See attached file.
Schrödinger equation in the absence of an external potential is:
i hbar d psi/dt = -hbar^2/(2m) d^2 psi/dx^2
Inserting in here psi(x,t) = A exp(i k x - i omega t) gives:
hbar omega = hbar^2 k^2/(2m) ---->
omega = hbar k^2/(2m)
Phase velocity = omega/k = hbar k/(2m) = (using p = hbar k) = p/(2m) = v/2
Group velocity = d omega/dk =hbar k/m =p/m = v
The group velocity, not the phase velocity, corresponds to the particle velocity. To see why that is the case, consider constructing a wave packet that is localized in space and then see how fast the peak of the wave moves. The peak of the wave is ...
A detailed explanation is given.