Quantum Corrections to Classical Approximations

DMS-0200235项目标题：经典近似的量子修正PI：Laszlo Erdos，佐治亚理工学院摘要：该提案包含三个相关项目，研究复杂量子问题的经典近似中的残余量子效应。第一个方案考虑了大原子托马斯-费米理论中自生磁场的修正。由变分原理得到的最佳磁场在空间上是不均匀的。第二个项目旨在从多粒子通过弱耦合库仑力相互作用的量子动力学中推导出非线性自洽演化方程。粒子的量子统计决定了标度和限制经典方程(Hartree或Vlasov)。最后一个项目研究了随机环境中非相互作用电子在扩散过程中的长时间动力学。在较短的动力学时间尺度上，较早地建立了玻耳兹曼方程。目的是确定从玻尔兹曼方程得到的经典扩散系数的量子修正，并验证著名的电导标度理论的一些预言。带电粒子的物理支配所有的电学现象。从理论上看，这些粒子可以用多体量子力学精确地描述。然而，在实践中，量子物理的基本方程-薛定谔方程太复杂了。典型的电子器件包含大量的电子，即使用当前的计算机技术也不可能描述它们的精确微观行为。薛定谔方程通常被更简单的方程取代，这些方程在计算上更可行。这些方程不包含关于复杂电子系统的所有信息，但它们可以足够的精度描述某些感兴趣的量。本文研究了三个复杂的量子系统：(I)自生磁场中的大原子；(Ii)具有弱相互作用的多个带电粒子；(Iii)不纯介质中的单电子。目标是找到正确的近似方程，并严格证明它们在一定的极限内与薛定谔理论一致。试探性地，其中一些方程可以根据经典力学猜测出来。然而，量子力学可能会修改经典的图景。所提出的工作确定了这三个基本模型的经典描述中的量子修正效应。

DMS-0200235Project title: Quantum corrections to classical approximationsPI: Laszlo Erdos, Georgia Institute of TechnologyAbstract:The proposal contains three related projects that investigateresidual quantum effects in classical approximationsof complex quantum problems. The first project considersthe correction due to the self-generated magnetic field in the Thomas-Fermi theory of large atoms. The optimal magnetic field obtained from a variationalprinciple is spatially inhomogeneous. The second project aims atderiving nonlinear self-consistent evolution equations from the quantum dynamics of many particlesinteracting via weakly coupled Coulomb force. The quantum statistics of theparticles determine the scaling and the limitingclassical equation (Hartree or Vlasov). The last projectstudies the long time dynamics of noninteractingelectrons in a random environment in the diffusive regime.On the shorter kinetic time scale the Boltzmannequation has been established earlier.The goal is to determine the quantum corrections to theclassical diffusivity obtained from the Boltzmann equationand verify some predictions of the celebrated scaling theoryof conductance.The physics of charged particles governs all electric phenomena.From theoretical point of view, these particles can be accurately described using many-body quantum mechanics. In practice, however, the fundamental equation of quantumphysics, the Schr\"odinger equation, is too complicated. Typical electronicdevices contain a huge number of electrons and it is impossible todescribe their precise microscopic behavior even with the current computer technology.The Schr\"odinger equation is usually replaced withmuch simpler equations that are computationally more feasible.These equations do not contain all information about the complexelectronic system, but they may describe certain quantitiesof interest with a sufficient precision. The proposalstudies three complex quantum systems: (i) a large atom in a self-generated magnetic field; (ii) many charged particles with a weak interaction;(iii) a single electron in an impure medium.The goal is to find the correct approximating equationsand to justify rigorously that they are consistent with the Schr\"odinger theory in certain limits.Heuristically, some of these equations can be guessed based uponclassical mechanics. Quantum mechanics, however, may modifythe classical picture. The proposed work identifies the quantum correction effects in the classical description of these three basic models.