Explain, in a few lines, how the Pauli matrices $\boldsymbol{\sigma}=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)$ with

$$\sigma_{1}=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right), \quad \sigma_{2}=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right), \quad \sigma_{3}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)$$

are used to represent angular momentum operators with respect to basis states $|\uparrow\rangle$ and $|\downarrow\rangle$ corresponding to spin up and spin down along the 3 -axis. You should state clearly which properties of the matrices correspond to general features of angular momentum and which are specific to spin half.

Consider two spin-half particles labelled A and B, each with its spin operators and spin eigenstates. Find the matrix representation of

$$\boldsymbol{\sigma}^{(\mathrm{A})} \cdot \boldsymbol{\sigma}^{(\mathrm{B})}=\sigma_{1}^{(\mathrm{A})} \sigma_{1}^{(\mathrm{B})}+\sigma_{2}^{(\mathrm{A})} \sigma_{2}^{(\mathrm{B})}+\sigma_{3}^{(\mathrm{A})} \sigma_{3}^{(\mathrm{B})}$$

with respect to a basis of two-particle states $|\uparrow\rangle_{\mathrm{A}}|\uparrow\rangle_{\mathrm{B}},|\downarrow\rangle_{\mathrm{A}}|\uparrow\rangle_{\mathrm{B}},|\uparrow\rangle_{\mathrm{A}}|\downarrow\rangle_{\mathrm{B}},|\downarrow\rangle_{\mathrm{A}}|\downarrow\rangle_{\mathrm{B}}$. Show that the eigenvalues of the matrix are $1,1,1,-3$ and find the eigenvectors.

What is the behaviour of each eigenvector under interchange of $\mathrm{A}$ and $\mathrm{B}$ ? If the particles are identical, and there are no other relevant degrees of freedom, which of the two-particle states are allowed?

By relating $\left(\boldsymbol{\sigma}^{(\mathrm{A})}+\boldsymbol{\sigma}^{(\mathrm{B})}\right)^{2}$ to the operator discussed above, show that your findings are consistent with standard results for addition of angular momentum.