Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 4
Текст из файла (страница 4)
. . . . . . . . . . . . . . 2183.3.2Time-Dependent Frequency . . . . . . . . . . . . . . . . . . 225Summing Spectral Representation of Green Function . . . . . . . . . 228Wronski Construction for Periodic and Antiperiodic Green Functions 230Time Evolution Amplitude in Presence of Source Term . . . . . . . 231Time Evolution Amplitude at Fixed Path Average .
. . . . . . . . . 235External Source in Quantum-Statistical Path Integral . . . . . . . . 2363.8.1Continuation of Real-Time Result . . . . . . . . . . . . . . 2373.8.2Calculation at Imaginary Time . . . . . . . . . . . . . . . . 241Lattice Green Function . . . . . . . . . . . . . . . . . . . . .
. . . . 248Correlation Functions, Generating Functional, and Wick Expansion 2483.10.1 Real-Time Correlation Functions . . . . . . . . . . . . . . . 251Correlation Functions of Charged Particle in Magnetic Field . . . . . 253Correlation Functions in Canonical Path Integral . . .
. . . . . . . . 2543.12.1 Harmonic Correlation Functions . . . . . . . . . . . . . . . 2553.12.2 Relations between Various Amplitudes . . . . . . . . . . . . 2573.12.3 Harmonic Generating Functionals . . . . . . . . . . . . . . . 258Particle in Heat Bath . . . . . . . . . . . . . . . . . . . . . . . . . . 261Heat Bath of Photons . .
. . . . . . . . . . . . . . . . . . . . . . . . 265Harmonic Oscillator in Ohmic Heat Bath . . . . . . . . . . . . . . . 267Harmonic Oscillator in Photon Heat Bath . . . . . . . . . . . . . . . 270Perturbation Expansion of Anharmonic Systems . . .
. . . . . . . . 271Rayleigh-Schrödinger and Brillouin-Wigner Perturbation Expansion 275Level-Shifts and Perturbed Wave Functions from Schrödinger Equation280Calculation of Perturbation Series via Feynman Diagrams . . . . . . 281Perturbative Definition of Interacting Path Integrals . . . . .
. . . . 286Generating Functional of Connected Correlation Functions . . . . . 2873.22.1 Connectedness Structure of Correlation Functions . . . . . . 2883.22.2 Correlation Functions versus Connected Correlation Functions2913.22.3 Functional Generation of Vacuum Diagrams . .
. . . . . . . 2933.22.4 Correlation Functions from Vacuum Diagrams . . . . . . . . 2973.22.5 Generating Functional for Vertex Functions. Effective Action2993.22.6 Ginzburg-Landau Approximation to Generating Functional 3043.22.7 Composite Fields . . . . . . . . . . . . . . . . . .
. . . . . . 305Path Integral Calculation of Effective Action by Loop Expansion . . 3063.23.1 General Formalism . . . . . . . . . . . . . . . . . . . . . . . 3063.23.2 Mean-Field Approximation . . . . . . . . . . . . . . . . . . 3073.23.3 Corrections from Quadratic Fluctuations . . . .
. . . . . . . 3113.23.4 Effective Action to Second Order in h̄ . . . . . . . . . . . . 3143.23.5 Finite-Temperature Two-Loop Effective Action . . . . . . . 3183.23.6 Background Field Method for Effective Action . . . . . . . 320Nambu-Goldstone Theorem . . . . .
. . . . . . . . . . . . . . . . . . 323Effective Classical Potential . . . . . . . . . . . . . . . . . . . . . . . 3253.25.1 Effective Classical Boltzmann Factor . . . . . . . . . . . . . 327H. Kleinert, PATH INTEGRALSxxi3.25.2 Effective Classical Hamiltonian . . . . . . . .
. . . . . .3.25.3 High- and Low-Temperature Behavior . . . . . . . . . .3.25.4 Alternative Candidate for Effective Classical Potential .3.25.5 Harmonic Correlation Function without Zero Mode . . .3.25.6 Perturbation Expansion . . . . . . . . . . . . . . . . . .3.25.7 Effective Potential and Magnetization Curves . . . . . .3.25.8 First-Order Perturbative Result . . . .
. . . . . . . . . .3.26 Perturbative Approach to Scattering Amplitude . . . . . . . . .3.26.1 Generating Functional . . . . . . . . . . . . . . . . . . .3.26.2 Application to Scattering Amplitude . . . . . . . . . . .3.26.3 First Correction to Eikonal Approximation . . . . . . .3.26.4 Rayleigh-Schrödinger Expansion of Scattering Amplitude3.27 Functional Determinants from Green Functions . . . . .
. . . . .Appendix 3A Matrix Elements for General Potential . . . . . . . . .Appendix 3B Energy Shifts for gx4 /4-Interaction . . . . . . . . . . .Appendix 3C Recursion Relations for Perturbation Coefficients . . .3C.1One-Dimensional Interaction x4 .
. . . . . . . . . . . . .3C.2General One-Dimensional Interaction . . . . . . . . . . .3C.3Cumulative Treatment of Interactions x4 and x3 . . . . .3C.4Ground-State Energy with External Current . . . . . . .3C.5Recursion Relation for Effective Potential . . . . . . . .3C.6Interaction r 4 in D-Dimensional Radial Oscillator . . . .3C.7Interaction r 2q in D Dimensions . . . .
. . . . . . . . . .3C.8Polynomial Interaction in D Dimensions . . . . . . . . .Appendix 3D Feynman Integrals for T 6= 0 . . . . . . . . . . . . . . .Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . ...........................4 Semiclassical Time Evolution Amplitude4.1 Wentzel-Kramers-Brillouin (WKB) Approximation . . . . .
. . . .4.2 Saddle Point Approximation . . . . . . . . . . . . . . . . . . . . .4.2.1Ordinary Integrals . . . . . . . . . . . . . . . . . . . . . .4.2.2Path Integrals . . . . . . . . . . . . . . . . . . . . . . . .4.3 Van Vleck-Pauli-Morette Determinant . . . . . . . . . . . . . . .
.4.4 Fundamental Composition Law for Semiclassical Time EvolutionAmplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.5 Semiclassical Fixed-Energy Amplitude . . . . . . . . . . . . . . . .4.6 Semiclassical Amplitude in Momentum Space . . . . . . . . . . .
.4.7 Semiclassical Quantum-Mechanical Partition Function . . . . . . .4.8 Multi-Dimensional Systems . . . . . . . . . . . . . . . . . . . . . .4.9 Quantum Corrections to Classical Density of States . . . . . . . .4.9.1One-Dimensional Case . . . . . . .
. . . . . . . . . . . . .4.9.2Arbitrary Dimensions . . . . . . . . . . . . . . . . . . . .4.9.3Bilocal Density of States . . . . . . . . . . . . . . . . . . .4.9.4Gradient Expansion of Tracelog of Hamiltonian Operator ...........................329330332333334335337339339340340341343349350352352355355357359362363363363366368. 368. 373. 373.
376. 383..........386388391393398403403405406408xxii4.9.5Local Density of States on Circle . . . . . . . . . . . . . . .4.9.6Quantum Corrections to Bohr-Sommerfeld Approximation .4.10 Thomas-Fermi Model of Neutral Atoms . . . . . . . .
. . . . . . . .4.10.1 Semiclassical Limit . . . . . . . . . . . . . . . . . . . . . . .4.10.2 Self-Consistent Field Equation . . . . . . . . . . . . . . . .4.10.3 Energy Functional of Thomas-Fermi Atom . . . . . . . . . .4.10.4 Calculation of Energies . . . . . . . . . . . . . . . . . . . .4.10.5 Virial Theorem . . . .
. . . . . . . . . . . . . . . . . . . . .4.10.6 Exchange Energy . . . . . . . . . . . . . . . . . . . . . . . .4.10.7 Quantum Correction Near Origin . . . . . . . . . . . . . . .4.10.8 Systematic Quantum Corrections to Thomas-Fermi Energies4.11 Classical Action of Coulomb System . . . . . . . . . . . . . . . . . .4.12 Semiclassical Scattering .
. . . . . . . . . . . . . . . . . . . . . . . .4.12.1 General Formulation . . . . . . . . . . . . . . . . . . . . . .4.12.2 Semiclassical Cross Section of Mott Scattering . . . . . . . .Appendix 4A Semiclassical Quantization for Pure Power Potentials . . .Appendix 4B Derivation of Semiclassical Time Evolution Amplitude . .Notes and References . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .4124144154164174194214244244264284324414414454464484525 Variational Perturbation Theory4545.1 Variational Approach to Effective Classical Partition Function . . . 4545.2 Local Harmonic Trial Partition Function . . . . . .
. . . . . . . . . 4555.3 Optimal Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . 4605.4 Accuracy of Variational Approximation . . . . . . . . . . . . . . . . 4615.5 Weakly Bound Ground State Energy in Finite-Range Potential Well 4635.6 Possible Direct Generalizations . . . . . . . . . . . .
. . . . . . . . . 4655.7 Effective Classical Potential for Anharmonic Oscillator . . . . . . . 4665.8 Particle Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4725.9 Extension to D Dimensions . . . . . . . . . . . . . . . . . . . . . . . 4755.10 Application to Coulomb and Yukawa Potentials . . . . . . . . . . . . 4775.11 Hydrogen Atom in Strong Magnetic Field . . . . . . . .
. . . . . . . 4805.11.1 Weak-Field Behavior . . . . . . . . . . . . . . . . . . . . . . 4845.11.2 Effective Classical Hamiltonian . . . . . . . . . . . . . . . . 4855.12 Variational Approach to Excitation Energies . . . . . . . . . . . . . 4885.13 Systematic Improvement of Feynman-Kleinert Approximation . . . .
. 4925.14 Applications of Variational Perturbation Expansion . . . . . . . . . 4945.14.1 Anharmonic Oscillator at T = 0 . . . . . . . . . . . . . . . . 4955.14.2 Anharmonic Oscillator for T > 0 . . . . . . . . . . . . . . . 4975.15 Convergence of Variational Perturbation Expansion . . . . . . . .
. 5015.16 Variational Perturbation Theory for Strong-Coupling Expansion . . 5085.17 General Strong-Coupling Expansions . . . . . . . . . . . . . . . . . . 5115.18 Variational Interpolation between Weak and Strong-Coupling Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5145.19 Systematic Improvement of Excited Energies . .
. . . . . . . . . . . 515H. Kleinert, PATH INTEGRALSxxiii5.20 Variational Treatment of Double-Well Potential . . . . . . . . . . . .5.21 Higher-Order Effective Classical Potential for Nonpolynomial Interactions . . . . . . . . . . . . . . . . . . . . . . . . . .
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
