Chaos and Stability in Quantum Mechanics

量子力学中的混沌与稳定性

• 批准号: 9706256
• 负责人: James Howland
• 金额: $ 7.23万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706256&HistoricalAwards=false
• 关键词: Chaos; Stability; Quantum; Mechanics

9706256 Howland Howland will study (1.) stability and chaos in discrete quantum systems driven by time-dependent perturbations; (2.) the asymptotics of resonance widths in multiscale perturbation problems; and (3.) the occurrence of twisted bundles in the Born-Oppenheimer approximation for molecular Hamiltonians. Chaos Theory, one of the most important new fields of mathematics to arise in the last few decades, is concerned with systems which can exhibit a very erratic motion. The most common example is the turbulent flow of water from a pipe when a faucet is turned on hard. A famous early example concerned computer models of the weather, where the occurrence of chaotic behavior makes it impossible to predict the weather more than a few days in advance. Perhaps the most important discovery has been simply the realization that, far from being rare, chaotic behavior is not only very common, but in some sense, is actually the most common type of behavior. The systems usually studied in Chaos Theory are described by Classical Mechanics, the theory initiated by Isaac Newton which describes ordinary objects like those we experience. Microscopic systems like atoms, molecules, and electrons are not at all like the objects of ordinary experience, and are described by a strange theory known as Quantum Mechanics, which was developed in the first quarter of this century. The main thrust of the present project is to study Chaos Theory in the context of atomic and molecular systems. This is quite difficult because, since Classical Mechanics and Quantum Mechanics are so very different, there is no agreement on exactly what it is that one wants to study. This is still a field where most experts feel that some sort of new ideas and approaches will be needed before the problem is understood. An example of the type of system that is of interest is a Hydrogen atom in a laser field. As simple as this seems physically, it is quite a difficult mathematical problem. In the present project, simple models of such a system will be studied in order to try to come to some understanding of how they work.

9706256 豪兰 豪兰将研究（1.） 含时微扰驱动的离散量子系统的稳定性和混沌;（2.）多尺度扰动问题中共振宽度的渐近性;（3.）在玻恩-奥本海默近似下的分子哈密顿中出现扭曲束。 混沌理论是近几十年来出现的最重要的数学新领域之一，它研究的是可以表现出非常不规则运动的系统。最常见的例子是当水龙头用力打开时，水从管道中湍流出来。一个著名的早期例子涉及天气的计算机模型，其中混沌行为的发生使得不可能提前几天预测天气。也许最重要的发现是简单地认识到，混沌行为远非罕见，它不仅非常常见，而且在某种意义上，实际上是最常见的行为类型。通常在混沌理论中研究的系统由经典力学描述，该理论由艾萨克·牛顿发起，描述了我们所经历的普通物体。 像原子、分子和电子这样的微观系统与普通经验中的对象完全不同，它们由一种被称为量子力学的奇怪理论来描述，这种理论是在本世纪的前25年发展起来的。本项目的主旨是在原子和分子系统的背景下研究混沌理论。这是相当困难的，因为经典力学和量子力学是如此的不同，对于人们想要研究的到底是什么没有一致的意见。这仍然是一个领域，大多数专家认为，在理解这个问题之前，需要一些新的想法和方法。感兴趣的系统类型的一个例子是激光场中的氢原子。尽管这在物理上看起来很简单，但它是一个相当困难的数学问题。在本项目中，将研究这种系统的简单模型，以试图了解它们是如何工作的。