MANY-BODY THEORY EXPOSED! : Propagator Description of Quantum Mechanics in Many-Body Systems

Standard textbooks on the many-body problem do not include a wealth of valuable experimental data, in particular recent results from direct knockout reactions, which are directly related to the single-particle propagator in many-body theory. In this indispensable book, the comparison with experimental data is incorporated from the start, making the abstract concept of propagators vivid and comprehensible. The discussion of numerical calculations using propagators or Green's functions, also absent from current textbooks, is presented in this book. Much of the material has been tested in the classroom and the introductory chapters allow a seamless connection with a one-year graduate course in quantum mechanics. While the majority of books on many-body theory deal with the subject from the viewpoint of condensed matter physics, this book also emphasizes finite systems and should be of considerable interest to researchers in nuclear, atomic, and molecular physics. A unified treatment of many different many-body systems is presented using the approach of self-consistent Green's functions. Several topics, not available in other books, in particular the description of atomic Bose-Einstein condensates, have been included. The coverage proceeds in a systematic way from elementary concepts, such as second quantization and mean-field properties, to a more advanced but self-contained presentation of the physics of atoms, molecules, nuclei, nuclear and neutron matter, electron gas, quantum liquids, atomic Bose-Einstein and fermion condensates, and pairing correlations in finite and infinite 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