  The principle of least action is arguably the most fundamental principle in
physics as it can be used to derive the equations of motion in various branches
of physics. However, this principle has not been experimentally demonstrated at
the quantum level because the propagators for Feymann's path integrals have
never been observed. The propagator is a fundamental concept and contains
various significant properties of a quantum system in path integral
formulation, so its experimental observation is itself essential in quantum
mechanics. Here we theoretically propose and experimentally observe single
photons' propagators based on the method of directly measuring quantum
wave-functions. Furthermore, we obtain the classical trajectories of the single
photons in free space and in a harmonic trap based on the extremum of the
observed propagators, thereby experimentally demonstrating the quantum
principle of least action. Our work paves the way for experimentally exploring
fundamental problems of quantum theory in the formulation of path integrals.
