A simple quantum dot: numerical and variational solutions

TL;DR

This paper investigates a simple quantum dot with a crossing trough geometry, comparing numerical and variational methods to accurately determine bound state energies.

Contribution

It introduces a novel variational approach and reviews numerical methods for solving the quantum dot problem with a crossing trough geometry.

Findings

Numerical method yields a bound state energy of 0.659606 units.

Variational approach provides a lower bound energy of 0.6812 units.

The variational solution is the most accurate analytical estimate to date.

Abstract

We describe a simple quantum dot that consists of two crossed troughs. As such there is no potential well; nonetheless this geometry gives rise to a bound state, centred around the point at which these troughs cross one another. In this paper we review existing numerical methods to solve this problem, and highlight one which we feel is particularly elegant and, in this case, provides the most accurate solution to the problem. The bound state is well-contained on the scale of the trough width, and yields a bound state energy of $0.659606$ in units of the minimum continuum state energy. This method also motivates a simple variational solution which yields the lowest energy known to date ($0.6812$ in the same units) to arise out of an analytical variational solution.