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34 3j, 6j, 9j SymbolsProperties

§34.7 Basic Properties: 9j Symbol


§34.7(i) Special Case

34.7.1 {j11j12j13j21j22j13j31j310}=(-1)j12+j21+j13+j31((2j13+1)(2j31+1))12{j11j12j13j22j21j31}.

§34.7(ii) Symmetry

The 9j symbol has symmetry properties with respect to permutation of columns, permutation of rows, and transposition of rows and columns; these relate 72 independent 9j symbols. Even (cyclic) permutations of either columns or rows, as well as transpositions, leave the 9j symbol unchanged. Odd permutations of columns or rows introduce a phase factor (-1)R, where R is the sum of all arguments of the 9j symbol.

For further symmetry properties of the 9j symbol see Edmonds (1974, pp. 102–103) and Varshalovich et al. (1988, §10.4.1).

§34.7(iii) Recursion Relations

For recursion relations see Varshalovich et al. (1988, §10.5).

§34.7(iv) Orthogonality

34.7.2 j12j34(2j12+1)(2j34+1)(2j13+1)(2j24+1){j1j2j12j3j4j34j13j24j}{j1j2j12j3j4j34j13j24j}=δj13,j13δj24,j24.

§34.7(v) Generating Functions

For generating functions for the 9j symbol see Biedenharn and van Dam (1965, p. 258, eq. (4.37)).

§34.7(vi) Sums

34.7.3 j13j24(-1)2j2+j24+j23-j34(2j13+1)(2j24+1){j1j2j12j3j4j34j13j24j}{j1j3j13j4j2j24j14j23j}={j1j2j12j4j3j34j14j23j}.

This equation is the sum rule. It constitutes an addition theorem for the 9j symbol.

34.7.4 (j13j23j33m13m23m33){j11j12j13j21j22j23j31j32j33}=mr1,mr2,r=1,2,3(j11j12j13m11m12m13)(j21j22j23m21m22m23)(j31j32j33m31m32m33)×(j11j21j31m11m21m31)(j12j22j32m12m22m32).
34.7.5 j(2j+1){j11j12jj21j22j23j31j32j33}{j11j12jj23j33j}=(-1)2j{j21j22j23j12jj32}{j31j32j33jj11j21}.