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[资源] From Classical To Quantum Mechanics

Contents
Preface page xiii
Acknowledgments xvi
Part I From classical to wave mechanics 1
1 Experimental foundations of quantum theory 3
1.1 The need for a quantum theory 3
1.2 Our path towards quantum theory 6
1.3 Photoelectric effect 7
1.4 Compton effect 11
1.5 Interference experiments 17
1.6 Atomic spectra and the Bohr hypotheses 22
1.7 The experiment of Franck and Hertz 26
1.8 Wave-like behaviour and the Bragg experiment 27
1.9 The experiment of Davisson and Germer 33
1.10 Position and velocity of an electron 37
1.11 Problems 41
Appendix 1.A The phase 1-form 41
2 Classical dynamics 43
2.1 Poisson brackets 44
2.2 Symplectic geometry 45
2.3 Generating functions of canonical transformations 49
2.4 Hamilton and Hamilton–Jacobi equations 59
2.5 The Hamilton principal function 61
2.6 The characteristic function 64
2.7 Hamilton equations associated with metric tensors 66
2.8 Introduction to geometrical optics 68
2.9 Problems 73
Appendix 2.A Vector fields 74
vii
viii Contents
Appendix 2.B Lie algebras and basic group theory 76
Appendix 2.C Some basic geometrical operations 80
Appendix 2.D Space–time 83
Appendix 2.E From Newton to Euler–Lagrange 83
3 Wave equations 86
3.1 The wave equation 86
3.2 Cauchy problem for the wave equation 88
3.3 Fundamental solutions 90
3.4 Symmetries of wave equations 91
3.5 Wave packets 92
3.6 Fourier analysis and dispersion relations 92
3.7 Geometrical optics from the wave equation 99
3.8 Phase and group velocity 100
3.9 The Helmholtz equation 104
3.10 Eikonal approximation for the scalar wave equation 105
3.11 Problems 114
4 Wave mechanics 115
4.1 From classical to wave mechanics 115
4.2 Uncertainty relations for position and momentum 128
4.3 Transformation properties of wave functions 131
4.4 Green kernel of the Schr¨ odinger equation 136
4.5 Example of isometric non-unitary operator 142
4.6 Boundary conditions 144
4.7 Harmonic oscillator 151
4.8 JWKB solutions of the Schrödinger equation 155
4.9 From wave mechanics to Bohr–Sommerfeld 162
4.10 Problems 167
Appendix 4.A Glossary of functional analysis 167
Appendix 4.B JWKB approximation 172
Appendix 4.C Asymptotic expansions 174
5 Applications of wave mechanics 176
5.1 Reflection and transmission 176
5.2 Step-like potential; tunnelling effect 180
5.3 Linear potential 186
5.4 The Schr¨ odinger equation in a central potential 191
5.5 Hydrogen atom 196
5.6 Introduction to angular momentum 201
5.7 Homomorphism between SU(2) and SO(3) 211
5.8 Energy bands with periodic potentials 217
5.9 Problems 220
Contents ix
Appendix 5.A Stationary phase method 221
Appendix 5.B Bessel functions 223
6 Introduction to spin 226
6.1 Stern–Gerlach experiment and electron spin 226
6.2 Wave functions with spin 230
6.3 The Pauli equation 233
6.4 Solutions of the Pauli equation 235
6.5 Landau levels 239
6.6 Problems 241
Appendix 6.A Lagrangian of a charged particle 242
Appendix 6.B Charged particle in a monopole field 242
7 Perturbation theory 244
7.1 Approximate methods for stationary states 244
7.2 Very close levels 250
7.3 Anharmonic oscillator 252
7.4 Occurrence of degeneracy 255
7.5 Stark effect 259
7.6 Zeeman effect 263
7.7 Variational method 266
7.8 Time-dependent formalism 269
7.9 Limiting cases of time-dependent theory 274
7.10 The nature of perturbative series 280
7.11 More about singular perturbations 284
7.12 Problems 293
Appendix 7.A Convergence in the strong resolvent sense 295
8 Scattering theory 297
8.1 Aims and problems of scattering theory 297
8.2 Integral equation for scattering problems 302
8.3 The Born series and potentials of the Rollnik class 305
8.4 Partial wave expansion 307
8.5 The Levinson theorem 310
8.6 Scattering from singular potentials 314
8.7 Resonances 317
8.8 Separable potential model 320
8.9 Bound states in the completeness relationship 323
8.10 Excitable potential model 324
8.11 Unitarity of the Möller operator 327
8.12 Quantum decay and survival amplitude 328
8.13 Problems 335
x Contents
Part II Weyl quantization and algebraic methods 337
9 Weyl quantization 339
9.1 The commutator in wave mechanics 339
9.2 Abstract version of the commutator 340
9.3 Canonical operators and the Wintner theorem 341
9.4 Canonical quantization of commutation relations 343
9.5 Weyl quantization and Weyl systems 345
9.6 The Schr¨ odinger picture 347
9.7 From Weyl systems to commutation relations 348
9.8 Heisenberg representation for temporal evolution 350
9.9 Generalized uncertainty relations 351
9.10 Unitary operators and symplectic linear maps 357
9.11 On the meaning of Weyl quantization 363
9.12 The basic postulates of quantum theory 365
9.13 Problems 372
10 Harmonic oscillators and quantum optics 375
10.1 Algebraic formalism for harmonic oscillators 375
10.2 A thorough understanding of Landau levels 383
10.3 Coherent states 386
10.4 Weyl systems for coherent states 390
10.5 Two-photon coherent states 393
10.6 Problems 395
11 Angular momentum operators 398
11.1 Angular momentum: general formalism 398
11.2 Two-dimensional harmonic oscillator 406
11.3 Rotations of angular momentum operators 409
11.4 Clebsch–Gordan coefficients and the Regge map 412
11.5 Postulates of quantum mechanics with spin 416
11.6 Spin and Weyl systems 419
11.7 Monopole harmonics 420
11.8 Problems 426
12 Algebraic methods for eigenvalue problems 429
12.1 Quasi-exactly solvable operators 429
12.2 Transformation operators for the hydrogen atom 432
12.3 Darboux maps: general framework 435
12.4 SU(1,1) structures in a central potential 438
12.5 The Runge–Lenz vector 441
12.6 Problems 443
Contents xi
13 From density matrix to geometrical phases 445
13.1 The density matrix 446
13.2 Applications of the density matrix 450
13.3 Quantum entanglement 453
13.4 Hidden variables and the Bell inequalities 455
13.5 Entangled pairs of photons 459
13.6 Production of statistical mixtures 461
13.7 Pancharatnam and Berry phases 464
13.8 The Wigner theorem and symmetries 468
13.9 A modern perspective on the Wigner theorem 472
13.10 Problems 476
Part III Selected topics 477
14 From classical to quantum statistical mechanics 479
14.1 Aims and main assumptions 480
14.2 Canonical ensemble 481
14.3 Microcanonical ensemble 482
14.4 Partition function 483
14.5 Equipartition of energy 485
14.6 Specific heats of gases and solids 486
14.7 Black-body radiation 487
14.8 Quantum models of specific heats 502
14.9 Identical particles in quantum mechanics 504
14.10 Bose–Einstein and Fermi–Dirac gases 516
14.11 Statistical derivation of the Planck formula 519
14.12 Problems 522
Appendix 14.A Towards the Planck formula 522
15 Lagrangian and phase-space formulations 526
15.1 The Schwinger formulation of quantum dynamics 526
15.2 Propagator and probability amplitude 529
15.3 Lagrangian formulation of quantum mechanics 533
15.4 Green kernel for quadratic Lagrangians 536
15.5 Quantum mechanics in phase space 541
15.6 Problems 548
Appendix 15.A The Trotter product formula 548
16 Dirac equation and no-interaction theorem 550
16.1 The Dirac equation 550
16.2 Particles in mutual interaction 554
16.3 Relativistic interacting particles. Manifest covariance 555
16.4 The no-interaction theorem in classical mechanics 556
16.5 Relativistic quantum particles 563
xii Contents
16.6 From particles to fields 564
16.7 The Kirchhoff principle, antiparticles and QFT 565
References 571
Index 588
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