Coulomb Scattering

As an illustration of the usefulness of the two-component theory, we calculate the scattering of a charged spin zero particle from a fixed Coulomb potential

which comes from a point charge Ze fixed at the origin.

Because of the fact that the two-component KG theory satisfies a first order differential equation, we may use the formalism for time-dependent perturbation theory developed in Sec. 3.1. The first order S-matrix element is [compare with Eq. (3.53)]

where |i) and |/) are initial and final KG free particle states with momenta ki and kj. The interaction Hamiltonian in this equation, Hi, must be expressed in the interaction representation, just as we did in Sec. 3.1 in our study of electromag-netism [recall Eqs. (3.4) and (3.5)]. For a pure Coulomb interaction, V = e A = 0, and the Schrodinger representation of Hi can be deduced from Eq. (4.38). In the interaction picture, it becomes

where fn is the reduced matrix element, which for this example becomes (q = ki - k}, and \q\ = q)

Jo r iqr

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