Preface ....................................................... XII
1 Introduction ................................................. 1
1.1 Origins ................................................. 2
1.2 Measurements ............................................ 6
Measurement involves disturbance ........................ 7
Heisenberg microscope ................................... 7
Ideal measurements ...................................... 8
Summary ................................................ 10
1.3 Probability amplitudes ................................. 11
Two-slit interference .................................. 12
1.4 Quantum states ......................................... 14
Observables ............................................ 14
Complete sets of amplitudes ............................ 14
Vector spaces and their duals .......................... 16
The energy representation .............................. 19
Polarisation of photons ................................ 20
1.5 Summary ................................................ 23
Problems ............................................... 24
2 Operators, measurement and time evolution ................... 25
2.1 Operators .............................................. 25
Functions of operators ................................. 29
Commutators ............................................ 29
2.2 Evolution in time ...................................... 31
Evolution of expectation values ........................ 32
2.3 The position representation ............................ 34
Hamiltonian of a particle .............................. 37
Wavefunction for well-defined momentum ................. 38
The uncertainty principle .............................. 39
Dynamics of a free particle ............................ 41
Back to two-slit interference .......................... 43
Generalisation to three dimensions ..................... 44
Probability current .................................... 45
The virial theorem ..................................... 46
2.4 Summary ................................................ 47
Problems ............................................... 48
3 Oscillators ................................................. 52
3.1 Stationary states of a harmonic oscillator ............. 52
3.2 Dynamics of oscillators ................................ 56
Anharmonic oscillators ................................. 58
Problems ............................................... 61
4 Transformations and observables ............................. 66
4.1 Transforming kets ...................................... 66
Translating kets ....................................... 67
Continuous transformations and generators .............. 69
The rotation operator .................................. 71
Discrete transformations ............................... 72
The parity operator .................................... 72
Mirror operators ....................................... 73
4.2 Transformations of operators ........................... 74
The parity operator .................................... 76
Mirror operators ....................................... 78
4.3 Symmetries and conservation laws ....................... 79
4.4 The Heisenberg picture ................................. 81
4.5 What is the essence of quantum mechanics? .............. 83
Problems ............................................... 85
5 Motion in step potentials ................................... 88
5.1 Square potential well .................................. 88
Limiting cases ......................................... 91
Infinitely deep well ................................... 91
Infinitely narrow well ................................. 92
5.2 A pair of square wells ................................. 93
Ammonia ................................................ 96
The ammonia maser ...................................... 97
5.3 Scattering of free particles ........................... 99
The scattering cross-section .......................... 102
Tunnelling through a potential barrier ................ 103
Scattering by a classically allowed region ............ 104
Resonant scattering ................................... 106
The Breit-Wigner cross-section ........................ 109
5.4 How applicable are our results? ....................... 113
5.5 Summary ............................................... 115
Problems .............................................. 117
6 Composite systems .......................................... 123
6.1 Composite systems ..................................... 124
Collapse of the wavefunction .......................... 128
Operators for composite systems ....................... 129
Development of entanglement ........................... 131
Einstein-Podolski-Rosen experiment .................... 132
Bell's inequality ..................................... 134
6.2 Quantum computing ..................................... 137
6.3 The density operator .................................. 144
Reduced density operators ............................. 148
Shannon entropy ....................................... 151
6.4 Thermodynamics ........................................ 154
6.5 Measurement ........................................... 158
Problems .............................................. 162
7 Angular momentum ........................................... 167
7.1 Eigenvalues of Jz and J2 .............................. 168
Rotation spectra of diatomic molecules ................ 171
7.2 Spin and orbital angular momentum ..................... 174
Orbital angular momentum .............................. 174
L as the generator of circular translations ........... 176
Spectra of and L2 and Lz .............................. 177
Spin angular momentum ................................. 177
7.3 Physics of spin ....................................... 180
Spin-half matrices .................................... 181
Spin-one matrices ..................................... 182
7.4 Orbital angular-momentum eigenfunctions ............... 191
The Stern-Gerlach experiment .......................... 183
Stern-Gerlach experiment with spin-one atoms .......... 186
Precession in a magnetic field ........................ 187
The classical Umit .................................... 188
Orbital angular momentum and parity ................... 196
Orbital angular momentum and kinetic energy ........... 196
Legendre polynomials .................................. 198
7.5 Three-dimensional harmonic oscillator ................. 199
7.6 Addition of anguleu momenta ........................... 205
Case of two spin-half systems ......................... 209
Case of spin-one and spin-half ........................ 211
The classical limit ................................... 212
Problems .............................................. 213
8 Hydrogen ................................................... 219
8.1 Gross structure of hydrogen ........................... 220
Emission-line spectra ................................. 226
Radial eigenfunctions ................................. 226
Shielding ............................................. 231
Expectation values for r-k ............................ 234
8.2 Fine structure and beyond ............................. 236
Spin-orbit coupling ................................... 236
Hyperfine structure ................................... 241
Problems .............................................. 243
9 Motion in a magnetic field ................................. 247
9.1 Hamiltonian for motion in a magnetic field ............ 248
Classical equations of motion ......................... 248
9.2 Gauge transformations ................................. 250
Probability current ................................... 251
9.3 Landau levels ......................................... 252
Displacement of the gyrocentre ........................ 255
9.4 Aharonov-Bohm effect .................................. 257
Problems .............................................. 259
10 Perturbation theory ........................................ 262
10.1 Time-independent perturbations ........................ 263
Quadratic Stark effect ................................ 265
Linear Stark effect and degenerate perturbation
theory ................................................ 266
Effect of an external magnetic field .................. 269
Paschen-Back effect ................................... 271
Zeeman effect ......................................... 271
10.2 Variational principle ................................. 273
10.3 Time-dependent perturbation theory .................... 275
Fermi golden rule ..................................... 276
Radiative transition rates ............................ 277
Selection rules ....................................... 282
Problems .............................................. 284
11 Helium and the periodic table .............................. 290
11.1 Identical particles ................................... 290
Generalisation to the case of N identical particles ... 291
Pauli exclusion principle ............................. 292
Electron pairs ........................................ 293
11.2 Gross structure of helium ............................. 295
Gross structure from perturbation theory .............. 296
Application of the variational principle to helium .... 298
Excited states of heUum ............................... 299
Electronic configurations and spectroscopic terms ..... 302
Spectrum of helium .................................... 303
11.3 The periodic table .................................... 303
From lithium to argon ................................. 303
The fourth and fifth periods .......................... 308
Problems .............................................. 309
12 Adiabatic principle ........................................ 312
12.1 Derivation of the adiabatic principle ................. 313
12.2 Application to kinetic theory ......................... 315
12.3 Application to thermodynamics ......................... 317
12.4 The compressibility of condensed matter ............... 318
12.5 Covalent bonding ...................................... 320
A model of a covalent bond ............................ 320
Molecular dynamics .................................... 322
Dissociation of molecules ............................. 323
12.6 The WKBJ approximation ................................ 324
Problems .............................................. 325
13 Scattering theory .......................................... 327
13.1 The scattering operator ............................... 328
Perturbative treatment of the scattering operator ..... 330
13.2 The S-matrix .......................................... 332
The ie prescription ................................... 332
Expanding the S-matrix ................................ 334
The scattering ampUtude ............................... 336
13.3 Cross-sections and scattering experiments ............. 339
The optical theorem ................................... 342
13.4 Scattering electrons ofF hydrogen ..................... 344
13.5 Partial wave expansions ............................... 346
Scattering at low energy .............................. 351
13.6 Resonant scattering ................................... 353
Breit-Wigner resonances ............................... 356
Radioactive decay ..................................... 356
Problems .............................................. 358
Appendices
A The laws of probability .................................... 361
В Ceirtesian tensors ......................................... 363
С Fourier series and transforms .............................. 365
D Operators in classical statistical mechanics ............... 367
E Lie groups and Lie algebras ................................ 369
F The hidden symmetry of hydrogen ............................ 370
G Lorentz covariant equations ................................ 372
H Thomas precession .......................................... 376
I Matrix elements for a dipole-dipole interaction ............ 378
J Selection rule for j ....................................... 380
К Restrictions on scattering potentials ...................... 381
Index ......................................................... 383
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