The hamiltonian of a spin in a magnetic field is given by a(bbysy+b, s), where α and the three components of b are all constants. (a) compute the energies and corresponding eigenstates of this system (b) let by-0 and b = 0. (i) use the energies and eigenstates for this case to determine the time evolution ψ(t) of a state with initial condition ψ(0) (ii) calculate the expectation values 〈s) and (s) as a function of tine. (iii) use a taylor expansion to show that the time-evolution operator u(t) exp(-iht/h) can be written as (e)s()i - isin(ut)x, where i is the identity operator, x (2/h)s, and w ab/2. hint: use the pauli matrix representation of the spin operator and show that