The operator corresponding to a rotation through an angle $\theta$ about an axis $\mathbf{n}$, where $\mathbf{n}$ is a unit vector, is

$$U(\mathbf{n}, \theta)=e^{i \theta \mathbf{n} \cdot \mathbf{J} / \hbar}$$

If $U$ is unitary show that $\mathbf{J}$ must be hermitian. Let $\mathbf{V}=\left(V_{1}, V_{2}, V_{3}\right)$ be a vector operator such that

$$U(\mathbf{n}, \delta \theta) \mathbf{V} U(\mathbf{n}, \delta \theta)^{-1}=\mathbf{V}+\delta \theta \mathbf{n} \times \mathbf{V} .$$

Work out the commutators $\left[J_{i}, V_{j}\right]$. Calculate

$$U(\hat{\mathbf{z}}, \theta) \mathbf{V U}(\hat{\mathbf{z}}, \theta)^{-1}$$

for each component of $\mathbf{V}$.

If $|j m\rangle$ are standard angular momentum states determine $\left\langle j m^{\prime}|U(\hat{\mathbf{z}}, \theta)| j m\right\rangle$ for any $j, m, m^{\prime}$ and also determine $\left\langle\frac{1}{2} m^{\prime}|U(\hat{\mathbf{y}}, \theta)| \frac{1}{2} m\right\rangle$.

$\left[\right.$ Hint $\left.: J_{3}|j m\rangle=m \hbar|j m\rangle, J_{+}\left|\frac{1}{2}-\frac{1}{2}\right\rangle=\hbar\left|\frac{1}{2} \frac{1}{2}\right\rangle, J_{-}\left|\frac{1}{2} \frac{1}{2}\right\rangle=\hbar\left|\frac{1}{2}-\frac{1}{2}\right\rangle \cdot\right]$