Towards a satisfactory formulation of the quantum Langevin equation
Hiroshi Hasegawa, John R. Klauder, M. Lakshmanan · Journal of Physics A Mathematical and General · 1985
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, (A 0 , A 0 *)=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.