Description: 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 $g_{munu}(x)=2M[V(x)-E]delta_{munu}$. 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 $rho(x;E)=$.