- C An Electron Moves In A Magnetic Field Directed Along The Z Axis So That The Hamiltonian Is H Gubb H S Determine 1 (135.06 KiB) Viewed 44 times
(c) An electron moves in a magnetic field directed along the z-axis, so that the Hamiltonian is H = (gµBB/ħ)s. Determine the spin operators in the Heisenberg picture at time t, sx(1), sy(1), s-(1), in terms of the corresponding operators at t=0. [The Heisenberg equation is in- do dt = [0, H].] [6 marks] (d) At the time t=0 the x-component of this electron's spin was measured, and found to be ħ/2. Determine the expectation values of the three spin components at a later time. Describe the motion of the spin vector. [6 marks] (e) Show that the uncertainty in the spin components is related to their expectation values. Hence determine the times, if any, at which a measurement of spin along the x, y, or z axes would give a completely predictable outcome, explaining your reasoning. [The uncertainty in an observable is AO = √(0²) - (0)², where (Ô) = (Ô) denotes the expectation value.] [6 marks] (f) Suppose that the electron spin had initially been pointing at 45 degrees to the z axis. By sketching the dynamics of the spin vector, or otherwise, determine whether there are any times at which the uncertainties in the spin components vanish in this case.