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

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.

Figures (5)

• Figure 1: Mesh Refinement Error plotted against energy state index $n$ for the Stadium and 5-Pointed Star billiards (semi-log scale).
• Figure 2: Localization of the wave function along classically UPOs as shown by Heller Heller-84.
• Figure 3: Contour Plots for the 5-Pointed Star billiard. The top plot corresponds to energy index $n = 49$, and the bottom plot corresponds to energy index $n = 56$.
• Figure 4: Contour Plots for the Stadium Billiard. The top plot corresponds to energy index $n = 18$, and the bottom plot corresponds to energy index $n = 102$.
• Figure 5: Contour plots for the stadium billiard. Top left: $n=217$. Top right: $n=229$. Bottom left: $n=298$. Bottom right: $n=293$.