Many-body theory exposed!: propagator description of quantum mechanics in many-body systems

Cover......Page 1
Half-title......Page 2
Title: Many-Body Theory Exposed!:  Propagator description of quantum mechanics in many-body systems by  Willem H Dickhoff & Dimitri Van Neck......Page 4
ISBN: 981256294X......Page 5
Dedication......Page 6
Preface......Page 8
Contents......Page 12
1.1 Some simple considerations......Page 20
1.2 Bosons and fermions......Page 22
1.3 Antisymmetric and symmetric two-particle states......Page 23
1.4 Some experimental consequences related to identical particles......Page 28
1.5 Antisymmetric and symmetric many-particle states......Page 30
1.6 Exercises......Page 34
2.1 Fermion addition and removal operators......Page 36
2.2 Boson addition and removal operators......Page 39
2.3 One-body operators in Fock space......Page 41
2.4 Two-body operators in Fock space......Page 43
2.5 Examples......Page 45
2.6 Exercises......Page 47
3.1 General results and the independent-particle model......Page 50
3.2 Electrons in atoms......Page 52
3.3 Nucleons in nuclei......Page 59
3.3.1 Empirical Mass Formula and Nuclear Matter......Page 66
3.4 Second quantization and isospin......Page 68
3.5 Exercises......Page 72
4.1 Symmetry considerations for two-particle states......Page 74
4.1.1 Free-particle states......Page 75
4.1.2 Pauli principle for two-particle states......Page 76
4.2 Two particles outside closed shells......Page 78
4.3 General discussion of two-body interactions......Page 82
4.4 Examples of relevant two-body interactions......Page 85
4.5 Exercises......Page 91
5.1 The Fermi gas at zero temperature......Page 92
5.2 Electron gas......Page 95
5.3 Nuclear and neutron matter......Page 98
5.4 Helium liquids......Page 100
5.5 Some statistical mechanics......Page 101
5.6.1 Bose-Einstein condensation in infinite systems......Page 103
5.6.2 Bose-Einstein condensation in traps......Page 106
5.6.3 Trapped bosons at finite temperature: thermodynamic considerations......Page 110
5.7.2 Fermion atoms in traps......Page 112
5.8 Exercises......Page 115
6.1 Time evolution and propagators......Page 116
6.2 Expansion of the propagator and diagram rules......Page 118
6.2.1 Diagram rules for the single-particle propagator......Page 119
6.3 Solution for discrete states......Page 123
6.4 Scattering theory using propagators......Page 126
6.4.1 Partial waves and phase shifts......Page 129
6.5 Exercises......Page 133
7. Single-particle propagator in the many-body system......Page 134
7.1 Fermion single-particle propagator......Page 135
7.2 Lehmann representation......Page 136
7.3 Spectral functions......Page 137
7.5 Propagator for noninteracting systems......Page 142
7.6 Direct knockout reactions......Page 144
7.7 Discussion of (e,2e) data for atoms......Page 147
7.8 Discussion of (e, e\'p) data for nuclei......Page 153
7.9 Exercises......Page 159
8.1 Time evolution in the interaction picture......Page 160
8.2 Perturbation expansion in the interaction......Page 162
8.3 Lowest-order contributions and diagrams......Page 164
8.4 Wick\'s theorem......Page 167
8.5 Diagrams......Page 173
8.6.1 Time-dependent version......Page 178
8.6.2 Energy formulation......Page 188
8.7 Exercises......Page 193
9. Dyson equation and self-consistent Green\'s functions......Page 194
9.1 Analysis of perturbation expansion, self-energy, and Dyson\'s equation......Page 196
9.2 Equation of motion method for propagators......Page 202
9.3 Two-particle propagator, vertex function, and self-energy......Page 204
9.4 Dyson equation and the vertex function......Page 209
9.5 Schrodinger-like equation from the Dyson equation......Page 213
9.6 Exercises......Page 215
10. Mean-field or Hartree-Fock approximation......Page 216
10.1.1 Derivation of the Hartree-Fock equations......Page 217
10.1.2 The Hartree-Fock propagator......Page 221
10.1.3 Variational content of the HF approximation......Page 225
10.1.4 HF in coordinate space......Page 228
10.1.5 Unrestricted and restricted Hartree-Fock......Page 229
10.2.1 Closed-shell configurations......Page 232
10.2.2 Comparison with experimental data......Page 235
10.2.3 Numerical details......Page 236
10.2.4 Computer exercise......Page 238
10.3.1 Molecular problems......Page 240
10.3.2 Hartree-Fock with a finite discrete basis set......Page 242
10.3.3 The hydrogen molecule......Page 244
10.4 Hartree-Fock in infinite systems......Page 250
10.5 Electron gas......Page 252
10.6 Nuclear matter......Page 256
10.7 Exercises......Page 258
11. Beyond the mean-field approximation......Page 260
11.1 The second-order self-energy......Page 261
11.2 Solution of the Dyson equation......Page 264
11.2.1 Diagonal approximation......Page 265
11.2.2 Link with perturbation theory......Page 269
11.2.3 Sum rules......Page 270
11.2.4 General (nondiagonal) self-energy......Page 272
11.3.1 Dispersion relations......Page 276
11.3.2 Behavior near the Fermi energy......Page 278
11.3.3 Spectral function......Page 280
11.4 Exact self-energy in infinite systems......Page 282
11.4.2 Self-energy and spectral function......Page 283
11.4.3 Quasiparticles......Page 284
11.4.4 Migdal-Luttinger theorem......Page 287
11.4.5 Quasiparticle propagation and lifetime......Page 288
11.5 Self-consistent treatment of \\Sigma^(2)......Page 289
11.5.1 Schematic model......Page 291
11.5.2 Nuclei......Page 293
11.5.3 Atoms......Page 294
11.6 Exercises......Page 296
12. Interacting boson systems......Page 298
12.1.1 Boson single-particle propagator......Page 299
12.1.2 Noninteracting boson propagator......Page 300
12.1.3 The condensate in an interacting Bose system......Page 301
12.1.4 Equations of motion......Page 303
12.2.1 Breakdown of Wick\'s theorem......Page 304
12.2.2 Equivalent fermion problem......Page 305
12.3.1 Derivation of the Hartree-Bose equation......Page 306
12.3.3 Physical interpretation......Page 308
12.3.4 Variational content......Page 309
12.3.5 Hartree-Bose expressions in coordinate space......Page 310
12.4.1 Pseudopotential......Page 311
12.4.2 Quick reminder of low-energy scattering......Page 313
12.4.3 The T-matrix......Page 316
12.4.4 Gross-Pitaevskii equation......Page 320
12.4.5 Confined bosons in harmonic traps......Page 321
12.4.6 Numerical solution of the GP equation......Page 328
12.4.7 Computer exercise......Page 330
12.5 Exercises......Page 332
13. Excited states in finite systems......Page 334
13.1 Polarization propagator......Page 335
13.2 Random Phase Approximation......Page 340
13.3 RPA in finite systems and the schematic model......Page 345
13.4 Energy-weighted sum rule......Page 351
13.5 Excited states in atoms......Page 355
13.6 Correlation energy and ring diagrams......Page 359
13.7 RPA in angular momentum coupled representation......Page 361
13.8 Exercises......Page 365
14.1 RPA in infinite systems......Page 366
14.2 Lowest-order polarization propagator in an infinite system......Page 371
14.3 Plasmons in the electron gas......Page 378
14.4.1 Correlation energy and the polarization propagator......Page 386
14.4.2 Correlation energy of the electron gas in RPA......Page 388
14.5 Response of nuclear matter with \\pi and \\rho-meson quantum numbers......Page 389
14.6 Excitations of a normal Fermi liquid......Page 400
14.7 Exercises......Page 415
15. Excited states in N ± 2 systems and in-medium scattering......Page 416
15.1 Two-time two-particle propagator......Page 417
15.1.1 Scattering of two particles in free space......Page 423
15.1.2 Bound states of two particles......Page 429
15.2 Ladder diagrams and short-range correlations in the medium......Page 432
15.2.1 Scattering of mean-field particles in the medium......Page 436
15.3 Cooper problem and pairing instability......Page 442
15.4 Exercises......Page 451
16. Dynamical treatment of the self-energy in infinite systems......Page 454
16.1 Diagram rules in uniform systems......Page 455
16.2.1 Electron self-energy in the G^W^ approximation......Page 459
16.2.2 Electron self-energy in the GW approximation......Page 467
16.2.3 Energy per particle of the electron gas......Page 475
16.3.1 Ladder diagrams and the self-energy......Page 477
16.3.2 Spectral function obtained from mean-field input......Page 479
16.3.3 Self-consistent spectral functions......Page 485
16.3.4 Saturation properties of nuclear matter......Page 488
16.4 Exercises......Page 500
17. Dynamical treatment of the self-energy in finite systems......Page 502
17.1.1 Second-order effects with G-matrix interactions......Page 504
17.1.2 Inclusion of collective excitations in the self-energy......Page 507
17.2 Self-consistent pphh RPA in finite systems......Page 515
17.3 Short-range correlations in finite nuclei......Page 524
17.4 Properties of protons in nuclei......Page 538
17.5 Exercises......Page 541
18.1 The Bose gas......Page 542
18.2 Bogoliubov prescription......Page 544
18.2.1 Particle-number nonconservation......Page 546
18.2.2 The chemical potential......Page 548
18.2.3 Propagator......Page 550
18.3 Bogoliubov perturbation expansion......Page 553
18.4 Hugenholtz-Pines theorem......Page 561
18.5 First-order results......Page 566
18.6 Dilute Bose gas with repulsive forces......Page 569
18.7 Canonical transformation for the Bose gas......Page 573
18.8 Exercises......Page 577
19.1.1 The He-II phase......Page 580
19.1.2 Phenomenological descriptions......Page 582
19.2.1 Inclusive scattering......Page 586
19.2.2 Asymptotic 1/Q expansion of the structure function......Page 589
19.3.1 The bosonic Bogoliubov transformation......Page 595
19.3.2 Bogoliubov prescription for nonuniform systems......Page 604
19.3.3 Bogoliubov-de Gennes equations......Page 605
19.4 Number-conserving approach......Page 608
19.5 Exercises......Page 609
20. In-medium interaction and scattering of dressed particles......Page 610
20.1 Propagation of dressed particles in wave-vector space......Page 611
20.2 Propagation of dressed particles in coordinate space......Page 619
20.3 Scattering of particles in the medium......Page 627
20.4 Exercises......Page 636
21. Conserving approximations and excited states......Page 638
21.1 Equations of motion and conservation laws......Page 639
21.1.1 The field picture......Page 640
21.1.2 Equations of motion in the field picture......Page 642
21.1.3 Conservation laws and approximations......Page 646
21.2 Linear response and extensions of RPA......Page 648
21.2.1 Brief encounter with functional derivatives......Page 649
21.2.2 Linear response and functional derivatives......Page 650
21.3 Ward-Pitaevskii relations for a Fermi liquid......Page 653
21.4.1 Hartree-Fock and the RPA approximation......Page 659
21.4.2 Second-order self-energy and the particle-hole interaction......Page 660
21.4.3 Extension of the RPA including second-order terms......Page 662
21.4.4 Practical ingredients of ERPA calculations......Page 665
21.4.5 Ring diagram approximation and the polarization propagator......Page 670
21.5 Excited states in nuclei......Page 673
21.6 Exercises......Page 681
22.1 General considerations......Page 682
22.2 Anomalous propagators in the Fermi gas......Page 685
22.3 Diagrammatic expansion in a superconducting system......Page 687
22.4 The BCS gap equation......Page 694
22.5 Canonical BCS transformation......Page 702
22.6.1 Superconductivity in metals......Page 707
22.6.3 Superfluidity in neutron stars......Page 710
22.7 Inhomogeneous systems......Page 711
22.8 Exact solutions of schematic pairing problems......Page 716
22.8.1 Richardson-Gaudin equations......Page 720
22.9 Exercises......Page 721
A.1 Schrodinger picture......Page 722
A.2 Interaction picture......Page 723
A.3 Heisenberg picture......Page 727
Appendix B: Practical results from angular momentum algebra......Page 730
Bibliography......Page 736
Index......Page 748