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 = -5w(t) . o such that a cos nt w(t) = w |asin It 1 where w absorbs all necessary constants, o 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 (1) of the Pauli operator oz, i.e. [w(t = 0)) = [1). What is the spin-flip probability, i.e. the probability of measuring |4) after time t? (a) (20 points) The fact that the Hamiltonian depends on time t complicates the situation. However, it is possible to simplify the Schrodinger equation for the state lu(t)) by performing the following unitary transformation: 10 (t ) ) = e- 20= 12(t)). Using this transformation, derive the simplified Schrodinger equation for the state lo(t)). (b) (10 points) Write down the unitary operator that connects |o(t)) at some time t with the initial state |o(t = 0)). (c) (10 points) To find |(t)), observe that lo(t = 0))=[(t = 0)) = [1). Find out the probability of measuring |4) after time t. Check that your answer makes sense when t = 0 or a = 0. Please show your work but you can omit any irrelevant expressions whenever you can justify doing that