A simple quantum dot: Numerical and variational solutions.
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| Title: | A simple quantum dot: Numerical and variational solutions. |
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| Authors: | Walsh, Connor M.1 (AUTHOR) cmwalsh@ualberta.ca, MacPherson, Ian1 (AUTHOR), Joseph, Davidson Noby1 (AUTHOR), Kabra, Suyash1 (AUTHOR), Toor, Ripanjeet Singh1 (AUTHOR), Protter, Mason1 (AUTHOR), Marsiglio, Frank1,2 (AUTHOR) fm3@ualberta.ca |
| Source: | American Journal of Physics. Jun2026, Vol. 94 Issue 6, p465-475. 11p. |
| Subjects: | Bound states, Finite differences, Physics education, Variational approach (Mathematics), Quantum dot devices, Matrix mechanics, Numerical analysis |
| 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, centered 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. Editor's Note: The one-dimensional potential well is a standard exercise in quantum mechanics, familiar to every student as an introduction to bound states and quantization. This paper extends that paradigm to a 2D situation in which two perpendicular troughs cross, forming a simple quantum dot defined purely by geometry rather than by a conventional closed well. Classically, no bound state is expected; nevertheless, quantum mechanically, the intersection region localizes a quantum particle and supports a true bound state. This problem is also used to illustrate and put into practice several standard methods such as matrix mechanics, finite differences, and mode matching. Appropriate for advanced quantum mechanics classes as well as for courses dealing with numerical methods in this field. [ABSTRACT FROM AUTHOR] |
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| Database: | Engineering Source |
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