Mathematical and Numerical Methods for Complex Quantum Systems

Semiclassical Computational Methods for Quantum Dynamics with Band-crossings

Shi Jin

University of Wisconsin-Madison

We develop semiclassical models and multiscale computational methods for quantum dynamics with non-adiabatic effects. Applications of such methods include surface hopping, Schrodinger equation with periodic potentials, elastic and electromagnetic waves with polarizations, and graphene. We use the Wigner transform to derive these models. The key idea is to evolve the dynamics of the entire Wigner matrices, which contain important non-adiabatic terms, not just the diagonal projections corresponding to the eigenstates of the Hamiltonians. We also develop multiscale computational methods based on these models and numerical examples will be used to show the validity of these models in captuing the quantum transitions at the crossing-points.