.3 Time-of-Flight Measurements......Page 235
8.3.4 Other Examples......Page 239
8.3.5 Simultaneous Measurement of Time and Energy......Page 241
8.4 The Larmor Clock......Page 242
8.4.1 Büttiker’s Analysis of the Larmor Clock......Page 243
8.4.2 Generalizations and Relations to Other Approaches......Page 246
8.4.3 Relation to Phase Times......Page 249
8.5 Other Clocks......Page 250
8.5.1 The Faraday Clock......Page 251
8.5.2 The Micromaser (Rabi) Clock......Page 252
8.5.3 Irreversible Stopwatch Models......Page 254
8.6 Simple “Time-Dependent” Clocks: The Kick-Clock......Page 257
8.7 Decoherence in Time......Page 260
References......Page 264
9.1 Introduction......Page 268
9.2 The Scattering Matrix and the Local Larmor Clock......Page 270
9.3 Absorption and Emission of Particles: Injectivities and Emissivities......Page 275
9.4 Potential Perturbations......Page 278
9.5 Generalized Bardeen Formulae......Page 279
9.6 Voltage Probe and Inelastic Scattering......Page 280
9.7 AC Conductance of Mesoscopic Conductors......Page 282
9.8 Transition from Capacitive to Inductive Response......Page 283
9.9 Partial Density of States Matrix......Page 285
9.10 Discussion......Page 288
References......Page 289
10.1 Introduction......Page 291
10.2 Kijowski’s Time-of-Arrival Distribution......Page 292
10.3 POVMs......Page 293
10.3.1 Maximally Symmetric Operators: Momentum on the Half-Line......Page 296
10.3.2 Naimark’s Dilation Theorem......Page 299
10.4 The POVM of the Aharonov-Bohm Time Operator......Page 301
10.5.1 Periodic Systems......Page 306
10.5.2 Grot, Rovelli and Tate......Page 308
10.6 Arrival States......Page 309
10.7 Times of Arrival for Identical Particles......Page 312
10.8 Conclusions and Outlook......Page 314
References......Page 315
11.1 Time Operators Versus Real Measurements......Page 317
11.2.1 Techniques for Measuring Arrival Time......Page 319
11.2.2 Superluminal Arrival Times......Page 322
11.3.1 Theoretical Proposals......Page 325
11.3.2 Some Experimental Examples......Page 329
11.4.1 Interpretational Aspects......Page 331
11.4.2 Potential Experimental Tests......Page 333
11.5 Conclusion......Page 334
References......Page 335
12.1 An Overview of Theoretical Models of Tunneling in the Electromagnetic Framework......Page 338
12.1.1 Frustrated Total Re.ection: An Electromagnetic Model......Page 347
12.1.2 More About the Stochastic Model......Page 349
12.1.3 The Procedure of Stevens and the Role of the Forerunner......Page 353
12.2.1 Delay Time Through a Rectangular Potential Barrier......Page 357
12.2.2 Penetration Time Inside a Barrier of Infinite Length......Page 364
12.3 Delay Time Measurements in a Diffraction Experiment......Page 366
12.4 Tunneling Time in Frustrated Total Re.ection......Page 372
12.5 Concluding Remarks......Page 377
Appendix: Complex Time Analysis......Page 378
References......Page 379
13.1.1 The Quantum State......Page 381
13.1.2 The Two-State Vector......Page 382
Single Forward-Evolving Quantum State......Page 383
The Two-State Vector......Page 384
A Single Backward-Evolving Quantum State......Page 385
13.1.4 The Generalized Two-State Vector......Page 386
13.2.2 The Aharonov-Bergmann-Lebowitz Rule......Page 387
13.2.3 Three-Boxes Example......Page 388
13.2.4 The Failure of the Product Rule......Page 389
13.2.5 Ideal Measurements Performed on a System Described by Generalized Two-State Vector......Page 390
13.3.1 Introduction......Page 391
13.3.2 Examples: Measurements of Spin Components......Page 394
13.3.3 Weak Measurements Which Are Not Really Weak......Page 397
13.3.4 Relations Between Weak and Strong Measurements......Page 402
13.3.5 Experimental Realizations of Weak Measurements......Page 404
13.4.1 Introduction......Page 405
13.4.2 A Peculiar Mathematical Identity......Page 406
13.4.4 Quantum Gravitational Time Machine......Page 408
13.4.5 The Probability of the Success of the Quantum Time Machine......Page 411
13.4.6 Time Translation to the Past and to the Future......Page 412
13.4.7 Experimental Realization of the Quantum Time-Translation Machine?!......Page 413
13.5.1 Time-Symmetric Aspects......Page 414
13.5.2 The Time-Asymmetry......Page 415
13.5.3 If Measurements Are Time-Asymmetric, How Is It That the Outcomes of Measurements Are Time-Symmetric?......Page 416
13.5.4 Counterfactual Interpretation of the ABL Rule......Page 417
13.6.1 Protective Measurement of a Single Quantum State......Page 419
13.6.2 Protective Measurement of a Two-State Vector......Page 420
References......Page 422
Index......Page 425