Elastic Scattering of Photons by a Hydrogen Atom

An exact analytic expression is derived for the Kramers-Heisenberg matrix element describing the elastic scattering of photons by a hydrogen atom in the dipole approximation. The method followed consists in writing the matrix element in terms of the Coulomb-field Green's function in momentum space and using for this an integral representation originally derived by Schwinger. Various integrations then yield the matrix element in terms of a hypergeometric function of the Gauss type ${}_2{}^{}{}_{}{}^{}F_1^{}{}_{}{}^{}$ with parameters and variable depending on the photon energy. Different limiting cases are considered. Finally, a very accurate numerical computation of the result is reported. The procedure used in the computation was to sum the series expansions of the hypergeometric functions occurring in the different equivalent conveniently chosen forms of the matrix element. The results presented cover all values of the photon energy.