Define the spin raising and spin lowering operators $S_{+}$and $S_{-}$. Show that

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

where $S_{z}|s, \sigma\rangle=\hbar \sigma|s, \sigma\rangle$ and $S^{2}|s, \sigma\rangle=s(s+1) \hbar^{2}|s, \sigma\rangle$.

Two spin- $\frac{1}{2}$ particles, with spin operators $\mathbf{S}^{(1)}$ and $\mathbf{S}^{(2)}$, have a Hamiltonian

$$H=\alpha \mathbf{S}^{(1)} \cdot \mathbf{S}^{(2)}+\mathbf{B} \cdot\left(\mathbf{S}^{(1)}-\mathbf{S}^{(2)}\right)$$

where $\alpha$ and $\mathbf{B}=(0,0, B)$ are constants. Express $H$ in terms of the two particles' spin raising and spin lowering operators $S_{\pm}^{(1)}, S_{\pm}^{(2)}$ and the corresponding $z$-components $S_{z}^{(1)}$, $S_{z}^{(2)}$. Hence find the eigenvalues of $H$. Show that there is a unique groundstate in the limit $B \rightarrow 0$ and that the first excited state is triply degenerate in this limit. Explain this degeneracy by considering the action of the combined spin operator $\mathbf{S}^{(1)}+\mathbf{S}^{(2)}$ on the energy eigenstates.