### Free Particle Propagator from the Path Integral Formulation

**Free Particle Kernel from the Path Integral Formulation**

In my post on the deriving the non-relativistic path integral, I motivated our discussion by the question of finding the transition probability amplitude of a particle existing in a position eigenstate, $|x_a\rangle$, to a new position eigenstate, $|x_b\rangle$. The power of the path integral, however, comes not in finding the transition probability amplitude, but in finding the Green's function (or "kernel") that describes the propagation of some arbitrary quantum mechanical system under some arbitrary Hamiltonian to a different state (by state, I am actually referring to the notion of a "state" in quantum mechanics). I must reiterate the last sentence because it is important - once we find the propagator, all we need to do is specify the initial state (the initial conditions). The boundary conditions are already taken care of - they are tucked away in the Hermitian nature of the operators that we deal …