The spin operators obey the commutation relations $\left[S_{i}, S_{j}\right]=i \hbar \epsilon_{i j k} S_{k}$. Let $|s, \sigma\rangle$ be an eigenstate of the spin operators $S_{z}$ and $\mathbf{S}^{2}$, with $S_{z}|s, \sigma\rangle=\sigma \hbar|s, \sigma\rangle$ and $\mathbf{S}^{2}|s, \sigma\rangle=s(s+1) \hbar^{2}|s, \sigma\rangle$. Show that

$$S_{\pm}|s, \sigma\rangle=\sqrt{s(s+1)-\sigma(\sigma \pm 1)} \hbar|s, \sigma \pm 1\rangle,$$

where $S_{\pm}=S_{x} \pm i S_{y}$. When $s=1$, use this to derive the explicit matrix representation

$$S_{x}=\frac{\hbar}{\sqrt{2}}\left(\begin{array}{lll} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right)$$

in a basis in which $S_{z}$ is diagonal.

A beam of atoms, each with spin 1 , is polarised to have spin $+\hbar$ along the direction $\mathbf{n}=(\sin \theta, 0, \cos \theta)$. This beam enters a Stern-Gerlach filter that splits the atoms according to their spin along the $\hat{\mathbf{z}}$-axis. Show that $N_{+} / N_{-}=\cot ^{4}(\theta / 2)$, where $N_{+}$(respectively, $N_{-}$) is the number of atoms emerging from the filter with spins parallel (respectively, anti-parallel) to $\hat{\mathbf{z}}$.