A wavefunction description of stochastic-mechanics tion: Fokker-Planck derivation, dissipastationary dynamics and numerical approximations

Abstract

A nonlinear Schr"odinger equation describing how a quantum particle interacts with its surrounding reservoir is derived from the Wigner--\linebreak Fokker--Planck equation (WFPE) via stochastic (Nelsonian) mechanical techniques. This model can be reduced just to a logarithmic Schr"odinger equation (LSE) through a suitable gauge transformation that allows to explore its steady state dynamics and makes its mathematical and numerical analysis more tractable. The transient behaviour of the standard deviation from the mean position associated with its solutions is also studied numerically and compared with that stemming from the WFPE.