The shift in the energy levels in Example 4.6 can be understood from classical electrodynamics. Consider the case where initially no current flows in the solenoid. Now imagine slowly increasing the current.
(a) Calculate (from classical electrodynamics) the emf produced by the changing flux and show that the rate at which work is done on the charge confined to the ring can be written
dΦdW=−q2πω,
where ω is the angular velocity of the particle.
(b) Calculate the z component of the mechanical angular momentum,
Lmechanical=r×mv=r×(p−qA),(4.231)
for a particle in the state ψn in Example 4.6. Note that the mechanical angular momentum is not quantized in integer multiples of ℏ!
(c) Show that your result from part (a) is precisely equal to the rate at which the stationary state energies change as the flux is increased: dEn/dΦ.