  We consider a driven quantum harmonic oscillator strongly coupled to a heat
bath. Starting from the exact quantum Langevin equation, we use a Green's
function approach to determine the corresponding semiclassical equation for the
Wigner phase space distribution. In the limit of high friction, we apply
Brinkman's method to derive the quantum Smoluchowski equation for the
probability distribution in position space. We further determine the range of
validity of the equation and discuss the special case of a Brownian parametric
oscillator.
