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 鲁什考伊 本研究是关于多粒子系统的数学分析问题。 主要的焦点是原子和分子的束缚态哈密顿与磁场中的系统的新重点。 特别是，一维模型将用于研究原子在强磁场中的最大负电离。 在非交换概率空间上，我们也将使用单调黎曼度量来研究一般量子系统的相对熵的收缩，并考虑一些相关的算子不等式。 由于真实的原子和分子存在于普通的三维空间中，核电荷具有公认的整数值，因此具有分数或渐近无限核电荷的一维或二维系统的模型似乎有些人为和深奥。 然而，真实的原子并不是孤立存在的，而是存在于从微观计算机芯片到恒星大小不等的各种材料中。 电子在这种复杂材料中的行为可以通过使用分数核电荷或假设它们被限制在一维或二维来精确地建模。 例如，半导体科学家最近制造了称为“量子点”的微观材料，其行为类似于具有分数核电荷的二维原子。 在另一个极端，人们发现在极强磁场中的原子，例如在中子星星表面上发现的原子，其行为就好像电子被限制在一维空间中。 因此，建议研究的系统具有广泛的适用性。 这个提议的另一部分是关于将一类重要的熵不等式推广到量子系统。 熵及其相关的泛函在经济学、统计学、种群生物学、信息论和物理学等不同领域有着重要的应用。 鉴于最近的进展， 量子计算，这里提出的量子力学熵的工作预计将对信息论以及物理学产生影响。