Towards a satisfactory formulation of the quantum Langevin equation

Abstract

The quantum mechanical Langevin equation for a damped harmonic oscillator with operator-valued-noise, da_t/dt=(-i omega - gamma )A_t+ lambda w(t), is reinvestigated. By assuming the canonical commutation relation at initial time, (A₀, A₀∗)=1, and a Hermitian operator for the Gaussian quantum noise w(t), it is shown that Streater's procedure to satisfy (A_t,A_t ∗)=1 for all later time yields an improved result: the noise operator w(t) can be determined universally and normalised such that, by choosing lambda ²=2 gamma ( beta h(cross) omega )^-1, the (symmetrised) power spectrum of w(t) is equal to the physically significant form 1/2 beta h(cross) omega coth(1/2 beta h(cross) omega ), where beta is the inverse temperature of the heat bath, thus ensuring the beta -KMS correlation functions for any pair of operators.