The Particle in a Box Warrants an Examination

The particle in a box is a simple model that has a classical Hamiltonian $H=p^2$ (using $2m=1$), with a limited coordinate space, $-b<q<b$, where $0<b<\infty$. Using canonical quantization, this example has been fully studied thanks to its simplicity, and it is a common example for beginners to understand. Despite its repeated analysis, there is a feature that puts the past results into question. In addition to pointing out the quantization issue, the procedures of affine quantization can lead to a proper quantization that nesaeccsrily points toward more complicated eigenfunctions and eigenvalues, which deserve to be solved.