Proof that Half-Harmonic Oscillators become Full-Harmonic Oscillators after the "Wall Slides Away"

TL;DR

This paper demonstrates that using affine quantization, a half-harmonic oscillator effectively becomes a full-harmonic oscillator as the wall slides away, revealing all eigenfunctions through numerical analysis.

Contribution

It introduces affine quantization as a method to recover full harmonic oscillator eigenfunctions from a half-harmonic oscillator setup.

Findings

Affine quantization recovers even eigenfunctions.

Numerical simulations support the transition to full harmonic oscillator.

Half-harmonic oscillator behaves like a full one under affine quantization.

Abstract

Normally, the half-harmonic oscillator is active when $x>0$ and absent when $x<0$. From a canonical quantization perspective, this leads to odd eigenfunctions being present while even eigenfunctions are absent. In that case, only the usual odd eigenfunctions will appear if the wall slides to negative infinity. However, if an affine quantization is used, sliding the wall away shows that all the odd and even eigenfunctions are encountered, exactly like any full-harmonic oscillator. We provide numerical support for this.