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34 3j, 6j, 9j SymbolsProperties

§34.7 Basic Properties: 9⁒j Symbol

  1. Β§34.7(i) Special Case
  2. Β§34.7(ii) Symmetry
  3. Β§34.7(iii) Recursion Relations
  4. Β§34.7(iv) Orthogonality
  5. Β§34.7(v) Generating Functions
  6. Β§34.7(vi) Sums

Β§34.7(i) Special Case

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

Β§34.7(ii) Symmetry

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

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

Β§34.7(iii) Recursion Relations

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

Β§34.7(iv) Orthogonality

34.7.2 βˆ‘j12⁒j34(2⁒j12+1)⁒(2⁒j34+1)⁒(2⁒j13+1)⁒(2⁒j24+1)⁒{j1j2j12j3j4j34j13j24j}⁒{j1j2j12j3j4j34j13β€²j24β€²j}=Ξ΄j13,j13′⁒δj24,j24β€².

Β§34.7(v) Generating Functions

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

Β§34.7(vi) Sums

34.7.3 βˆ‘j13⁒j24(βˆ’1)2⁒j2+j24+j23βˆ’j34⁒(2⁒j13+1)⁒(2⁒j24+1)⁒{j1j2j12j3j4j34j13j24j}⁒{j1j3j13j4j2j24j14j23j}={j1j2j12j4j3j34j14j23j}.

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

34.7.4 (j13j23j33m13m23m33)⁒{j11j12j13j21j22j23j31j32j33}=βˆ‘mr⁒1,mr⁒2,r=1,2,3(j11j12j13m11m12m13)⁒(j21j22j23m21m22m23)⁒(j31j32j33m31m32m33)Γ—(j11j21j31m11m21m31)⁒(j12j22j32m12m22m32).
34.7.5 βˆ‘jβ€²(2⁒jβ€²+1)⁒{j11j12jβ€²j21j22j23j31j32j33}⁒{j11j12jβ€²j23j33j}=(βˆ’1)2⁒j⁒{j21j22j23j12jj32}⁒{j31j32j33jj11j21}.