Wave function of a Brownian particle

Abstract

Using the Caldirola-Kanai Hamiltonian, we study the time evolution of the wave function of a particle whose classical motion is governed by the Langevin equation. We show, in particular, that if the initial wave function is Gaussian, then (i) it remains Gaussian for all times, (ii) its width grows, approaching a finite value when t->infinity, and (iii) its center describes a Brownian motion, and so the uncertainty in the position of the particle grows without limit.