Study on Physical Properties of Quantum Singularity

本研究旨在为量子力学中的奇点提供一个连贯的框架，从而寻求奇点在各种情况下可能产生的物理效应。为此，我们主要研究了三个主题：(1)量子奇点存在下许多粒子系统的统计性质；(2)量子奇点和量子可积系统；(3)量子奇点和量子信息。首先，对于(1)我们研究了由玻色子/费米子粒子组成的系统，这些粒子被限制在一维的量子阱中，其中量子奇点被放置在阱的中心，作为隔壁。不同的量子压力取决于奇点的性质，这也取决于温度和粒子的数量以及它们的统计数据。我们发现了一个奇特的粒子数和温度满足的标度律，这提出了一种在低温条件下通过观察标度律来检测粒子性质的新方法。关于(2)，我们考虑了N = 3 Calogero模型在势中具有广义奇点，并找到了一组新的解。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。我们还研究了量子纠缠的物理效应，利用博弈论作为一个试验场。我们已经成功地提供了一个完整的二人博弈框架，其中所有可能的量子策略都可以处理，而纠缠是透明的。由此产生的纳什均衡也可以很容易地分类，因此人们可以观察到量子策略优于量子纠缠的经典对应策略。

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.