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.
Mathematical relation to Clebsch–Gordan coefficients.
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 (and every corresponding ) 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 :
Definitional relation to Clebsch–Gordan coefficients.
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 formula_5 is the zero-angular-momentum state (formula_6). 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 formula_5 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.
Selection rules.
The Wigner 3-"j" symbol is zero unless all these conditions are satisfied:
Symmetry properties.
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 formula_11 quantum numbers () also gives a phase:
The 3-"j" symbols also have so-called Regge symmetries, which are not due to permutations or time reversal. These symmetries are:
With the Regge symmetries, the 3-"j" symbol has a total of 72 symmetries. These are best displayed by the definition of a Regge symbol, which is a one-to-one correspondence between it and a 3-"j" symbol and assumes the properties of a semi-magic square:
whereby the 72 symmetries now correspond to 3! row and 3! column interchanges plus a transposition of the matrix. These facts can be used to devise an effective storage scheme.
Orthogonality relations.
A system of two angular momenta with magnitudes and can be described either in terms of the uncoupled basis states (labeled by the quantum numbers and ), or the coupled basis states (labeled by and ). The 3-"j" symbols constitute a unitary transformation between these two bases, and this unitarity implies the orthogonality relations
The "triangular delta" is equal to 1 when the triad ("j"1, "j"2, "j"3) satisfies the triangle conditions, and is zero otherwise. The triangular delta itself is sometimes confusingly called a "3-"j" symbol" (without the "m") in analogy to 6-"j" and 9-"j" symbols, all of which are irreducible summations of 3-"jm" symbols where no variables remain.
Relation to spherical harmonics.
The 3-"jm" symbols give the integral of the products of three spherical harmonics
with formula_19, formula_20 and formula_21 integers.
Relation to integrals of spin-weighted spherical harmonics.
Similar relations exist for the spin-weighted spherical harmonics if formula_22:
Asymptotic expressions.
For formula_25 a non-zero 3-"j" symbol is
where formula_27, and formula_28 is a Wigner function. Generally a better approximation obeying the Regge symmetry is given by 
where formula_30.
Metric tensor.
The following quantity acts as a metric tensor in angular-momentum theory and is also known as a "Wigner 1-jm symbol":
It can be used to perform time reversal on angular momenta.
Other properties.
where "P" are Legendre polynomials.
Relation to Racah -coefficients.
Wigner 3-"j" symbols are related to Racah -coefficients by a simple phase: