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
W ( j 1 j 2 J j 3 ; J 12 J 23 ) ≡ ⟨ ( j 1 , ( j 2 j 3 ) J 23 ) J | ( ( j 1 j 2 ) J 12 , j 3 ) J ⟩ ( 2 J 12 + 1 ) ( 2 J 23 + 1 ) . {\displaystyle W(j_{1}j_{2}Jj_{3};J_{12}J_{23})\equiv {\frac {\langle (j_{1},(j_{2}j_{3})J_{23})J|((j_{1}j_{2})J_{12},j_{3})J\rangle }{\sqrt {(2J_{12}+1)(2J_{23}+1)}}}.}
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 j 1 {\displaystyle \mathbf {j} _{1}} and j 2 {\displaystyle \mathbf {j} _{2}} is the construction of simultaneous eigenfunctions of J 2 {\displaystyle \mathbf {J} ^{2}} and J z {\displaystyle J_{z}} , where J = j 1 + j 2 {\displaystyle \mathbf {J} =\mathbf {j} _{1}+\mathbf {j} _{2}} , as explained in the article on Clebsch–Gordan coefficients. The result is
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