Wavefunctions for the regular pentagonal quantum box and microstrip antenna

We solve for the quantum wavefunctions for the two-dimensional regular pentagonal quantum box and microstrip antenna with respective Dirichlet and Neumann boundary conditions, as for the equilateral triangle, square, and circular box and antenna [1]. We broke the pentagon into five equivalent isosceles triangles, with one of them having its unequal side parallel to the y-axis, along which either boundary condition was imposed. The wavefunctions satisfy the Schrödinger equation inside the box or antenna. The general wavefunctions are characterized by two quantum numbers n and m. For the square and the equilateral triangle, both m = 1,2,3,… and n= 1,2,3,...are unbounded. But unlike the square and the equilateral triangle, only n=1,2,3,… is unbounded but m is an integer ranging from 0 to 5, with the 0 value depending upon the boundary condition. Exact forms for the normalized wavefunctions and color figures of the entire wavefunctions are shown. Pentagonal microstrip antennas of the high-temperature superconductor Bi₂Sr₂CaCu₂O_8+δ (Bi2212) have been studied for their terahertz emission properties.

[1] J. R. Rain et al., PhysRevA.104.062205 (2021).