The analytical evaluation of one-dimensional Gaussian path-integrals

摘要

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(tb, ta) exp[iScl(b, a)/ħ] where F(tb, ta) is a coefficient solely depending on the initial time-instant ta and the final time-instant tb, and Scl(b, a) denotes the classical action. Section 3 is devoted to the calculation of a simple formula for the factor F(tb, ta) 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(xb,tb;xa,ta)=∫+∞−∞K(xb,tb;x,t)K(x,t;xa,ta)dx satisfied by quantummechanical Green's functions, constitute important steps in the theoretical development. At the end, F(tb, ta) 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.