The components of $\boldsymbol{\sigma}=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)$ are $2 \times 2$ hermitian matrices obeying

$$\left[\sigma_{i}, \sigma_{j}\right]=2 i \varepsilon_{i j k} \sigma_{k} \quad \text { and } \quad(\mathbf{n} \cdot \boldsymbol{\sigma})^{2}=1$$

for any unit vector $\mathbf{n}$. Show that these properties imply

$$(\mathbf{a} \cdot \boldsymbol{\sigma})(\mathbf{b} \cdot \boldsymbol{\sigma})=\mathbf{a} \cdot \mathbf{b}+i(\mathbf{a} \times \mathbf{b}) \cdot \boldsymbol{\sigma}$$

for any constant vectors a and $\mathbf{b}$. Assuming that $\theta$ is real, explain why the matrix $U=\exp (-i \mathbf{n} \cdot \boldsymbol{\sigma} \theta / 2)$ is unitary, and show that

$$U=\cos (\theta / 2)-i \mathbf{n} \cdot \boldsymbol{\sigma} \sin (\theta / 2)$$

Hence deduce that

$$U \mathbf{m} \cdot \boldsymbol{\sigma} U^{-1}=\mathbf{m} \cdot \boldsymbol{\sigma} \cos \theta+(\mathbf{n} \times \mathbf{m}) \cdot \boldsymbol{\sigma} \sin \theta$$

where $\mathbf{m}$ is any unit vector orthogonal to $\mathbf{n}$.

Write down an equation relating the matrices $\sigma$ and the angular momentum operator $\mathbf{S}$ for a particle of spin one half, and explain briefly the significance of the conditions $(*)$. Show that if $|\chi\rangle$ is a state with spin 'up' measured along the direction $(0,0,1)$ then, for a certain choice of $\mathbf{n}, U|\chi\rangle$ is a state with spin 'up' measured along the direction $(\sin \theta, 0, \cos \theta)$.