On the exactness of the Semi-Classical Approximation for   Non-Relativistic One Dimensional Propagators
  For one dimensional non-relativistic quantum mechanical problems, we investigate the conditions for all the position dependence of the propagator to be in its phase, that is, the semi-classical approximation to be exact. For velocity independent potentials we find that:   (i) the potential must be quadratic in space, but can have arbitrary time dependence.   (ii) the phase may be made proportional to the classical action, and the magnitude (``fluctuation factor'') can also be found from the classical solution.   (iii) for the driven harmonic oscillator the fluctuation factor is independent of the driving term. 
Ibrahim Semiz and Koray Duztas