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arXiv:2603.24864

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

Daniel Pierce, Renuka Rajapakse

quant-phSubmitted Mar 25, 2026• Announced Mar 27, 2026
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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.

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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.
Scarring in quantum systems with classical chaotic dynamics is one of the most remarkable phenomena in modern physics. Previous works were concerned mostly with nonrelativistic quantum systems described by the Schrödinger equation. The question remains outstanding of whether truly relativistic quantum particles that obey the Dirac equation can scar. A significant challenge is the lack of a general method for solving the Dirac equation in closed domains of arbitrary shape. In this paper, we develop a numerical framework for obtaining complete eigensolutions of massless fermions in general two-dimensional confining geometries. The key ingredients of our method are the proper handling of the boundary conditions and an efficient discretization scheme that casts the original equation in a matrix representation. The method is validated by (1) comparing the numerical solutions to analytic results for a geometrically simple confinement and (2) verifying that the calculated energy level-spacing statistics of integrable and chaotic geometries agree with the known results. Solutions of the Dirac equation in a number of representative chaotic geometries establish firmly the existence of scarring of Dirac fermions.
A simple expansion method for numerically calculating the energy levels and the corresponding wave functions of a quantum particle in a two-dimensional infinite potential well with arbitrary shape (quantum billiard) is presented. The method permits the study of quantum billiards in an introductory quantum mechanics course. According to the method, wave functions inside the billiard are expressed in terms of an expansion of a complete set of orthonormal functions defined in a surrounding rectangle for which the Dirichlet boundary conditions apply, while approximating the billiard boundary by a potential energy step of a sufficiently large size. Numerical implementations of the method are described and applied to determine the energies and wave functions for quarter-circle, circle, and triangle billiards. Finally, the expansion method is applied to investigate the quantum signatures of chaos in a classically chaotic generic-triangle billiard.
We present quasi-analytical and numerical calculations of Gaussian wave packet solutions of the Schr\\"odinger equation for two-dimensional infinite well and quantum billiard problems with equilateral triangle, square, and circular footprints. These cases correspond to N=3, N=4, and NtoinftyN \\to \\inftyNtoinfty regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wavefunctions are given in terms of familiar special functions. For the first two systems, we obtain closed form expressions for the expansion coefficients for localized Gaussian wavepackets in terms of the eigenstates of the particular geometry. For the circular case, we discuss numerical approaches. We use these results to discuss the short-time, quasi-classical evolution in these geometries and the structure of wave packet revivals. We also show how related half-well problems can be easily solved in each of the three cases.
We present quasianalytical and numerical calculations of Gaussian wave packet solutions of the Schrödinger equation for two-dimensional infinite well and quantum billiard problems with equilateral triangle, square, and circular footprints. These cases correspond to N=3, N=4, and N→∞ regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wave functions are given in terms of familiar special functions. For the first two systems, we obtain closed form expressions for the expansion coefficients for localized Gaussian wave packets in terms of the eigenstates of the particular geometry. For the circular case, we discuss numerical approaches. We use these results to discuss the short-time, quasiclassical evolution in these geometries and the structure of wave packet revivals. We also show how related half-well problems can be easily solved in each of the three cases.
Quantum scar is an intriguing phenomenon in quantum or wave dynamics that the wavefunction takes an exceptionally large value around an unstable periodic orbit. It has attracted much attention and advances the understanding of the semiclassical quantization. Most of previous researches involving quantum scars focus on hard-wall quantum billiards. Here we investigate the quantum billiard with a smooth confinement potential which possesses complex classical dynamics. We demonstrate that the semiclassical quantization approach works well for both the stable and unstable classical periodic orbit, besides the fact that the shape of the orbits varies as the energy increases or even the stability switches. The recurrence rule of the quantum scars in this complex solf-wall billiard differs from that of the hard-wall nonrelativistic quantum billiard, such as being equally spaced in energy instead of being equally spaced in the square root of energy. These results implement the previous knowledge and may be used for understanding the measurements of density of states and transport properties in two-dimensional electron systems with random long-range impurities.
Using the fact that the energy eigenstates of the equilateral triangle infinite well (or billiard) are available in closed form, we examine the connections between the energy eigenvalue spectrum and the classical closed paths in this geometry, using both periodic orbit theory and the short-term semi-classical behavior of wave packets. We also discuss wave packet revivals and show that there are exact revivals, for all wave packets, at times given by Trev=9mua2/4hbarpiT_{rev} = 9 mu a^2/4hbar piTrev​=9mua2/4hbarpi where aaa and mumumu are the length of one side and the mass of the point particle respectively. We find additional cases of exact revivals with shorter revival times for zero-momentum wave packets initially located at special symmetry points inside the billiard. Finally, we discuss simple variations on the equilateral (60circ−60circ−60circ60^{circ}-60^{circ}-60^{circ}60circ−60circ−60circ) triangle, such as the half equilateral (30circ−60circ−90circ30^{circ}-60^{circ}-90^{circ}30circ−60circ−90circ) triangle and other `foldings', which have related energy spectra and revival structures.