Mathematical Studies in Quantum Mechanics

这个项目主要集中在研究分子量子力学中出现的数学问题。分子中的原子核有很大的质量。因此，它们的运动可以用经典力学来近似描述。相比之下，电子质量非常小，电子迅速调整其量子力学状态，以响应慢得多的原子核运动。这种直觉是Born-Oppenheimer近似的基础。Hagedorn教授、他的研究生和他的合作者将研究Born-Oppenheimer近似不能足够准确地描述所有感兴趣的现象的情况。例如，他们计划开发精确的“表面跳跃模型”，以包含电子进行量子力学跃迁的可能性。在化学和物理文献中已经提出了这种类型的模型，但它们并不是从第一性原理推导出来的。这个项目的主要目标是严格地推导和实现这样一个模型。电子和原子核在分子中的运动可以用薛定谔方程的解来很好地描述。从来没有人解过分子的薛定谔方程，而我们的化学和分子物理的理论知识是从近似到解的。这个项目的主要目标是研究几个常用的近似，并开发对化学家和物理学家有用的改进的近似。除了理论上的理解，这种基础研究的潜在用途是合成新的化学化合物，设计新的催化剂，或设计新的药物。

This project concentrates primarily on the investigation ofmathematical problems that arise in the study of molecular quantummechanics. The nuclei in a molecule have large masses. Consequently, theirmotion is approximately described by classical mechanics. By comparison,electron masses are very small, and electrons adjust their quantummechanical states rapidly in response to the much slower motion of thenuclei. This intuition is the basis for Born-Oppenheimer approximations.Professor Hagedorn, his graduate students, and his collaborators willstudy situations where Born-Oppenheimer approximations are notsufficiently accurate to describe all the phenomena of interest. Forexample, they plan to develop accurate "surface hopping models" to includethe possibility of electrons making quantum mechanical transitions. Modelsof this type have been proposed in the chemistry and physics literature,but they have not been derived from first principles. The main goal ofthis project is rigorously to derive and implement such a model.Motion of electrons and nuclei in molecules are well described bysolutions to the Schrodinger equation. No one has ever solved theSchrodinger equation for a molecule, and our theoretical knowledge ofchemistry and molecular physics has come from approximations to solutions.The principal goals of this project are to study the several commonly usedapproximations and to develop improved approximations that will be usefulto chemists and physicists. In addition to theoretical understanding,potential uses of this basic research are the synthesis of new chemicalcompounds, the design of new catalysts, or the design of new drugs.