Retardation in the Elastic Scattering of Photons by Atomic Hydrogen

The matrix element for the elastic scattering of photons by atomic hydrogen is evaluated analytically in closed form in the nonrelativistic case including retardation. The method of integration is similar to the one used previously by one of the authors for the dipole approximation. It is based on the possibility of expressing the matrix element in terms of the Green's function for the Coulomb field, for which the momentum-space integral representation derived by Schwinger and others is then used. Some integrations yield the matrix element in terms of essentially two hypergeometric functions of Appell's type $F_1$, with variables and parameters depending on the photon energy. A discussion of the result is given.