Study on Physical Properties of Quantum Singularity

这项研究旨在为量子力学中的奇点提供一个连贯的框架，从而寻求它们在不同情况下可能的物理效应。为此，我们主要研究了三个主题：(1)存在量子奇点的多粒子系统的统计性质，(2)量子奇性与量子可积系统，(3)量子奇性与量子信息。首先，我们研究了一维量子势垒中由玻色子/费米子粒子组成的系统，其中量子奇异性被放置在势垒的中心作为隔壁。不同的量子压强取决于奇点的性质，奇点的性质也取决于温度、粒子的数量和它们的统计数据。我们发现了一个由粒子数和温度满足的奇怪的标度律，这为通过观察低温区域的标度律来检测粒子的性质提供了一种新的方法。关于(2)，我们考虑了位势具有广义奇性的N=3 Calogero模型，并在那里找到了一组新的解。这清楚地表明，物理性质确实根据奇点的论述而改变。最后，对于(3)，我们提出了一个利用量子奇点(量子算盘)进行量子计算的简单模型。我们还利用博弈论作为试验场，研究了量子纠缠的物理效应。我们已经成功地提供了一个两人博弈的完整框架，在这个框架中，所有可能的量子策略都可以处理，但纠缠是透明的。由此产生的纳什均衡也可以很容易地分类，由此人们可以观察到由于量子纠缠而导致的量子策略相对于经典策略的优越性。

The present study aims at providing a coherent framework for singularities in quantum mechanics and thereby seek their possible physical effects in various situations. To this end, we have investigated primarily on the three topics: (1) statistical properties of many particle systems in the presence of quantum singularity, (2) quantum singularity and quantum integrable systems, and (3) quantum singularity and quantum information.First, for (1) we have studied systems consisting of bosonic/fermionic particles confined in a quantum well in one dimension, where a quantum singularity is placed at the center of the well acting as a partition wall. Different quantum pressures emerge depending on the nature of the singularity, which is also dependent on the temperature and the number of the particles and their statistics. We have found a curious scaling law satisfied by the particle number and temperature, which suggests a novel method of detecting the nature of particles by observation of the scaling law in low temperature regimes. Concerning (2), we have considered the N = 3 Calogero model with generalized singularity in the potential, and found a new set of solutions there. This shows explicitly that physical properties do change according to the treatise of the singularity. Finally, for (3) we have proposed a simple model of quantum computation by means of quantum singularities (quantum abacus). We have also investigated the physical effects of quantum entanglement by using game theory as a testing ground. We have succeeded to provide a complete framework of two-player games in which all possible quantum strategies can be treated and yet the entanglement is transparent. The resultant Nash equilibrium can also be classified easily whereby one can observe the superiority of quantum strategies over the classical counterparts due to quantum entanglement.