Physics: Action and Wave Function

I believe action, S,  is defined as integral of KE - PE over time or integral L over time. There appears to be an intimate relationship between the wave function and the action. A sample wave function can be A exp[(-h/i)S(omega t -kx)] or something similar. A function of this type satisfies Schroedinger's equation in that the second spatial derivative  gives KE and the first time derivative gives total E as long as E =hbar [ Plancks constant /2 pi] omega.

Since the probability of a system being in a state is proportional to the wave function* dot  wavefunction, the question is why the probability density is related so closely to the action. L can be thought of as two times the work input from the field into the particles KE in going from some reference point to the current position.