The angular momentum operators $\mathbf{J}^{(1)}$ and $\mathbf{J}^{(2)}$ refer to independent systems, each with total angular momentum one. The combination of these systems has a basis of states which are of product form $\left|m_{1} ; m_{2}\right\rangle=\left|1 m_{1}\right\rangle\left|1 m_{2}\right\rangle$ where $m_{1}$ and $m_{2}$ are the eigenvalues of $J_{3}^{(1)}$ and $J_{3}^{(2)}$ respectively. Let $|J M\rangle$ denote the alternative basis states which are simultaneous eigenstates of $\mathbf{J}^{2}$ and $J_{3}$, where $\mathbf{J}=\mathbf{J}^{(1)}+\mathbf{J}^{(2)}$ is the combined angular momentum. What are the possible values of $J$ and $M$ ? Find expressions for all states with $J=1$ in terms of product states. How do these states behave when the constituent systems are interchanged?

Two spin-one particles $A$ and $B$ have no mutual interaction but they each move in a potential $V(\mathbf{r})$ which is independent of spin. The single-particle energy levels $E_{i}$ and the corresponding wavefunctions $\psi_{i}(\mathbf{r})(i=1,2, \ldots)$ are the same for either $A$ or $B$. Given that $E_{1}<E_{2}<\ldots$, explain how to construct the two-particle states of lowest energy and combined total spin $J=1$ for the cases that (i) $A$ and $B$ are identical, and (ii) $A$ and $B$ are not identical.

[You may assume $\hbar=1$ and use the result $\left.J_{\pm}|j m\rangle=\sqrt{(j \mp m)(j \pm m+1)}|j m \pm 1\rangle .\right]$