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Is there a general way to express the singlet state? For instance, is this form $$ \frac{1}{\sqrt{2}}\left(\vert \uparrow \rangle \vert \downarrow \rangle - \vert \downarrow \rangle \vert \uparrow \rangle\right) $$ general, or valid only for two 1/2 spin particles?

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It is possible to construct singlet states for two particles of any identical spin or angular momentum $j$.

In general: \begin{align} \vert 00\rangle = \sum_{m_1 m_2} C^{00}_{jm_1;j,-m_1} \vert j,m_1\rangle \vert j,-m_1\rangle \end{align} where $C^{JM}_{j_1m_1;j_2m_2}$ is a Clesbsh-Gordan coefficient.

Furthermore note that singlet states are not necessarily antisymmetric.

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