- A Show That The Operators 2 15 Y 2 15 1 1 2 6 9 Obey The Spin Commutation Relations And That They 1 (113.41 KiB) Viewed 41 times
(a) Show that the operators = 2(15). ³y = 2(15). $ = 1/(1-2). -(6-9). obey the spin commutation relations, and that they also correspond to spin-½. State the basis being used, and give the physical meaning of the two components of the spinor (6) in this basis. [6 marks] (b) An electron moves in a constant magnetic field, such that its spin is described by the Hamiltonian H = (guBB₂/h)s, with guaB > 0. Using the Heisenberg equation, or otherwise, show that the expectation values of the spin operators are given by (5-(t)) = A cos((21) + B sin(r) (sy(1)) = -A sin(2) + B cos(2), where A and B are constants. Determine the angular frequency 2. [The Heisenberg equation, giving the time-dependence of an operator O, is do id= [0,H).] [6 marks] (c) Determine the behaviour of the expectation value (s()), and comment on why this behaviour occurs. [2 marks] (d) Suppose that at time /-0 the electron has (₂)=-1/2, Deduce the corresponding spinor, and hence the values of (sx) and (sy), at /-0, explaining your reasoning. Hence determine the expectation values of the spin components at a subsequent time . [4 marks] 3. (e) At time r=0 the z-component of the spin of an electron is measured, and found to be -ħ/2. At a time = x/2 the z-component of the spin is measured again. Determine the possible outcomes of this measurement and, if there is more than one, the corresponding probabilities. [2 marks] (f) A spin is prepared as in part (e), but the measurement of the z-component is instead performed at time=/(202). Determine the possible outcomes of this measurement and, if there is more than one, the associated probabilities. [2 marks] (g) Suppose that at time /-0 the x-component of the spin was measured. At some later time the z-component of the spin is measured. What are the possible outcomes of this measurement, and with what probabilities? [3 marks] 3.