Spin 3/2 along an arbitrary direction

[Samuel Williams]

Homework Statement

Consider a particle of spin 3/2. Find the matrix for the component of the spin along a unit vector
with arbitrary direction n. Find its eigenvalues and eigenvectors.

Homework Equations

I know that the general spin operator is
\begin{equation}
\widehat{S} = a\cdot \widehat{S}_x + b\cdot \widehat{S}_y + c\cdot \widehat{S}_z
\end{equation}
and that
\begin{equation}
\hat{n} = (\sin{\phi}\cos{\theta},\sin{\phi}\sin{\theta},\cos{\phi})
\end{equation}

The Attempt at a Solution

Now
\begin{equation}
\widehat{S}_x \stackrel{\cdot}{=} \frac{\hbar}{2}\begin{pmatrix}0 & \sqrt3 & 0 & 0 \\ \sqrt3 & 0 & 2 & 0 \\ 0 & 2 & 0 & \sqrt3 \\ 0 & 0 & \sqrt3 & 0\end{pmatrix}\, ,\
\widehat{S}_y \stackrel{\cdot}{=} \frac{\hbar}{2i}\begin{pmatrix}0 & \sqrt3 & 0 & 0 \\ -\sqrt3 & 0 & 2 & 0 \\ 0 & -2 & 0 & \sqrt3 \\ 0 & 0 & -\sqrt3 & 0\end{pmatrix}\, ,\
\widehat{S}_z \stackrel{\cdot}{=} \hbar\begin{pmatrix}3/2 & 0 & 0 & 0 \\ 0 & 1/2 & 0 & 0 \\ 0 & 0 & -1/2 & 0 \\ 0 & 0 & 0 & -3/2\end{pmatrix}\ .
\end{equation}

So then we get
\begin{equation}
\hat{n}\cdot \widehat{S} \stackrel{\cdot}{=} \frac{\hbar}{2}(\sin{\phi}\cos{\theta})\begin{pmatrix}0 & \sqrt3 & 0 & 0 \\ \sqrt3 & 0 & 2 & 0 \\ 0 & 2 & 0 & \sqrt3 \\ 0 & 0 & \sqrt3 & 0\end{pmatrix}\ + \frac{\hbar}{2i}(\sin{\phi}\sin{\theta})\begin{pmatrix}0 & \sqrt3 & 0 & 0 \\ -\sqrt3 & 0 & 2 & 0 \\ 0 & -2 & 0 & \sqrt3 \\ 0 & 0 & -\sqrt3 & 0\end{pmatrix}\ + \hbar(\cos{\phi})\begin{pmatrix}3/2 & 0 & 0 & 0 \\ 0 & 1/2 & 0 & 0 \\ 0 & 0 & -1/2 & 0 \\ 0 & 0 & 0 & -3/2\end{pmatrix}\ .
\end{equation}

Now, I know I should determine the secular equation (characteristic equation) after adding the matrices and find the eigenvectors and eigenvalues from there, but I keep getting long and complicated equations that don't seem to factor. Is there any shorter method or any other way to do it?

 

[blue_leaf77]

A less space-consuming calculation is done by using the braket notation instead of matrices. In the basis of the eigenstates of $S_z$, the braket version of $S$ is
$$
S = \sum_{m=-3/2}^{3/2} \sum_{m'=-3/2}^{3/2} |m\rangle \langle m| S |m'\rangle \langle m'|
$$
Write the eigenvalue problem $S|u_1\rangle = \lambda |u_1\rangle$ where $|u_1\rangle = a|3/2\rangle + b|1/2\rangle + c|-1/2\rangle + d|-3/2\rangle$. You are then to find $\lambda$'s and $a$, $b$, $c$, and $d$ for each one value of $\lambda$.