J 0, centrifugal coupling and helicity decoupling

The cross section for a reactive scattering process is given by [13] 

The important point to note is that there are three summations. These are over the helicity quantum numbers A, and over the total angular momentum quantum number J. 

J is a good quantum number. This means that it is a strictly conserved quantity and that calculations can be carried out separately for each J quantum number. A is the quantum number for the projection of the total angular momentum on the body-fixed z axis. This is not a good quantum number in the sense that the dissociation dynamics of the system mixes up, or couples, different A values. The different A values are coupled by the centrifugal coupling terms in the hamiltonian operator, eq. (5.1). 

Our wavepackets are written as (R,r,7,t). This indicates that we have a separate wavepacket for each value of A, and that these wavepackets are coupled together. 

To evaluate a cross section we need to perform calculations of the reactive scattering S-matrix for many J values. In a semi-classical approach J is related to the orbital angular momentum of the relative motion of the collision partners. Approximately, 

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