= Spin-one singlet state
{title2=$|0,0\rangle=(|1,-1\rangle-|0,0\rangle+|-1,1\rangle)/\sqrt3$}
In the product basis $|m_1,m_2\rangle$ for two spin-one particles, total magnetic quantum number zero allows only $|1,-1\rangle$, $|0,0\rangle$ and $|-1,1\rangle$. Applying the total <ladder operators> to $\alpha|1,-1\rangle+\beta|0,0\rangle+\gamma|-1,1\rangle$ gives
$$
S_+|\psi\rangle=\sqrt2\hbar[(\alpha+\beta)|1,0\rangle+(\beta+\gamma)|0,1\rangle].
$$
A <singlet state> is annihilated by $S_+$ and $S_-$, so $\alpha=\gamma=-\beta$. <Wavefunction normalization> gives the displayed state, up to an overall phase. Here the ket on the left of the title formula has total-spin labels, while its right-hand kets have individual magnetic labels. The state is symmetric under exchanging the particles, although no exchange constraint is needed for distinguishable particles.
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