The ‘Miracle’ of Applicability? The Curious Case of the Simple Harmonic Oscillator

The paper discusses to what extent the conceptual issues involved in solving the simple harmonic oscillator model fit Wigner’s famous point that the applicability of mathematics borders on the miraculous. We argue that although there is ultimately nothing mysterious here, as is to be expected, a careful demonstration that this is so involves unexpected difficulties. Consequently, through the lens of this simple case we derive some insight into what is responsible for the appearance of mystery in more sophisticated examples of the Wigner problem.