  It is shown that general solutions of the free-particle Schroedinger equation
can be mapped onto solutions of the Schroedinger equation for the harmonic
oscillator. This is done in such a way that the time evolution of a free
particle subjected to a sudden transition to a harmonic potential can be
described by a simple coordinate transformation applied at the transition time.
This procedure is computationally more efficient than either state-projection
or propagator techniques. A concatenation of the map and its inverse allows us
to map from one harmonic oscillator to another with a different spring
constant.
