Let $\mathbf{J}=\left(J_{1}, J_{2}, J_{3}\right)$ and $|j m\rangle$ denote the standard angular-momentum operators and states so that, in units where $\hbar=1$,

$$\mathbf{J}^{2}|j m\rangle=j(j+1)|j m\rangle, \quad J_{3}|j m\rangle=m|j m\rangle$$

Show that $U(\theta)=\exp \left(-i \theta J_{2}\right)$ is unitary. Define

$$J_{i}(\theta)=U(\theta) J_{i} U^{-1}(\theta) \quad \text { for } i=1,2,3$$

and

$$|j m\rangle_{\theta}=U(\theta)|j m\rangle \text {. }$$

Find expressions for $J_{1}(\theta), J_{2}(\theta)$ and $J_{3}(\theta)$ as linear combinations of $J_{1}, J_{2}$ and $J_{3}$. Briefly explain why $U(\theta)$ represents a rotation of $\mathbf{J}$ through angle $\theta$ about the 2-axis.

Show that

$$J_{3}(\theta)|j m\rangle_{\theta}=m|j m\rangle_{\theta}$$

Express $|10\rangle_{\theta}$ as a linear combination of the states $|1 m\rangle, m=-1,0,1$. By expressing $J_{1}$ in terms of $J_{\pm}$, use $(*)$ to determine the coefficients in this expansion.

A particle of spin 1 is in the state $|10\rangle$ at time $t=0$. It is subject to the Hamiltonian

$$H=-\mu \mathbf{B} \cdot \mathbf{J}$$

where $\mathbf{B}=(0, \mathbf{B}, 0)$. At time $t$ the value of $J_{3}$ is measured and found to be $J_{3}=0$. At time $2 t$ the value of $J_{3}$ is measured again and found to be $J_{3}=1$. Show that the joint probability for these two values to be measured is

$$\frac{1}{8} \sin ^{2}(2 \mu B t) .$$

[The following result may be quoted: $\left.J_{\pm}|j m\rangle=\sqrt{(j \mp m)(j \pm m+1)}|j m \pm 1\rangle .\right]$