Angular-momentum integrals integral evaluation

In order to evaluate the matrix element in Eq. (4.151), (p S p0), we must calculate three-dimensional integrals. In the following we show, however, that the matrix element can be reduced to a sum over one-dimensional S -matrix elements. This is obtained via an expansion of the momentum eigenstates (R p) in a basis where we can use that the angular momentum of the relative motion is conserved. 

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. 

The integrals in Eqs. (149) and (150) can be evaluated as follows. We slightly deviate from a strictly classical analysis by quantizing the orbital angular momentum I and the diatom rotational angular momentum j. That is, we assign... 

The first step in reducing the MBPT expressions into a form suitable for numerical evaluation is a decomposition of the Coulomb integrals Uy w into sums of products of angular momentum coupling coefficients and radial integrals. To accomplish this decomposition, we first expand the kernel of the Coulomb integrals l/ri2 as... 

The generalized coordinate q and its conjugate momentum p vary periodically, and the line integral is evaluated over one cycle. (A similar quantization condition on angular momentum led to the famous Bohr postulate L = nh for the hydrogen atom in the old quantum theory [16].) For vibrational motion subject to a potential U K) in a diatomic, the classical energy is E = p llp -f- U R). The integral in (4.80) then translates into... 

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