The analytical evaluation of one-dimensional Gaussian path-integrals
type: Item
instance of: scholarly article
title: The analytical evaluation of one-dimensional Gaussian path-integrals
zbMATH Open document ID: 0631.28004
DOI: 10.1016/0377-0427(88)90316-0
author: C. C. Grosjean
author: Marc J. Goovaerts
published in: Journal of Computational and Applied Mathematics
publication date: 1988-01-01T00:00:00Z
review text: In the introductory Section 1, it is outlined how path-integrals have made their appearance in quantum mechanics through Feynman's formalism. In Section 2, it is recalled that Gaussian path-integrals, defined as those in which the action is an integral whose integrand is a polynomial of at most the second degree in every dynamical variable involved, can be cast into the form \(F(t_ b,t_ a)\exp [iS_{cl}(b,a)/\hslash]\) where \(F(t_ b,t_ a)\) is a coefficient solely depending on the initial time- instant \(t_ a\) and the final time-instant \(t_ b\), and \(S_{cl}(b,a)\) denotes the classical action. Section 3 is devoted to the calculation of a simple formula for the factor \(F(t_ b,t_ a)\) in the case of a single particle in one Euclidean dimension whereby the most general quadratic form of the Lagrange function is considered. The method is based upon successive transformations of various related auxiliary Gaussian path-integrals which are introduced. Applying Feynman's theorem and making use of the well-known property \[ K(x_ b,t_ b;x_ a,t_ a)=\int^{+\infty}_{-\infty}K(x_ b,t_ b;x,t)K(x,t;x_ a,t_ a)dx \] satisfied by quantummechanical Green's functions, constitute important steps in the theoretical development. At the end, \(F(t_ b,t_ a)\) is expressed solely in terms of two functions which one encounters in the description of the classical particle trajectory. In Section 4, four examples of practical application are given. The last one concerns the linear harmonic oscillator with time-dependent period and damping, subject to a time-dependent perturbative force.
Mathematics Subject Classification ID: 28C20
Mathematics Subject Classification ID: 81S40
zbMATH DE Number: 4025756
zbMATH Keywords: Feynman's formulation of quantum mechanics
zbMATH Keywords: Gaussian path-integrals
zbMATH Keywords: quadratic form of the Lagrange function
zbMATH Keywords: quantummechanical Green's functions
zbMATH Keywords: classical particle trajectory
MaRDI profile type: Publication
full work available at URL: 0377-0427(88)90316-0
OpenAlex ID: W2036769646
cites work: Q5569065
cites work: Integration in Functional Spaces and its Applications in Quantum Physics
cites work: Q3714308
Recommended article: A general formula for the calculation of Gaussian path-integrals in two and three Euclidean dimensions
Recommended article: Path integral treatment for the one-dimensional Natanzon potentials
Recommended article: Q3396687
Recommended article: Q5429937
Recommended article: A translation theorem for the generalised analytic Feynman integral associated with Gaussian paths
Recommended article: On exact evaluation of path integrals
Recommended article: Path-integral approach to the statistical physics of one-dimensional random systems
Recommended article: Simple explicit formulas for Gaussian path integrals with time-dependent frequencies
Recommended article: Q3589397
Recommended article: Q4954550
description: scientific article; zbMATH DE number 4025756