Numerik und Asymptotik mikroskopischer und makroskopischer Gleichungen für Quantensysteme

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Quantum mechanical systems appearing in quantum optics (lasers), semiconductors, electro-magnetic and acoustic wave propagation can be described by a hierarchy of physical models that differ in mathematical complexity and incorporate phenomena on various time and length scales. It is proposed to investigate the mathematical interplay of these models by rigorously deducing simplified (hydrodynamic- and drift-diffusiontype) models from quantum kinetic equations. Understanding these scaling limits gives important information for the range of validity (and the limitations) of reduced models.The second main aim is to develop efficient numerical methods for these equations (Wigner-Fokker-Planck, Madelung, Schrödinger-type, quantum drift-diffusion) in order to simulate quantum waveguides and radio-transmission problems in 2D. This includes a careful treatment of the boundary conditions.The ultimate goal is a numerical comparison of various models on a quantum diode. This will allow to identify regions of such a semiconductor device where simplified (and hence numerically cheaper) models can be used, and areas where the full quantum kinetic model has to be employed.