Quantum Langevin Equation

Abstract

It is shown (by means of a perturbation series) that for a class of potentials $V\left(x\right)\mathrm{}$ the stationary distribution of the solution $x\left(t\right)\mathrm{}$ of the quantum Langevin equation approaches in the weak-coupling limit ($f\to 0$) the quantum mechanical canonical distribution of the displacement of the oscillator, subject to the potential $V\left(x\right)\mathrm{}$, if and only if $E\left(t\right)\mathrm{}$ is the operator version of the purely random Gaussian process so that, in particular, higher symmetrized averages ${〈E\left(t_1\right)\cdots E\left(t_n\right)〉}_s$ are expressible in terms of pair correlations, in the usual way.