Rm. Cavalcanti, WAVE-FUNCTION OF A BROWNIAN PARTICLE, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 58(5), 1998, pp. 6807-6809
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].