Wave functions for the regular pentagonal two-dimensional quantum box and thin microstrip antenna

TL;DR

This paper derives exact, analytically tractable wave functions for a two-dimensional regular pentagonal box under Dirichlet boundary conditions and for a thin regular pentagonal microstrip antenna under Neumann boundary conditions. The authors develop an isosceles-triangle decomposition and enforce pentagonal symmetry to obtain wave functions that depend on two quantum numbers $n\ge1$ and $0\le m\le5$, with a single effective description arising from continuity constraints across rotated sectors. They provide explicit expressions for $\Psi^{(e)}_{nm}$ and $\Psi^{(o)}_{nm}$, derive the energies $E_{n,m}$, and show that the normalization constant $N$ is independent of $(n,m)$, enabling color-plot Visualizations of the modes. The work supports mode analysis in pentagonal THz devices and hints at potential power enhancements by geometric modifications (e.g., slits), with practical relevance to Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ THz emitters. All mathematical notation is presented with proper LaTeX-style delimiters for clarity and reuse in computational or SEO contexts.

Abstract

The general wave functions for the two-dimensional regular pentagonal quantum box and thin microstrip antenna are derived. As for the square, equilateral triangular, and circular disk-shaped boxes and antennas, there are two quantum nunbers $n$ and $m$. In those cases, $n\ge1 $ and $m\ge 0$ are both unlimited non-negative integers of any value. For the regular pentagon, only $n\ge1 $ is an unlimited positive quantum number, but $m_{\rm min}\le m\le 5$, where $m_{\rm min}=0$ for the pentagonal microstrip antenna with Neumann boundary conditions and $m_{\rm min}=1$ for the pentagonal quantum box with Dirichlet boundary conditions. Color-coded pictures of the wave functions for the regular pentagonal quantum box and microstrip antenna are presented for all allowed $m$ values and for $1\le n\le 2$ and for the microstrip antenna for all allowed $m$ values and $n=3$.

Figures (30)

• Figure 1: Plot of the boundary of a regular pentagon with sides of length $a$ inscribed inside a circle of radius $\alpha$ with its corners labelled $A, B, C, D, E$, where corner $A$ is the far left corner and the other four corners are labelled in a consecutive clockwise fashion as pictured.
• Figure 2: Bar code indicating the numerical values of the colors in all of the normalized wave function plots.
• Figure 3: Plot of the normalized wave function from Eqs. (3) and (7) for the regular pentagonal quantum box with $m=1, n=1$.
• Figure 4: Plot of the normalized wave function from Eqs. (3) and (7) for the regular pentagonal quantum box with $m=2, n=1$.
• Figure 5: Plot of the normalized wave function from Eqs. (3) and (7) for the regular pentagonal quantum box with $m=3, n=1$.