Quantumdots

TL;DR

This paper reviews recent rigorous results on quantum dots, focusing on asymptotic ground state theories, novel mathematical features, and the emergence of classical electrostatic descriptions in large magnetic fields.

Contribution

It introduces a Thomas-Fermi type theory for quantum dots and explores the transition to classical electrostatics under strong magnetic fields, with new mathematical insights.

Findings

Thomas-Fermi theory is asymptotically correct for large N and B.

Large magnetic fields lead to a classical electrostatic limit.

Discrepancies between quantum and classical energies are characterized.

Abstract

Atomic-like systems in which electronic motion is two dimensional are now realizable as ``quantum dots''. In place of the attraction of a nucleus there is a confining potential, usually assumed to be quadratic. Additionally, a perpendicular magnetic field $B$ may be present. We review some recent rigorous results for these systems. We have shown that a Thomas-Fermi type theory for the ground state is asymptotically correct when $N$ and $B$ tend to infinity. There are several mathematically and physically novel features. 1. The derivation of the appropriate Lieb-Thirring inequality requires some added effort. 2. When $B$ is appropriately large the TF ``kinetic energy'' term disappears and a peculiar ``classical'' continuum electrostatic theory emerges. This is a two dimensional problem, but with a three dimensional Coulomb potential. 3. Corresponding to this continuum theory is a discrete ``classical'' electrostatic theory. The former provides an upper bound and the latter a lower bound to the true quantum energy; the problem of relating the two classical energies offers an amusing exercise in electrostatics.