On the j z-conserving coupled states approximation: Magnetic transitions and angular distributions in rotating and fixed frames

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Abstract
Recently Shimoni and Kouri have pointed out that a careful treatment of the jz‐conserving coupled states (CS) approximation results in a body frame T‐matrix TJ(jλ‖j0λ0) which is not diagonal in λ,λ0. In addition they have shown that previous investigations of the CS did not optimally identify the body frame T‐matrix. In this paper, we explore the consequences of these observations. The exact T‐matrix is obtained in the R13‐ and P‐helicity frames, as well as in an uncoupled spaceframe (USF) representation. The resulting exact expressions for these T‐matrices are in terms of certain integrals, IJl(jλ‖j0λ0), introduced earlier by Shimoni and Kouri. By obtaining the CS approximation to these integrals, we are able to derive the preferred CS approximation in the R‐ and P‐helicity and USF representations. We then employ the resulting CS T‐matrices to derive the differential scattering amplitude and cross section in the various possible reference frames. The result is a unified treatment of these quantities. We are then able to demonstrate the equivalence of the CS approximation to the R‐ and P‐helicity amplitudes. In addition, we show explicitly that the CS approximate degeneracy averaged differential cross section is frame independent. The CS approximation to the USF equation provides a rigorous basis for the original derivation of the CS method as given by McGuire and Kouri. In particular, our treatment shows that when the L2 operator is approximated by an eigenvalue form l (l+1) h/2 (as was suggested first by McGuire and Kouri), there is no longer any difference between the BF and USF in the dynamical equations (for the wavefunction or amplitude density). Any differences are strictly kinematic in origin, and are the source of the λ transitions which occur in the BF CS approximation. In the USF, since there are no rotational kinematic effects, there are no magnetic transitions in the CS approximation. Thus, the name jz‐conserving coupled states is appropriate in two senses. First, in the USF, jz is conserved in the CS approximation. Second, even though jz is not conserved in the BF CS‐approximation T‐matrices, jz‐conservation does occur so far as the dynamics are concerned; i.e., the BF amplitude density and wavefunction both conserve jz in the CS approximation.