Wigner 3-j symbols

In quantum mechanics, the Wigner 3-j symbols, also called 3-jm symbols, are an alternative to Clebsch–Gordan coefficients for the purpose of adding angular momenta. While the two approaches address exactly the same physical problem, the 3-j symbols do so more symmetrically, and thus have greater and simpler symmetry properties than the Clebsch-Gordan coefficients.

The 3-j symbols are given in terms of the Clebsch-Gordan coefficients by

The j 's and m 's are angular momentum quantum numbers, i.e., every j (and every corresponding m) is either a nonnegative integer or half-odd-integer. The exponent of the sign factor is always an integer, so it remains the same when transposed to the left hand side, and the inverse relation follows upon making the substitution m₃ → −m₃:

The C-G coefficients are defined so as to express the addition of two angular momenta in terms of a third:

The 3-j symbols, on the other hand, are the coefficients with which three angular momenta must be added so that the resultant is zero:

Here, ${\displaystyle |0\,0\rangle }$ is the zero angular momentum state (${\displaystyle j=m=0}$). It is apparent that the 3-j symbol treats all three angular momenta involved in the addition problem on an equal footing, and is therefore more symmetrical than the C-G coefficient.

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