Schrodinger Equation versus Classical Action Approach to Finding a Propagator (Oscillator Example)

 In (1), a method for finding the propagator of a generalized quantum oscillator directly from the time-dependent Schrodinger equation is presented. This method assumes the form Q(t)exp[i (B(t)xx + A(t)yy + C(t)xy) ] which is inserted into the Schrodinger equation. The equation is broken into purely imaginary and real parts. Time dependent coefficients of xx, xy and yy are found leading to four differential equations which may be solved to ultimately yield a full solution for the propagator.

    In (2), it is suggested that the propagator is given by  Q(t) exp(i Integral (t=0 to Tf) Lc dt1)).

In order to use this approach, a solution to the classical Lagrangian Lc is found and x written as:

XA(Tf-t)+YB(t-0)  where X and Y are the initial and final x positions (and are numbers). Thus, dx/dt may be computed, and the classical action integral (t=0 to Tf) Lc dt) found through integration.

    In this note, we consider the quantum oscillator and try to demonstrate the two approaches are the same. A key point is that -id/dt (partial) in the Schrodinger equation really represents -i d/dTf (partial).