  For a non-relativistic particle, we consider the evolution of wave functions
that consist of polynomial segments, usually joined smoothly together. These
spline wave functions are compact (that is, they are initially zero outside a
finite region), but they immediately extend over all available space as they
evolve. The simplest splines are the square and triangular wave functions in
one dimension, but very complicated splines have been used in physics. In
general the evolution of such spline wave functions can be expressed in terms
of antiderivatives of the propagator; in the case of a free particle or an
oscillator, all the evolutions are expressed exactly in terms of Fresnel
integrals. Some extensions of these methods to two and three dimensions are
discussed.
