  We show how the quantum analog of the Fokker-Planck equation for describing
Brownian motion can be obtained as the diffusive limit of the quantum linear
Boltzmann equation. The latter describes the quantum dynamics of a tracer
particle in a dilute, ideal gas by means of a translation-covariant master
equation. We discuss the type of approximations required to obtain the
generalized form of the Caldeira-Leggett master equation, along with their
physical justification. Microscopic expressions for the diffusion and
relaxation coefficients are obtained by analyzing the limiting form of the
equation in both the Schroedinger and the Heisenberg picture.
