On the phase space quantization of the linearly damped harmonic oscillator

The quantal form of the classical Liouville equation will be investigated on the basis of a recently developed generalized Hamiltonian theory. The essential novelty inthat theory comprises the use of complex classical coordinates and momenta. We first show how for the nondissipative harmonic oscillator driven by an external classical force, the theory leads to the correct well-known quantum analogue of the classical Liouville equation. We then generalize this procedure to include frictional phenomena for which the novel theory has been observed to be particularly suited. The resulting quantal master equation for the simple linearly damped harmonic oscillator demonstrates that one cannot expect to find a proper quantum mechanical description of dissipative systems in terms of a single Schrödinger wave function. The master equation will then be transformed into its Wigner representation, providing a convinient form for discussion. The diffusion coefficients occuring in the resultant Fokker-Planck equation will be seen to be intimately connected with the survival of Heisenberg's uncertainty principle for dissipative systems. Apart from conceptual elegance, the present approach has superiority to a previous one in at least three aspects: i) there is no need to introduce ad-hoc quantal noise operators, ii) the above mentioned diffusion coefficients are specified and emerge in a natural way, and iii) the present approach has the important advantage of easy extension to more general systems.