Quantum Corrections to Classical Approximations

项目名称：经典近似的量子修正[ei: Laszlo Erdos， Georgia Institute of technology]摘要：该提案包含三个相关的项目，研究复杂量子问题经典近似中的残余量子效应。第一个项目考虑由于大原子的托马斯-费米理论自生磁场的修正。由变分原理得到的最优磁场在空间上是非均匀的。第二个项目旨在从许多粒子通过弱耦合库仑力相互作用的量子动力学中推导非线性自洽演化方程。粒子的量子统计决定了尺度和限制经典方程（Hartree或Vlasov）。最后一个项目研究了扩散状态下随机环境中非相互作用电子的长时间动力学。在较短的动力学时间尺度上，玻尔兹曼方程建立得较早。目标是确定从玻尔兹曼方程得到的经典扩散率的量子修正，并验证著名的电导标度理论的一些预测。带电粒子的物理学支配着一切电现象。从理论上讲，这些粒子可以用多体量子力学来精确描述。然而，在实践中，量子物理学的基本方程，薛定谔方程，太复杂了。典型的电子设备包含大量的电子，即使用目前的计算机技术也不可能精确地描述它们的微观行为。薛定谔方程通常被计算上更可行的更简单的方程所取代。这些方程不包含有关复杂电子系统的所有信息，但它们可以以足够的精度描述感兴趣的某些量。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）具有弱相互作用的许多带电粒子；（iii）不纯介质中的单个电子。目标是找到正确的近似方程，并严格证明它们在某些极限下与薛定谔理论一致。启发式地说，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这一经典图景。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.