Express the spin operator $\mathbf{S}$ for a particle of spin $\frac{1}{2}$ in terms of the Pauli matrices $\boldsymbol{\sigma}=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)$ where

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

Show that $(\mathbf{n} \cdot \boldsymbol{\sigma})^{2}=\mathbb{I}$ for any unit vector $\mathbf{n}$ and deduce that

$$e^{-i \theta \mathbf{n} \cdot \mathbf{S} / \hbar}=\mathbb{I} \cos (\theta / 2)-i(\mathbf{n} \cdot \boldsymbol{\sigma}) \sin (\theta / 2) .$$

The space of states $V$ for a particle of spin $\frac{1}{2}$ has basis states $|\uparrow\rangle,|\downarrow\rangle$ which are eigenstates of $S_{3}$ with eigenvalues $\frac{1}{2} \hbar$ and $-\frac{1}{2} \hbar$ respectively. If the Hamiltonian for the particle is $H=\frac{1}{2} \alpha \hbar \sigma_{1}$, find

$$e^{-i t H / \hbar}|\uparrow\rangle \quad \text { and } \quad e^{-i t H / \hbar}|\downarrow\rangle$$

as linear combinations of the basis states.

The space of states for a system of two spin $\frac{1}{2}$ particles is $V \otimes V$. Write down explicit expressions for the joint eigenstates of $\mathbf{J}^{2}$ and $J_{3}$, where $\mathbf{J}$ is the sum of the spin operators for the particles.

Suppose that the two-particle system has Hamiltonian $H=\frac{1}{2} \lambda \hbar\left(\sigma_{1} \otimes \mathbb{I}-\mathbb{I} \otimes \sigma_{1}\right)$ and that at time $t=0$ the system is in the state with $J_{3}$ eigenvalue $\hbar$. Calculate the probability that at time $t>0$ the system will be measured to be in the state with $\mathbf{J}^{2}$ eigenvalue zero.