(i) In units where $\hbar=1$, angular momentum states $|j m\rangle$ obey

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

Use the algebra of angular momentum $\left[J_{i}, J_{j}\right]=i \epsilon_{i j k} J_{k}$ to derive the following in terms of $J^{2}, J_{\pm}=J_{1} \pm i J_{2}$ and $J_{3}$ : (a) $\left[J^{2}, J_{i}\right]$; (b) $\left[J_{3}, J_{\pm}\right]$; (c) $\left[J^{2}, J_{\pm}\right]$.

(ii) Find $J_{+} J_{-}$in terms of $J^{2}$ and $J_{3}$. Thus calculate the quantum numbers of the state $J_{\pm}|j m\rangle$ in terms of $j$ and $m$. Derive the normalisation of the state $J_{-}|j m\rangle$. Therefore, show that

$$\left\langle j j-1\left|J_{+}^{j-1} J_{-}^{j}\right| j j\right\rangle=\sqrt{A}(2 j-1) !$$

finding $A$ in terms of $j$.

(iii) Consider the combination of a spinless particle with an electron of spin $1 / 2$ and orbital angular momentum 1. Calculate the probability that the electron has a spin of $+1 / 2$ in the $z$-direction if the combined system has an angular momentum of $+1 / 2$ in the $z$-direction and a total angular momentum of $+3 / 2$. Repeat the calculation for a total angular momentum of $+1 / 2$.