Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 3
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Holm, D. Johnston, and P. Kornilovitch have all contributed with constructive criticism. Dr. U.Eckern from Karlsruhe University clarified some points in the path integral derivation of the Fokker-Planck equation in Chapter 18. Useful comments are due to Dr.P.A. Horvathy, Dr. J. Whitenton, and to my colleague Prof. W. Theis. Their carefulreading uncovered many shortcomings in the first draft of the manuscript. Specialthanks go to Dr. W.
Janke with whom I had a fertile collaboration over the yearsand many discussions on various aspects of path integration.Thanks go also to my secretary S. Endrias for her help in preparing themanuscript in LATEX, thus making it readable at an early stage, and to U. Grimmfor drawing the figures.Finally, and most importantly, I am grateful to my wife Dr. Annemarie Kleinertfor her inexhaustible patience and constant encouragement.H.
KleinertBerlin, January 1990H. Kleinert, PATH INTEGRALSContents1 Fundamentals1.1 Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . .1.2 Relativistic Mechanics in Curved Spacetime . . . . . . . . . .1.3 Quantum Mechanics . . . . . . . . . . . . . . . . . . .
. . . .1.3.1Bragg Reflections and Interference . . . . . . . . . .1.3.2Matter Waves . . . . . . . . . . . . . . . . . . . . . .1.3.3Schrödinger Equation . . . . . . . . . . . . . . . . .1.3.4Particle Current Conservation . . . . . . . . . . . . .1.4 Dirac’s Bra-Ket Formalism . . . . . . . .
. . . . . . . . . . .1.4.1Basis Transformations . . . . . . . . . . . . . . . . .1.4.2Bracket Notation . . . . . . . . . . . . . . . . . . . .1.4.3Continuum Limit . . . . . . . . . . . . . . . . . . . .1.4.4Generalized Functions . . . . . . . . . . . .
. . . . .1.4.5Schrödinger Equation in Dirac Notation . . . . . . .1.4.6Momentum States . . . . . . . . . . . . . . . . . . .1.4.7Incompleteness and Poisson’s Summation Formula .1.5 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . .1.5.1Uncertainty Relation . . . . . . . . . . . . . . . . . .1.5.2Density Matrix and Wigner Function . . . . . .
. . .1.5.3Generalization to Many Particles . . . . . . . . . . .1.6 Time Evolution Operator . . . . . . . . . . . . . . . . . . . .1.7 Properties of 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 Approximationxvii........................................................................................................................................11101112131517181921222425272931323334343840424346485257606868697071xviii1.16.5 Scattering Amplitude from Time Evolution Amplitude1.16.6 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 Fresnel Integral . . . . . . . . . . . .Appendix 1C The Asymmetric Top . . . . . . . . . . . . . . . . . .Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . ...............................737377777882848586882 Path Integrals — Elementary Properties and Simple Solutions2.1 Path Integral Representation of Time Evolution Amplitudes . .2.1.1Sliced Time Evolution Amplitude .
. . . . . . . . . . . .2.1.2Zero-Hamiltonian Path Integral . . . . . . . . . . . . . .2.1.3Schrödinger Equation for Time Evolution Amplitude . .2.1.4Convergence of Sliced Time Evolution Amplitude . . . .2.1.5Time Evolution Amplitude in Momentum Space . . . . .2.1.6Quantum Mechanical Partition Function . . . . . . . . .2.1.7Feynman’s Configuration Space Path Integral .
. . . . .2.2 Exact Solution for Free Particle . . . . . . . . . . . . . . . . . .2.2.1Direct Solution . . . . . . . . . . . . . . . . . . . . . . .2.2.2Fluctuations around Classical Path . . . . . . . . . . . .2.2.3Fluctuation Factor . . . . . . .
. . . . . . . . . . . . . .2.2.4Finite Slicing Properties of Free-Particle Amplitude . . .2.3 Exact Solution for Harmonic Oscillator . . . . . . . . . . . . . .2.3.1Fluctuations around Classical Path . . . . . . . . . . . .2.3.2Fluctuation Factor . . . . . . . . . . . . . . . . . . . . .2.3.3The iη-Prescription and Maslov-Morse Index . . . . . .2.3.4Continuum Limit . .
. . . . . . . . . . . . . . . . . . . .2.3.5Useful Fluctuation Formulas . . . . . . . . . . . . . . . .2.3.6Oscillator Amplitude on Finite Time Lattice . . . . . . .2.4 Gelfand-Yaglom Formula . . . . . . . . . . . . . . . . . . . . . .2.4.1Recursive Calculation of Fluctuation Determinant . . . .2.4.2Examples . .
. . . . . . . . . . . . . . . . . . . . . . . .2.4.3Calculation on Unsliced Time Axis . . . . . . . . . . . .2.4.4D’Alembert’s Construction . . . . . . . . . . . . . . . .2.4.5Another Simple Formula . . . . . . . . . . . . . . . . . .2.4.6Generalization to D Dimensions . . . . . . . . . . . . .2.5 Harmonic Oscillator with Time-Dependent Frequency .
. . . . .2.5.1Coordinate Space . . . . . . . . . . . . . . . . . . . . . .2.5.2Momentum Space . . . . . . . . . . . . . . . . . . . . .2.6 Free-Particle and Oscillator Wave Functions . . . . . . . . . . . .2.7 General Time-Dependent Harmonic Action . . . . . . . . . . . .................................................................909090929393959798102102103105111112113114115116118119121121122123125126128128128131133134H.
Kleinert, PATH INTEGRALSxix2.82.92.102.112.122.132.142.15Path Integrals and Quantum Statistics . . . . . . . . . . . . . . . . . 136Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138Quantum Statistics of Harmonic Oscillator . . . . . . . . . . . . .
. 144Time-Dependent Harmonic Potential . . . . . . . . . . . . . . . . . . 148Functional Measure in Fourier Space . . . . . . . . . . . . . . . . . . 152Classical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Calculation Techniques on Sliced Time Axis. Poisson Formula .
. . 156Field-Theoretic Definition of Harmonic Path Integral by AnalyticRegularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1592.15.1 Zero-Temperature Evaluation of Frequency Sum . . . . . . . 1602.15.2 Finite-Temperature Evaluation of Frequency Sum . . . . . .
1632.15.3 Tracelog of First-Order Differential Operator . . . . . . . . 1652.15.4 Gradient Expansion of One-Dimensional Tracelog . . . . . . 1672.15.5 Duality Transformation and Low-Temperature Expansion . 1672.16 Finite-N Behavior of Thermodynamic Quantities . . . . . . .
. . . . 1732.17 Time Evolution Amplitude of Freely Falling Particle . . . . . . . . . 1752.18 Charged Particle in Magnetic Field . . . . . . . . . . . . . . . . . . 1782.18.1 Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1782.18.2 Gauge Properties . . . . . . . . . . . .
. . . . . . . . . . . . 1802.18.3 Time-Sliced Path Integration . . . . . . . . . . . . . . . . . 1802.18.4 Classical Action . . . . . . . . . . . . . . . . . . . . . . . . 1822.18.5 Translational Invariance . . . . . . . . . . . . . . . . . . . . 1832.19 Charged Particle in Magnetic Field plus Harmonic Potential . . . . . 1842.20 Gauge Invariance and Alternative Path Integral Representation . . 1862.21 Velocity Path Integral .
. . . . . . . . . . . . . . . . . . . . . . . . . 1872.22 Path Integral Representation of Scattering Matrix . . . . . . . . . . 1892.22.1 General Development . . . . . . . . . . . . . . . . . . . . . 1892.22.2 Improved Formulation . . . . . . . . . . . . . . . . . . . . . 1912.22.3 Eikonal Approximation to Scattering Amplitude . . . . .
. 1922.23 Heisenberg Operator Approach to Time Evolution Amplitude . . . . 1932.23.1 Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . 1932.23.2 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 1952.23.3 Charged Particle in Magnetic Field . . . . . . . . . .
. . . . 196Appendix 2A Baker-Campbell-Hausdorff Formula and Magnus Expansion 200Appendix 2B Direct Calculation of Time-Sliced Oscillator Amplitude . . 202Appendix 2C Derivation of Mehler Formula . . . . . . . . . . . . . . . . 204Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2053 External Sources, Correlations, and Perturbation Theory3.1 External Sources . . . . . . . . . . . . . . . . . . . . . .
. .3.2 Green Function of Harmonic Oscillator . . . . . . . . . . .3.2.1Wronski Construction . . . . . . . . . . . . . . . .3.2.2Spectral Representation . . . . . . . . . . . . . . .3.3 Green Functions of First-Order Differential Equation . . . .....................208. 208. 212. 212. 216. 218xx3.43.53.63.73.83.93.103.113.123.133.143.153.163.173.183.193.203.213.223.233.243.253.3.1Time-Independent Frequency . . .
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