Rigorous Studies in Quantum Mechanics

这个项目主要集中在分子量子力学中出现的数学问题的调查。 因为原子核有很大的质量，它们的运动可以用经典力学很好地近似。 电子的质量要小得多，它们的运动是真正的量子力学。 电子的运动受缓慢运动的原子核的影响，并且它可以用量子绝热近似很好地描述。 这种直觉是玻恩-奥本海默近似的基础。 哈格多恩教授、他的研究生和他的合作者将研究波恩-奥本海默近似不足以准确描述所有感兴趣现象的情况。例如，他们计划开发精确的“表面跳跃模型”，以包括电子进行非绝热跃迁的可能性。 这种类型的模型已经在化学和物理学文献中提出，但它们不是从第一原理推导出来的。 该项目的主要目标是严格推导和实施这些模型。 此外，还将研究其他几个与量子力学密切相关的问题，包括半经典和绝热近似。电子和原子核在分子中的运动极其复杂，化学和分子物理的所有理论知识都来自近似。 这个项目的主要目标是提供这些近似更好的理解，并开发新的和改进的近似。 除了提供更好的理论理解外，这项研究的主旨是获得对化学家和分子物理学家有用的新技术。

This project concentrates primarily on the investigation of mathematical problems that arise in molecular quantum mechanics. Because nuclei have large masses, their motion is well approximated by classical mechanics. Electrons have much smaller mass, and their motion is truly quantum mechanical. The electronic motion is influenced by the more slowly moving nuclei, and it is well described by the quantum adiabatic approximation. This intuition is the basis for Born-Oppenheimer approximations. Professor Hagedorn, his graduate students, and his collaborators will study situations where Born-Oppenheimer approximations are not sufficiently accurate to describe all the phenomena of interest. For example, they plan to develop accurate "surface hopping models" to include the possibility of electrons making non-adiabatic transitions. Models of this type have been proposed in the chemistry and physics literature, but they have not been derived from first principles. The main goal of this project is rigorously to derive and implement such models. In addition, several other closely related questions involving semiclassical and adiabatic approximations to quantum mechanics will be studied.The motion of electrons and nuclei in molecules is extremely complicated, and all theoretical knowledge of chemistry and molecular physics has come from approximations. The primary goal of this project is to provide greater understanding of these approximations and to develop new and improved approximations. In addition to providing greater theoretical understanding, the thrust of this research is to derive new techniques that will be useful for chemists and molecular physicists.