Quantum Maupertuis Principle

Abstract

According to the Maupertuis principle, the movement of a classical particle in an external potential $V(x)$ can be understood as the movement in a curved space with the metric $g_{\mu \nu }(x)=2M[V(x)-E]{\delta }_{\mu \nu }$. We show that the principle can be extended to the quantum regime, i.e., we show that the wave function of the particle follows a Schrödinger equation in curved space where the kinetic operator is formed with the {\it Weyl--invariant Laplace-Beltrami} operator. As an application, we use DeWitt's recursive semiclassical expansion of the time-evolution operator in curved space to calculate the semiclassical expansion of the particle density $\rho (x;E)=<x|\delta (E-\hat{H})|x>$.

4 pages