In its geometric form, the Maupertuis Principle states that the movement of a
classical particle in an external potential V(x) can be understood as a free
movement in a curved space with the metric gμν(x) = 2M[V(x) - E]δμν. We extend
this principle to the quantum regime by showing that the wavefunction of the
particle is governed by a Schrödinger equation of a free particle moving
through curved space. The kinetic operator is the Weyl-invariant
Laplace–Beltrami operator. On the basis of this observation, we calculate the
semiclassical expansion of the particle density