(a) Let $|j m\rangle$ be standard, normalised angular momentum eigenstates with labels specifying eigenvalues for $\mathbf{J}^{2}$ and $J_{3}$. Taking units in which $\hbar=1$,

$$J_{\pm}|j m\rangle=\{(j \mp m)(j \pm m+1)\}^{1 / 2}|j m \pm 1\rangle .$$

Check the coefficients above by computing norms of states, quoting any angular momentum commutation relations that you require.

(b) Two particles, each of spin $s>0$, have combined spin states $|J M\rangle$. Find expressions for all such states with $M=2 s-1$ in terms of product states.

(c) Suppose that the particles in part (b) move about their centre of mass with a spatial wavefunction that is a spherically symmetric function of relative position. If the particles are identical, what spin states $|J 2 s-1\rangle$ are allowed? Justify your answer.

(d) Now consider two particles of spin 1 that are not identical and are both at rest. If the 3-component of the spin of each particle is zero, what is the probability that their total, combined spin is zero?