Answer Happy

3. The gyromagnetic factor of the electron, g, determines the relationship between the electron magnetic moment and the electron spin S. When e is the electron charge and m is the electron mass, e ū=95 2m An electron beam is allowed to interact with a constant magnetic field B = Bê via the Hamiltonian = 1 2 H= 2m (3 -ca) p- - B = where A is the vector potential. The electrons are confined to the ry-plane, and electron-electron interactions are ignored. (a) Show that the commutation relations [, H] =ihang (Vy, H] = -ihwu are satisfied, where v = (-A)/m is the gauge-invariant velocity operator and w= : eB/m is the electron's cyclotron frequency. (Hint: The Landau gauge Ā= (0, Br, 0) can be used.) (b) Derive a set of coupled equations that describe the time evolution of the two expectation values Ci(t) and C2(t), defined as Ci(t) = (S202 + Syvy), Cz(t) = (S. V, - Syur) If g = 2 as the Dirac equation predicts, show that Ci(t) and C2(t) do not change with time.