It is widely known in quantum mechanics that solutions of the Schrödinger 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 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.

中文翻译：

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谐振子等效于自由粒子和艾森哈特提升

在量子力学中众所周知，线性势的薛定谔方程 (SE) 的解与自由 SE 的解一一对应。这种对应的物理原因是爱因斯坦的等效原理。通常不那么广为人知的是，具有谐波势的薛定谔方程的解也可以映射到自由薛定谔方程的解。这种等效的物理理解不像等效原理那样精确。我们展示了一张几何图，它将上述两个等价与 Eisenhart 度量的一个约束联系起来。