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

TL;DR

This paper derives the wave functions for a regular pentagonal quantum box and microstrip antenna, revealing unique quantum number constraints and providing visualizations for various modes, expanding understanding of pentagonal quantum systems.

Contribution

It presents the first explicit derivation of wave functions for the regular pentagonal quantum box and microstrip antenna, highlighting differences from other polygonal shapes.

Findings

Wave functions for the pentagonal quantum box and antenna are explicitly derived.

Quantum number constraints differ from other polygonal geometries.

Color-coded wave function visualizations are provided for multiple modes.

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 $n=3$.