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Study of Quantum Billiard System with Pointlike Obstacles

点状障碍物量子台球系统研究

基本信息

批准号：
10640396
负责人：
CHEON Taksu
金额：
$ 1.79万
依托单位：
Kochi University of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 2000
项目状态：
已结题

项目摘要

We have investigated the mathematical and physical properties of one- and two-dimensional quantum billiard system with pointlike obstacles. We have found out that there are number of intriguing features in this seemingly innocent very simple system. On the two-dimensional system. which has been known to posses "chaotic" quantum level statistics, we have obtained the analytical expressions for the condition for the appearance of the chaotic spectra. On one-dimensional system with pointlike defect, it has been known that there exists a"second-class" of point interaction that causes the discontinuity of the wavefunction itself as opposed to the usual deltafunction interaction that causes the discontinuity in the derivative of the wavefunction. Very little has been known, however, on the physical realization and also on the phvsical properties of this second-class point interaction. which has been mostly thought as a mathematical curiosity. We have made a extensive study on this object clarifying1) how to realize this second-class point interaction out of experimentally realizable local potential in its short-range limit,2) the appearance of the "level anholonomy" in the system with the generalized point interaction that comprizes the delta-function and second-class point interactions,3) the existence of duality (in the sense of equivalence with strong and weak coupling reversed) between the fermionic manybody system with second-class point interaction and the bosonic system with delta-function point interaction, and4) the existence of the nontrivial topologyT^2xS^2 in the parameter space of generalized point interaction, which is behind the above mentioned exotic features.Our results could be considered as a model case of the global analysis of a family of quantum systems, and thus should be useful for designing and tailoring quantum systems for particular properties for the "quantum engineering" purpose in the near future.

我们研究了一维和二维点状障碍物量子台球系统的数学和物理性质。我们发现，在这个看似无辜、非常简单的系统中，有许多耐人寻味的特征。在二维系统上。在已知具有“混沌”量子能级统计的情况下，我们得到了混沌光谱出现条件的解析表达式。在具有点状缺陷的一维系统中，已知存在导致波函数本身不连续的“第二类”点相互作用，而不是通常的导致波函数导数不连续的三角函数相互作用。然而，人们对这种第二类点相互作用的物理实现和物理性质知之甚少。这在很大程度上被认为是数学上的好奇。我们对这一问题进行了广泛的研究，阐明了1)如何在实验上可实现的局域势的短程极限内实现这种二级点相互作用，2)广义点相互作用系统中“能级不完整”的出现，3)具有第二类点相互作用的费米子多体系统和具有三角函数点相互作用的玻色子系统之间存在对偶(在强弱耦合颠倒的等价意义下)，以及4)在广义点相互作用的参数空间中非平凡拓扑T^2xs^2的存在性。这就是上述奇异特征背后的原因。我们的结果可以被视为一类量子系统的全球分析的模型案例，因此在不久的将来应该有助于设计和定制具有特定性质的量子系统，以达到“量子工程”的目的。