  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.
