  We consider the expansion of wave packets governed by the free Schr\"odinger
equation. This seemingly simple task plays an important role in simulations of
various quantum experiments and in particular in the field of matter-wave
interferometry. The initial tight confinement results in a very fast expansion
of the wave function at later times which significantly complicates an
efficient and precise numerical evaluation. In many practical cases the
expansion time is too short for the validity of the stationary phase
approximation and too long for an efficient application of Fourier
collocation-based methods. We develop an alternative method based on a
discretization of the free-particle propagator. This simple approach yields
highly accurate results which readily follows from the exceptionally fast
convergence of the trapezoidal rule approximation of integrals involving smooth
and rapidly decaying functions. We discuss and analyze our approach in detail
and demonstrate how to estimate the numerical error in the one-dimensional
setting. Furthermore, we show that by exploiting the separability of the
Green's function, the numerical effort of the multi-dimensional approximation
is considerably reduced. Our method is very fast, highly accurate, and easy to
implement on modern hardware.
