The proposer proposes to develop efficient numerical methods for several problems in quantum dynamics. Specifically, the following three topics are selected: Semiclassical methods for quantumscattering, surface hopping, and Bloch-decomposition based computational methods for quantum dynamics in periodic lattice. These are challenging computational issues that involve high frequency waves and quantum-classical coupling. Mathematical and computational methods can play important roles to enhance our understanding as well as our ability to simulate these problems.
These problems arise in solid state physics, semiconductor device modeling, Bose-Eistein Condensations (BEC), solar energy, and functional materials such as graphane, thus the developed computational methods will have a wide range of applications, some of which of significant national interests. Some of the research results will provide excellent additions for future graduate courses in applied mathematics and scientific computing, thus will help to improve the graduate curriculum in applied mathematics in order to better train future graduate students in modern applied mathematics.