Preface ...................................................... XIII
1 Origin of Main Quantum Concepts .............................. 1
1.1 Light: Waves or Particles? .............................. 1
1.2 Planck Constant, Beginning of the Quantum Era ........... 2
1.3 Photons ................................................. 3
1.4 Spectroscopy and Stability of Atoms ..................... 5
1.5 Bohr Postulates ......................................... 6
1.6 Hydrogen Atom ........................................... 9
1.7 Correspondence Principle ............................... 15
1.8 Spatial Quantization ................................... 18
1.9 Spin ................................................... 20
1.10 De Broglie Waves ....................................... 21
2 Wave Function and the Simplest Problems ..................... 25
2.1 Free Motion ............................................ 25
2.2 Probability Density and Current ........................ 26
2.3 Superposition Principle and Uncertainty ................ 29
2.4 Potential Wall ......................................... 30
2.5 Potential Barrier ...................................... 31
2.6 Barrier Penetration .................................... 35
2.7 Tunneling .............................................. 36
3 Bound States ................................................ 43
3.1 Potential Box .......................................... 43
3.2 Orthogonality and Completeness ......................... 45
3.3 Delta-Function* ........................................ 46
3.4 Time Evolution ......................................... 49
3.5 Shallow Well and Quantum Halo .......................... 52
3.6 Resonances ............................................. 59
3.7 Level Density .......................................... 60
3.8 Periodic Boundary Conditions ........................... 62
3.9 Counting Levels in a Smooth Potential .................. 63
4 Dynamical Variables ......................................... 67
4.1 Momentum Representation ................................ 67
4.2 Introducing Operators .................................. 70
4.3 Commutators ............................................ 72
4.4 Eigenfunctions and Eigenvalues ......................... 74
4.5 Momentum as a Translation Generator .................... 75
4.6 Introduction to Groups ................................. 78
4.7 Orbital Momentum as a Rotation Generator ............... 79
4.8 Transformation of Operators ............................ 81
5 Uncertainty Relations ....................................... 85
5.1 Uncertainty in Wave Mechanics .......................... 85
5.2 Simple Examples ........................................ 87
5.3 Complementarity and Probability ........................ 91
5.4 Wave Packet: Propagation ............................... 94
5.5 Spreading of a Wave Packet ............................. 96
5.6 Estimates with Uncertainty Relations .................. 100
5.7 Classification of Molecular Excitations ............... 104
5.8 Level Width ........................................... 107
5.9 Line Width and Mossbauer Effect ....................... 109
5.10 Virtual Processes and Relativistic Effects ............ 112
5.11 Spatial Quantization Revisited ........................ 114
6 Hilbert Space and Operators ................................ 119
6.1 Probability Amplitude ................................. 119
6.2 Superposition and Interference ........................ 120
6.3 State Vectors ......................................... 123
6.4 Geometry of Hilbert Space* ............................ 124
6.5 Linear Operators* ..................................... 128
6.6 Hermitian Operators* .................................. 130
6.7 Properties of Hermitian Operators* .................... 132
6.8 Diagonalization* ...................................... 134
6.9 Basis Transformations* ................................ 136
6.10 Continuous Transformations and Generators* ............ 138
6.11 Projection Operators* ................................. 140
6.12 Operators of Observables .............................. 142
6.13 Simultaneous Measurability ............................ 144
6.14 Quantifying Uncertainty Relations ..................... 146
7 Quantum Dynamics ........................................... 153
7.1 Hamiltonian and Schrodinger Equation .................. 153
7.2 Single-Particle Hamiltonian ........................... 155
7.3 Continuity Equation ................................... 161
7.4 Wigner Distribution ................................... 165
7.5 Heisenberg Picture .................................... 167
7.6 Operator Dynamics ..................................... 168
7.7 Virial Theorem ........................................ 171
7.8 Survival Probability .................................. 173
7.9 Sum Rules ............................................. 174
7.10 Conservation Laws ..................................... 178
7.11 Path Integral Formulation ............................. 180
7.12 Relation to Classical Mechanics ....................... 183
7.13 Back to the Schrodinger Picture ....................... 184
8 Discrete Symmetries ........................................ 187
8.1 Time-Reversal Invariance .............................. 187
8.2 Time-Reversal Transformation of Operators ............. 189
8.3 Inversion and Parity .................................. 191
8.4 Scalars and Pseudoscalars, Vectors and
Pseudovectors ......................................... 192
8.5 Parity Conservation ................................... 193
8.6 Symmetry of a Crystal Lattice ......................... 197
8.7 Quasimomentum and Bloch Functions ..................... 198
8.8 Energy Bands .......................................... 201
8.9 Symmetry of Molecules ................................. 203
8.10 More Group Theory: Conjugate Classes* ................. 206
8.11 Group Representations* ................................ 207
8.12 Orthogonality and Completeness* ....................... 209
8.13 Characters* ........................................... 212
9 One-Dimensional Motion: Continuum .......................... 217
9.1 Eigenvalue Problem .................................... 217
9.2 Continuous Spectrum ................................... 218
9.3 Degeneracy in the Continuum ........................... 221
9.4 Transfer Matrix ....................................... 224
9.5 Delay Time ............................................ 225
9.6 Uniform Field ......................................... 228
9.7 Airy and Bessel Functions* ............................ 229
9.8 Asymptotic Behavior* .................................. 232
9.9 Asymptotics of the Airy Function* ..................... 234
9.10 Green Function for One-Dimensional Scattering ......... 237
9.11 Potential as Perturbation ............................. 241
9.12 Quasistationary States ................................ 244
10 Variational Approach and Diagonalization ................... 247
10.1 Variational Principle ................................. 247
10.2 Direct Variational Method ............................. 249
10.3 Diagonalization in a Truncated Basis .................. 251
10.4 Two-State System ...................................... 252
10.5 Level Repulsion and Avoided Crossing .................. 254
10.6 Time Evolution of a Two-State System .................. 257
10.7 Bright State and Fragmentation ........................ 259
10.8 Collective States ..................................... 261
10.9 Lanczos Algorithm ..................................... 265
11 Discrete Spectrum and Harmonic Oscillator .................. 267
11.1 One-Dimensional Bound States .......................... 267
11.2 Linear Harmonic Oscillator ............................ 269
11.3 Hermite Polynomials* .................................. 275
11.4 Harmonic Oscillator in Plane: Separation of
Variables ............................................. 278
11.5 Isotropic Oscillator .................................. 280
11.6 Solving the Problem in Polar Coordinates .............. 282
11.7 Ladder Construction ................................... 285
11.8 Creation and Annihilation Operators ................... 286
11.9 Operator Solution for the Harmonic Oscillator ......... 288
12 Coherent and Squeezed States ............................... 293
12.1 Introducing Coherent States ........................... 293
12.2 Displacements in the Phase Plane ...................... 294
12.3 Properties of Coherent States ......................... 296
12.4 Coherent States of the Harmonic Oscillator ............ 298
12.5 Linear Source ......................................... 299
12.6 Semiclassical Limit, Number of Quanta and the Phase ... 302
12.7 Pairwise Source ....................................... 304
12.8 Squeezed States ....................................... 307
12.9 More about Squeezed States............................. 310
13 Introducing Magnetic Field ................................. 315
13.1 Magnetic Field in Classical Mechanics ................. 315
13.2 Quantum Formulation and Gauge Invariance .............. 317
13.3 Are Electromagnetic Potentials Observable? ............ 320
13.4 Landau Levels: Energy Spectrum ........................ 321
13.5 Landau Levels: Degeneracy and Wave Functions .......... 323
13.6 Quantum Hall Effect ................................... 327
13.7 Arbitrary Dispersion Law .............................. 331
13.8 Symmetric Gauge ....................................... 335
13.9 Coherent States in the Magnetic Field ................. 336
14 Macroscopic Quantum Coherence .............................. 339
14.1 Ideas of Macroscopic Coherence ........................ 339
14.2 Macroscopic Wave Function ............................. 340
14.3 Hydrodynamic Description .............................. 341
14.4 Dynamics of the Macroscopic Coherent State ............ 344
14.5 Josephson Effects ..................................... 346
14.6 Quantization of Circulation and Quantum Vortices ...... 349
14.7 Magnetic Fluxoid Quantization and London
Electrodynamics ....................................... 353
15 Semiclassical (WKB) Approximation .......................... 357
15.1 Heuristic Introduction ................................ 357
15.2 Semiclassical Approximation ........................... 360
15.3 Asymptotic Expansion .................................. 363
15.4 Stationary Phase ...................................... 364
15.5 Matching Conditions ................................... 365
15.6 Bohr-Sommerfeld Quantization .......................... 369
15.7 Semiclassical Matrix Elements ......................... 371
15.8 Solutions in the Complex Plane* ....................... 373
15.9 Going Around the Complex Plane* ....................... 376
15.10 Connection Formulae Revisited* ....................... 378
15.11 Close Turning Points* ................................ 379
15.12 Path Integral Approach ............................... 383
16 Angular Momentum and Spherical Functions ................ 387
16.1 Angular Momentum as a Generator of Rotations .......... 387
16.2 Spin .................................................. 389
16.3 Angular Momentum Multiplets ........................... 390
16.4 Matrix Elements of Angular Momentum ................... 396
16.5 Realization of the Algebra for Orbital Momentum ....... 399
16.6 Constructing a Set of Spherical Functions* ............ 401
16.7 Simplest Properties of Spherical Functions* ........... 403
16.8 Scalars and Vectors* .................................. 404
16.9 Second Rank Tensors* .................................. 408
16.10 Spherical Functions and Legendre Polynomials* ........ 410
16.11 Angular Momentum in an External Field ................ 414
17 Motion in a Central Field .................................. 417
17.1 Reduction to the One-Body Problem ..................... 417
17.2 Separation of Angular Variables ....................... 420
17.3 Radial Part of the Schrodinger Equation ............... 422
17.4 Free Motion ........................................... 426
17.5 Plane and Spherical Waves ............................. 430
17.6 Spherical Well ........................................ 432
17.7 Short-Range Potential ................................. 435
17.8 Adding the Second Center .............................. 436
17.9 Three-Dimensional Harmonic Oscillator ................. 439
18 Hydrogen Atom .............................................. 445
18.1 Bound States .......................................... 445
18.2 Ground State .......................................... 447
18.3 Discrete Spectrum ..................................... 450
18.4 Operator Solution ..................................... 458
18.5 On the Way to Precision Spectroscopy .................. 460
18.6 Solution in Parabolic Coordinates* .................... 462
18.7 Continuum States ...................................... 463
19 Stationary Perturbations ................................... 469
19.1 Introduction .......................................... 469
19.2 Perturbation Theory With No Degeneracy ................ 470
19.1 Convergence ........................................... 474
19.4 Case of Close Levels .................................. 477
19.5 Adiabatic Approximation ............................... 478
19.6 Molecular Ion of Hydrogen ............................. 482
19.7 Interactions of Atoms at Large Distances .............. 486
20 Spin 1/2 ................................................... 489
20.1 SU(2) Group ........................................... 489
20.2 Spin 1/2: Algebra ..................................... 490
20.3 Spinors ............................................... 494
20.4 Magnetic Resonance .................................... 499
20.5 Time-Reversal Transformation and Kramers Theorem ...... 501
20.6 Time-Conjugate States ................................. 503
20.7 Spinors as Qubits ..................................... 504
21 Finite Rotations and Tensor Operators ...................... 509
21.1 Matrices of Finite Rotations .......................... 509
21.2 Spherical Functions as Matrix Elements of Finite
Rotations ............................................. 511
21.3 Addition Theorem* ..................................... 514
21.4 Transformation of Operators ........................... 516
21.5 Introduction to Selection Rules ....................... 518
21.6 Electromagnetic Multipoles ............................ 519
22 Angular Momentum Coupling .................................. 523
22.1 Two Subsystems ........................................ 523
22.2 Decomposition of Reducible Representations ............ 525
22.3 Two Particles of Spin 1/2 ............................. 528
22.4 Tensor Operators and Selection Rules Revisited ........ 532
22.5 Applying to Electromagnetic Multipoles ................ 533
22.6 Vector Coupling of Angular Momenta .................... 534
22.7 Wigner-Eckart Theorem ................................. 538
22.8 Vector Model .......................................... 539
22.9 Electric Dipole Moment and Anapole Moment ............. 541
22.10 Clebsch-Gordan Series* ............................... 543
23 Fine and Hyperfine Structure ............................ 545
23.1 Spin-Orbit Coupling ................................... 545
23.2 Spin-Orbit Splitting .................................. 547
23.3 Hydrogen Fine Structure ............................... 550
23.4 Fine Structure in Complex Atoms ....................... 553
23.5 Magnetic Moment with Spin-Orbit Coupling .............. 555
23.6 Magnetic Hyperfine Structure .......................... 558
23.7 Example: One Valence Electron ......................... 560
23.8 Quadrupole Hyperfine Structure ........................ 562
24 Atom in a Static Field .................................. 567
24.1 Polarizability in a Static Electric Field ............. 567
24.2 Stark Effect .......................................... 569
24.3 Polarizability of the Hydrogen Atom ................... 570
24.4 Stark Effect in the Hydrogen Atom ..................... 572
24.5 Non-uniform Electric Field and Additional Comments .... 573
24.6 Classical Zeeman Effect ............................... 574
24.7 A Quantum System in a Magnetic Field .................. 575
24.8 Normal Quantum Zeeman Effect .......................... 576
24.9 Anomalous Quantum Zeeman Effect ....................... 578
24.10 Stronger Magnetic Field .............................. 579
24.11 Diamagnetism ......................................... 581
24.12 Towards Really Strong Magnetic Fields ................ 583
References ................................................. 587
Further Readings ........................................... 591
Index ......................................................... 597
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