Quantum Corrections to Classical Approximations

DMS-0200235项目名称：经典近似的量子修正PI：Laszlo Erdos，格鲁吉亚理工学院摘要：该提案包含三个相关的项目，研究复杂量子问题的经典近似中的剩余量子效应。第一个方案是对大原子的E-F理论中自生磁场的修正。由变分原理得到的最佳磁场是空间非均匀的。第二个项目的目标是从弱耦合库仑力作用下多粒子相互作用的量子动力学出发，导出非线性自洽演化方程。粒子的量子统计决定了标度和极限经典方程（Hartree或Vlasov）。最后一个项目研究了在随机环境中扩散区中不相互作用的电子的长时间动力学。在较短的动力学时间尺度上，Boltzmann方程已经建立。目标是确定由Boltzmann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某些预言。带电粒子的物理支配着所有的电现象。从理论上讲，我们认为，在电子的运动过程中，电子的运动是相互作用的结果，而电子的运动是相互作用的结果。我们认为，电子的运动是相互作用的结果，电子的运动是相互作用的结果，电子的运动是相互作用的结果这些粒子可以用多体量子力学精确描述。 然而，实际上，量子物理学的基本方程，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精确地描述它们的微观行为。薛定谔方程通常被更简单、计算上更可行的方程所取代。这些方程并不包含复杂电子系统的所有信息，但它们可以足够精确地描述某些感兴趣的量。该方案研究了三个复杂的量子系统：（i）自生磁场中的大原子，（ii）多个相互作用较弱的带电粒子，（iii）多个相互作用较弱的带电粒子，（iv）多个相互作用较弱的带电粒子。（iii）不纯介质中的单电子.目的是找到正确的近似方程，并严格证明它们在一定范围内与薛定谔理论是一致的.启发式地，这些方程中的一些可以根据经典力学来猜测。然而，量子力学可能会改变经典图景。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校正效应。

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.