Fun with Quantum Dots

TL;DR

This paper explores the properties of quantum dots with parabolic confinement, deriving key features like magic numbers and symmetries using combinatoric principles and a novel basis of states, avoiding perturbation theory.

Contribution

Introduces a simple hypothesis and a new basis for quantum dots that simplifies analysis of their qualitative features without perturbation theory.

Findings

Derivation of magic numbers and symmetries from combinatorics

A new basis simplifies the analysis of quantum dots

Qualitative features are obtained without perturbation theory

Abstract

We consider quantum dots with a parabolic confining potential. The qualitative features of such mesoscopic systems as functions of the total number of electrons N and their total angular momentum J, e.g. magic numbers, overall symmetries etc., are derived solely from combinatoric principles. The key is one simple hypothesis about such quantum dots yielding a basis of states (different from the usual single electron states one starts with) which is extremely easy to handle. Within this basis all qualitative features are already present without the need of any perturbation theory.