A quantum particle in a high-symmetry two-dimensional box

We present contour-plot representations of the low-energy wave functions for a quantum particle in a two-dimensional infinite well potential exhibiting perfect $C_{\infty}$ (disk), $C_{2v}$, (rectangular), $C_{3v}$ (equilateral triangular), or $C_{4v}$ (square) point group symmetry. The rotationally-invariant $C_{\infty}$-allowed wave functions have the integer quantum numbers $n\ge1$. For the rectangular box, all wave functions with $n,n'\ge1$ are allowed, and each one is an allowed representation of the $C_{2v}$ point group. However, for the equilateral-triangular and square boxes, some quantum numbers have to be eliminated, as the wave functions to which they correspond cannot be made into representations of the respective $C_{3v}$ or $C_{4v}$ point groups. For the equilateral triangular box, only $|n-n'|=3p$ are allowed, where $p\ge0$ for the wave functions even about the three mirror planes, and $p\ge1$ for wave functions odd about the three mirror planes. For the square box, $|n-n'|=2p$ are allowed, where for $p\ne0$, only the sum and difference of the two degenerate wave functions are representations of the $C_{4v}$ point group.