Fluctuation Term in Quantum Oscillator Propagator

In (1), the quantum oscillator propagator is obtained from the integration of the classical Lagrangian over time multiplied by a fluctuation term. In particular, a classical solution of x(t) is found from the Lagrangian i.e. xc(tf,t, ti) such that x(t=tf)=xf and x(t=ti)=xi. The Lagrangian is then expanded about xc(tf,t,ti)+ delta x(tf,t,ti). Thus, there is a term Lc[xc(tf,t,ti) d/dt xc(tf,t,ti))] which may be integrated over time yielding a main factor of exp(i Integral dt Lc) ((1)). A second factor is obtained by evaluating the delta x (tf,t,ti) term which involves some computation. In this note, we argue that the second fator, which is purely a function of time, may be obtained by inserting ((1)) into the time-dependent Schrodinger equation. Furthermore, we argue that in some cases, this approach also dictates part of the form of Integral dt Lc. We apply this approach to a quantum oscillator and a generalized quantum oscillator