8.4.4 Boundary Conditions

The Wigner-Poisson system has a classical analog, the Vlasov-Poisson equation, whose quantization is given by the Wigner-Poisson equation. However, to quantize the system correctly, one also has to use a correct quantum mechanical formulation of the boundary conditions. The various formulations found in the literature on device simulation all appear to be incorrect in the general case. A physically correct approach is discussed in [BMP⁺00], [SBM93].

For the Wigner simulation we use inflow boundary conditions which are modeled by Dirichlet type boundary conditions in $(x,k)$-space. In the coherent case these are only an approximation to the absorbing boundary conditions as applied in the quantum transmitting boundary method. More specifically: in the Wigner model the inflow is prescribed as the Wigner equilibrium distribution, and outgoing waves appear to be partially reflected at the boundaries. Improved boundary conditions were suggested in [RFK89].