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arXiv · 2511.16053

A simple quantum dot: numerical and variational solutions

Connor Walsh·Ian MacPherson·Davidson Joseph·Suyash Kabra·Ripanjeet Singh Toor·Mason Protter·Frank Marsiglio
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Abstract

We describe a simple quantum dot that consists of two crossed two-dimensional troughs. As such there is no potential well; nonetheless, this geometry gives rise to a bound state, centred on the point at which these troughs cross one another. This problem is interesting both because the existence of a bound state may surprise students and because it can be solved using a variety of computational techniques, including matrix mechanics, finite differences, and mode matching. We present these methods and show how the mode-matching method in this case provides the most accurate solution to the problem. Additionally, the mode-matching method can be used to generate a simple wave function that yields the lowest energy known to date to arise out of an analytical variational solution for this problem.

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Connor Walsh, Ian MacPherson, Davidson Joseph, Suyash Kabra, Ripanjeet Singh Toor, Mason Protter, Frank Marsiglio. 2025-11-20. A simple quantum dot: numerical and variational solutions. https://doi.org/10.1119/5.0275290

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