3-j symbol

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. The 3-j symbols are given in terms of the Clebsch–Gordan coefficients by The j and m components 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, and the inverse relation follows upon making the substitution m3 → −m3: The summation is performed over those integer values k for which the argument of each factorial in the denominator is non-negative, i.e. summation limits K and N are taken equal: the lower one the upper one Factorials of negative numbers are conventionally taken equal to zero, so that the values of the 3j symbol at, for example, or are automatically set to zero. The CG 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 is the zero-angular-momentum state (). 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 CG coefficient. Since the state is unchanged by rotation, one also says that the contraction of the product of three rotational states with a 3-j symbol is invariant under rotations. The Wigner 3-j symbol is zero unless all these conditions are satisfied: A 3-j symbol is invariant under an even permutation of its columns: An odd permutation of the columns gives a phase factor: Changing the sign of the quantum numbers (time reversal) also gives a phase: The 3-j symbols also have so-called Regge symmetries, which are not due to permutations or time reversal.