Post-BO scheme process description

Consider the venerable problem of electron capture in H - - H(ls) collisions. The standard model describes an encounter in which nucleus T, with charge Zj and rest mass M, is incident on an atom (or ion) consisting of an electron e bound to nucleus 2 of charge Zi (and a total rest mass m + M). 

In the impact parameter model (IPM), it is assumed that the position of beam-2 remains fixed while beam-1 moves in a straight line with a constant velocity v [18,19]. 

For scattering situations, the beams (represented with local frames) are in motion with respect to specific laboratory frames. Given a beam system, a quantum state is defined in its corresponding inertial frame electrons and nuclei states will be characterized by the total mass that depends on the relative velocity according to special relativity theory. These frames evolve in real (laboratory) space. The energy levels for the basis states are defined with respect to their rest mass energy (m + 2M)c when measured from the laboratory, the levels will be shifted by M(vi-Vi) for beam 1 and + M)(v2-V2) for beam 2. The classical beam momenta appear in the phases. 

To compare better with the IPM approaeh, we use a truneated off-diagonal basis for the Hilbert space. The simple produet for a proton in beam 1 and a bonnd hydrogen atom state in beam 2 is = klm) 2 nlm), where the index k indicates the kinetie energy above the ionization limit for the proton beam 1. In the case where the protons are in beam 2 and the bonnd hydrogen atoms in beam 1, the corresponding produet function would be = 11 nlm) 2 klm). (The notation uses the superindex 

The asymptotic channel 1 is represented by a set of non-zero eoeffieients among and zero for all After the interaction, one must 

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