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

TL;DR

This paper applies the Finite Element Method to quantum billiards of various shapes to approximate energy spectra, validate numerical accuracy, and explore quantum scarring phenomena in complex geometries.

Contribution

It demonstrates the use of FEM in Wolfram Mathematica for modeling quantum billiards and investigates quantum scarring in geometries lacking closed-form solutions.

Findings

FEM results closely match analytical solutions for known geometries.

Numerical methods effectively approximate energy spectra for complex shapes.

Quantum scarring may occur at high energy levels in certain geometries.

Abstract

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.