3.14.1 Suppose we have a particle in a wavepacket, where the spatial wavefunction at some time t is (r) = A(r) exp (ik ·r). Here, A(r) is a function that varies very slowly in space compared to the function exp (ik ·r), describing the envelope of the wavepacket.

(i)   Given that the particle current density is given by  show that  where p is the (vector) expectation value of the momentum.

(ii)  With similar approximations, evaluate the expectation value of the energy on the assumption that the potential energy is constant in space.

(iii) Hence, show that the velocity of the probability density corresponds to the velocity we would expect classically.