Techniques and Applications of Path Integration
Courier Dover Publications, 10/ott/2012 - 448 pagine
A book of techniques and applications, this text defines the path integral and illustrates its uses by example. It is suitable for advanced undergraduates and graduate students in physics; its sole prerequisite is a first course in quantum mechanics. For applications requiring specialized knowledge, the author supplies background material.
The first part of the book develops the techniques of path integration. Topics include probability amplitudes for paths and the correspondence limit for the path integral; vector potentials; the Ito integral and gauge transformations; free particle and quadratic Lagrangians; properties of Green's functions and the Feynman-Kac formula; functional derivatives and commutation relations; Brownian motion and the Wiener integral; and perturbation theory and Feynman diagrams.
The second part, dealing with applications, covers asymptotic analysis and the calculus of variations; the WKB approximation and near caustics; the phase of the semiclassical amplitude; scattering theory; and geometrical optics. Additional topics include the polaron; path integrals for multiply connected spaces; quantum mechanics on curved spaces; relativistic propagators and black holes; applications to statistical mechanics; systems with random impurities; instantons and metastability; renormalization and scaling for critical phenomena; and the phase space path integral.
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amplitude analytic continuation appears applications argument black hole boundary conditions Brownian motion calculation caustic change of variables classical mechanics classical path coefficient conjugate point consider constant contribution critical points defined derivative dimension eigenvalues electromagnetic field energy evaluated expansion exponent fact factor field theory finite follows formula free particle functional integral Gaussian integrals given gives Green’s function Hamiltonian harmonic oscillator Hence homotopy infinite instanton interaction Lagrangian Laplace’s method limit Math mathematical matrix momentum NOTES obtained operator parameter partition function phase space phase space path phase transition Phys physical problem proof propagator quadratic Lagrangian quantity quantization quantum mechanics R. P. Feynman result satisfies Schrodinger equation Section singularity solution space path integral stationary phase approximation statistical mechanics sum over paths theorem tion trajectory vanishes variation wave function Wiener integral winding number WKB approximation yields zero