Quantum Maupertuis Principle

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 gμν(x) = 2M [V (x) − E]δμν . 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 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 ρ(x;E) = 〈x|δ(E − Ĥ)|x〉.