Quantum treatment of Brownian motion and influence of dissipation on diffusion in dynamically disordered systems

Abstract

The time-dependent hamiltonian formulation of the Langevin equation is used as a starting point for a quantum treatment of the motion of a free Brownian particle. From an exact solution of the time-dependent Schrödinger equation for the density-matrix in one dimension we obtain the mean square displacement, 〈x ²(t)〉, of the Brownian particle, as well as the mean displacement induced by a uniform electric fieldE(t). While quantum effects are significant for time intervals up to the frictional relaxation time, the long time results are identical to those obtained directly from the solution of the Langevin equation. Next, we analyse in a similar way the motion of an electron in a dynamically disordered continuum where the effect of a classical friction force (dissipation) is taken into account. The friction effect is described using the phenomenological time-dependent hamiltonian inferred from the Langevin equation and the potential fluctuations are assumed to have a generalized gaussian white-noise form. The final result for 〈x ² (t)〉 shows a time-dependence similar to that obtained for the case of Brownian motion. In particular, it corresponds to diffusive behavior at long times, in contrast to thet ³-dependence obtained in a recent study for the case where friction is absent.