Wavelets and Basis Set Optimization for Molecular and Other Few-Body Quantum Calculations

分子和其他少体量子计算的小波和基集优化

• 批准号: 0070879
• 负责人: Robert Littlejohn
• 金额: $ 19.5万
• 依托单位: University of California-Berkeley
• 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: 2000-08-15 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070879&HistoricalAwards=false
• 关键词: Wavelets; Basis; Set; Optimization; Molecular

0070879Littlejohn Quantum mechanics is the fundamental theory which describes the behavior of atoms, molecules and nuclei. This work concerns relatively simple, or ``few-body,'' systems. In some cases it is possible to use quantum theory and computer calculations to predict the outcome of molecular reactions (one example of an application) which are difficult or impossible to test in the laboratory, with ultimately an important impact on biology, medicine and environmental sciences, among other fields. The application of quantum theory involves finding certain wave functions, which in practice are expressed in terms of more elementary waves (the basis wave functions). The problem of finding an optimal basis is an old one in quantum theory, but in recent years there has arisen a host of new ideas relevant to this question from a variety of fields, including pure and applied mathematics, physics, chemistry and engineering. The object of this work is to integrate these new developments, to search for and exploit fundamental, unifying principles, and to apply them to basis set selection in few-body quantum physics. The new ideas include the following. The first is wavelets, arelatively recent development in applied mathematics, which has already had an important impact on the processing, transmission and storage of signals and data. A second concerns phase space or semiclassical methods, which are called ``microlocal analysis'' in the mathematics literature. These methods are characterized by a consideration of position and momentum together, an unusual and relatively unfamiliar point of view in quantum mechanics where the Heisenberg uncertainly principle limits the simultaneous knowledge of these quantities. A third concerns the mathematics of abstract, higher dimensional spaces in few-body quantum problems, the so-called ``geometrical'' methods. Such methods have been a very active area of mathematics in recent years and are well known in certain areas of physics. They have not, however, been exploited much in few-body quantum mechanics, although they are vital for understanding the nature of the ``internal'' spaces of few-body problems. Finally, this work will take the corpus of methods which are in current use in atomic, molecular and nuclear physics for basis set selection, compare them, integrate them, search for generalizations and improvements, and apply them.

[00:07 . 87]利特尔约翰量子力学是描述原子、分子和原子核行为的基本理论。这项工作涉及相对简单的或“少体”系统。在某些情况下，可以使用量子理论和计算机计算来预测分子反应的结果（一个应用的例子），这些反应在实验室中很难或不可能测试，最终对生物学，医学和环境科学以及其他领域产生重要影响。量子理论的应用涉及找到某些波函数，这些波函数实际上是用更基本的波（基波函数）来表示的。寻找最优基的问题是量子理论中的一个老问题，但近年来，从纯数学和应用数学、物理、化学和工程等各个领域出现了许多与此问题相关的新思想。这项工作的目的是整合这些新的发展，寻找和利用基本的、统一的原则，并将它们应用于少体量子物理的基集选择。新思想包括以下内容。第一个是小波，应用数学中相对较新的发展，它已经对信号和数据的处理、传输和存储产生了重要影响。第二种涉及相空间或半经典方法，在数学文献中称为“微局部分析”。这些方法的特点是同时考虑了位置和动量，这在量子力学中是一种不寻常的、相对陌生的观点，在量子力学中，海森堡不确定性原理限制了对这些量的同时认识。第三种方法涉及抽象、高维空间的数学，即所谓的“几何”方法。这些方法是近年来数学中非常活跃的一个领域，在物理学的某些领域也很有名。尽管它们对于理解小体问题的“内部”空间的本质是至关重要的，但它们在小体量子力学中还没有被充分利用。最后，本工作将采用目前原子、分子和核物理学中使用的各种方法进行基集选择，对它们进行比较、整合，寻找概括和改进，并加以应用。