Modeling Quantum Billiards with the Finite Element Method in Wolfram Mathematica

An electron in Quantum Confinement takes on a discrete energy spectrum which is defined based on the solution to the Schrödinger 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 paper, a simple application of the Finite Element Method (FEM) in Wolfram Mathematica is presented and compared to other numerical methods that are applied to Quantum Billiards. Its accuracy is assessed by examining the unbounded limit of the energy states for a polygon with an increasing number of sides. The FEM results closely match analytical solutions for known potentials, demonstrating its high accuracy. Quantum Scarring candidates are examined for irregular boundaries.