How Does Spin Interaction with a Magnetic Field Affect Quantum States?

[forceface]

Homework Statement

The problem is stated on page 10 and it is number 9. http://panda.unm.edu/pandaweb/graduate/prelims/QM_F11.pdf

Homework Equations

I know how to write S_y in the z basis for either spin up or down and I have all the spinors.

The Attempt at a Solution

I tried solving for |ψ(t)> by using B_o as the hamiltonian but that seems to give me an answer that a is single solution and I was expecting a column vector as a solution. If I got a column vector I would just take the inner product of |<S_y|ψ(t)>|² and that would give me the probability. With this said I feel like I am approaching this problem wrong and I have to take into account that fact that the magnetic field is in the z direction.

 

[Oxvillian]

Hmm what you say above doesn't make much sense to me because $S_y$ and $H$ are supposed to be matrices.