  In quantum physics the free particle and the harmonically trapped particle
are arguably the most important systems a physicist needs to know about. It is
little known that, mathematically, they are one and the same. This knowledge
helps us to understand either from the viewpoint of the other. Here we show
that all general time-dependent solutions of the free-particle Schroedinger
equation can be mapped to solutions of the Schroedinger equation for harmonic
potentials, both the trapping oscillator and the inverted 'oscillator'. This
map is fully invertible and therefore induces an isomorphism between both types
of system, they are equivalent. A composition of the map and its inverse allows
us to map from one harmonic oscillator to another with a different spring
constant and different center position. The map is independent of the state of
the system, consisting only of a coordinate transformation and multiplication
by a form factor, and can be chosen such that the state is identical in both
systems at one point in time. This transition point in time can be chosen
freely, the wave function of the particle evolving in time in one system before
the transition point can therefore be linked up smoothly with the wave function
for the other system and its future evolution after the transition point. Such
a cut-and-paste procedure allows us to describe the instantaneous changes of
the environment a particle finds itself in. Transitions from free to trapped
systems, between harmonic traps of different spring constants or center
positions, or, from harmonic binding to repulsive harmonic potentials are
straightforwardly modelled. This includes some time-dependent harmonic
potentials. The mappings introduced here are computationally more efficient
than either state-projection or harmonic oscillator propagator techniques
conventionally employed when describing instantaneous (non-a 1,920 characters
