Nonlinear problems in quantum semiconductor modeling

A model hierarchy of macroscopic models for quantum semiconductors is prese nted. Furthermore, a review on recent results obtained for two classes of t hese macroscopic models: the quantum hydrodynamic and the quantum drift-dif fusion equations is given. In particular, the existence of solutions and th eir qualitative behavior is studied and a numerical scheme for the quantum drift-diffusion model is derived. Finally, this scheme is applied to the tr ansient simulation of a resonant tunneling diode.