Faddeev-Watson equations

The Faddeev-Watson equations have also provided new physical understanding in those cases where the interaction potential is given by a sum of pair-potentials. In molecular reactions however, the particles are atoms or ions and their internal structure play an essential role. As a result, the interaction potential is dominated by a three-body term whenever it is expressed only as a function of the positions of the three bodies, which would seem to invalidate the conceptual advantages of the equations. A solution to this problem has recently been presented (Micha, 1971, 1972), which may be simply described for three hydrogen atoms. A pair of ground-state hydrogens 

Quantitative results for high-energy processes may be obtained from a multiple-collision expansion. Indicating with I k the free total state in channel k (a product of relative, internal and atomic spin states), the transition matrix element M31 for A + BC - AB + C is given by 

The first term corresponds to the spectator-stripping model, which arises naturally in the formalism. The second term is a displacement contribution in which atom B moves undisturbed, and so on. The first term may be conveniently calculated (Micha and McGuire, 1972) in the momentum representation, which gives 

The Faddeev-Watson equations are suitable to the study of permutational symmetry for identical nuclei. This has been done (Micha, 1974) for the three cases in which (1) C = B (2) C = A and (3) B = A and C = A , to obtain transition amplitudes for direct, atom-exchange and dissociative processes. Nuclear spin variables were included, and amplitudes were found by successively reexpressing symmetrized amplitudes in terms of unsymmetrized ones, reducing nuclear-spin dependences and uncoupling the equations required for calculations. For example in case (2), the terms in the total wavefunction 

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