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

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.