The broadening of one-dimensional Gaussian wave packets is presented in all textbooks on quantum mechanics. It is used as an example to elucidate Heisenberg's uncertainty relation. The behaviour on a lattice is drastically different if the amplitude and/or phase of the wave packet varies on the scale of the lattice constant. Important examples are very narrow wavepackets or wavepackets with an average velocity comparable to the maximum velocity on the lattice. The analytical and numerical results for the time dependence of wave packets on a one-dimensional lattice are presented.

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