The Particle in a Box Warrants an Examination

Abstract

The particle in a box is a simple model that has a classical Hamiltonian H = p2 (using 2m = 1), with a limited coordinate space, -b q b, where 0 b < ∞. 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 necessarily points toward more complicated eigenfunctions and eigenvalues, which deserve to be solved.