量子力学、统计学、聚合物物理学和金融市场中的路径积分:第1分册(第5版)

内容概要

Hagen Kleinert（H.克莱尼特,德国)是国际知名学者，在数学和物理学界享有盛誉。

书籍目录

Preface

Preface to Fourth Edition

Preface to Third Edition

Preface to Second Edition

Preface to First Edition

1 Fundamentals

1.1 Classical Mechanics

1.2 Relativistic Mechanics in Curved Spacetime

1.3 Quantum Mechanics

1.3.1 Bragg Reflections and Interference

1.3.2 Matter Waves

1.3.3 SchrSdinger Equation

1.3.4 Particle Current Conservation

1.4 Dirac's Bra-Ket Formalism

1.4.1 Basis Transformations

1.4.2 Bracket Notation

1.4.3 Continuum Limit

1.4.4 Generalized Functions

1.4.5 SchrSdinger Equation in Dirac Notation

1.4.6 Momentum States

1.4.7 Incompleteness and Poisson's Summation Formula

1.5 Observables

1.5.1 Uncertainty Relation

1.5.2 Density Matrix and Wigner Function

1.5.3 Generalization to Many Particles

1.6 Time Evolution Operator

1.7 Properties of the Time Evolution Operator

1.8 Heisenberg Picture of Quantum Mechanics

1.9 Interaction Picture and Perturbation Expansion

1.10 Time Evolution Amplitude

1.11 Fixed-Energy Amplitude

1.12 Free-Particle Amplitudes

1.13 Quantum Mechanics of General Lagrangian Systems

1.14 Particle on the Surface of a Sphere

1.15 Spinning Top

1.16 Scattering

1.16.1 Scattering Matrix

1.16.2 Cross Section

1.16.3 Born Approximation

1.16.4 Partial Wave Expansion and Eikonal Approximation

1.16.5 Scattering Amplitude from Time Evolution Amplitude

1.16.6 Lippmann-Schwinger Equation

1.17 Classical and Quantum Statistics

1.17.1 Canonical Ensemble

1.17.2 Grand-Canonical Ensemble

1.18 Density of States and Tracelog

Appendix 1A Simple Time Evolution Operator

Appendix 1B Convergence of the Fresnel Integral

Appendix 1C The Asymmetric Top

Notes and References

Path Integrals -- Elementary Properties and Simple Solutions

2.1 Path Integral Representation of Time Evolution Amplitudes

2.1.1 Sliced Time Evolution Amplitude

2.1.2 Zero-Hamiltonian Path Integral

2.1.3 SchrSdinger Equation for Time Evolution Amplitude

2.1.4 Convergence of of the Time-Sliced Evolution Amplitude

2.1.5 Time Evolution Amplitude in Momentum Space

2.1.6 Quantum-Mechanical Partition Function

2.1.7 Feynman's Configuration Space Path Integral

2.2 Exact Solution for the Free Particle

2.2.1 Direct Solution

2.2.2 Fluctuations around the Classical Path

2.2.3 Fluctuation Factor

2.2.4 Finite Slicing Properties of Free-Particle Amplitude

2.3 Exact Solution for Harmonic Oscillator

2.3.1 Fluctuations around the Classical Path

2.3.2 Fluctuation Factor

2.3.3 The/q-Prescription and Maslov-Morse Index

2.3.4 Continuum Limit

2.3.5 Useful Fluctuation Formulas

2.3.6 Oscillator Amplitude on Finite Time Lattice

2.4 Gelfand-Yaglom Formula

2.4.1 Recursive Calculation of Fluctuation Determinant

2.4.2 Examples

2.4.3 Calculation on Unsliced Time Axis

2.4.4 D'Alembert's Construction

2.4.5 Another Simple Formula

2.4.6 Generalization to D Dimensions

2.5 Harmonic Oscillator with Time-Dependent Frequency

2.5.1 Coordinate Space

2,5.2 Momentum Space

2.6 Free-Particle and Oscillator Wave Functions

2.7 General Time-Dependent Harmonic Action

2.8 Path Integrals and Quantum Statistics

2.9 Density Matrix

2.10 Quantum Statistics of the Harmonic Oscillator

2.11 Time-Dependent Harmonic Potential

2.12 Functional Measure in Fourier Space

2.13 Classical Limit

2.14 Calculation Techniques on Sliced Time Axis via the Poisson Formula

2.15 Field-Theoretic Definition of Harmonic Path Integrals by Analytic

Regularization

2.15.1 Zero-Temperature Evaluation of the Frequency Sum

2.15.2 Finite-Temperature Evaluation of the Frequency Sum

2.15.3 Quantum-Mechanical Harmonic Oscillator

2.15.4 Tracelog of the First-Order Differential Operator

2.15.5 Gradient Expansion of the One-Dimensional Tracelog

2.15.6 Duality Transformation and Low-Temperature Expansion

2.16 Finite-N Behavior of Thermodynamic Quantities

2.17 Time Evolution Amplitude of Freely Falling Particle

2.18 Charged Particle in Magnetic Field

2.18.1 Action

2.18.2 Gauge Properties

2.18.3 Time-Sliced Path Integration

2.18.4 Classical Action

2.18.5 Translational Invariance

2.19 Charged Particle in Magnetic Field plus Harmonic Potential

2.20 Gauge Invariance and Alternative Path Integral Representation

2.21 Velocity Path Integral

2.22 Path Integral Representation of the Scattering Matrix

2.22.1 General Development

2.22.2 Improved Formulation

2.22.3 Eikonal Approximation to the Scattering Amplitude

2.23 Heisenberg Operator Approach to Time Evolution Amplitude

2.23.1 Free Particle

2.23.2 Harmonic Oscillator

2.23.3 Charged Particle in Magnetic Field

3 External Sources, Correlations, and Perturbation Theory

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