Chaos and Stability in Quantum Mechanics

9706256 Howland Howland将研究(1)由时间相关扰动驱动的离散量子系统的稳定性和混沌；(2)多尺度微扰问题共振宽度的渐近性；(3)分子哈密顿量的Born-Oppenheimer近似中扭束的出现。混沌理论是近几十年来出现的最重要的数学新领域之一，它关注的是表现出非常不稳定运动的系统。最常见的例子是当水龙头使劲打开时，水管里的水流湍急。一个著名的早期例子是关于天气的计算机模型，其中混沌行为的发生使得无法提前几天预测天气。也许最重要的发现仅仅是认识到，混沌行为不仅不罕见，而且非常普遍，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是由经典力学描述的，经典力学是由艾萨克·牛顿创立的，它描述的是像我们所经历的那样的普通物体。像原子、分子和电子这样的微观系统完全不同于日常经验的对象，它们是由一种被称为量子力学的奇怪理论描述的，这种理论是在本世纪头25年发展起来的。本项目的主要目的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不同，对于一个人究竟想要研究什么，并没有达成一致。在这个领域，大多数专家认为，在理解这个问题之前，还需要一些新的想法和方法。我们感兴趣的系统类型的一个例子是激光场中的氢原子。这在物理上看似简单，却是一个相当难的数学问题。在目前的项目中，将研究这样一个系统的简单模型，以便试图对它们的工作原理有所了解。

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.