  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.
