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34 3j, 6j, 9j SymbolsProperties

§34.2 Definition: 3⁒j Symbol

The quantities j1,j2,j3 in the 3⁒j 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 |jrβˆ’js|≀jt≀jr+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=βˆ’jr,βˆ’jr+1,…,jrβˆ’1,jr,

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 3⁒j symbol is zero. When both conditions are satisfied the 3⁒j symbol can be expressed as the finite sum

34.2.4 (j1j2j3m1m2m3)=(βˆ’1)j1βˆ’j2βˆ’m3⁒Δ⁑(j1⁒j2⁒j3)⁒((j1+m1)!⁒(j1βˆ’m1)!⁒(j2+m2)!⁒(j2βˆ’m2)!⁒(j3+m3)!⁒(j3βˆ’m3)!)12Γ—βˆ‘s(βˆ’1)ss!⁒(j1+j2βˆ’j3βˆ’s)!⁒(j1βˆ’m1βˆ’s)!⁒(j2+m2βˆ’s)!⁒(j3βˆ’j2+m1+s)!⁒(j3βˆ’j1βˆ’m2+s)!,


34.2.5 Δ⁑(j1⁒j2⁒j3)=((j1+j2βˆ’j3)!⁒(j1βˆ’j2+j3)!⁒(βˆ’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.


34.2.6 (j1j2j3m1m2m3)=(βˆ’1)j2βˆ’m1+m3⁒(j1+j2+m3)!⁒(j2+j3βˆ’m1)!Δ⁑(j1⁒j2⁒j3)⁒(j1+j2+j3+1)!⁒((j1+m1)!⁒(j3βˆ’m3)!(j1βˆ’m1)!⁒(j2+m2)!⁒(j2βˆ’m2)!⁒(j3+m3)!)12Γ—F23⁑(βˆ’j1βˆ’j2βˆ’j3βˆ’1,βˆ’j1+m1,βˆ’j3βˆ’m3;βˆ’j1βˆ’j2βˆ’m3,βˆ’j2βˆ’j3+m1;1),

where F23 is defined as in Β§16.2.

For alternative expressions for the 3⁒j 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).