Mathematical Studies in Quantum Mechanics

量子力学的数学研究

• 批准号: 0071692
• 负责人: George Hagedorn
• 金额: $ 14.3万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0071692&HistoricalAwards=false
• 关键词: Mathematical; Studies; Quantum; Mechanics

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.

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。分子中的原子核质量很大。因此，它们的运动可以用经典力学近似地描述。相比之下，电子的质量非常小，而且电子会迅速调整其量子力学状态，以响应原子核的慢得多的运动。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-奥本海默近似法不足以准确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”，以包括电子进行量子力学跃迁的可能性。这种类型的模型已经在化学和物理文献中提出，但它们不是从第一原理推导出来的。这个项目的主要目标是严格地推导和实现这样一个模型。薛定谔方程的解很好地描述了分子中电子和原子核的运动。从来没有人解出过一个分子的薛定谔方程，我们的化学和分子物理理论知识都来自于近似解。该项目的主要目标是研究几种常用的近似方法，并开发对化学家和物理学家有用的改进近似方法。除了理论理解，这一基础研究的潜在用途是合成新的化合物，设计新的催化剂，或设计新的药物。