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We consider the expansion of wave packets governed by the free Schr\"odingerequation. This seemingly simple task plays an important role in simulations ofvarious quantum experiments and in particular in the field of matter-waveinterferometry. The initial tight confinement results in a very fast expansionof the wave function at later times which significantly complicates anefficient and precise numerical evaluation. In many practical cases theexpansion time is too short for the validity of the stationary phaseapproximation and too long for an efficient application of Fouriercollocation-based methods. We develop an alternative method based on adiscretization of the free-particle propagator. This simple approach yieldshighly accurate results which readily follows from the exceptionally fastconvergence of the trapezoidal rule approximation of integrals involving smoothand rapidly decaying functions. We discuss and analyze our approach in detailand demonstrate how to estimate the numerical error in the one-dimensionalsetting. Furthermore, we show that by exploiting the separability of theGreen's function, the numerical effort of the multi-dimensional approximationis considerably reduced. Our method is very fast, highly accurate, and easy toimplement on modern hardware.

The discrete Green's function method for wave packet expansion via the free Schr\"odinger equation