Exact solutions for a two-electron quantum dot model in a magnetic field and application to more complex systems

TL;DR

This paper presents exact solutions for a two-electron quantum dot in a magnetic field, explores their application to multi-electron systems, and investigates related physical properties and theoretical implications.

Contribution

It provides exact analytical and numerical solutions for two-electron quantum dots and extends these solutions to three-electron systems and quantum dot lattices, highlighting their physical and theoretical significance.

Findings

Exact solutions reduce to a single radial Schrödinger equation.

Analytical solutions exist for specific oscillator frequencies.

Exact solutions for three electrons in certain limits.

Abstract

We discussed exact solutions of the Schroedinger equation for a two-dimensional parabolic confinement potential in a homogeneous external magnetic field. It turns out that the two-electron system is exactly solvable in the sense, that the problem can be reduced to numerically solving one radial Schroedinger equation. For a denumerably infinite set of values of the effective oscillator frequency $\tilde{\omega}=\sqrt{\omega_0^2+(\omega_c/2)^2}$ (where $\omega_0$ is the frequency of the harmonic confinement potential and $\omega_c$ is the cyclotron frequency of the magnetic field) even analytical solutions can be given. Our solutions for three electrons are exact in the strong - and the weak correlation limit. For quantum dot lattices with Coulomb-correlations between the electrons in different dots exact solutions are given, provided the lattice constant is large compared with the dot diameters. We are investigating basic physical properties of these solutions like the formation and distortion of Wigner molecules, the dependence of the correlation strength from $\omega_0$ and $\omega_c$, and we show that in general there is no exact Kohn- Sham system for the semi-relativistic Current-Density-Functional Theory.