Past exam of the physics A course of the University of Cambridge / 2013 / Paper 1 / Section A / 2

The free particle dispersion relation is $\hslash \omega ={\hslash }^2{k}^2/(2m)$. HenceThe phase velocity follows constant-phase surfaces of a plane wave. The group velocity follows the envelope of a narrow-band wave packet and equals its classical velocity $p/m$. Dispersion can spread the packet; the phase speed is not the particle's transport velocity. In three dimensions ${\mathbf{v}}_g=\hslash \mathbf{k}/m$ and the phase speed quoted is along $\mathbf{k}$.