3. A spin-1/2 particle is in a magnetic field B = By: + By (cos(wt) + sin(wt)) . (2) (a) Let [(t)) = are 2| 1) + agetwt/?| |) (3) Write the Schroedinger equation for [1()), and use this to write the differential equation that the coefficients ey must satisfy. (Do not solve the Schroedinger equation at this point). (b) From (a), find the wave function in the rotating frame, e.g. find [9(t)) = ar(t)] 1) + az(t)] 1) (4) if the particle starts in the state | [} at time = 0. Hint: One way to do this is to take your differential equation, and write it in the form 2 (ox9) = 17 (i) where H is a 2 x 2 matrix, which you can think of as the Hamiltonian in the rotating frame. A general form for the eigenvectors of this matrix is given in Eq. 3.57 (see discussion starting at 3.55) in the book. Write your answer in terms of the angle 4, defined by tanf = w:'j: () Use your result from (c) to find [|(t)). (d) Compute the probability of finding the system in the state |[4) as a function of time, in both [(t)) and [(t)). Which if these answers gives you the actual probability of finding your spin in the state |[4) if you measure it in the lab