Modeling Quantum Billiards with the Finite Element Method: Searching for Quantum Scarring Candidates

Quantum Physics [Submitted on 25 Mar 2026] Modeling Quantum Billiards with the Finite Element Method: Searching for Quantum Scarring Candidates Daniel Pierce, Renuka Rajapakse An electron in quantum confinement takes on a discrete energy spectrum which is defined based on the solution to the Schrodinger Equation for a given potential. Well defined closed-form energy spectra are known for the particle in a box, circular potential, quarter circle potential, and an equilateral triangle. A closed-form solution for more complex shapes may not be known, but numerical methods can be used to find an approximate solution. In this research, an application of the Finite Element Method (FEM) in Wolfram Mathematica is presented and applied to Quantum Billiards with a variety of geometries. To assess the accuracy of the method, the computed energy states are analyzed in the limit of a polygon with an increasing number of sides, the numerical results are validated against analytical solutions for geometries with known exact forms, and a standard convergence test is conducted. The FEM results closely match analytical solutions for known potentials, demonstrating its high accuracy. For high energy index n, quantum scarring may emerge for certain geometries. The nature of quantum scarring and its presence in the computed models is also investigated qualitatively. Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2603.24864 [quant-ph]   (or arXiv:2603.24864v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2603.24864 Focus to learn more Submission history From: Renuka Rajapakse [view email] [v1] Wed, 25 Mar 2026 23:09:45 UTC (3,797 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-03 References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)