I need some help with this question.

'.' At time t = 0, the state of an electron spin is IIMt = 0)) = I +)n with 9 = E and (I) = Z. The system is allowed to time-evolve in a uniform magnetic eld B = 302. [I a) ' Write the initial state in the usual z-basis i.e. |1.b(0)) = (b ), and find a and b. . . - . . . 1' Suggestion: Leave the factor el'l' = 81le 4 in that form; don't rewrite it as 51. b) Follow the usual procedure to find |1,b(t)) (in the usual Sz-basis). c) What is the probability that the electron will be measured to have spin up in the y-direction after a time t? Make a graph of this probability as a function of time. It is always useful to check your answers this is a probability, so convince yourself (and us) that your answer always falls in the allowed range for probabilities. d) Will the electron ever get back to its original starting state? (If so, when? If not, why not?) e) ' Calculate the expectation values of the three spin operators, (3:), (g), and (5:) as a function of time. (Show all your work.) Draw graphs of these quantities as a function of time and use the graphs to describe in your own words the behavior of the average spin vector (3'). From (5:), you can easily find the expectation value of the energy, (Q)