Equivalence of a harmonic oscillator to a free particle and Eisenhart lift

TL;DR

This paper explores the geometric relationship between harmonic oscillator solutions and free particle solutions in quantum mechanics, revealing a unified perspective through Eisenhart lift and a specific metric constraint.

Contribution

It introduces a geometric framework linking harmonic oscillator and free particle solutions via Eisenhart lift, extending the understanding of quantum equivalences.

Findings

Harmonic oscillator solutions can be mapped to free particle solutions.

A geometric picture using Eisenhart metric explains the equivalence.

Unified understanding of quantum solutions through geometry.

Abstract

It is widely known in quantum mechanics that solutions of the Schr\"{o}inger equation (SE) for a linear potential are in one-to-one correspondence with the solutions of the free SE. The physical reason for this correspondence is Einstein's principle of equivalence. What is usually not so widely known is that solutions of the Schr\"{o}dinger equation with harmonic potential can also be mapped to the solutions of the free Schr\"{o}dinger equation. The physical understanding of this equivalence is not known as precisely as in the case of the equivalence principle. We present a geometric picture that will link both of the above equivalences with one constraint on the Eisenhart metric.