Consider a joint eigenstate of $\mathbf{J}^{2}$ and $J_{3},|j m\rangle$. Write down a unitary operator $U(\mathbf{n}, \theta)$ for rotation of the state by an angle $\theta$ about an axis with direction $\mathbf{n}$, where $\mathbf{n}$ is a unit vector. How would a state with zero orbital angular momentum transform under such a rotation?

What is the relation between the angular momentum operator $\mathbf{J}$ and the Pauli matrices $\boldsymbol{\sigma}$ when $j=\frac{1}{2}$ ? Explicitly calculate $(\mathbf{J} \cdot \mathbf{a})^{2}$, for an arbitrary real vector $\mathbf{a}$, in this case. What are the eigenvalues of the operator $\mathbf{J} \cdot \mathbf{a}$ ? Show that the unitary rotation operator for $j=\frac{1}{2}$ can be expressed as

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

Starting with a state $\left|\frac{1}{2} m\right\rangle$ the component of angular momentum along a direction $\mathbf{n}^{\prime}$, making and angle $\theta$ with the $z$-axis, is susequently measured to be $m^{\prime}$. Immediately after this measurement the state is $\left|\frac{1}{2} m^{\prime}\right\rangle_{\theta}$. Write down an eigenvalue equation for $\left|\frac{1}{2} m^{\prime}\right\rangle_{\theta}$ in terms of $\mathbf{n}^{\prime} \cdot \mathbf{J}$. Show that the probability for measuring an angular momentum of $m^{\prime} \hbar$ along the direction $\mathbf{n}^{\prime}$ is, assuming $\mathbf{n}^{\prime}$ is in the $x-z$ plane,

$$\left|\left\langle\frac{1}{2} m \mid \frac{1}{2} m^{\prime}\right\rangle_{\theta}\right|^{2}=\left|\left\langle\frac{1}{2} m|U(\mathbf{y}, \theta)| \frac{1}{2} m^{\prime}\right\rangle\right|^{2},$$

where $\mathbf{y}$ is a unit vector in the $y$-direction. Using $(*)$ show that the probability that $m=+\frac{1}{2}, m^{\prime}=-\frac{1}{2}$ is of the form

$$A+B \cos ^{2} \frac{\theta}{2}$$

determining the integers $A$ and $B$ in the process.

[Assume $\hbar=1$. The Pauli matrices are

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