From Classical to Quantum Mechanics

Half-title......Page 2
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Contents......Page 8
Preface......Page 14
Acknowledgments......Page 17
Part I From classical to wave mechanics......Page 18
1.1 The need for a quantum theory......Page 20
1.2 Our path towards quantum theory......Page 23
1.3 Photoelectric effect......Page 24
1.4 Compton effect......Page 28
1.5 Interference experiments......Page 34
1.6 Atomic spectra and the Bohr hypotheses......Page 39
1.7 The experiment of Franck and Hertz......Page 43
1.8 Wave-like behaviour and the Bragg experiment......Page 44
1.9 The experiment of Davisson and Germer......Page 50
1.10 Position and velocity of an electron......Page 54
Appendix 1.A The phase 1-form......Page 58
2 Classical dynamics......Page 60
2.1 Poisson brackets......Page 61
2.2 Symplectic geometry......Page 62
2.3 Generating functions of canonical transformations......Page 66
2.3.1 Time-dependent Hamiltonian formalism......Page 67
2.3.2 Dynamical time......Page 70
2.3.3 Various generating functions......Page 71
2.3.4 An example: particle in a repulsive potential......Page 74
2.3.5 The harmonic oscillator......Page 75
2.4 Hamilton and Hamilton–Jacobi equations......Page 76
2.5.1 Free particle on a line......Page 78
2.5.2 One-dimensional harmonic oscillator......Page 79
2.5.3 Time-dependent Hamiltonian......Page 80
2.6 The characteristic function......Page 81
2.7 Hamilton equations associated with metric tensors......Page 83
2.8 Introduction to geometrical optics......Page 85
2.8.1 Variational principles......Page 89
2.9 Problems......Page 90
Appendix 2.A Vector fields......Page 91
Appendix 2.B Lie algebras and basic group theory......Page 93
Euclidean and rotation group......Page 94
Galilei group......Page 95
Lorentz and Poincaré groups......Page 96
Appendix 2.C Some basic geometrical operations......Page 97
Appendix 2.E From Newton to Euler–Lagrange......Page 100
3.1 The wave equation......Page 103
3.2 Cauchy problem for the wave equation......Page 105
3.3.1 Wave equations in spherical polar coordinates......Page 107
3.4 Symmetries of wave equations......Page 108
3.6 Fourier analysis and dispersion relations......Page 109
3.6.1 The symbol of difierential operators......Page 110
3.6.2 Dispersion relations......Page 111
3.7 Geometrical optics from the wave equation......Page 116
3.8 Phase and group velocity......Page 117
3.9 The Helmholtz equation......Page 121
3.10 Eikonal approximation for the scalar wave equation......Page 122
3.11 Problems......Page 131
4.1 From classical to wave mechanics......Page 132
4.1.1 Continuity equation......Page 134
4.1.2 Physical interpretation of the wave function......Page 137
4.1.3 Mean values......Page 142
4.2 Uncertainty relations for position and momentum......Page 145
4.2.1 Uncertainty relations in relativistic systems......Page 147
4.3 Transformation properties of wave functions......Page 148
4.3.1 Direct approach to the transformation properties of the Schrödinger equation......Page 150
4.3.2 Width of the wave packet......Page 151
4.4 Green kernel of the Schrödinger equation......Page 153
4.4.1 Free particle......Page 157
4.5 Example of isometric non-unitary operator......Page 159
4.6 Boundary conditions......Page 161
4.6.1 Particle confined by a potential......Page 165
4.6.2 Improper eigenfunctions......Page 167
4.7.1 One-dimensional oscillator......Page 168
4.7.2 Hermite polynomials......Page 170
4.8 JWKB solutions of the Schrödinger equation......Page 172
4.8.2 Example: alpha-decay......Page 177
4.9 From wave mechanics to Bohr–Sommerfeld......Page 179
4.9.1 Quantization of Keplerian motion......Page 180
4.9.2 Harmonic oscillator......Page 181
4.9.3 Rotator in a plane......Page 182
Appendix 4.A Glossary of functional analysis......Page 184
Appendix 4.B JWKB approximation......Page 189
Appendix 4.C Asymptotic expansions......Page 191
5.1 Reflection and transmission......Page 193
5.2 Step-like potential: tunnelling effect......Page 197
5.2.1 Step-like potential......Page 198
5.2.2 Tunnelling effect......Page 201
5.3 Linear potential......Page 203
5.4 The Schrödinger equation in a central potential......Page 208
5.5 Hydrogen atom......Page 213
5.5.1 A simpler derivation of the Balmer formula......Page 217
5.6 Introduction to angular momentum......Page 218
5.6.1 Lie algebra of O(3) and associated vector fields......Page 219
5.6.2 Quantum definition of angular momentum......Page 220
5.6.3 Harmonic polynomials and spherical harmonics......Page 222
5.6.4 Back to central potentials in R......Page 226
5.7 Homomorphism between SU(2) and SO(3)......Page 228
5.8 Energy bands with periodic potentials......Page 234
5.9 Problems......Page 237
Appendix 5.A Stationary phase method......Page 238
Appendix 5.B Bessel functions......Page 240
6.1 Stern–Gerlach experiment and electron spin......Page 243
6.2 Wave functions with spin......Page 247
6.2.1 Addition of orbital and spin angular momentum......Page 249
6.3 The Pauli equation......Page 250
6.4 Solutions of the Pauli equation......Page 252
6.5 Landau levels......Page 256
6.6 Problems......Page 258
Appendix 6.B Charged particle in a monopole field......Page 259
7.1 Approximate methods for stationary states......Page 261
7.1.1 Rayleigh–Schrödinger expansion......Page 264
7.1.3 Remark on quasi-stationary states......Page 266
7.2 Very close levels......Page 267
7.3 Anharmonic oscillator......Page 269
7.4 Occurrence of degeneracy......Page 272
7.5 Stark effect......Page 276
7.6 Zeeman effect......Page 280
7.7 Variational method......Page 283
7.8 Time-dependent formalism......Page 286
7.8.1 Harmonic perturbations......Page 289
7.9 Limiting cases of time-dependent theory......Page 291
7.9.1 Adiabatic switch on and off of the perturbation......Page 294
7.9.2 Perturbation suddenly switched on......Page 295
7.10.1 Regular perturbation theory......Page 297
7.10.2 Asymptotic perturbation theory......Page 298
7.10.3 Spectral concentration......Page 299
7.11 More about singular perturbations......Page 301
7.11.1 The Harrell method......Page 302
7.11.2 Extension to other singular potentials......Page 304
7.11.3 Concluding remarks......Page 308
7.12 Problems......Page 310
Appendix 7.A Convergence in the strong resolvent sense......Page 312
8.1 Aims and problems of scattering theory......Page 314
8.2 Integral equation for scattering problems......Page 319
8.3 The Born series and potentials of the Rollnik class......Page 322
8.4 Partial wave expansion......Page 324
8.5 The Levinson theorem......Page 327
8.6 Scattering from singular potentials......Page 331
8.7 Resonances......Page 334
8.8 Separable potential model......Page 337
8.9 Bound states in the completeness relationship......Page 340
8.10 Excitable potential model......Page 341
8.11 Unitarity of the Möller operator......Page 344
8.12.1 Law of radioactive decay: Poisson distribution......Page 345
8.12.2 Quantum decay transitions. The survival amplitude......Page 347
8.12.3 Decay amplitude under a Lorentz transformation......Page 350
8.12.4 Quantum mechanics in dual spaces......Page 351
8.13 Problems......Page 352
Part II Weyl quantization and algebraic methods......Page 354
9.1 The commutator in wave mechanics......Page 356
9.2 Abstract version of the commutator......Page 357
9.3 Canonical operators and the Wintner theorem......Page 358
9.4 Canonical quantization of commutation relations......Page 360
9.5.2 Unitary equivalence......Page 362
9.5.3 Weyl quantization......Page 363
9.6 The Schrödinger picture......Page 364
9.7 From Weyl systems to commutation relations......Page 365
9.8 Heisenberg representation for temporal evolution......Page 367
9.9 Generalized uncertainty relations......Page 368
9.9.1 Time–energy uncertainty relation......Page 371
9.10 Unitary operators and symplectic linear maps......Page 374
9.10.2 Rotations......Page 375
9.10.3 Harmonic oscillator......Page 378
9.11 On the meaning of Weyl quantization......Page 380
9.12 The basic postulates of quantum theory......Page 382
9.12.1 Rigged Hilbert spaces......Page 386
Position and momentum operators......Page 387
9.13 Problems......Page 389
10.1 Algebraic formalism for harmonic oscillators......Page 392
10.2 A thorough understanding of Landau levels......Page 400
10.3 Coherent states......Page 403
10.4 Weyl systems for coherent states......Page 407
10.5 Two-photon coherent states......Page 410
10.6 Problems......Page 412
11.1 Angular momentum: general formalism......Page 415
11.1.1 Algebraic method for the spectrum......Page 416
11.1.2 Representations......Page 420
11.1.3 Hilbert space......Page 422
11.2 Two-dimensional harmonic oscillator......Page 423
11.2.1 Introduction of different bases......Page 424
11.3 Rotations of angular momentum operators......Page 426
11.4 Clebsch–Gordan coefficients and the Regge map......Page 429
11.5 Postulates of quantum mechanics with spin......Page 433
11.6 Spin and Weyl systems......Page 436
11.7 Monopole harmonics......Page 437
11.8 Problems......Page 443
12.1 Quasi-exactly solvable operators......Page 446
12.2 Transformation operators for the hydrogen atom......Page 449
12.3 Darboux maps: general framework......Page 452
12.4 SU(1, 1) structures in a central potential......Page 455
12.5 The Runge–Lenz vector......Page 458
12.6 Problems......Page 460
13 From densitymatrix to geometrical phases......Page 462
13.1 The density matrix......Page 463
13.2 Applications of the density matrix......Page 467
13.3 Quantum entanglement......Page 470
13.4 Hidden variables and the Bell inequalities......Page 472
13.5 Entangled pairs of photons......Page 476
13.6 Production of statistical mixtures......Page 478
13.7 Pancharatnam and Berry phases......Page 481
13.7.1 More concerning non-integrable phases......Page 484
13.8 The Wigner theorem and symmetries......Page 485
13.9 A modern perspective on the Wigner theorem......Page 489
13.10 Problems......Page 493
Part III Selected topics......Page 494
14 From classical to quantum statistical mechanics......Page 496
14.1 Aims and main assumptions......Page 497
14.2 Canonical ensemble......Page 498
14.3 Microcanonical ensemble......Page 499
14.4 Partition function......Page 500
14.5 Equipartition of energy......Page 502
14.6 Specific heats of gases and solids......Page 503
14.7 Black-body radiation......Page 504
14.7.1 The Kirchhoff laws......Page 505
14.7.2 Stefan and displacement laws......Page 507
14.7.3 The Planck model......Page 511
14.7.4 The contributions of Einstein......Page 516
14.8 Quantum models of specific heats......Page 519
14.9 Identical particles in quantum mechanics......Page 521
14.9.1 An application: helium atom......Page 527
14.10 Bose–Einstein and Fermi–Dirac gases......Page 533
14.11 Statistical derivation of the Planck formula......Page 536
Appendix 14.A Towards the Planck formula......Page 539
15.1 The Schwinger formulation of quantum dynamics......Page 543
15.2 Propagator and probability amplitude......Page 546
15.2.2 Green kernel in the energy representation......Page 549
15.3 Lagrangian formulation of quantum mechanics......Page 550
15.4 Green kernel for quadratic Lagrangians......Page 553
15.4.1 Harmonic oscillator......Page 556
15.4.2 Free particle......Page 557
15.5 Quantum mechanics in phase space......Page 558
15.5.1 Operators vs. phase-space functions......Page 561
15.5.2 Quantum tomograms......Page 563
Appendix 15.A The Trotter product formula......Page 565
16.1 The Dirac equation......Page 567
16.1.1 Non-relativistic limit of the Dirac equation and antiparticles......Page 569
16.2 Particles in mutual interaction......Page 571
16.3 Relativistic interacting particles. Manifest covariance......Page 572
16.4 The no-interaction theorem in classical mechanics......Page 573
16.5 Relativistic quantum particles......Page 580
16.6 From particles to fields......Page 581
16.7 The Kirchhoff principle, antiparticles and QFT......Page 582
16.7.1 Time and space inversions in quantum mechanics......Page 584
16.7.2 The spin-statistics connection......Page 586
References......Page 588
Index......Page 605