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This article has to do with the derivation and solution of the Fokker--Planckequation associated to the momentum-independent Wigner function, of a particlesubjected to a harmonic external field and immersed in a ohmic thermal bath ofquantum harmonic oscillators. The strategy employed is a simplified version ofthe phenomenological approach of Schramm, Jung and Grabert of interpreting theoperators as c-numbers to derive the adjoint equation arising from a twofoldtransformation of the Wigner function of the entire phase space. Thestatistical properties of the random noise comes from the integral functionaltheory of Grabert, Schramm and Ingold. By means of a single Wignertransformation, it is found a simpler master quantum equation than that ofmentioned before. The Wigner function reproduces the known results of theclassical limit. This allowed to rewrite the underdamped classical Langevinequation as a first order stochastic differential equation with time-dependentdrift and diffusion terms.

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