2. A spin-1/2 (quantum two-level system) is subjected to a periodic-in-time time-dependent mag- netic field in the ry-plane and a constant magnetic field in the z-direction. The Hamiltonian of this system can be written as H = -4w(#) - o such that a cos !t w(!) = w | asin fit where w absorbs all necessary constants, or is the ratio of the rotating in plane magnetic field to the field component along the z-direction. ? is the angular frequency of the rotating field. The system is initially prepared in the eigenstate (t) of the Pauli operator oz, i.e. [(t - 0)) = It). What is the spin-flip probability, i.e. the probability of measuring (4) after time ? (a) (20 points) The fact that the Hamiltonian depends on time { complicates the situation. However, it is possible to simplify the Schrodinger equation for the state [o(t)) by performing the following unitary transformation: ()). Using this transformation, derive the simplified Schrodinger equation for the state (off)) (b) (10 points) Write down the unitary operator that connects off)) at some time t with the initial state lo(f = 0)). (c) (10 points) To find (v(t)), observe that loft - O) To(1 - 0)) = It). Find out the probability of measuring |4) after time t. Check that your answer makes sense when f = 0 or o - 0. Please show your work but you can omit any irrelevant expressions whenever you can justify doing that