Wave functions for high-symmetry, thin microstrip antennas and two-dimensional quantum boxes

TL;DR

This paper explores the symmetry properties of wave functions in high-symmetry two-dimensional microstrip antennas and quantum boxes, visualizing their patterns and discussing implications for terahertz emission.

Contribution

It classifies wave functions in high-symmetry 2D structures into symmetry-based classes and visualizes their patterns, highlighting the role of degeneracy in terahertz emission.

Findings

Wave functions form distinct symmetry classes in high-symmetry 2D structures.

Degeneracy patterns are invariant under point group operations.

Neumann boundary conditions influence terahertz emission power.

Abstract

For a spinless quantum particle in a one-dimensional box or an electromagnetic wave in a one-dimensional cavity, the respective Dirichlet and Neumann boundary conditions both lead to non-degenerate wave functions. However, in two spatial dimensions, the symmetry of the box or microstrip antenna is an important feature that has often been overlooked in the literature. In the high-symmetry cases of a disk, square, or equilateral triangle, the wave functions for each of those two boundary conditions are grouped into two distinct classes, which are one- and two-dimensional representations of the respective point groups, $C_{\infty v}$, $C_{4v}$, and $C_{3v}$. Here we present visualizations of representative wave functions for both boundary conditions and both one- and two-dimensional representations of those point groups. For the one-dimensional representations, color contour plots of the wave functions are presented. For the two-dimensional representations, the infinite degeneracies are presented as common nodal points and/or lines, the patterns of which are invariant under all operations of the respective point group. The wave functions with the Neumann boundary conditions have important consequences for the coherent terahertz emission from the intrinsic Josephson junctions in the high-temperature superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$: the enhancement of the output power from electromagnetic cavity resonances is only strong for wave functions that are not degenerate.