Description of the 6−j and the 9−j Symbols in Terms of Small Numbers of 3−j Symbols

Abstract

The 6−j symbol (the Racah coefficient) is described in a form of linear combination of 3−j symbols (Clebsch-Gordan coefficients). Deduction is based on the quasi-spin formalism combined with the state operator formalism. The result is used to express the 9−j symbol, expanded usually in terms of three 6−j symbols, as a weighted sum over two 3−j symbols. The obtained expressions facilitate rapid evaluation of the 6−j and the 9−j symbols. An asymptotic expansion of the 9−j symbol is discussed.