Question 1 (20 points) We start with a system in the following state which is a super- position of harmonic oscillator states (v) = C [(1 + 2)|0) + 3|1) + (2)] where the oscillation frequency is w. a) Find the normalization constant C' (3 points). b) Find the expectation value for the ladder operator (a) (3 points). c) Find the expectation value for the ladder operator (at) (3 Points). d) Find the expectation value (H), where H is the harmonic oscillator Hamiltonian (3 points)? e) Find the time evolution for the state ly(t))? (3 points) Question 2 (8 points) Consider a system with total angular momentum J = 1. a) What values can the z projection quantum number m; take on and write down all the eigenstates in the form (j, m;) (3 points)? b) For a wavefunction () in the m; = 1 state (|1, 1) ) find Joly) (Hint: recall Jx = 1/2(J+ + J_) (3 points)? c) Now our system is coupled to a nuclear spin I = 3/2 where the total angular momentum for the whole system is now given by F = J + I. What are the possible values for the z projection quantum number mp now (2 points)? Question 3 (7 points) Suppose a spin 1/2 system is initialized to an eigenstate of Sx, 120) = 12+) =1/V2(12+) + 12->). a) At time t = 0 we apply a magnetic field along the z direction such that the Hamiltonian is now H = woSz. Determine the time evolution of the system |(t)) (4 points). b) Find the probability of measuring the system in (z+) as a function of time (3 points)