We consider a quantum particle moving in a harmonic exterior potential and
lineally coupled to a heat bath of quantum oscillators. Caldeira and Legett
derived the Fokker-Planck equation with friction for the Wigner distributi
on of the particle in the large-temperature limit; however, their (nonrigor
ous) derivation was not free of criticism, especially since the limiting eq
uation is not of Lindblad form. Tn this paper we recover the correct Form o
f their result in a rigorous way. We also point out that the source of the
diffusion is physically restrictive under this scaling. We investigate the
model at a fixed temperature and in the large-time limit, where the origin
of the diffusion is a cumulative effect of many resonant collisions. We obt
ain a heat equation with a friction term for the radial process in phase sp
ace and we prove the Einstein relation in this case.