6-j symbol

The symbol in parentheses is called Wigner 9- j symbol. For more angular momenta one can define 12 - j,... symbols. 

Coupling of three or more angular momenta Racah algebra, Wigner 6-j and 9-j symbols... 

When there are four angular momenta to couple together, there are again several alternative coupling schemes and as before the basis functions in any two schemes are related. In this case, the relationship defines a Wigner 9-j symbol ... 

Note that the 9- j symbol, like the 6 -j symbol, is independent of m and therefore makes no reference to the projection quantum numbers. 

Equation (5.136) results from this equation when we set Tk2 A2) and equation (5.138) follows when T4 (A ) = 1 and k =0. Another important result arises when k =k2 = k and K = 0. In this case Tjj is just the scalar product Tk(A ) Tk(A2), apart from a phase and a normalisation factor, see equation (5.110). In this case, we can use equation (5.94) to replace the 9-j symbol by a 6-j symbol and so arrive at the important result... 

The coefficients for the transformation from the LS representation to the JJ representation are proportional to the 9-j symbols. The transformation is... 

The 9-j symbol is multiplied by (-1), where p is the sum of all 9 arguments, on exchanging any two rows or coliunns. It is unchanged by an even permutation of rows or columns, or by a reflection about a diagonal. When one argument of the 9- j symbol is zero, it reduces to a 6- j symbol ... 

Note that the Wigner 3-j, 6-j and 9-j symbols are simply numbers when evaluated for particular values of the angular momentum symbols. There are numerous tabulations and programs available which allow their evaluation and the program Mathematica can handle 3-j and 6-j symbols analytically. 

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