Let $|j, m\rangle$ denote the normalised joint eigenstates of $\mathbf{J}^{2}$ and $J_{3}$, where $\mathbf{J}$ is the angular momentum operator for a quantum system. State clearly the possible values of the quantum numbers $j$ and $m$ and write down the corresponding eigenvalues in units with $\hbar=1$.

Consider two quantum systems with angular momentum states $\left|\frac{1}{2}, r\right\rangle$ and $|j, m\rangle$. The eigenstates corresponding to their combined angular momentum can be written as

$$|J, M\rangle=\sum_{r, m} C_{r m}^{J M}\left|\frac{1}{2}, r\right\rangle|j, m\rangle,$$

where $C_{r m}^{J M}$ are Clebsch-Gordan coefficients for addition of angular momenta $\frac{1}{2}$ and $j$. What are the possible values of $J$ and what is a necessary condition relating $r, m$ and $M$ in order that $C_{r m}^{J M} \neq 0$ ?

Calculate the values of $C_{r m}^{J M}$ for $j=2$ and for all $M \geqslant \frac{3}{2}$. Use the sign convention that $C_{r m}^{J J}>0$ when $m$ takes its maximum value.

A particle $X$ with spin $\frac{3}{2}$ and intrinsic parity $\eta_{X}$ is at rest. It decays into two particles $A$ and $B$ with spin $\frac{1}{2}$ and spin 0 , respectively. Both $A$ and $B$ have intrinsic parity $-1$. The relative orbital angular momentum quantum number for the two particle system is $\ell$. What are the possible values of $\ell$ for the cases $\eta_{X}=+1$ and $\eta_{X}=-1$ ?

Suppose particle $X$ is prepared in the state $\left|\frac{3}{2}, \frac{3}{2}\right\rangle$ before it decays. Calculate the probability $P$ for particle $A$ to be found in the state $\left|\frac{1}{2}, \frac{1}{2}\right\rangle$, given that $\eta_{X}=+1$.

What is the probability $P$ if instead $\eta_{X}=-1$ ?

[Units with $\hbar=1$ should be used throughout. You may also use without proof

$$\left.J_{-}|j, m\rangle=\sqrt{(j+m)(j-m+1)}|j, m-1\rangle .\right]$$