The need to model the quantum effects in semiconductor devices such as resonance tunneling diodes and quantum dots has lead to an intense study of the Wigner-Poisson (WP) and Schroedinger-Poisson (SP) systems. In this work we present the mathematical analysis of several related models for these systems. These include: a time-dependent model of dissipation in (SP), a quasi-linear (SP) system, a study of the stationary (WP)-(SP) problem with a discussion of the quantum analogue of classical BGK modes and a proof of existence of eigenfunctions for (SP) with periodic boundary conditions, and an examination of the stationary Wigner equations with inflow" boundary conditions. Finally, a proposed numerical scheme for the stationary (SP) system with Boltzmann distribution functions is shown along with its corresponding Bloch equation.