Fokker-Planck equation of the reduced Wigner function associated to an Ohmic quantum Langevin dynamics

This article has to do with the derivation and solution of the Fokker--Planck equation associated to the momentum-independent Wigner function, of a particle subjected to a harmonic external field and immersed in a ohmic thermal bath of quantum harmonic oscillators. The strategy employed is a simplified version of the phenomenological approach of Schramm, Jung and Grabert of interpreting the operators as c-numbers to derive the adjoint equation arising from a twofold transformation of the Wigner function of the entire phase space. The statistical properties of the random noise comes from the integral functional theory of Grabert, Schramm and Ingold. By means of a single Wigner transformation, it is found a simpler master quantum equation than that of mentioned before. The Wigner function reproduces the known results of the classical limit. This allowed to rewrite the underdamped classical Langevin equation as a first order stochastic differential equation with time-dependent drift and diffusion terms.