Semiclassical asymptotics for open chaotic systems
开放混沌系统的半经典渐近
基本信息
- 批准号:EP/E015336/1
- 负责人:
- 金额:$ 29.28万
- 依托单位:
- 依托单位国家:英国
- 项目类别:Research Grant
- 财政年份:2007
- 资助国家:英国
- 起止时间:2007 至 无数据
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
The study of the quantum mechanical properties of open chaotic systems brings together the fields of fractal geometry and semiclassical asymptotics, giving rise to problems that are of fundamental mathematical importance and that are closely linked with scientific and technological developments concerning microelectronic, nanoelectronic and microlasing devices. Quantum mechanical properties at the semiclassical boundary with classical mechanics that were established decades ago for closed systems, from which particles cannot escape, are still far from being understood in open systems. For example, in closed systems there are formulae allowing one to count the number of quantum states (Weyl's law) and a detailed description of what the corresponding quantum wavefunctions look like. In open systems these remain outstanding unsolved problems. The long-term goal of our research programme is to develop a mathematically consistent semiclassical theory that will ultimately solve these problems.Scientifically and technologically this is a pressing issue. There are now a very considerable number of experiments starting to probe electronic and lasing devices on the mesoscopic and nanoscopic scale. Specifically, recent experimental developments have made it possible to construct clean mesoscopic and nanoscopic devices in which electronic transport is ballistic, i.e. not influenced by impurities. In these systems the corresponding classical dynamics has a controlling influence on the quantum properties and so semiclassical methods are essential. Another situation in which a semiclassical description is needed is in the theory of microlasers, which are currently the focus of considerable experimental and technological interest. Devices such as those just described are likely to form the basis for major new technologies. Yet we still lack many fundamental mathematical tools needed to probe them theoretically; tools that will be needed if they are to be computer-designed. One of the key reasons for this is that these systems are open, rather than closed. The central problem we face in developing a semiclassical theory of open quantum systems is that the classical motion in them, which must be used to form the skeleton of such a theory, is dominated by fractal structures. We need a fundamentally new mathematical approach. It is the goal of our research programme to understand how a semiclassical theory can be built on this fractal skeleton.
对开放混沌系统的量子力学性质的研究将分形几何学和半经典渐近学结合在一起,产生了具有基本数学重要性的问题,这些问题与微电子、纳米电子和微激光器件方面的科学和技术发展密切相关。在半经典边界上的量子力学性质与几十年前为封闭系统建立的经典力学,粒子无法逃脱,在开放系统中仍然远远没有被理解。例如,在封闭系统中,有一些公式允许人们计算量子态的数量(外尔定律),并详细描述了相应的量子波函数。在开放系统中,这些仍然是悬而未决的问题。我们研究计划的长期目标是发展一个数学上一致的半经典理论,最终解决这些问题。现在有相当多的实验开始探测电子和激光器件的介观和纳米尺度。具体而言,最近的实验发展已经使得有可能构建清洁的介观和纳米级器件,其中电子传输是弹道式的,即不受杂质的影响。在这些系统中,相应的经典动力学对量子性质具有控制性影响,因此半经典方法是必不可少的。另一种需要半经典描述的情况是在微激光器理论中,这是目前相当大的实验和技术兴趣的焦点。像刚才描述的那些装置很可能构成主要新技术的基础。然而,我们仍然缺乏许多基本的数学工具来从理论上探索它们;如果要用计算机设计它们,就需要这些工具。其中一个关键原因是这些系统是开放的,而不是封闭的。我们在发展开放量子系统的半经典理论时所面临的中心问题是,其中的经典运动(必须用来形成这种理论的骨架)是由分形结构支配的。我们需要一种全新的数学方法。我们研究计划的目标是了解如何在这个分形骨架上建立半经典理论。
项目成果
期刊论文数量(5)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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