On the averaged quantum dynamics by white‐noise hamiltonians with and without dissipation

Abstract

AbstractExact results are derived on the averaged dynamics of a class of random quantum‐dynamical systems in continuous space. Each member of the class is characterized by a Hamiltonian which is the sum of two parts. While one part is deterministic, time‐independent and quadratic, the Weyl‐Wigner symbol of the other part is a homogeneous Gaussian random field which is delta correlated in time, but smoothly correlated in position and momentum. The averaged dynamics of the resulting white‐noise system is shown to be a monotone mixing increasing quantum‐dynamical semigroup. Its generator is computed explicitly. Typically, in the course of time the mean energy of such a system grows linearly to infinity. In the second part of the paper an extended model is studied, which, in addition, accounts for dissipation by coupling the white‐noise system linearly to a quantum‐mechanical harmonic heat bath. It is demonstrated that, under suitable assumptions on the spectral density of the heat bath, the mean energy then saturates for long times.