Two states $\left|j_{1} m_{1}\right\rangle_{1},\left|j_{2} m_{2}\right\rangle_{2}$, with angular momenta $j_{1}, j_{2}$, are combined to form states $|J M\rangle$ with total angular momentum

$$J=\left|j_{1}-j_{2}\right|,\left|j_{1}-j_{2}\right|+1, \ldots, j_{1}+j_{2} .$$

Write down the state with $J=M=j_{1}+j_{2}$ in terms of the original angular momentum states. Briefly describe how the other combined angular momentum states may be found in terms of the original angular momentum states.

If $j_{1}=j_{2}=j$, explain why the state with $J=0$ must be of the form

$$|00\rangle=\sum_{m=-j}^{j} \alpha_{m}|j m\rangle_{1}|j-m\rangle_{2}$$

By considering $J_{+}|00\rangle$, determine a relation between $\alpha_{m+1}$ and $\alpha_{m}$, hence find $\alpha_{m}$.

If the system is in the state $|j j\rangle_{1}|j-j\rangle_{2}$ what is the probability, written in terms of $j$, of measuring the combined total angular momentum to bero?

[Standard angular momentum states $|j m\rangle$ are joint eigenstates of $\mathbf{J}^{2}$ and $J_{3}$, obeying

$$J_{\pm}|j m\rangle=\sqrt{(j \mp m)(j \pm m+1)}|j m \pm 1\rangle$$

Units in which $\hbar=1$ have been used throughout.]