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34 3j, 6j, 9j SymbolsProperties

§34.3 Basic Properties: 3j Symbol


§34.3(i) Special Cases

When any one of j1,j2,j3 is equal to 0,12, or 1, the 3j symbol has a simple algebraic form. Examples are provided by

34.3.1 (jj0m-m0) =(-1)j-m(2j+1)12,
34.3.2 (jj1m-m0) =(-1)j-m2m(2j(2j+1)(2j+2))12,
34.3.3 (jj1m-m-11) =(-1)j-m(2(j-m)(j+m+1)2j(2j+1)(2j+2))12,

For these and other results, and also cases in which any one of j1,j2,j3 is 32 or 2, see Edmonds (1974, pp. 125–127).

Next define

34.3.4 J=j1+j2+j3.

Then assuming the triangle conditions are satisfied

34.3.5 (j1j2j3000)={0,J odd,(-1)12J((J-2j1)!(J-2j2)!(J-2j3)!(J+1)!)12(12J)!(12J-j1)!(12J-j2)!(12J-j3)!,J even.


34.3.6 (j1j2j1+j2m1m2-m1-m2)=(-1)j1-j2+m1+m2×((2j1)!(2j2)!(j1+j2+m1+m2)!(j1+j2-m1-m2)!(2j1+2j2+1)!(j1+m1)!(j1-m1)!(j2+m2)!(j2-m2)!)12,
34.3.7 (j1j2j3j1-j1-m3m3)=(-1)j1-j2-m3((2j1)!(-j1+j2+j3)!(j1+j2+m3)!(j3-m3)!(j1+j2+j3+1)!(j1-j2+j3)!(j1+j2-j3)!(-j1+j2-m3)!(j3+m3)!)12.

Again it is assumed that in (34.3.7) the triangle conditions are satisfied.

§34.3(ii) Symmetry

Even permutations of columns of a 3j symbol leave it unchanged; odd permutations of columns produce a phase factor (-1)j1+j2+j3, for example,

34.3.8 (j1j2j3m1m2m3) =(j2j3j1m2m3m1)=(j3j1j2m3m1m2),
34.3.9 (j1j2j3m1m2m3) =(-1)j1+j2+j3(j2j1j3m2m1m3).


34.3.10 (j1j2j3m1m2m3) =(-1)j1+j2+j3(j1j2j3-m1-m2-m3),
34.3.11 (j1j2j3m1m2m3) =(j112(j2+j3+m1)12(j2+j3-m1)j2-j312(j3-j2+m1)+m212(j3-j2+m1)+m3),
34.3.12 (j1j2j3m1m2m3) =(12(j1+j2-m3)12(j2+j3-m1)12(j1+j3-m2)j3-12(j1+j2+m3)j1-12(j2+j3+m1)j2-12(j1+j3+m2)).

Equations (34.3.11) and (34.3.12) are called Regge symmetries. Additional symmetries are obtained by applying (34.3.8)–(34.3.10) to (34.3.11)) and (34.3.12). See Srinivasa Rao and Rajeswari (1993, pp. 44–47) and references given there.

§34.3(iii) Recursion Relations

In the following three equations it is assumed that the triangle conditions are satisfied by each 3j symbol.

34.3.13 ((j1+j2+j3+1)(-j1+j2+j3))12(j1j2j3m1m2m3)=((j2+m2)(j3-m3))12(j1j2-12j3-12m1m2-12m3+12)-((j2-m2)(j3+m3))12(j1j2-12j3-12m1m2+12m3-12),
34.3.14 (j1(j1+1)-j2(j2+1)-j3(j3+1)-2m2m3)(j1j2j3m1m2m3)=((j2-m2)(j2+m2+1)(j3-m3+1)(j3+m3))12(j1j2j3m1m2+1m3-1)+((j2-m2+1)(j2+m2)(j3-m3)(j3+m3+1))12(j1j2j3m1m2-1m3+1),
34.3.15 (2j1+1)((j2(j2+1)-j3(j3+1))m1-j1(j1+1)(m3-m2))(j1j2j3m1m2m3)=(j1+1)(j12-(j2-j3)2)12((j2+j3+1)2-j12)12(j12-m12)12(j1-1j2j3m1m2m3)+j1((j1+1)2-(j2-j3)2)12((j2+j3+1)2-(j1+1)2)12((j1+1)2-m12)12(j1+1j2j3m1m2m3).

For these and other recursion relations see Varshalovich et al. (1988, §8.6). See also Micu (1968), Louck (1958), Schulten and Gordon (1975a), Srinivasa Rao and Rajeswari (1993, pp. 220–225), and Luscombe and Luban (1998).

§34.3(iv) Orthogonality

34.3.16 m1m2(2j3+1)(j1j2j3m1m2m3)(j1j2j3m1m2m3) =δj3,j3δm3,m3,
34.3.17 j3m3(2j3+1)(j1j2j3m1m2m3)(j1j2j3m1m2m3) =δm1,m1δm2,m2,
34.3.18 m1m2m3(j1j2j3m1m2m3)(j1j2j3m1m2m3) =1.

In the summations (34.3.16)–(34.3.18) the summation variables range over all values that satisfy the conditions given in (34.2.1)–(34.2.3). Similar conventions apply to all subsequent summations in this chapter.

§34.3(v) Generating Functions

For generating functions for the 3j symbol see Biedenharn and van Dam (1965, p. 245, Eq. (3.42) and p. 247, Eq. (3.55)).

§34.3(vi) Sums

For sums of products of 3j symbols, see Varshalovich et al. (1988, pp. 259–262).

§34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics

For the polynomials Pl see §18.3, and for the function Yl,m see §14.30.

34.3.19 Pl1(cosθ)Pl2(cosθ)=l(2l+1)(l1l2l000)2Pl(cosθ),
34.3.20 Yl1,m1(θ,ϕ)Yl2,m2(θ,ϕ)=l,m((2l1+1)(2l2+1)(2l+1)4π)12(l1l2lm1m2m)Yl,m(θ,ϕ)¯(l1l2l000),
34.3.21 0πPl1(cosθ)Pl2(cosθ)Pl3(cosθ)sinθdθ=2(l1l2l3000)2,
34.3.22 02π0πYl1,m1(θ,ϕ)Yl2,m2(θ,ϕ)Yl3,m3(θ,ϕ)sinθdθdϕ=((2l1+1)(2l2+1)(2l3+1)4π)12(l1l2l3000)(l1l2l3m1m2m3).

Equations (34.3.19)–(34.3.22) are particular cases of more general results that relate rotation matrices to 3j symbols, for which see Edmonds (1974, Chapter 4). The left- and right-hand sides of (34.3.22) are known, respectively, as Gaunt’s integral and the Gaunt coefficient (Gaunt (1929)).