Quantum Particle in a Regular Polygon by Finite Element Method
The two-dimensional Schrödinger equation for a particle confined to a regular -gon (with ) can be written (in atomic units )[more]
with the Dirichlet boundary condition (on the boundary of the polyhedron). Analytic solutions are known for , and (equilateral triangle, square and disk, respectively). Accurate numerical solutions can be obtained for all by the finite element method. This entails discretizing the -gon and applying the Mathematica algorithm NDEigensystem. Comparing with the known analytic solutions shows that the first eight computed eigenvalues are accurate to better than .01%.
Consider the cases to (equilateral triangle to regular octagon) and (disk). The size of each -gon is adjusted so that it has an area of 1. The lowest eigenvalue represents the ground state, which is labeled . The second and third eigenvalues are equal. The degenerate first excited states are labeled as and . Continuing, we find , , , . Only for the pentagon () and hexagon () do the levels deviate from this pattern.
You can display a contour plot for any of the eight eigenstates of any -gon. You can also choose to display multiple plots for all eigenstates of a selected -gon or the states of all -gons for a given quantum number.
The energy-level diagram shows the energy spectrum for all -gons. As increases, the spectrum approaches that for the disk ().[less]
Test for particle in a square. Exact solution:
Comparison with finite element method:
(), (), (), (), (, ).
 Wikipedia. "Finite Element Method." (Jul 1, 2022). en.wikipedia.org/wiki/Finite_element_method.