On the Schrödinger-Langevin Equation

Abstract

It is shown that the Heisenberg‐Langevin equation can be used to derive a Schrödinger equation for a Brownian particle interacting with a thermal environment. The equation derived is $$\begin{array}{c}{\mathrm{i\hslash (\partial \psi /\partial \; t)=-(\hslash }}^2{\text{/2m)∇}}^2{\mathrm{\psi +V\psi +V}}_R\text{ψ+[(ℏ f/2im)\hspace{0.17em}}\ln \mathrm{(\psi /\psi *)+W(t)]\psi (}\mathrm{r}\mathrm{,\; t),}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} W(t)=−(ℏ f/2im)∫ψ*\hspace{0.17em}}\ln \mathrm{(\psi /\psi *)\psi \; d}\mathrm{r},\end{array}$$ where f is the friction constant and V_R is a random potential.