Quantum Mechanics: An Introduction

Cover......Page 1
Half Title......Page 2
Title Page......Page 4
Copyright Page......Page 5
Contents......Page 6
PREFACE......Page 16
AUTHOR......Page 20
1.1 WHAT YOU WILL LEARN IN THIS BOOK......Page 22
1.2 WHAT YOU SHOULD KNOW ABOUT LINEAR ALGEBRA......Page 26
1.3 THE ESSENTIALS OF PROBABILITY THEORY......Page 30
1.4 PHILOSOPHICAL INTRODUCTION......Page 35
1.4.1 The Essentials of Quantum Mechanics......Page 36
1.4.2 Unhappening......Page 40
1.4.4 What is Probability?......Page 41
1.5 A NOTE ON MATHEMATICS......Page 43
1.7 ACKNOWLEDGMENTS......Page 44
1.8 EXERCISES......Page 45
2.1 INTRODUCTION AND MOTIVATION......Page 48
2.2 THE AMMONIA MOLECULE AS A TWO STATE SYSTEM......Page 49
2.3 PHYSICAL QUANTITIES AS MATRICES......Page 51
2.4 OPERATIONS AS MATRICES......Page 55
2.5 THE EIGENVALUES OF A MATRIX......Page 57
2.6 THE BORN RULE AND UNCERTAINTY......Page 59
2.6.1 Born’s Rule in Pictures......Page 61
2.7 WHICH OPERATORS ARE DIAGONALIZABLE?......Page 63
2.7.1 Vectors, Functions, and Diagonal Matrices......Page 65
2.8 QUANTUM DYNAMICS OF THE AMMONIA MOLECULE......Page 66
2.9 SUMMARY......Page 73
2.10 SKIP THIS UNLESS YOU ARE INTERESTED IN PHILOSOPHY......Page 74
2.11 FOUR STATE SYSTEMS......Page 77
2.12 FURTHER EXERCISES......Page 81
3.1 INTRODUCTION......Page 84
3.2.1 No Experimental Definition of Infinity......Page 85
3.2.2 Free Particles on an Infinite Continuous Line......Page 89
3.2.3 Primer on the Fourier Transform and Fourier Series......Page 98
3.3 LAGRANGIAN AND HAMILTONIAN MECHANICS: A REMINDER OR A PRIMER......Page 100
3.4 FURTHER EXERCISES......Page 104
4.2 A MORE MATHEMATICAL DESCRIPTION OF ONE-DIMENSIONAL MOTION......Page 110
4.3 CONTINUUM OR DELTA FUNCTION NORMALIZATION......Page 114
4.4 TIME EVOLUTION OF A FREE PARTICLE WAVE FUNCTION......Page 117
4.5 THE DOUBLE SLIT EXPERIMENT......Page 120
4.6 A WORKED EXERCISE......Page 123
4.7 FURTHER EXERCISES......Page 128
5.2 QUANTIZING THE SIMPLE HARMONIC OSCILLATOR......Page 134
5.3 QUANTIZATION OF FIELDS AND WAVE–PARTICLE DUALITY......Page 142
5.4 FERMI–DIRAC STATISTICS......Page 148
5.5 EXERCISES......Page 150
6.1 INTRODUCTION......Page 154
6.2 SEPARABLE HILBERT SPACES......Page 155
6.3 UNITARY TRANSFORMATIONS, UNITARY MATRICES, AND CHANGES OF BASIS......Page 156
6.4 NORMAL OPERATORS ARE DIAGONALIZABLE OPERATORS......Page 159
6.5 CONTINUUM EIGENVALUES AND UNBOUNDED OPERATORS......Page 160
6.6 SUMMARY......Page 162
6.7 DIRECT SUMS AND TENSOR PRODUCTS......Page 163
6.8 THE GENERALIZED UNCERTAINTY RELATION......Page 164
6.9 SYMMETRIES AND CONSERVATION LAWS......Page 165
6.9.1 Dynamical Symmetry Groups......Page 168
6.9.2 Projective Representations of Symmetry Groups......Page 170
6.9.3 Examples......Page 173
6.10 EXERCISES......Page 181
7.1 INTRODUCTION......Page 186
7.2 UNITS AND SYMMETRIES......Page 187
7.3 IRREDUCIBLE REPRESENTATIONS OF THE COMMUTATION RELATIONS......Page 189
7.4 ADDITION OF ANGULAR MOMENTA......Page 196
7.5 THE HAMILTONIAN OF SPHERICALLY SYMMETRIC POTENTIALS......Page 197
7.6 THE PERIODIC TABLE......Page 202
7.7 THE SPIN OF THE ELECTRON......Page 207
7.8 SPIN PRECESSION AND SPIN RESONANCE......Page 211
7.9 STERN–GERLACH EXPERIMENTS......Page 212
7.10 EXERCISES......Page 213
8.2 POSITIVE ENERGY EIGENFUNCTIONS AND SCATTERING AMPLITUDES......Page 218
8.3 ANALYSIS OF THE COULOMB SCATTERING AMPLITUDES......Page 221
8.4 EXERCISES......Page 225
9.2 THE LORENTZ FORCE AND LANDAU’S SOLUTION......Page 228
9.3 EXERCISES......Page 233
10.1 INTRODUCTION......Page 236
10.2 COUNTING STATES AND COLLECTIVE COORDINATES......Page 237
10.3 COLLECTIVE COORDINATES......Page 238
10.4 INTERFERENCE OF CLASSICAL HISTORIES......Page 241
10.5 SCHRÖDINGER’S CAT AND SCHRÖDINGER’S BOMB......Page 243
10.6 THE EINSTEIN–PODOLSKY–ROSEN (EPR) PARADOX, BELL’S THEOREM, AND OTHER ODDITIES......Page 249
10.7 THE RELATION BETWEEN THEORY AND EXPERIMENT IN QM......Page 255
10.8 EXERCISES......Page 257
11.1 INTRODUCTION......Page 260
11.2 THE BORN–OPPENHEIMER APPROXIMATION......Page 261
11.3 COLLECTIVE COORDINATES, ROTATIONAL LEVELS, AND PHONONS......Page 263
11.3.1 Water......Page 264
11.3.2 Phonons......Page 265
11.4 THE HYDROGEN MOLECULAR ION......Page 266
11.5 DENSITY FUNCTIONAL THEORY......Page 268
11.6 ELEMENTS OF THE THEORY OF CRYSTALLINE SOLIDS......Page 271
11.7 BAND STRUCTURE......Page 273
11.7.1 A Simple Model......Page 276
11.8 THE FERMI LIQUID THEORY OF CONDUCTORS......Page 277
11.9 EXERCISES......Page 280
12.1 INTRODUCTION......Page 282
12.2 QUANTUM FIELD THEORY OF FERMIONS......Page 284
12.3 STATISTICAL MECHANICS OF BOSONS AND FERMIONS......Page 285
12.4 THE PARTITION FUNCTION......Page 286
12.5 THE LOW TEMPERATURE LIMIT......Page 292
12.6 STATISTICAL MECHANICS OF A FERMI LIQUID......Page 296
12.7 PLANCK’S RADIATION LAW......Page 298
12.8 EXERCISES......Page 299
13.2 BRILLOUIN–WIGNER PERTURBATION THEORY FOR NONDEGENERATE LEVELS......Page 302
13.3 DEGENERATE PERTURBATION THEORY......Page 306
13.5.1 The Stark Effect......Page 308
13.5.2 The Zeeman Effect......Page 310
13.5.3 Fine Structure and the Weak-Field Zeeman Effect......Page 311
13.5.4 Coulomb Expectation Values of Powers of R......Page 313
13.5.5 A Three-Dimensional Example......Page 315
13.5.6 Degenerate Perturbation Theory in a Macroscopic System......Page 316
13.6 EXERCISES......Page 319
14.2 DYSON’S FORMULA......Page 322
14.4 TRANSITION AMPLITUDES......Page 324
14.5 ELECTROMAGNETIC RADIATION FROM ATOMS......Page 325
14.6 INCOHERENT PERTURBATIONS AND RADIATIVE DECAY......Page 327
14.7 SELECTION RULES......Page 330
14.8 EXERCISES......Page 331
15.2 ADIABATIC ENERGY EIGENSTATES......Page 336
15.3 THE BERRY PHASE......Page 339
15.4 THE AHARONOV–BOHM EFFECT......Page 341
15.5 ANYONS, FERMIONS, AND THE SPIN STATISTICS THEOREM......Page 343
15.6 EXERCISES......Page 347
16.1 INTRODUCTION......Page 350
16.2 GENERAL FORMALISM......Page 351
16.3 POTENTIAL SCATTERING......Page 352
16.4 THE BORN APPROXIMATION......Page 353
16.5 PHASE SHIFT ANALYSIS......Page 354
16.5.1 The Effective Range Approximation......Page 355
16.6 RESONANCES......Page 356
16.7 A PARTIALLY WORKED EXERCISE: THE & SHELL POTENTIAL......Page 357
16.8 EXERCISES......Page 360
17.2 THE JWKB EXPANSION......Page 362
17.3 THE JWKB APPROXIMATION IN ONE DIMENSION......Page 364
17.4 COMMENTS ON THE SOLUTIONS......Page 365
17.5 THE JWKB APPROXIMATION FOR THE PROPAGATOR......Page 366
17.6 THE JWKB APPROXIMATION FOR ENERGY LEVELS......Page 367
17.7 THE JWKB APPROXIMATION TO THE WAVE FUNCTIONS OF EIGENSTATES......Page 369
17.7.1 Examples......Page 371
17.8 THE DECAY OF METASTABLE STATES......Page 373
17.9 MORE EXAMPLES......Page 375
17.10 THE JWKB APPROXIMATION FOR PHASE SHIFTS......Page 377
17.11 EXERCISES......Page 378
18.2 GENERAL PROPERTIES......Page 382
18.3 EXAMPLES......Page 383
18.4 THE HARTREE AND HARTREE–FOCK APPROXIMATIONS......Page 386
18.5 THE LANCZOS METHOD......Page 391
18.6 EXERCISES......Page 393
19.2 TWO DERIVATIONS OF THE PATH INTEGRAL FORMULA......Page 396
19.3 THE PATH INTEGRAL FOR A HARMONIC OSCILLATOR......Page 401
19.4 MORE GENERAL POTENTIALS......Page 406
19.5 PATH INTEGRALS AT FINITE TEMPERATURE......Page 407
19.6 PATH INTEGRALS AND THE JWKB APPROXIMATION......Page 409
19.7 PATH INTEGRALS FOR SPIN AND OTHER DISCRETE VARIABLES......Page 410
19.8 FERMIONS AND GRASSMANN INTEGRATION......Page 412
19.9 FURTHER EXERCISES......Page 417
20.1 INTRODUCTION......Page 420
20.2 QUANTUM INFORMATION......Page 421
20.3 PAGE’S THEOREM REDUX, MONOGAMY, AND CLONING......Page 423
20.4 QUANTUM KEY DISTRIBUTION......Page 425
20.5 QUANTUM TELEPORTATION......Page 427
20.6 GATES FOR QUANTUM COMPUTERS......Page 429
20.7 COMPUTATIONAL COMPLEXITY......Page 431
20.8 CAN WE BUILD A QUANTUM COMPUTER?......Page 434
20.9 EXERCISES......Page 435
CHAPTER 21 L’Envoi: Relativistic Quantum Field Theory......Page 438
APPENDIX A: Interpretations of Quantum Mechanics......Page 444
APPENDIX B: The Dirac Delta Function......Page 468
APPENDIX C: Noether’s Theorem......Page 470
APPENDIX D: Group Theory......Page 474
APPENDIX E: Laguerre Polynomials......Page 482
APPENDIX F: Summary of Dirac Notation and Linear Algebra......Page 484
APPENDIX G: Answers to Selected Problems......Page 490
Bibliography......Page 544
Index......Page 556