Analysis of Multi-Particle Systems

多粒子系统分析

• 批准号: 9706981
• 负责人: Mary Beth Ruskai
• 金额: $ 6.07万
• 依托单位: University of Massachusetts Lowell
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-15 至 2000-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706981&HistoricalAwards=false
• 关键词: Analysis; Multi; Particle; Systems

9706981 Ruskai This research is concerned with problems in the mathematical analysis of multi-particle systems. The main focus is on bound states of atomic and molecular Hamiltonians with a new emphasis on systems in magnetic fields. In particular, one-dimensional models will be used to study the maximum negative ionization of atoms in strong magnetic fields. The contraction of relative entropy for general quantum systems will also be studied using monotone Riemannian metrics on non-commutative probability spaces, and some related operator inequalities will be considered. Because real atoms and molecules exist in ordinary 3-dimensional space with well established integer values for the nuclear charges, models of systems in one or two dimensions with fractional, or asymptotically infinite, nuclear charge may seem somewhat artificial and esoteric. However, real atoms do not exist in isolation, but within various kinds of materials ranging in size from microscopic computer chips to stars. The behavior of electrons within such complex materials may be accurately modelled by using a fractional nuclear charge or by supposing that they are confined to one or two dimensions. For example, semiconductor scientists have recently manufactured microscopic materials called "quantum dots" which behave like two-dimensional atoms with fractional nuclear charge. At the other extreme, one finds that atoms in extremely strong magnetic fields, such as those found on the surface of a neutron star, behave as if the electrons were confined to one dimension. Thus, the systems proposed for study have a wide range of applicability. The other part of this proposal is concerned with the generalization of an important class of entropy inequalities to quantum systems. Entropy and the related functionals have significant applications in such diverse fields as economics, statistics, population biology, information theory, and physics. In view of recent advances in quantum co mputing, the work on quantum mechanical entropy proposed here is expected to have an impact on information theory as well as physics.

9706981 Ruskai这项研究与多粒子系统的数学分析中的问题有关。 主要重点是原子和分子哈密顿量的结合状态，对磁场中的系统有了新的重视。 特别是，一维模型将用于研究强磁场中原子的最大负电离。 一般量子系统的相对熵的收缩也将使用单调riemannian指标在非共同概率空间上进行研究，并将考虑一些相关的操作员不平等。 由于真实原子和分子存在于具有核电的整数良好整数值的普通3维空间中，因此具有一个或两个维度的系统模型，具有分数或渐近无限的无限，核电荷似乎在某种程度上是人为的和深奥的。 但是，实际原子不是孤立存在的，而是在各种材料中，从微观计算机芯片到恒星。 可以通过使用分数核电荷或假设将它们局限于一个或两个维度来准确地建模电子中电子的行为。 例如，半导体科学家最近制造了称为“量子点”的微观材料，其行为就像二维原子具有分数核电。 在另一个极端情况下，人们发现在极强的磁场中的原子，例如在中子恒星表面发现的原子，就像电子被局限于一个维度一样。 因此，提出的研究系统具有广泛的适用性。 该提案的另一部分与对量子系统的一系列熵不平等的概括有关。 熵和相关功能在经济学，统计，种群生物学，信息理论和物理等各种领域中具有重要的应用。 鉴于量子co的最新进展，预计此处提出的量子机械熵的工作将对信息理论和物理学产生影响。