 Solution

ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2013/paper-1/section-a/2/solution

ℏ ω = ℏ^{2} k^{2} / (2 m)

. Hence

v_{ph} = ω / k = ℏ k / (2 m), v_{g} = d ω / d k = ℏ k / m .

The 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

v_{g} = ℏ k / m

and the phase speed quoted is along

k

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