Multiscale Computational Methods for Semiclassical Schroedinger Equations with Non-Adiabatic Effects

该项目解决了现代科学计算中的一些基本问题：多尺度建模和仿真，这在纳米技术、通信、材料科学和其他科学技术领域发挥着重要作用。该研究将为涉及多尺度量子-经典耦合的系统开发最先进的计算方法。正在开发的方法在固态物理，半导体器件建模，量子化学和材料科学中出现的问题具有广泛的应用。一些研究生将通过这些研究活动得到培训。一些研究成果也将纳入研究生课程，以更好地培养下一代现代应用数学研究人员。研究人员将开发有效的半经典和多尺度计算方法来解决一些具有非绝热效应的量子动力学问题。非绝热效应是重要的，因为它们对应于不同势能表面之间的量子跃迁，这对描述化学反应和半导体中的量子动力学行为很重要。具体来说，该项目将研究周期晶格中的表面跳变和量子动力学。在绝热情况下，在半经典极限中只需要考虑Wigner矩阵的对角项，这通常会得到经典的Liouville方程。为了解释非绝热效应，人们还需要遵循维格纳矩阵的非对角线项的动力学，以便充分描述不同能量表面之间的量子跃迁。这些产生耦合的、非齐次的、振荡的偏微分方程，将通过高斯光束和多尺度方法进行数值求解。

The project addresses some fundamental issues in scientific computation in the modern age: multiscale modeling and simulation, which play essential roles in nanotechnology, communications, material sciences, and other areas of science and technology. The research will develop state-of-the-art computational methods for systems involving multiscale quantum-classical coupling. The methods under development have a wide range of applications to problems arising in solid-state physics, semiconductor device modeling, quantum chemistry, and materials science. Several graduate students will be trained through these research activities. Some of the research results also will be incorporated in the graduate curriculum to better train the next generation of researchers in modern applied mathematics.The investigator will develop efficient semiclassical and multiscale computational methods for some problems in quantum dynamics with non-adiabatic effects. The non-adiabatic effects are important since they correspond to quantum transitions between different potential energy surfaces that are important to describe quantum dynamic behavior in chemical reactions and semiconductors. Specifically, the project will investigate surface hopping and quantum dynamics in periodic lattices. In the adiabatic cases only the diagonal entries of the Wigner matrix need to be considered in the semiclassical limit, which often results in classical Liouville equations. To account for the non-adiabatic effects, one needs to also follow the dynamics of the off-diagonal entries of the Wigner matrix in order to adequately describe the quantum transition between different energy surfaces. These yield coupled, inhomogeneous, oscillatory partial differential equations that will be numerically solved by Gaussian beam and multiscale methods.