Let $\boldsymbol{\sigma}=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)$ be a set of Hermitian operators obeying

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

where $\mathbf{n}$ is any unit vector. Show that $(*)$ implies that

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

for any vectors a and $\mathbf{b}$. Explain, with reference to the properties $(*)$, how $\boldsymbol{\sigma}$ can be related to the intrinsic angular momentum $\mathbf{S}$ for a particle of spin $\frac{1}{2}$.

Show that the operators $P_{\pm}=\frac{1}{2}(1 \pm \mathbf{n} \cdot \boldsymbol{\sigma})$ are Hermitian and obey

$$P_{\pm}^{2}=P_{\pm}, \quad P_{+} P_{-}=P_{-} P_{+}=0$$

Show how $P_{\pm}$can be used to write any state $|\chi\rangle$ as a linear combination of eigenstates of $\mathbf{n} \cdot \boldsymbol{\sigma}$. Use this to deduce that if the system is in a normalised state $|\chi\rangle$ when $\mathbf{n} \cdot \boldsymbol{\sigma}$ is measured, then the results $\pm 1$ will be obtained with probabilities

$$\| P_{\pm}|\chi\rangle \|^{2}=\frac{1}{2}(1 \pm\langle\chi|\mathbf{n} \cdot \boldsymbol{\sigma}| \chi\rangle)$$

If $|\chi\rangle$ is a state corresponding to the system having spin up along a direction defined by a unit vector $\mathbf{m}$, show that a measurement will find the system to have spin up along $\mathbf{n}$ with probability $\frac{1}{2}(1+\mathbf{n} \cdot \mathbf{m})$.