Quantum Maupertuis Principle

TL;DR

This paper extends Maupertuis' classical principle to quantum mechanics, demonstrating that a quantum particle's wave function obeys a Schrödinger equation in curved space with a Weyl-invariant Laplace-Beltrami operator, and applies semiclassical methods to analyze particle density.

Contribution

It introduces a quantum Maupertuis principle linking quantum wave functions to curved space Schrödinger equations with Weyl-invariant operators, a novel extension of classical concepts.

Findings

Wave function follows a Schrödinger equation in curved space.

Uses DeWitt's expansion to compute semiclassical particle density.

Establishes a geometric framework for quantum dynamics in potential fields.

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 Schr\"odinger 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 $\rho(x;E)=<x|\delta(E-\hat H)|x>$.