The Dirac Coulomb Green’s function and its application to relativistic Rayleigh scattering

Abstract

The Dirac Coulomb Green’s function is obtained in both coordinate and momentum space. The Green’s function in coordinate space is obtained by the eigenfunction expansion method in terms of the wave functions obtained by Wong and Yeh. The result is simpler than those obtained previously by other authors, in that the radial part for each component contains one term only instead of four terms. Our Green’s function reduces to the Schrödinger Green’s function upon some simple conditions, chiefly by neglecting the spin and replacing λ by l. The Green’s function in momentum space is obtained as the Fourier transform of the coordinate space Green’s function, and is expressed in terms of basically three types of functions: (1) FA (α; β1 β2 β3; γ1 γ2 γ3; z1 z2 z3), (2) the hypergeometric function, and (3) spherical harmonics. The matrix element for Rayleigh scattering, or elastic Compton scattering, from relativistically bound electrons is then obtained in analytically closed form. The matrix element is written basically in terms of the coordinate space Dirac Coulomb Green’s function. The technique used in the evaluation of the matrix element is based on the calculation of the momentum space Dirac Coulomb Green’s function. Finally the relativistic result is compared with the nonrelativistic result.