Quantum Mechanics

Quantum Mechanics Third Edition......Page 1
Front Flap......Page 2
Numerical Values Of Some Physical Quantities......Page 3
Half-Title......Page 4
International Series In Pure And Applied Physics......Page 6
Title-Page......Page 8
Copyright......Page 9
Dedication......Page 10
Preface......Page 12
Contents......Page 14
Chapter 1 The Physical Basis Of Quantum Mechanics......Page 24
Inadequacy Of Classical Mechanics......Page 25
Summary Of Principle Experiments And Inferences......Page 26
Bohr-Sommerfeld quantization rules......Page 27
Conceptual difficulties......Page 28
Quantum-mechanical viewpoint......Page 29
Uncertainty principle......Page 30
Complementarity principle......Page 31
Localization experiment......Page 32
Momentum determination experiment......Page 34
Analysis of diffraction experiment......Page 35
5 Wave Packets in Space and Time......Page 37
Space packets......Page 38
Time packets......Page 39
Wave formalism......Page 40
Problems......Page 41
Chapter 2 The Schrodinger Wave Equation......Page 42
Need for a wave equation......Page 43
Extension to three dimensions......Page 45
Inclusion of forces......Page 46
Statistical interpretation......Page 47
Normalization of psi......Page 48
Probability current density......Page 49
Expectation value......Page 50
Ehrenfest's theorem......Page 51
Separation of the wave equation......Page 53
Significance of the separation constant E......Page 54
Boundary conditions for infinite potential energy......Page 55
Energy eigenvalues in one dimension......Page 57
Discrete energy levels......Page 58
9 One-dimensional Square Well Potential......Page 60
Perfectly rigid walls......Page 61
Finite potential step......Page 62
Energy levels......Page 64
Parity......Page 65
A simplified solution......Page 66
Problems......Page 67
Chapter 3 Eigenfunctions and Eigenvalues......Page 68
Expansion in eigenfunctions......Page 69
Normalization in a box......Page 70
Orthonormality of energy eigenfunctions......Page 71
Expansion in energy eigenfunctions......Page 73
Probability function and expectation value......Page 74
Form of the eigenfunctions......Page 76
Box normalization......Page 77
The Dirac delta-function......Page 78
Normalization in terms
of the delta-function......Page 79
Closure......Page 80
Expansion in momentum eigenfunctions......Page 81
Probability function and expectation value......Page 82
The minimum uncertainty product......Page 83
Form of the minimum packet......Page 84
Momentum expansion coefficients......Page 85
Change with time of a minimum
packet......Page 86
Classical limit......Page 87
Problems......Page 88
13 Linear Harmonic Oscillator......Page 89
Asymptotic behavior......Page 90
Energy levels......Page 91
Hermite polynomials......Page 92
Harmonic-oscillator wave functions......Page 94
Correspondence with classical theory......Page 95
Oscillating wave packet......Page 97
14 Spherically Symmetric Potentials in Three Dimensions......Page 99
Separation of the wave equation......Page 100
Legendre polynomials......Page 101
Spherical
harmonics......Page 102
Angular momentum......Page 104
15 Three-dimensional Square Well Potential......Page 106
Interior solutions for arbitrary l......Page 107
Exterior solutions for arbitrary l......Page 109
Energy levels......Page 110
Reduced mass......Page 111
Asymptotic behavior......Page 113
Energy levels......Page 114
Laguerre polynomials......Page 115
Hydrogen-atom wave functions......Page 116
Degeneracy......Page 117
Separation in parabolic coordinates......Page 118
Energy levels......Page 120
Problems......Page 121
Chapter 5 Continuous Eigenvalues: Collision Theory......Page 123
Asymptotic behavior......Page 124
Scattering coefficients......Page 125
Scattering of a wave packet......Page 127
18 Collisions in Three Dimensions......Page 128
Scattering cross section......Page 133
Relations between angles in the laboratory
and center-of-mass systems......Page 134
Dependence on gamma......Page 136
Asymptotic behavior......Page 137
Normalization......Page 138
19 Scattering by Spherically Symmetric Potentials......Page 139
Asymptotic behavior......Page 140
Differential cross section......Page 141
Total elastic cross
section......Page 143
Calculation of delta-l (Phase Shift)......Page 144
Relation between signs of
delta-l and V(r)......Page 145
Ramsauer-Townsend effect......Page 146
Scattering by a perfectly rigid sphere......Page 147
Scattering by a square well potential......Page 149
Resonance scattering......Page 150
20 Scattering by Complex Potentials......Page 152
Conservation of probability......Page 153
Complex phase shifts......Page 154
Asymptotic relations......Page 156
Generalized optical theorem......Page 158
Optical theorem......Page 160
21 Scattering by a Coulomb Field......Page 161
Parabolic coordinates......Page 162
Confluent hypergeometric function......Page 163
Scattering cross section and normalization......Page 164
Solution in spherical
coordinates......Page 165
Modified coulomb field......Page 167
Classical limit for a pure coulomb field......Page 168
Problems......Page 169
Chapter 6 Matrix Formulation of Quantum Mechanics......Page 171
Matrix addition and multiplication......Page 172
Null, unit, and constant
matrices......Page 173
Hermitian
and unitary matrices......Page 174
Transformation and diagonalization of
matrices......Page 175
Matrices of infinite rank......Page 176
Unitary matrix W......Page 178
Transformation of the hamiltonian with W......Page 179
Transformation of the hamiltonian with U......Page 180
Representations of operators......Page 182
A useful
identity......Page 183
Row and column matrices......Page 184
Hilbert space......Page 186
Dirac's bra and ket notation......Page 187
Projection operators......Page 189
24 Equations of Motion......Page 190
Schrodinger picture......Page 191
Heisenberg picture......Page 193
Interaction picture......Page 194
Classical lagrangian and hamiltonian equations of motion......Page 196
Poisson brackets and commutator brackets......Page 198
Quantization of a classical system......Page 199
Evaluation of commutator brackets......Page 200
Velocity and acceleration of a charged particle......Page 201
The Lorentz force......Page 202
25 Matrix Theory of the Harmonic Oscillator......Page 203
Energy representation......Page 204
Raising and lowering operators......Page 205
Matrices for a, x, and p......Page 206
Coordinate representation......Page 207
Problems......Page 208
Chapter 7 Symmetry in Quantum Mechanics......Page 210
Unitary displacement operator......Page 211
 Symmetry and degeneracy......Page 213
The group concept......Page 214
Time displacement......Page 216
Proper rotation group......Page 217
Geometrical isomorphism......Page 218
Infinitesimal rotations......Page 219
Spin of a vector particle......Page 220
Commutation relations for the generators......Page 222
Choice of a representation......Page 223
Values of m, f(j), and lambda-sub-m......Page 224
Angular momentum matrices......Page 226
Spin angular momentum......Page 227
Covering group......Page 228
Unitary and special unitary groups in two dimensions......Page 229
The groups U(n) and SU(n)......Page 230
Generators of U(n) and SU(n)......Page 231
The SU(3) group......Page 232
Representation in terms of coordinates and momenta......Page 233
28 Combination of Angular Momentum States and Tensor Operators......Page 235
Eigenvalues of the total angular momentum......Page 236
Clebsch-Gordan coefficients......Page 237
Construction procedure......Page 238
Some particular coefficients......Page 240
Irreducible tensor operators......Page 242
Product of tensor operators......Page 243
Wigner-Eckart theorem......Page 245
Space inversion......Page 247
Unitary inversion operator......Page 248
Inverted states and operators......Page 249
Time reversal......Page 250
Antilinear operators......Page 251
Antiunitary operators......Page 252
T for a zero spin particle......Page 253
T for a nonzero spin particle......Page 254
Systems of several particles......Page 255
Reality of eigenfunctions......Page 256
30 Dynamical Symmetry......Page 257
Classical Kepler problem......Page 258
Hydrogen Atom......Page 259
The O(4) group......Page 260
Energy levels of hydrogen......Page 261
Classical isotropic oscillator......Page 262
Quantum isotropic oscillator......Page 264
Problems......Page 265
Chapter 8 Approximation Methods for Bound States......Page 267
Nondegenerate case......Page 268
First-order perturbation......Page 269
Second-order perturbation......Page 270
Degenerate case......Page 271
Removal of degeneracy in second order......Page 273
Zeeman effect without
electron spin......Page 274
First-order Stark effect in hydrogen......Page 275
Occurrence of permanent electric dipole moments......Page 276
Expectation value of the energy......Page 278
Application to excited states......Page 279
Ground state of helium......Page 280
Electron interaction energy......Page 281
van der Waals interaction......Page 282
Perturbation calculation......Page 284
Variation calculation......Page 285
Second-order Stark effect in hydrogen......Page 286
Polarizability of hydrogen......Page 288
Method of Dalgarno and Lewis......Page 289
Interaction of a hydrogen atom and a point charge......Page 290
34 The WKB Approximation......Page 291
Classical limit......Page 292
Approximate solutions......Page 293
Asymptotic nature of the
solutions......Page 294
Solution near a turning point......Page 295
Linear turning point......Page 296
Connection at the turning point......Page 297
Energy levels of a potential well......Page 298
Special boundary conditions......Page 300
Tunneling through a barrier......Page 301
35 Methods for Time-dependent Problems......Page 302
Time-dependent perturbation theory......Page 303
Interaction picture......Page 304
Harmonic perturbation......Page 305
Transition probability......Page 306
Ionization of a hydrogen atom......Page 308
Density of final states......Page 309
Ionization probability......Page 310
Second-order perturbation......Page 311
Adiabatic approximation......Page 312
Choice of phases......Page 313
Connection with perturbation theory......Page 314
Sudden approximation......Page 315
Disturbance of an oscillator......Page 317
Problems......Page 318
36 The Scattering Matrix......Page 321
Green's functions and propagator......Page 323
Free-particle Green's functions......Page 324
Integral equation for psi......Page 325
Use of the advanced Green's function......Page 327
Differential equation for the Green's functions......Page 328
Symbolic relations......Page 329
Application to scattering......Page 330
Unitarity of the S matrix......Page 332
Symmetry properties of the S matrix......Page 333
Transition matrix......Page 335
Transition probability......Page 336
Scattering cross section......Page 337
Green's functions for stationary case......Page 338
Green's functions as inverse operators......Page 340
Stationary propagator......Page 341
Free-particle propagator......Page 342
Scattering amplitude......Page 343
Ingoing waves......Page 344
Angular momentum representation......Page 345
Born approximation......Page 347
Validity of the Born approximation......Page 348
Scattering from two potentials......Page 349
Distorted wave Born approximation......Page 350
Partial wave analysis of the DWBA......Page 351
Scatterer with internal degrees of freedom......Page 353
Elastic and inelastic cross sections......Page 354
Electron scattering from
hydrogen......Page 356
Production of a cloud chamber track......Page 358
Second-order perturbation theory......Page 359
Evaluation of the second-order matrix element......Page 360
Eikonal approximation......Page 362
Scattering amplitude and cross section......Page 364
Perfect absorber......Page 366
39 Analytic Properties and Dispersion Relations......Page 367
Radial solutions......Page 368
Jost function......Page 370
Enhancement factor......Page 371
Bound states......Page 372
Dispersion relations for the Jost
function......Page 373
Effect of bound states......Page 374
Levinson's theorem......Page 376
Effective range......Page 377
Forward scattering amplitude......Page 379
Dispersion relation for T(E)......Page 380
Subtracted dispersion relation......Page 382
Problems......Page 383
Chapter 10 Identical Particles and Spin......Page 385
Physical meaning of identity......Page 386
Symmetric and antisymmetric wave functions......Page 387
Construction from unsymmetrized functions......Page 388
The symmetric group......Page 389
Distinguishability of identical particles......Page 390
The
exclusion principle......Page 391
Connection with statistical mechanics......Page 392
Collisions of identical particles......Page 393
Connection between spin and statistics......Page 394
Spin matrices and eigenfunctions......Page 395
Collisions of identical particles......Page 396
Electron spin functions......Page 397
The helium atom......Page 399
Spin functions for three electrons......Page 400
Expectation value and projection operator......Page 401
Density operator......Page 402
Equations of motion......Page 403
Projection operator for a spin ½particle......Page 404
Polarization vector for a spin s particle......Page 405
Precession of the polarization vector......Page 406
43 Rearrangement Collisions......Page 407
Alternative expression for the T matrix element......Page 408
T matrix element for rearrangements......Page 409
Presence of a core interaction......Page 410
Elimination of the core term......Page 411
Exchange collisions of electrons with hydrogen......Page 413
Relation between amplitude and matrix element......Page 414
Effects of identity and spin......Page 416
Exchange collisions with helium......Page 417
Problems......Page 418
Chapter 11 Semiclassical Treatment of Radiation......Page 420
Maxwell's equations......Page 421
Plane electromagnetic waves......Page 422
Use of perturbation theory......Page 423
Transition probability......Page 424
Interpretation in terms of absorption and emission......Page 426
Electric dipole transitions......Page 427
Forbidden transitions......Page 428
Classical radiation field......Page 429
Radiated energy......Page 431
Dipole radiation......Page 432
Angular momentum......Page 433
Dipole case......Page 434
Conversion from classical to quantum theory......Page 436
Planck distribution formula......Page 437
Line breadth......Page 438
Selection rules for a single particle......Page 439
Conservation of angular momentum......Page 440
Selection rules for many-particle systems......Page 441
Photoelectric effect......Page 443
Cross section for the atomic photoelectric effect......Page 444
Improvement on the Born approximation......Page 445
Problems......Page 446
47 Approximations in Atomic Structure......Page 447
Central-field approximation......Page 448
Periodic system of the elements......Page 449
Thomas-Fermi statistical model......Page 450
Evaluation of the potential......Page 453
Hartree's self-consistent fields......Page 454
Connection with the variation method......Page 455
Corrections to the central-field approximation......Page 456
LS coupling scheme......Page 457
48 The Alkali Atoms......Page 459
Doublet separation......Page 460
Doublet intensity......Page 461
Weak-field case......Page 463
Strong-field case......Page 465
Quadratic Zeeman effect......Page 466
49 Molecules......Page 468
Classification of energy levels......Page 469
Wave equation......Page 470
The hydrogen molecule......Page 472
Potential-energy function......Page 473
The Morse potential......Page 474
Energy levels......Page 476
Effect of nuclear identity......Page 477
General properties of nuclei......Page 478
Two-nucleon interaction......Page 479
Neutron-proton system......Page 480
Arbitrary shape of potential......Page 481
Relations for the
phase shift......Page 482
Effective range......Page 483
Exchange operators......Page 485
Problems......Page 486
Chapter 13 Relativistic Wave Equations......Page 489
Free particle......Page 490
Electromagnetic potentials......Page 491
Separation of the equation......Page 492
Energy levels in a coulomb field......Page 493
Free-particle equation......Page 495
Matrices for alpha and beta......Page 496
Free-particle solutions......Page 498
Charge and current densities......Page 500
Electromagnetic potentials......Page 501
Spin angular momentum......Page 503
Approximate reduction; spin-orbit energy......Page 504
Separation of the equation......Page 506
The hydrogen atom......Page 508
Negative energy states......Page 510
Problems......Page 511
Chapter 14 The Quantization of Wave Fields......Page 513
Coordinates of the field......Page 514
Classical lagrangian equation......Page 515
Functional derivative......Page 517
Classical hamiltonian equations......Page 518
Quantum equations for the field......Page 519
Fields with more than one component......Page 520
55 Quantization of the Nonrelativistic Schrodinger Equation......Page 521
Classical lagrangian and hamiltonian equations......Page 522
Quantum equations......Page 523
The N representation......Page 525
Creation, destruction, and number
operators......Page 526
Anticommutation relations......Page 527
Physical
implications of anticommutation......Page 529
Representation of the anticommuting a-sub-k operators......Page 530
56 Electromagnetic Field in Vacuum......Page 531
Lagrangian equations......Page 532
Hamiltonian equations......Page 533
Quantum equations......Page 534
Commutation relations for E and H......Page 536
Plane wave representation......Page 537
Quantized field energy......Page 539
Quantized field momentum.
A(r,t) in the plane wave representation......Page 541
Commutation relations
at different times......Page 542
Lagrangian and hamiltonian equations......Page 544
Elimination of phi......Page 545
Quantization of the fields......Page 548
Perturbation theory of the interparticle interaction......Page 549
Einstein-Bose case......Page 550
Radiation theory......Page 551
Transition probability for absorption......Page 553
Transition probability for emission......Page 554
Problems......Page 556
Index......Page 558
Back Cover......Page 568