
Many-body Theory Exposed!
Propagator Description Of Quantum Mechanics In Many-body Systems
By:Â Dimitri V Y Van Neck, Willem Hendrik Dickhoff
Hardcover | 21 April 2005
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752 Pages
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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.
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| Preface | p. vii |
| Identical particles | p. 1 |
| Some simple considerations | p. 1 |
| Bosons and fermions | p. 3 |
| Antisymmetric and symmetric two-particle states | p. 4 |
| Some experimental consequences related to identical particles | p. 9 |
| Antisymmetric and symmetric many-particle states | p. 11 |
| Exercises | p. 15 |
| Second quantization | p. 17 |
| Fermion addition and removal operators | p. 17 |
| Boson addition and removal operators | p. 20 |
| One-body operators in Fock space | p. 22 |
| Two-body operators in Fock space | p. 24 |
| Examples | p. 26 |
| Exercises | p. 28 |
| Independent-particle model for fermions in finite systems | p. 31 |
| General results and the independent-particle model | p. 31 |
| Electrons in atoms | p. 33 |
| Nucleons in nuclei | p. 40 |
| Empirical Mass Formula and Nuclear Matter | p. 47 |
| Second quantization and isospin | p. 49 |
| Exercises | p. 53 |
| Two-particle states and interactions | p. 55 |
| Symmetry considerations for two-particle states | p. 55 |
| Free-particle states | p. 56 |
| Pauli principle for two-particle states | p. 57 |
| Two particles outside closed shells | p. 59 |
| General discussion of two-body interactions | p. 63 |
| Examples of relevant two-body interactions | p. 66 |
| Exercises | p. 72 |
| Noninteracting bosons and fermions | p. 73 |
| The Fermi gas at zero temperature | p. 73 |
| Electron gas | p. 76 |
| Nuclear and neutron matter | p. 79 |
| Helium liquids | p. 81 |
| Some statistical mechanics | p. 82 |
| Bosons at finite T | p. 84 |
| Bose-Einstein condensation in infinite systems | p. 84 |
| Bose-Einstein condensation in traps | p. 87 |
| Trapped bosons at finite temperature: thermodynamic considerations | p. 91 |
| Fermions at finite T | p. 93 |
| Noninteracting fermion systems | p. 93 |
| Fermion atoms in traps | p. 93 |
| Exercises | p. 96 |
| Propagators in one-particle quantum mechanics | p. 97 |
| Time evolution and propagators | p. 97 |
| Expansion of the propagator and diagram rules | p. 99 |
| Diagram rules for the single-particle propagator | p. 100 |
| Solution for discrete states | p. 104 |
| Scattering theory using propagators | p. 107 |
| Partial waves and phase shifts | p. 110 |
| Exercises | p. 114 |
| Single-particle propagator in the many-body system | p. 115 |
| Fermion single-particle propagator | p. 116 |
| Lehmann representation | p. 117 |
| Spectral functions | p. 118 |
| Expectation values of operators in the correlated ground state | p. 121 |
| Propagator for noninteracting systems | p. 123 |
| Direct knockout reactions | p. 125 |
| Discussion of (e, 2e) data for atoms | p. 128 |
| Discussion of (e, e[prime]p) data for nuclei | p. 134 |
| Exercises | p. 140 |
| Perturbation expansion of the single-particle propagator | p. 141 |
| Time evolution in the interaction picture | p. 141 |
| Perturbation expansion in the interaction | p. 143 |
| Lowest-order contributions and diagrams | p. 145 |
| Wick's theorem | p. 148 |
| Diagrams | p. 154 |
| Diagram rules | p. 159 |
| Time-dependent version | p. 159 |
| Energy formulation | p. 169 |
| Exercises | p. 174 |
| Dyson equation and self-consistent Green's functions | p. 175 |
| Analysis of perturbation expansion, self-energy, and Dyson's equation | p. 177 |
| Equation of motion method for propagators | p. 183 |
| Two-particle propagator, vertex function, and self-energy | p. 185 |
| Dyson equation and the vertex function | p. 190 |
| Schrodinger-like equation from the Dyson equation | p. 194 |
| Exercises | p. 196 |
| Mean-field or Hartree-Fock approximation | p. 197 |
| The Hartree-Fock formalism | p. 198 |
| Derivation of the Hartree-Fock equations | p. 198 |
| The Hartree-Fock propagator | p. 202 |
| Variational content of the HF approximation | p. 206 |
| HF in coordinate space | p. 209 |
| Unrestricted and restricted Hartree-Fock | p. 210 |
| Atoms | p. 213 |
| Closed-shell configurations | p. 213 |
| Comparison with experimental data | p. 216 |
| Numerical details | p. 217 |
| Computer exercise | p. 219 |
| Molecules | p. 221 |
| Molecular problems | p. 221 |
| Hartree-Fock with a finite discrete basis set | p. 223 |
| The hydrogen molecule | p. 225 |
| Hartree-Fock in infinite systems | p. 231 |
| Electron gas | p. 233 |
| Nuclear matter | p. 237 |
| Exercises | p. 239 |
| Beyond the mean-field approximation | p. 241 |
| The second-order self-energy | p. 242 |
| Solution of the Dyson equation | p. 245 |
| Diagonal approximation | p. 246 |
| Link with perturbation theory | p. 250 |
| Sum rules | p. 251 |
| General (nondiagonal) self-energy | p. 253 |
| Second order in infinite systems | p. 257 |
| Dispersion relations | p. 257 |
| Behavior near the Fermi energy | p. 259 |
| Spectral function | p. 261 |
| Exact self-energy in infinite systems | p. 263 |
| General considerations | p. 264 |
| Self-energy and spectral function | p. 264 |
| Quasiparticles | p. 265 |
| Migdal-Luttinger theorem | p. 268 |
| Quasiparticle propagation and lifetime | p. 269 |
| Self-consistent treatment of [sigma superscript (2)] | p. 270 |
| Schematic model | p. 272 |
| Nuclei | p. 274 |
| Atoms | p. 275 |
| Exercises | p. 277 |
| Interacting boson systems | p. 279 |
| General considerations | p. 280 |
| Boson single-particle propagator | p. 280 |
| Noninteracting boson propagator | p. 281 |
| The condensate in an interacting Bose system | p. 282 |
| Equations of motion | p. 284 |
| Perturbation expansions and the condensate | p. 285 |
| Breakdown of Wick's theorem | p. 285 |
| Equivalent fermion problem | p. 286 |
| Hartree-Bose approximation | p. 287 |
| Derivation of the Hartree-Bose equation | p. 287 |
| Hartree-Bose ground-state energy | p. 289 |
| Physical interpretation | p. 289 |
| Variational content | p. 290 |
| Hartree-Bose expressions in coordinate space | p. 291 |
| Gross-Pitaevskii equation for dilute systems | p. 292 |
| Pseudopotential | p. 292 |
| Quick reminder of low-energy scattering | p. 294 |
| The T-matrix | p. 297 |
| Gross-Pitaevskii equation | p. 301 |
| Confined bosons in harmonic traps | p. 302 |
| Numerical solution of the GP equation | p. 309 |
| Computer exercise | p. 311 |
| Exercises | p. 313 |
| Excited states in finite systems | p. 315 |
| Polarization propagator | p. 316 |
| Random Phase Approximation | p. 321 |
| RPA in finite systems and the schematic model | p. 326 |
| Energy-weighted sum rule | p. 332 |
| Excited states in atoms | p. 336 |
| Correlation energy and ring diagrams | p. 340 |
| RPA in angular momentum coupled representation | p. 342 |
| Exercises | p. 346 |
| Excited states in infinite systems | p. 347 |
| RPA in infinite systems | p. 347 |
| Lowest-order polarization propagator in an infinite system | p. 352 |
| Plasmons in the electron gas | p. 359 |
| Correlation energy | p. 367 |
| Correlation energy and the polarization propagator | p. 367 |
| Correlation energy of the electron gas in RPA | p. 369 |
| Response of nuclear matter with [pi] and [rho] meson quantum numbers | p. 370 |
| Excitations of a normal Fermi liquid | p. 381 |
| Exercises | p. 396 |
| Excited states in N [plus or minus] 2 systems and in-medium scattering | p. 397 |
| Two-time two-particle propagator | p. 398 |
| Scattering of two particles in free space | p. 404 |
| Bound states of two particles | p. 410 |
| Ladder diagrams and short-range correlations in the medium | p. 413 |
| Scattering of mean-field particles in the medium | p. 417 |
| Cooper problem and pairing instability | p. 423 |
| Exercises | p. 432 |
| Dynamical treatment of the self-energy in infinite systems | p. 435 |
| Diagram rules in uniform systems | p. 436 |
| Self-energy in the electron gas | p. 440 |
| Electron self-energy in the G[superscript (0)] W[superscript (0)] approximation | p. 440 |
| Electron self-energy in the GW approximation | p. 448 |
| Energy per particle of the electron gas | p. 456 |
| Nucleon properties in nuclear matter | p. 458 |
| Ladder diagrams and the self-energy | p. 458 |
| Spectral function obtained from mean-field input | p. 460 |
| Self-consistent spectral functions | p. 466 |
| Saturation properties of nuclear matter | p. 469 |
| Exercises | p. 481 |
| Dynamical treatment of the self-energy in finite systems | p. 483 |
| Influence of collective excitations at low energy | p. 485 |
| Second-order effects with G-matrix interactions | p. 485 |
| Inclusion of collective excitations in the self-energy | p. 488 |
| Self-consistent pphh RPA in finite systems | p. 496 |
| Short-range correlations in finite nuclei | p. 505 |
| Properties of protons in nuclei | p. 519 |
| Exercises | p. 522 |
| Bogoliubov perturbation expansion for the Bose gas | p. 523 |
| The Bose gas | p. 523 |
| Bogoliubov prescription | p. 525 |
| Particle-number nonconservation | p. 527 |
| The chemical potential | p. 529 |
| Propagator | p. 531 |
| Bogoliubov perturbation expansion | p. 534 |
| Hugenholtz-Pines theorem | p. 542 |
| First-order results | p. 547 |
| Dilute Bose gas with repulsive forces | p. 550 |
| Canonical transformation for the Bose gas | p. 554 |
| Exercises | p. 553 |
| Boson perturbation theory applied to physical systems | p. 561 |
| Superfluidity in liquid [superscript 4]He | p. 561 |
| The He-II phase | p. 561 |
| Phenomenological descriptions | p. 563 |
| The dynamic structure function | p. 567 |
| Inclusive scattering | p. 567 |
| Asymptotic 1/Q expansion of the structure function | p. 570 |
| Inhomogeneous systems | p. 576 |
| The bosonic Bogoliubov transformation | p. 576 |
| Bogoliubov prescription for nonuniform systems | p. 585 |
| Bogoliubov-de Gennes equations | p. 586 |
| Number-conserving approach | p. 589 |
| Exercises | p. 590 |
| In-medium interaction and scattering of dressed particles | p. 591 |
| Propagation of dressed particles in wave-vector space | p. 592 |
| Propagation of dressed particles in coordinate space | p. 600 |
| Scattering of particles in the medium | p. 608 |
| Exercises | p. 617 |
| Conserving approximations and excited states | p. 619 |
| Equations of motion and conservation laws | p. 620 |
| The field picture | p. 621 |
| Equations of motion in the field picture | p. 623 |
| Conservation laws and approximations | p. 627 |
| Linear response and extensions of RPA | p. 629 |
| Brief encounter with functional derivatives | p. 630 |
| Linear response and functional derivatives | p. 631 |
| Ward-Pitaevskii relations for a Fermi liquid | p. 634 |
| Examples of conserving approximations | p. 640 |
| Hartree-Fock and the RPA approximation | p. 640 |
| Second-order self-energy and the particle-hole interaction | p. 641 |
| Extension of the RPA including second-order terms | p. 643 |
| Practical ingredients of ERPA calculations | p. 646 |
| Ring diagram approximation and the polarization propagator | p. 651 |
| Excited states in nuclei | p. 654 |
| Exercises | p. 662 |
| Pairing phenomena | p. 663 |
| General considerations | p. 663 |
| Anomalous propagators in the Fermi gas | p. 666 |
| Diagrammatic expansion in a superconducting system | p. 668 |
| The BCS gap equation | p. 675 |
| Canonical BCS transformation | p. 683 |
| Applications | p. 688 |
| Superconductivity in metals | p. 688 |
| Superfluid [superscript 3]He | p. 691 |
| Superfluidity in neutron stars | p. 691 |
| Inhomogeneous systems | p. 692 |
| Exact solutions of schematic pairing problems | p. 697 |
| Richardson-Gaudin equations | p. 701 |
| Exercises | p. 702 |
| Pictures in quantum mechanics | p. 703 |
| Schrodinger picture | p. 703 |
| Interaction picture | p. 704 |
| Heisenberg picture | p. 708 |
| Practical results from angular momentum algebra | p. 711 |
| Bibliography | p. 717 |
| Index | p. 729 |
| Table of Contents provided by Ingram. All Rights Reserved. |
ISBN: 9789812562944
ISBN-10: 981256294X
Published: 21st April 2005
Format: Hardcover
Language: English
Number of Pages: 752
Audience: College, Tertiary and University
Publisher: World Scientific Publishing Co Pte Ltd
Country of Publication: GB
Dimensions (cm): 23.42 x 16.38 x 4.37
Weight (kg): 1.21
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