Stochastic approach to generalized Schr{ö}dinger equation: A method of eigenfunction expansion

Abstract

Using a method of eigenfunction expansion, a stochastic equation is developed for the generalized Schr{ö}dinger equation with random fluctuations. The wave field $\psi$ is expanded in terms of eigenfunctions: $\psi ={\sum }_n{a}_n(t){\varphi }_n(x)$, with ${\varphi }_n$ being the eigenfunction that satisfies the eigenvalue equation $H_0{\varphi }_n={\lambda }_n{\varphi }_n$, where $H_0$ is the reference "Hamiltonian" conventionally called "unperturbed" Hamiltonian. The Langevin equation is derived for the expansion coefficient $a_n(t)$, and it is converted to the Fokker--Planck (FP) equation for a set $\{a_n\}$ under the assumption of the Gaussian white noise for the fluctuation. This procedure is carried out by a functional integral, in which the functional Jacobian plays a crucial role for determining the form of the FP equation. The analyses are given for the FP equation by adopting several approximate schemes.

17 pages