Clebsch–Gordan coefficients

Summary

Take two irreducible representations of $SU(2)$, or of $\mathfrak{su}(2)$. The Clebsch–Gordan coefficients are the entries of the change of basis that splits their tensor product into irreducible pieces. They express a state of total angular momentum $J$ through products of states of the two parts. They are central in quantum mechanics, where they describe the coupling of two angular momenta. The decomposition behind them is the rule $V_{j_1} \otimes V_{j_2} \cong V_{\abs{j_1 - j_2}} \oplus \dots \oplus V_{j_1 + j_2}$ for the representations of sl2, here labelled by spin, so that $V_j$ has dimension $2j + 1$.