WAVE-FUNCTION OF A BROWNIAN PARTICLE

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 m(x) double over dot + eta(x) over dot = F(t) . We show in particular that if the initial wave function is Gaussian, then (i) it remains Gaussian for all times, (ii) its width grows, app roaching a finite value when t-->infinity and (iii) its center describ es a Brownian motion and so the uncertainty in the position of the par ticle grows without limit. [S1063-651X(98)10111-3].