Dynamics by White-Noise Hamiltonians

Abstract

A new class of random quantum--dynamical systems in continuous space is introduced and studied in some detail. 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 and arbitrary, but smooth in position and momentum. Exact expressions for the time evolution of both(mixed) states and observables averaged over randomness are obtained.The differences between the quantum and the classical behavior are clearly exhibited.As a special case it is shown that, if the deterministic part corresponds to a particle subjected to a constant magnetic field,the spatial variance of the averaged state grows diffusively for long times independent of the initial state.