Equivalence of a harmonic oscillator to a free particle and Eisenhart lift
Abstract
It is widely known in quantum mechanics that solutions of the Schröinger equation (SE) for a linear potential are in onetoone 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 Schrödinger equation with harmonic potential can also be mapped to the solutions of the free Schrödinger 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.09523
 Bibcode:
 2021arXiv210609523D
 Keywords:

 Quantum Physics;
 High Energy Physics  Theory
 EPrint:
 28 pages, 1 figure, version to be published in Annals of Physics