Difficulties in simulating systems composed of classical and quantum particles lie in the treatment of the many-body interactions between quantum particles and the geometrical variety of configurations of classical particles. In order to overcome these difficulties, we have developed some numerical methods and applied them to simple cases. As for stationary states, the finite element method provides us with sufficient geometrical freedom.
Combined with the Kohn-Sham equation based on the density
functional theory, this method virtually satisfies our requirement. In order to investigate time-dependent phenomena, we apply the time-dependent Kohn-Sham equation. Adopting the finite difference method, we are able to follow the development of quantum many-body system. As an example, we estimate the effects of the potential height, the electric field, and many-body interactions in some transition processes in quantum wells coupled by a tunneling barrier. This example is important in itself in relation to semiconductor superlattices and also serves as a benchmark for quantum simulations, variety of geometry
corresponding to that of classical particles.