Quantum Mechanics with Basic Field Theory

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 19
Physical constants......Page 22
1.1 State vectors......Page 23
1.2 Operators and physical observables......Page 25
1.3 Eigenstates......Page 26
1.4 Hermitian conjugation and Hermitian operators......Page 27
1.5 Hermitian operators: their eigenstates and eigenvalues......Page 28
1.6 Superposition principle......Page 29
1.7 Completeness relation......Page 30
1.8 Unitary operators......Page 31
1.9 Unitary operators as transformation operators......Page 32
1.10 Matrix formalism......Page 34
1.11 Eigenstates and diagonalization of matrices......Page 38
1.11.1 Diagonalization through unitary operators......Page 39
1.12 Density operator......Page 40
1.13 Measurement......Page 42
1.14 Problems......Page 43
2.1 Continuous variables: X and P operators......Page 46
2.2 Canonical commutator [X,P]......Page 48
2.3 P as a derivative operator: another way......Page 51
2.4 X and P as Hermitian operators......Page 52
2.5 Uncertainty principle......Page 54
2.6.2 Bohr radius and ground-state energy of the hydrogen atom......Page 57
2.7 Space displacement operator......Page 58
2.8 Time evolution operator......Page 63
2.9.1 Dirac delta function......Page 66
Representations of the delta-function......Page 68
Three dimensions......Page 71
2.10 Problems......Page 74
3.1 Schrödinger picture......Page 77
3.2 Heisenberg picture......Page 79
3.3 Interaction picture......Page 81
3.4 Superposition of time-dependent states and energy–time uncertainty relation......Page 85
3.4.1 Virtual particles......Page 87
3.5 Time dependence of the density operator......Page 88
3.6 Probability conservation......Page 89
3.7 Ehrenfest\'s theorem......Page 90
3.8 Problems......Page 92
4.1 Free particle in one dimension......Page 95
4.2 Normalization......Page 97
4.3 Momentum eigenfunctions and Fourier transforms......Page 100
4.4 Minimum uncertainty wave packet......Page 101
4.5 Group velocity of a superposition of plane waves......Page 105
4.6 Three dimensions – Cartesian coordinates......Page 106
4.7 Three dimensions – spherical coordinates......Page 109
4.8 The radial wave equation......Page 113
4.9 Properties of Ylm(theta, phi)......Page 114
4.10 Angular momentum......Page 116
4.10.1 Angular momentum in classical physics......Page 118
4.11 Determining L2 from the angular variables......Page 119
4.12 Commutator [Li,Lj] and [L2,Lj]......Page 120
4.13 Ladder operators......Page 122
4.14 Problems......Page 124
5.1 Spin ½ system......Page 125
5.2 Pauli matrices......Page 126
5.3 The spin ½ eigenstates......Page 127
5.4 Matrix representation of sigmax and sigmay......Page 128
5.5 Eigenstates of sigmax and sigmay......Page 130
5.6 Eigenstates of spin in an arbitrary direction......Page 131
5.7 Some important relations for sigmai......Page 132
5.8 Arbitrary 2×2 matrices in terms of Pauli matrices......Page 133
5.9 Projection operator for spin ½ systems......Page 134
5.10 Density matrix for spin ½ states and the ensemble average......Page 136
5.12 Pauli exclusion principle and Fermi energy......Page 138
5.13 Problems......Page 140
6.1 Gauge invariance......Page 142
6.2 Quantum mechanics......Page 143
6.3 Canonical and kinematic momenta......Page 145
6.4 Probability conservation......Page 146
6.5 Interaction with the orbital angular momentum......Page 147
6.6 Interaction with spin: intrinsic magnetic moment......Page 148
6.7 Spin–orbit interaction......Page 150
6.8 Aharonov–Bohm effect......Page 151
6.9 Problems......Page 153
7.1 Experimental set-up and electron\'s magnetic moment......Page 155
7.2 Discussion of the results......Page 156
7.3 Problems......Page 158
8.1.1 Infinite potential barrier......Page 159
8.1.2 Finite potential barrier......Page 162
8.2 Delta-function potential......Page 167
8.3 Properties of a symmetric potential......Page 169
8.4 The ammonia molecule......Page 170
8.5 Periodic potentials......Page 173
8.6.1 Separable potentials in Cartesian coordinates......Page 178
8.6.2 Potentials depending on relative distance between particles......Page 179
8.6.3 Formalism for spherically symmetric potentials......Page 181
8.7.1 Spherical wall......Page 182
8.7.2 Finite barrier......Page 183
8.7.3 Square-well potential......Page 185
8.8 Hydrogen-like atom......Page 186
8.9 Problems......Page 192
9.1.1 Formalism in the Heisenberg representation......Page 196
a† and a as raising and lowering operators......Page 197
Determination of the energy eigenvalue, En......Page 198
a and a†, again......Page 199
Time dependence......Page 200
The wavefunctions......Page 201
9.1.2 Formalism in the Schrödinger picture......Page 203
9.2 Problems......Page 206
10.1 Eigenstates of the lowering operator......Page 209
10.2 Coherent states and semiclassical description......Page 214
10.3 Interaction of a harmonic oscillator with an electric field......Page 216
10.4.1 Some important identities involving exponentials of operators......Page 221
10.5 Problems......Page 222
11.1.1 Eigenstates |nx, ny> of H......Page 225
11.1.2 Eigenstates |n+, n-> of H and Lz......Page 226
11.2 Problems......Page 229
12.1.1 Basic equations......Page 230
12.1.3 Comparison with the two-dimensional oscillator and the degeneracy question......Page 232
12.2 Wavefunctions for the LLL......Page 234
12.3 Landau levels in Landau gauge......Page 236
12.4 Quantum Hall effect......Page 238
12.5 Wavefunction for filled LLLs in a Fermi system......Page 242
12.6 Problems......Page 243
13.1.1 Basic formalism......Page 245
13.1.2 Simple example......Page 247
13.1.4 More examples......Page 251
13.1.5 Level crossing and switching of eigenstates......Page 253
13.1.6 Relation between energy eigenstates and eigenvalues in two different frameworks......Page 255
13.2.1 Basic formalism......Page 256
13.2.2 Constant perturbation......Page 258
13.2.3 Mixing angles and Rabi’s formula......Page 259
13.2.4 Harmonic time dependence......Page 263
13.2.5 Ammonia maser......Page 266
13.3 Problems......Page 268
14.1 Constant magnetic field......Page 273
14.1.1 Initial spin in the z-direction......Page 274
14.1.2 Initial spin in the x-direction......Page 275
14.2 Spin precession......Page 276
14.3 Time-dependent magnetic field: spin magnetic resonance......Page 277
14.3.1 Initial spin in the z-direction......Page 278
14.3.2 Spin aligned with the magnetic field......Page 279
14.4 Problems......Page 280
15.2 The solar neutrino puzzle......Page 282
15.3 Neutrino oscillations......Page 285
15.4.1 Basic formalism......Page 287
15.4.2 Current conservation with complex potentials......Page 289
15.5 Oscillation and regeneration of stable and unstable systems......Page 291
15.6 Neutral K-mesons......Page 295
15.6.1 The K0…......Page 296
15.6.2 Regeneration of K-mesons......Page 297
15.7 Problems......Page 298
16.1 Basic formalism......Page 299
16.2.1 Energy levels......Page 303
16.2.2 Wavefunctions......Page 305
16.3 Second-order Stark effect......Page 306
16.4 Degenerate states......Page 309
16.5 Linear Stark effect......Page 311
16.6 Problems......Page 312
17.1 Basic formalism......Page 315
17.2 Harmonic perturbation and Fermi\'s golden rule......Page 318
17.3.1 Harmonic perturbation......Page 321
17.3.2 Free particle scattering......Page 323
17.4.1 Basic formalism......Page 325
17.5.1 Adiabatic perturbation......Page 332
17.5.2 Berry’s phase......Page 334
Example: Spin aligned with the magnetic field......Page 336
17.6 Problems......Page 337
18.1 Electron in an electromagnetic field: the absorption cross-section......Page 340
18.2 Photoelectric effect......Page 345
18.3 Coulomb excitations of an atom......Page 347
18.4 Ionization......Page 350
18.5 Thomson, Rayleigh, and Raman scattering in second-order perturbation......Page 353
18.5.1 Thomson scattering......Page 358
18.5.2 Rayleigh scattering......Page 359
18.6 Problems......Page 361
19.1 Reflection and transmission coefficients......Page 364
19.2 Infinite barrier......Page 366
19.3 Finite barrier with infinite range......Page 367
19.4 Rigid wall preceded by a potential well......Page 370
19.5 Square-well potential and resonances......Page 373
19.6 Tunneling......Page 376
19.7 Problems......Page 378
20.1 Formal solutions in terms of Green\'s function......Page 380
20.2 Lippmann–Schwinger equation......Page 382
20.3 Born approximation......Page 385
20.4 Scattering from a Yukawa potential......Page 386
20.5 Rutherford scattering......Page 387
20.6 Charge distribution......Page 388
20.7 Probability conservation and the optical theorem......Page 389
20.8 Absorption......Page 392
20.9.1 The optical theorem......Page 394
20.10.1 Basic Formalism......Page 396
20.10.2 The S-matrix......Page 399
20.11 Unitarity of the S-matrix and the relation between S and T......Page 400
20.12 Properties of the T-matrix and the optical theorem (again)......Page 404
20.13.1 Integrals involved in Green’s function calculations......Page 405
20.14 Problems......Page 406
21.1 Scattering amplitude in terms of phase shifts......Page 408
21.1.1 Comparing Rl for V = 0 and V = V(r)…......Page 413
21.2 X…......Page 414
21.3 Integral relations for…......Page 415
21.4.1 Same potentials......Page 417
Sign of deltal......Page 418
Ramsauer–Townsend effect......Page 419
Dependence of deltal on l......Page 420
Born approximation......Page 421
21.5.1 Square well......Page 422
Approximate properties of delta0......Page 423
21.5.2 Rigid sphere......Page 424
21.5.3 Absorption and inelastic amplitudes......Page 425
21.6 Problems......Page 427
22.1.1 Bound states......Page 429
22.1.2 Resonances......Page 431
Bound states for…......Page 432
22.1.3 Resonance as complex poles of the partial wave S-matrix......Page 434
22.2 Jost function formalism......Page 435
22.2.1 Zeros of F(k)......Page 438
22.2.2 Representation of F0(k) and S0(k) in terms of zeros and poles......Page 439
22.2.3 Residue of the pole......Page 440
22.2.4 Phase shift delta0(k)......Page 441
22.3 Levinson\'s theorem......Page 442
22.4 Explicit calculation of the Jost function for l = 0......Page 443
22.5 Integral representation of F0(k)......Page 446
22.6 Problems......Page 448
23.1 Relation between the time-evolution operator and the Green\'s function......Page 449
23.2 Stable and unstable states......Page 451
23.3 Scattering amplitude and resonance......Page 452
23.5 Two types of resonances......Page 453
23.6 The reaction matrix......Page 454
23.6.1 Relation between R- and T-matrices......Page 457
23.6.2 Relation between R and G......Page 458
23.6.3 Properties of Rss(E)......Page 461
23.7.1 Resonant photon–atom scattering......Page 464
23.7.2 Resonant electron–atom scattering......Page 467
23.8.1 Dispersion relations......Page 469
24.1.1 Introduction......Page 472
24.1.2 Energy levels of a particle trapped inside a potential......Page 476
24.1.3 Tunneling through a barrier......Page 477
24.2 Variational method......Page 480
24.2.1 Helium atom......Page 481
24.3 Eikonal approximation......Page 483
24.3.1 Absorption......Page 486
24.3.2 Perfect absorption......Page 487
24.4 Problems......Page 488
25.1 Euler--Lagrange equations......Page 491
25.2 N oscillators and the continuum limit......Page 493
25.3.1 Time evolution operator and Green’s function......Page 495
25.3.2 N-intervals......Page 498
25.4 Problems......Page 500
26.1 Rotation of coordinate axes......Page 501
26.1.1 Infinitesimal transformations......Page 503
26.2 Scalar functions and orbital angular momentum......Page 505
26.3 State vectors......Page 507
26.4 Transformation of matrix elements and representations of the rotation operator......Page 509
26.5 Generators of infinitesimal rotations: their eigenstates and eigenvalues......Page 511
26.6 Representations of J2 and Ji for j = 1/2 and j = 1......Page 516
26.7.1 Ladder operators and eigenfunctions......Page 517
26.7.2 Explicit expression for Ylm(theta, phi)......Page 519
26.7.3 Properties of Ylm(theta, phi)......Page 521
26.8 Problems......Page 523
27.1 Rotational symmetry......Page 524
27.2.1 Consequences of parity transformation......Page 527
27.3 Time reversal......Page 529
27.3.1 Correct form of T......Page 530
Orbital angular momentum......Page 532
27.4.1 Definition of a group......Page 533
Some important examples......Page 534
Lie algebra......Page 535
27.5.1 Spin j = 1/2......Page 536
27.5.2 Spin j = 1......Page 537
27.6 Problems......Page 538
28.1 Combining eigenstates: simple examples......Page 540
28.1.1 j1 = 1/2, j2 = 1/2......Page 541
28.2 Clebsch–Gordan coefficients and their recursion relations......Page 544
28.3 Combining spin ½ and orbital angular momentum l......Page 546
28.4.1 Table of Clebsch–Gordan coefficients......Page 549
28.5 Problems......Page 550
29.1 Irreducible spherical tensors and their properties......Page 551
29.2 The irreducible tensors: Ylm(theta, phi) and Dj(χ)......Page 555
29.3 Wigner–Eckart theorem......Page 558
29.4 Applications of the Wigner–Eckart theorem......Page 560
29.5.1 Constructing irreducible tensors......Page 563
29.5.3 Two spin ½ particles......Page 567
29.5.4 Higher systems......Page 569
29.6 Problems......Page 570
30.1 Definition of an entangled state......Page 571
30.2 The singlet state......Page 573
30.3 Differentiating the two approaches......Page 574
30.4 Bell\'s inequality......Page 575
30.5 Problems......Page 577
31.1 Lorentz transformation......Page 578
31.2 Contravariant and covariant vectors......Page 579
31.3 An example of a covariant vector......Page 582
31.4 Generalization to arbitrary tensors......Page 583
31.5 Relativistically invariant equations......Page 585
31.5.1 Electromagnetism......Page 586
31.5.2 Classical mechanics......Page 588
31.6.1 Infinitesimal Lorentz transformations: rotations and “pure” Lorentz transformations......Page 591
31.7 Problems......Page 594
32.1 Covariant equations in quantum mechanics......Page 597
32.2.1 Current conservation and negative energies......Page 598
32.3 Normalization of matrix elements......Page 600
32.4.1 Gauge invariance and zero mass......Page 601
32.5.1 Maxwell’s equation......Page 603
32.5.2 Klein–Gordon equation......Page 605
32.7.1 Klein–Gordon equation......Page 608
32.8.1 Coulomb potential......Page 609
32.8.2 Yukawa potential......Page 610
32.9 Scattering interpreted as an exchange of virtual particles......Page 611
32.9.2 Rutherford scattering as due to the exchange of a zero-mass particle (photon)......Page 612
32.9.3 Yukawa scattering as due to the exchange of a massive particle......Page 614
32.10.1 Hydrogen atom......Page 615
33.1 Basic formalism......Page 619
33.2 Standard representation and spinor solutions......Page 622
33.3 Large and small components of u(p)......Page 623
33.3.1 Positive-energy solution......Page 624
33.3.2 Negative-energy solutions......Page 626
33.4 Probability conservation......Page 627
33.5 Spin ½ for the Dirac particle......Page 629
34.1 Spin–orbit coupling......Page 633
34.2 K-operator for the spherically symmetric potentials......Page 635
34.2.1 Nonrelativistic limit......Page 637
34.3 Hydrogen atom......Page 638
34.4 Radial Dirac equation......Page 640
34.5 Hydrogen atom states......Page 645
34.6 Hydrogen atom wavefunction......Page 646
34.7.1 The commutator [K,H]......Page 648
34.7.2 Derivation of the spin–orbit term......Page 650
35.1 Covariant Dirac equation......Page 653
35.2 Properties of the gamma-matrices......Page 654
35.3 Charge–current conservation in a covariant form......Page 655
35.3.1 Derivation directly from the Dirac equation......Page 656
35.4 Spinor solutions: ur(p) and vr(p)......Page 657
35.5 Normalization and completeness condition for ur(p) and vr(p)......Page 658
35.5.1 Projection operators......Page 661
35.6 Gordon decomposition......Page 662
35.7 Lorentz transformation of the Dirac equation......Page 664
35.7.1 Bilinear covariant terms......Page 665
35.8.1 Further properties of gamma-matrices......Page 666
35.8.2 Trace of products of the form (gamma · a1gamma · a2 · · · )......Page 668
36.1 Charged particle Hamiltonian......Page 669
36.2 Deriving the equation another way......Page 672
36.3 Gordon decomposition and electromagnetic current......Page 673
36.4 Dirac equation with electromagnetic field and comparison with the Klein–Gordon equation......Page 675
36.5 Propagators: the Dirac propagator......Page 677
36.6.1 Rutherford scattering......Page 679
36.7.1 Trace properties of matrix elements and summation over spins......Page 683
37.1 Wavefunctions for identical particles......Page 685
37.2 Occupation number space and ladder operators......Page 686
37.3.1 Symmetric case......Page 688
37.3.2 Antisymmetric case......Page 689
37.4 Writing single-particle relations in multiparticle language: the operators, N, H, and P......Page 692
37.5 Matrix elements of a potential......Page 693
37.6 Free fields and continuous variables......Page 694
37.7 Klein–Gordon/scalar field......Page 696
37.7.1 Second quantization......Page 697
37.8 Complex scalar field......Page 700
37.9 Dirac field......Page 702
37.10 Maxwell field......Page 705
37.11 Lorentz covariance for Maxwell field......Page 709
37.12.1 Scalar field......Page 710
37.12.3 Dirac field......Page 711
37.13 Canonical quantization......Page 712
37.14 Casimir effect......Page 715
37.15 Problems......Page 719
38.1.1 One dimension......Page 721
38.1.2 Three dimensions......Page 723
38.1.4 Heavy nucleus......Page 725
38.2 Interacting electron gas......Page 726
38.2.1 Calculating Hel......Page 728
38.3 Phonons......Page 730
38.4 Electron–phonon interaction......Page 735
38.4.1 Photon exchange......Page 736
38.4.2 Phonon exchange......Page 738
39.1 Many-body system of half-integer spins......Page 741
39.2 Normal states…......Page 746
39.3.1 BCS ground state......Page 747
39.3.2 Excited states and the gap function......Page 748
39.4 BCS condensate in Green\'s function formalism......Page 749
39.5 Meissner effect......Page 754
39.5.1 Ginzburg–Landau equation......Page 755
39.6 Problems......Page 757
40.1.1 Ground state and quasiparticles......Page 758
40.2 Superfluidity......Page 762
40.3 Problems......Page 764
41.1 Basic structure......Page 765
41.2 Noether\'s theorem......Page 766
41.3.1 Klein–Gordon field......Page 768
Conserved current......Page 770
Conserved current......Page 771
41.4 Maxwell\'s equations and consequences of gauge invariance......Page 772
41.4.2 Maxwell’s equations with Dirac and scalar particles......Page 775
Gauge theory......Page 776
42.1 BCS mechanism......Page 777
42.2 Ferromagnetism......Page 778
42.3 SSB for discrete symmetry in classical field theory......Page 780
42.4 SSB for continuous symmetry......Page 782
42.5 Nambu–Goldstone bosons......Page 784
42.5.1 Examples......Page 786
42.6 Higgs mechanism......Page 787
43.1 Perturbation theory......Page 792
43.2 Feynman diagrams......Page 795
43.3 T(HI (x1)HI (x2)) and Wick’s theorem......Page 799
43.5 Cross-section for 1…......Page 805
43.7.1 Rutherford scattering (nonrelativistic)......Page 808
43.7.2 Mott scattering (Dirac)......Page 809
43.7.3 Møller scattering…......Page 810
43.8.1 Thomson scattering......Page 811
43.8.2 Compton scattering…......Page 812
43.8.3 Electron–positron annihilation, pair production…......Page 813
43.8.4 Electron–positron scattering, Bhabha scattering…......Page 814
44.1 Radiative corrections and renormalization......Page 815
44.2 Electron self-energy......Page 816
44.3.1 Ward Identity......Page 821
44.3.2 Ward identity, vertex function, and photon propagator......Page 822
44.3.3 Feynman integration technique......Page 823
44.3.4 Dimensional regularization......Page 824
45.2 Vertex function and the magnetic moment......Page 828
45.3 Calculation of the vertex function diagram......Page 830
45.4 Divergent part of the vertex function......Page 832
45.5 Radiative corrections to the photon propagator......Page 833
45.6 Divergent part of the photon propagator......Page 835
45.7 Modification of the photon propagator and photon wavefunction......Page 836
45.8 Combination of all the divergent terms: basic renormalization......Page 838
45.9.1 Anomalous magnetic moment......Page 839
45.9.2 Lamb shift......Page 840
45.10.1 The photon propagator integral......Page 843
45.10.2 Calculating…......Page 844
Relativistic quantum mechanics and quantum field theory......Page 847
Gauge theory and elementary particles......Page 848
Mathematical physics......Page 849
Index......Page 850