Expectation values of spin operators in changing magnetic field

[athrun200]

Homework Statement

Homework Equations

The Attempt at a Solution

I have totally no idea how to solve this question. But I find it somehow similar to the Larmor precession problem. Therefore I try to solve my problem by referring to that.
Are there any mistakes if I do it like this?

p.s. since the matrix elements are quite complicated, in order to save time, I use "c" to stand for cosine, "e" to stand for exponential that repeat over and over again.

 

[TSny]

It looks like you're taking the spin state of the particle at time t = 0 to be proportional to $\left(\begin{array}{c}1\\0\end{array}\right)$. That would correspond to spin up along the z-axis. But at t = 0 the spin state is an eigenstate of the Hamiltonian H for a magnetic field in the direction specified by $\theta$ and $\phi$. In particular, it's the state that corresponds to "spin up" along the $\theta$, $\phi$ direction (or, positive energy eigenvalue for H).