On the master equation and pure state representations for the quantized damped oscillator

Abstract

A general master equation for the linearly damped harmonic oscillator is investigated. First, and extended version is presented of Hasse's existence condition for a pure state representation by means of a nonlinear frictional Schrödinger equation with a nonhermitian, but norm conserving hamiltonian. Secondly, reconsidering Dekker's phase space quantization, invoking complex hamiltonians, it is shown (i) how it involves the mirror image system, (ii) that its formulation is terms of a diagonal complex number representation of the hamiltonian allows for a free but nonobservable phase parameter, and (iii) that it leads to specific diffusion coefficients in the master equation which in the long time limit exactly satisfy the pure state condition.