Digital Library of Mathematical Functions
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34 3j,6j,9j SymbolsProperties

§34.2 Definition: 3j Symbol

The quantities j1,j2,j3 in the 3j symbol are called angular momenta. Either all of them are nonnegative integers, or one is a nonnegative integer and the other two are half-odd positive integers. They must form the sides of a triangle (possibly degenerate). They therefore satisfy the triangle conditions

34.2.1 |jr-js|jtjr+js,

where r,s,t is any permutation of 1,2,3. The corresponding projective quantum numbers m1,m2,m3 are given by

34.2.2 mr=-jr,-jr+1,,jr-1,jr,
r=1,2,3,

and satisfy

34.2.3 m1+m2+m3=0.

See Figure 34.2.1 for a schematic representation.

See accompanying text
Figure 34.2.1: Angular momenta jr and projective quantum numbers mr, r=1,2,3. Magnify

If either of the conditions (34.2.1) or (34.2.3) is not satisfied, then the 3j symbol is zero. When both conditions are satisfied the 3j symbol can be expressed as the finite sum

34.2.4 (j1j2j3m1m2m3)=(-1)j1-j2-m3Δ(j1j2j3)((j1+m1)!(j1-m1)!(j2+m2)!(j2-m2)!(j3+m3)!(j3-m3)!)12×s(-1)ss!(j1+j2-j3-s)!(j1-m1-s)!(j2+m2-s)!(j3-j2+m1+s)!(j3-j1-m2+s)!,

where

34.2.5 Δ(j1j2j3)=((j1+j2-j3)!(j1-j2+j3)!(-j1+j2+j3)!(j1+j2+j3+1)!)12,

and the summation is over all nonnegative integers s such that the arguments in the factorials are nonnegative.

Equivalently,

34.2.6 (j1j2j3m1m2m3)=(-1)j2-m1+m3(j1+j2+m3)!(j2+j3-m1)!Δ(j1j2j3)(j1+j2+j3+1)!((j1+m1)!(j3-m3)!(j1-m1)!(j2+m2)!(j2-m2)!(j3+m3)!)12×F23(-j1-j2-j3-1,-j1+m1,-j3-m3;-j1-j2-m3,-j2-j3+m1;1),

where F23 is defined as in §16.2.

For alternative expressions for the 3j symbol, written either as a finite sum or as other terminating generalized hypergeometric series F23 of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).