Macroscopic Properties of Quantum Mechanical Systems

量子力学系统的宏观特性

• 批准号: 0601075
• 负责人: Thomas Kennedy
• 金额: --
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0601075&HistoricalAwards=false
• 关键词: Macroscopic; Properties; Quantum; Mechanical; Systems

Three aspects of systems with large numbers of quantum particles will be considered: Non-equilibrium transport properties, the behavior of bosonic particles at low temperatures, and properties of the discrete Laplacian. We will study transport processes that take place in thermal contacts or tunneling junctions between several macroscopically extended metals at different temperatures and chemical potentials. A goal is to better understand the Onsager relations, which claim a certain symmetry in the dependence of currents from thermodynamic forces. Systems of interacting bosons will be considered in the Feynman-Kac representation. We will investigate the links between the following three approaches to Bose-Einstein condensation: Feynman's notion of infinite cycles; Bogolubov's approximations that result in a gas of excitations with a linear dispersion relation; and Penrose and Onsager's notion of off-diagonal long-range order. Finally, we will study the sum of lowest eigenvalues of the discrete Laplacian on arbitrary finite subsets of the cubic lattice. It represents the ground state energy of non-interacting electrons. The goal is to exhibit the effects of the boundary.brbrThe quantum world is a strange one, and many physical experiments defy intuition. General knowledge of quantum mechanical systems and the ability to study them experimentally considerably benefit from theoretical investigations. This study in mathematical physics is centered on the fact that macroscopic quantities are given by averages over the contribution of a huge number of microscopic particles, and these averages can be computed with the tools of statistical mechanics. One aspect of this project is the development of the mathematical description of systems involving very many particles. Another aspect is the study of systems of bosonic particles, such as helium, whose low-temperature behavior has recently attracted much attention. We will consider geometric approaches where quantum particles are represented by space-time trajectories with arbitrary winding numbers. The goal is a deeper understanding of the Bose-Einstein condensation in systems of interacting particles.

大量量子粒子系统的三个方面将被考虑：非平衡输运性质，玻色子粒子在低温下的行为，以及离散拉普拉斯算子的性质。 我们将研究在不同温度和化学势下，几种宏观延伸金属之间的热接触或隧道结中发生的输运过程。 一个目标是更好地理解昂萨格关系，其中声称一定的对称性，从热力学力的电流的依赖。 相互作用玻色子系统将在费曼-卡茨表示中考虑。 我们将研究以下三种方法之间的联系玻色-爱因斯坦凝聚：费曼的无限循环的概念; Bogolubov的近似，导致气体的激发与线性色散关系;和彭罗斯和Onsager的概念的非对角长程秩序。 最后，我们将研究立方格的任意有限子集上的离散拉普拉斯算子的最低特征值之和。 它表示非相互作用电子的基态能量。 我们的目标是展示边界的影响。brbr量子世界是一个奇怪的世界，许多物理实验违背了直觉。 量子力学系统的一般知识和实验研究它们的能力大大受益于理论研究。 这个数学物理学的研究集中在这样一个事实上，即宏观量是由大量微观粒子的贡献的平均值给出的，这些平均值可以用统计力学的工具来计算。 该项目的一个方面是发展涉及非常多粒子的系统的数学描述。 另一个方面是玻色子系统的研究，如氦，其低温行为最近引起了人们的广泛关注。 我们将考虑几何方法，其中量子粒子由具有任意缠绕数的时空轨迹表示。 目标是更深入地了解相互作用粒子系统中的玻色-爱因斯坦凝聚。