  Quantum drift-diffusion equations for a two-dimensional electron gas with
spin-orbit interactions of Rashba type are formally derived from a collisional
Wigner equation. The collisions are modeled by a Bhatnagar-Gross-Krook-type
operator describing the relaxation of the electron gas to a local equilibrium
that is given by the quantum maximum entropy principle. Because of
non-commutativity properties of the operators, the standard diffusion scaling
cannot be used in this context, and a hydrodynamic time scaling is required. A
Chapman-Enskog procedure leads, up to first order in the relaxation time, to a
system of nonlocal quantum drift-diffusion equations for the charge density and
spin vector densities. Local equations including the Bohm potential are
obtained in the semiclassical expansion up to second order in the scaled Planck
constant. The main novelty of this work is that all spin components are
considered, while previous models only consider special spin directions.
