Reduced Lippmann—Schwinger equations

The 77 coupling representation is generally useful in the reduction of the Lippmann—Schwinger equations since it applies to situations where spin—orbit coupling is not negligible. The quantum numbers used in the representation are defined in table 7.1. Primed and double-primed quantities are used to distinguish different angular-momentum states. 

The channel state ik) is described by quantities that determine the differential cross section (6.55) and other experimental observables described 

The projectile state is represented by a partial-wave expansion (4.188). The quantum numbers of the projectile partial wave and the target state are coupled to total angular momentum P and parity n. The jy-coupling expansion of the potential matrix element is 

The T-matrix element is expanded similarly. The reduced V- and T-matrix elements are obtained by inverting (7.36) using the orthonormality relations (3.71) of the spherical harmonics and (3.89) of the Clebsch-Gordan coefficients. 

The reduced Lippmann—Schwinger equations are obtained by expanding all the amplitudes of (6.73) according to (7.36) and again using the orthonormality relations (3.71,3.89) to eliminate the integral over k and the sum over Clebsch—Gordan coefficients in the expansion of the projection operator 

---