  Using the essence of Feynman's path integral and the space-time geodesics, an
infinity of differentiable paths that follow the geometry of a continuous
geodesic are constructed, and a wave function is associated to each path as a
probability amplitude identical to the Feynman's probability amplitude for each
path. We prove that each probability amplitude obeys to the Schr\"odinger's
equation for a non relativistic physical system moving in a time independent
potential, starting from the Jacobi-Hamilton equation.
