Matrix mechanics of the infinite square well and the equivalence proofs of Schrödinger and von Neumann

Abstract

There are few simple examples of the formal equivalence of wave mechanics and matrix mechanics. The momentum matrix for a particle in an infinite square well is easy to calculate and rarely discussed in textbooks. We square this matrix to construct the energy levels and use the energy theorem of Fourier analysis to establish the wave-matrix connection. The key ingredients of the equivalence proofs of Schrödinger and von Neumann, such as the d/dx rule and the Riesz-Fischer theorem, find simple expression within the particle-in-a-box framework.