Quantum Corrections to Classical Approximations

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

DMS-0200235 Project title: Quantum corrections to classical approximations PI: Laszlo Erdos, Georgia Institute of Technology Abstract: The proposal contains three related projects that investigate residual quantum effects in classical approximations of complex quantum problems. The first project considers the correction due to the self-generated magnetic field in the Thomas-Fermi theory of large atoms. The optimal magnetic field obtained from a variational principle is spatially inhomogeneous. The second project aims at deriving nonlinear self-consistent evolution equations from the quantum dynamics of many particles interacting via weakly coupled Coulomb force. The quantum statistics of the particles determine the scaling and the limiting classical equation (Hartree or Vlasov). The last project studies the long time dynamics of noninteracting electrons in a random environment in the diffusive regime. On the shorter kinetic time scale the Boltzmann equation has been established earlier. The goal is to determine the quantum corrections to the classical diffusivity obtained from the Boltzmann equation and verify some predictions of the celebrated scaling theory of 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 quantum physics, the Schr\"odinger equation, is too complicated. Typical electronic devices contain a huge number of electrons and it is impossible to describe their precise microscopic behavior even with the current computer technology. The Schr\"odinger equation is usually replaced with much simpler equations that are computationally more feasible. These equations do not contain all information about the complex electronic system, but they may describe certain quantities of interest with a sufficient precision. The proposal studies 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 equations and to justify rigorously that they are consistent with the Schr\"odinger theory in certain limits. Heuristically, some of these equations can be guessed based upon classical mechanics. Quantum mechanics, however, may modify the classical picture. The proposed work identifies the quantum correction effects in the classical description of these three basic models.