Running into trouble with the time-dependent propagation of a wavepacket

The propagation in time of a wavepacket is a conceptually rich problem suitable to be studied in any introductory quantum mechanics course. This subject is covered analytically in most of the standard textbooks. Computer simulations have become a widespread pedagogical tool, easily implemented in computer labs and in classroom demonstrations. However, we have detected issues raising difficulties in the practical effectuation of these codes which are especially evident when discrete grid methods are used. One issue-relatively well known-appears at high incident energies, producing a wavepacket slower than expected theoretically. The other issue, which appears at low wavepacket energies, does not affect the time evolution of the propagating wavepacket proper, but produces dramatic effects on its spectral decomposition. The origin of the troubles is investigated, and different ways to deal with these issues are proposed. Finally, we show how this problem is manifested and solved in the practical case of the electronic spectra of a metal surface ionized by an ultrashort laser pulse.