I wish I could give you a detailed explanation but I have to go pretty soon...but here's a LaTeX document I wrote:
\documentclass[onecolumn, 10pt]{article}
\title{Quantum Mechanics II: Homework}
\maketitle
\begin{document}
4.26 (a) Check that the spin matrices
\begin{equation}
S_{z} = \frac{\hbar}{2}\left(\begin{array}{ccc}
1 & 0 \\
0 & -1 \end{array} \right)
\end{equation}
\newline
\begin{equation}
S_{x} = \frac{\hbar}{2}\left(\begin{array}{ccc}
0 & 1 \\
1 & 0 \end{array} \right)
\end{equation}
\newline
\begin{equation}
S_{y} = \frac{\hbar}{2}\left(\begin{array}{ccc}
0 & -i \\
i & 0 \end{array} \right)
\end{equation}
\newline
obey the fundamental communtation relations for angular momentum:
\begin{equation}
[S_{x}, S_{y}] = i\hbar S_{z}
\end{equation}
\newline
\begin{equation}
[S_{y}, S_{z}] = i\hbar S_{x}
\end{equation}
\newline
\begin{equation}
[S_{z}, S_{x}] = i\hbar S_{y}
\end{equation}
(b) Show that the Pauli spin matrices
\begin{equation}
\sigma_{x} = \left(\begin{array}{ccc}
0 & 1 \\
1 & 0 \end{array} \right)
\end{equation}
\newline
\begin{equation}
\sigma_{y} = \left(\begin{array}{ccc}
0 & -i \\
i & 0 \end{array} \right)
\end{equation}
\newline
\begin{equation}
\sigma_{z} = \left(\begin{array}{ccc}
1 & 0 \\
0 & -1 \end{array} \right)
\end{equation}
\newline
satisfy the product rule
\begin{equation}
\sigma _{j}\sigma _{k} = \delta _{jk} + i\sum_{l}\epsilon _{jkl}\sigma _{l}
\end{equation}
\newpage
4.29) (a) Find the eigenvalues and eigenspinors of $S_{y}$ \newline
(b) If you measure $S_{y}$ on a particle in the general state $\chi$ where
\begin{equation}
\chi = \left(\begin{array}{ccc} a\\b\end{array}\right) = \left(a\chi _{+} + b\chi _{-}\right)
\end{equation}
\newline
what values might you get, and what is the probability of each? $a$ and $b$ need not be real and check that the probabilities add up to 1 \newline
(c) If you measured $S^{2}_{y}$ what values might you get and with what probabilities?\newline
4.30) Construct the matrix $S_{r}$ representing the component of spin angular momentum along the arbitrary direction $\widehat{r}$
\begin{equation}
\widehat{r} = \sin \theta \cos \phi \widehat{i} + \sin \theta \sin \phi \widehat{j} + \cos \theta \widehat{k}
\end{equation}
Find the eigenvalues and (normalized) eigenspinors of $S_{r}$ \newline
\newline
4.33) An electron is at rest in an oscillating magnetic field \newline
\begin{equation}
\widehat{B} = \widehat{B}_{0} \cos \left(\omega t \right) \widehat{k}
\end{equation}
where $\widehat{B}_{0}$ and $\omega $ are constants. \newline
(a) Construct the Hamitonian matrix directly for this system.\newline
(b) The electron starts out (at \textit{t} = 0) in the spin up state with respect to the \textit{x} axis (that is: $\chi(\theta) = \chi^{x}_{+}$)\newline
Determine $\chi \left(t\right)$ at any subsequent time. \newline
(c) Find the probability of getting $\frac{-\hbar}{2}$ if you measure $S_{x}$\newline
(d) What is the minimum field required to force a complete flip in $S_{x}$?
\end{document}