Quantum Theory of Motion, The: An account of the De Broglie-Bohm causal interpretation of quantum mechanics

Cover......Page cover.djvu
Abstract......Page ad
Title Page......Page v
Contents......Page ix
Preface......Page xvii
1.1 The nature of the problem......Page p1
1.2 The wavefunction and the Schrödinger equation......Page p2
1.3 The completeness assumption......Page p7
1.4 Einstein\'s point of view......Page p11
1.5.1 De Broglie and Bohm......Page p15
1.5.2 What the great men said......Page p20
1.5.3 Some objections......Page p24
2.1 The need for a common language......Page p27
2.2.1 Hamilton\'s principal function......Page p29
2.2.2 The action function......Page p33
2.2.3 A single particle......Page p35
2.3.1 The nonuniqueness of S for a given mechanical problem......Page p36
2.3.2 The basic law of motion......Page p38
2.3.3 Multivalued trajectory fields......Page p40
2.4 The propagation of the S-function......Page p41
2.5.1 Conservation of probability......Page p45
2.5.2 Connection with Liouville\'s equation......Page p49
2.5.3 Pure and mixed states......Page p52
2.6.1 The wave equation of classical mechanics......Page p55
2.6.2 The potential step......Page p58
2.7 Generalization — internal potentials......Page p61
3.1 The basic postulates......Page p66
3.2.1 Reformulation of the Schrödinger equation......Page p68
3.2.2 Single-valuedness of the wavefunction......Page p70
3.2.3 Introduction of the particle......Page p72
3.2.4 What about the commutation relations?......Page p77
3.3.1 Context dependence......Page p78
3.3.2 Relative status of field and particle......Page p79
3.3.4 Effect of external potential on field and particle......Page p80
3.3.5 Factorizable and nonfactorizable wavefunctions......Page p82
3.3.6 Comparison with other field theories......Page p83
3.3.7 Are there quantum jumps?......Page p84
3.3.8 Trajectories do not cross, or pass through nodes......Page p85
3.3.10 Conditions for interference......Page p86
3.3.11 Generalization of the de Broglie relations......Page p87
3.4.2 Comparison with classical potentials......Page p89
3.4.4 Radar wave analogy......Page p90
3.5 The relation between particle properties and quantum mechanical operators......Page p91
3.6.1 Ensemble of identical systems......Page p95
3.6.2 Ensemble of particles......Page p97
3.6.3 A special assumption......Page p99
3.6.4 Ensemble of waves. The density matrix......Page p102
3.7.1 Eigenvalues and probabilities......Page p104
3.7.2 What are the hidden variables?......Page p106
3.8.1 Averages as individual properties......Page p108
3.8.2 Ensemble averages and quantum mechanical expectation values......Page p109
3.8.3 Ehrenfest\'s theorem......Page p111
3.9.1 Canonical formalism for the Schrödinger field......Page p113
3.9.2 Field conservation laws......Page p114
3.9.3 Particle conservation laws......Page p117
3.10 Hydrodynamic analogy......Page p120
3.11.1 Galilean invariance......Page p122
3.11.2 Gauge invariance. Introduction of the electromagnetic field......Page p124
3.12.1 The position representation and the Schrödinger picture......Page p127
3.12.2 The method of the causal interpretation......Page p129
3.13 Comparison with classical Hamilton-Jacobi theory......Page p131
Appendix B......Page p134
4.1.1 General properties......Page p136
4.1.2 Is the ground energy the lowest possible?......Page p138
4.2 Plane and spherical waves......Page p139
4.3 Superposition of plane waves......Page p141
4.4 Interference and the potential step......Page p146
4.5.1 Rotating plane waves and electron trajectories......Page p148
4.5.2 The stability of matter. Relation with Bohr orbits......Page p153
4.6 Wave packets......Page p156
4.7.1 Properties of the Gaussian packet......Page p158
4.7.2 Particle motion......Page p161
4.8 Gaussian packet in a uniform field......Page p164
4.9 Harmonic oscillator......Page p165
4.10 Nonspreading free packet......Page p167
4.11.1 Edge dislocation......Page p169
4.11.2 Screw dislocation......Page p170
5.1.1 Preliminary remarks......Page p173
5.1.2 Particle trajectories in the electron interferometer......Page p176
5.1.3 More detailed predictions than are contained in the wavefunction......Page p183
5.1.4 Relative ‘wave and particle knowledge’......Page p186
5.1.5 Delayed-choice experiments......Page p189
5.2.1 Effect of vector potential on charged particle trajectories......Page p190
5.2.2 Connection between the vector and quantum potentials......Page p195
5.2.3 Locality and nonlocality......Page p197
5.3 Tunnelling through a square barrier......Page p198
5.4.1 The neutron interferometer......Page p203
5.4.2 Beam attenuation effects......Page p209
5.5.1 Time of transit — a testable prediction beyond quantum mechanics?......Page p211
5.5.2 Age......Page p215
6.1 Conceptual and formal problems......Page p218
6.2 A state-dependent criterion for the classical limit......Page p224
6.3 Identical quantum and classical motions......Page p229
6.4.1 The WKB approximation......Page p231
6.4.2 Semiclassical wavefunctions......Page p234
6.4.3 Is classical mechanics the short-wave limit of wave mechanics?......Page p238
6.5.1 Stationary states......Page p239
6.5.2 Classical ensembles......Page p240
6.5.3 Einstein\'s critique......Page p243
6.6.1 Harmonic oscillator......Page p247
6.6.2 Free Gaussian packet......Page p251
6.7.1 Operators and mean values......Page p253
6.7.2 Ehrenfest\'s theorem......Page p254
6.7.3 Heisenberg\'s relations......Page p256
6.8.1 The principle of equivalence in quantum mechanics......Page p259
6.8.2 Breakdown of weak equivalence (b) in the WKB limit......Page p262
6.9 Remarks on the path integral approach......Page p266
6.10.1 The principle of linear superposition......Page p270
6.10.2 Remarks on chaos......Page p274
7.1.1 Definition of individual system......Page p277
7.1.2 Equations of motion......Page p278
7.1.3 Novel properties of the many-body system......Page p281
7.1.4 Identical but distinguishable particles......Page p283
7.1.5 Conservation laws......Page p285
7.2.1 Factorization of the wavefunction......Page p287
7.2.2 Factorization in other coordinates......Page p290
7.2.3 Identical particles......Page p291
7.2.4 The separation of two particles......Page p293
7.2.5 Particle properties......Page p294
7.2.6 Classical limit......Page p295
7.3.1 General results......Page p296
7.3.2 Plane waves......Page p298
7.4.1 The three statistics......Page p300
7.4.2 General remarks. The exclusion principle......Page p309
7.5.1 The collision of an electron with a hydrogen atom......Page p310
7.5.2 Pauli\'s objection......Page p314
7.6 The formation of a molecule......Page p315
7.7 Other approaches to a causal interpretation......Page p319
Appendix: The connection between Q and stationary perturbation theory......Page p321
8.1 Measurement in classical physics......Page p324
8.2 The measurement problem in quantum theory......Page p328
8.3.1 General remarks......Page p336
8.3.2 Stage 1: state preparation......Page p339
8.3.3 Derivation of Born\'s statistical postulate......Page p344
8.3.4 Stage 2: registration......Page p348
8.4.1 Single and joint measurements......Page p351
8.4.2 Prediction and retrodiction......Page p355
8.4.3 The Wigner function......Page p357
8.5 Heisenberg\'s relations......Page p359
8.6 Time-of-flight measurements......Page p366
8.7 Measuring the actual momentum and the wavefunction......Page p369
8.8.1 Can we detect empty waves?......Page p371
8.8.2 Can we tell which path and observe interference?......Page p374
9.1 Introduction......Page p379
9.2.1 The triad implied by a spinor......Page p380
9.2.2 Eulerian representation......Page p384
9.3.1 Physical model......Page p387
9.3.2 Equations of motion......Page p390
9.3.3 Is there a classical analogue of spin?......Page p394
9.4.1 A quantum spin flipper......Page p395
9.4.2 Application to neutron interferometry......Page p399
9.5 The Stern-Gerlach experiment......Page p404
9.6 Extension to many bodies......Page p416
9.7 Minimalist approach and problems with the Pauli theory......Page p420
10.1 Classical dynamics of rigid bodies......Page p424
10.2 The quantum rigid rotator......Page p427
10.3.1 One body......Page p432
10.3.2 Many bodies......Page p438
10.4.1 Wave equation......Page p439
10.4.2 Particle equations......Page p441
10.5.1 Free rotator......Page p443
10.5.2 Effect of a uniform magnetic field......Page p447
10.6.1 Bosons......Page p449
10.6.2 Fermionic analogue of the oscillator picture of boson fields......Page p451
10.6.3 The classical limit of a quantized fermion field......Page p455
11.1.1 Is quantum mechanics complete and local…......Page p458
11.1.2 … or incomplete and nonlocal?......Page p461
11.2 Nonlocal correlations in a double Stern-Gerlach experiment performed on a singlet state......Page p465
11.3 The problem of signalling......Page p471
11.4 More general treatment: the rigid rotator......Page p477
11.5.1 Bell\'s inequality......Page p479
11.5.2 Further remarks......Page p481
11.6.1 Meaning of nonlocality......Page p483
11.6.2 Quantitative description of propagation......Page p487
11.6.3 The shutter problem......Page p490
Appendix......Page p495
12.1 Problems with the extension to relativity. The Klein-Gordon equation......Page p498
12.2 Causal interpretation of the Dirac equation......Page p503
12.3.1 Elementary solutions......Page p509
12.3.2 The Klein paradox......Page p511
12.3.3 Zitterbewegung......Page p515
12.4.1 Space representation......Page p519
12.4.2 Normal mode representation......Page p525
12.5.1 Vacuum state. The Casimir effect......Page p528
12.5.2 Excited states and nonlocality......Page p531
12.5.3 Coherent states and the classical limit......Page p534
12.6.1 Are there photon trajectories?......Page p538
12.6.2 Energy flow in classical optics......Page p542
12.6.3 Energy flow in quantum optics......Page p547
12.6.4 Examples of mean energy flow......Page p552
12.6.5 Remarks on the detection process......Page p556
12.7.1 Single source......Page p558
12.7.2 Independent sources......Page p564
12.8 Quantum potential as the origin of mass?......Page p566
12.9 Beyond space-time-matter. Wavefunction of the universe......Page p567
References......Page p572
Index......Page p585