Semiclassical Transport Model for Thin Quantum Barriers

摘要

We present a time-dependent semiclassical transport model for mixed- state scattering with thin quantum barriers. The idea is to use a multiscale approach as a means of connecting regions for which a classical description of the system dynamics is valid across regions for which the classical description fails, such as when the gradient of the potential is undefined. We do this by first solving a stationary Schrodinger equation in the quantum region to obtain the scattering coefficients. These coefficients allow us to build an interface condition to the particle flux that bridges the quantum region, connecting the two classical regions. Away from the barrier, the problem maybe solved by traditional numerical methods. Therefore, the overall numerical cost is roughly the same as solving a classical barrier. We construct numerical methods based on this semiclassical approach and validate the mode using various numerical examples. In the one-dimensional case, we use a finite-volume methods that extends the Hamiltonian-preserving scheme introduced by Jin and Wen for a classical barrier. In the two-dimensional case, we consider a mesh-free particle method that can be computed efficiently and that may be extended to higher-dimensions. The semiclassical transport model is verified numerically by examining the convergence of the Schrodinger and the von Neumann equations to the semiclassical limit for several examples. Finally, we examine an extension of the model to coherent dynamics necessary for periodic crystalline and mesoscopic scale quantum barriers.