Racah W-coefficient

Racah's W-coefficients were introduced by Giulio Racah in 1942. These coefficients have a purely mathematical definition. In physics they are used in calculations involving the quantum mechanical description of angular momentum, for example in atomic theory.
The coefficients appear when there are three sources of angular momentum in the problem. For example, consider an atom with one electron in an s orbital and one electron in a p orbital. Each electron has electron spin angular momentum and in addition
the p orbital has orbital angular momentum (an s orbital has zero orbital angular momentum). The atom may be described by "LS" coupling or by "jj" coupling as explained in the article on angular momentum coupling. The transformation between the wave functions that correspond to these two couplings involves a Racah W-coefficient.
Apart from a phase factor, Racah's W-coefficients are equal to Wigner's 6-j symbols, so any equation involving Racah's W-coefficients may be rewritten using 6-"j" symbols. This is often advantageous because the symmetry properties of 6-"j" symbols are easier to remember.
Racah coefficients are related to recoupling coefficients by
Recoupling coefficients are elements of a unitary transformation and their definition is given in the next section. Racah coefficients have more convenient symmetry properties than the recoupling coefficients (but less convenient than the 6-"j" symbols).
Recoupling coefficients.
Coupling of two angular momenta formula_2 and formula_3 is the construction of simultaneous eigenfunctions of formula_4 and formula_5, where formula_6, as explained in the article on Clebsch–Gordan coefficients. The result is 
where formula_8 and formula_9.
Coupling of three angular momenta formula_2, formula_3, and formula_12, may be done by first coupling formula_2 and formula_3 to formula_15 and next coupling formula_15 and formula_12 to total angular momentum formula_18:
Alternatively, one may first couple formula_3 and formula_12 to formula_22 and next couple formula_2 and formula_22 to formula_18:
Both coupling schemes result in complete orthonormal bases for the formula_27 dimensional space spanned by
Hence, the two total angular momentum bases are related by a unitary transformation. The matrix elements of this unitary transformation are given by a scalar product and are known as recoupling coefficients. The coefficients are independent of formula_29 and so we have
The independence of formula_29 follows readily by writing this equation for formula_32 and applying the lowering operator formula_33 to both sides of the equation.
Algebra.
Let
be the usual triangular factor, then the Racah coefficient is a product
of four of these by a sum over factorials,
where
and
The sum over formula_41 is finite over the range
Relation to Wigner's 6-j symbol.
Racah's W-coefficients are related to Wigner's 6-j symbols, which have even more convenient symmetry properties
Cf. or