Angular integration theorem

The general matrix element over the term kets obeys the reduction (integration with respect to the angular momentum functions) according to the Wigner-Eckart theorem ... 

The evaluation of this integral by conventional methods (even for the special case m = m2 = m2 = 0 when the rotation matrices reduce to spherical harmonics) is extremely laborious. We shall use this result in the derivation of the Wigner-Eckart theorem and other angular momentum relationships later in this chapter. 

In an analogous fashion to the atomic Hartree-Fock equations, the angular variables can be separated and integrated out using the Wigner-Eckart theorem in the Dirac equation to yield a set of coupled differential equations depending on r (29). 

The simplest example of the Wigner—Eckart theorem is given by the Gaunt integral over three spherical harmonics, which is the matrix element for the transition between eigenstates m) and fm ) of a single orbital angular momentum observable due to a tensor operator Tj. We prefer to use the renormalised tensor operator C, which simplifies the expression. 

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