Wave function of a Brownian particle

Abstract

Using the Hamiltonian of Caldirola [Nuovo Cimento 18, 393 (1941)] and Kanai [Prog. Theor. Phys. 3, 440 (1948)], we study the time evolution of the wave function of a particle whose classical motion is governed by the Langevin equation [Formula Presented] 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 [Formula Presented] and (iii) its center describes a Brownian motion and so the uncertainty in the position of the particle grows without limit. © 1998 The American Physical Society.