Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 35A / b / Solution

Write $c=\cos (\theta /2)$ and $s=\sin (\theta /2)$. ThenThus $|\psi ⟩$ is unchanged. It is the spin-one-half singlet state, so joint rotations leave it invariant. Equivalently, direct use of the Pauli matrices showsIt is therefore an eigenstate of every Cartesian component of the combined spin, with eigenvalue zero.