Coulomb Green's Functions and the Furry Approximation

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Abstract
The Coulomb Green's function for the nonrelativistic Schrödinger equation is obtained in closed form starting from the partial‐wave expansion and using an integral representation for a product of two Whittaker functions with different arguments. The Neumann's series for Jv(kz) is required in evaluating the sum on states. Using the same methods, the Coulomb Green's functions for the Klein‐Gordon and iterated Dirac equations are obtained in closed form in the ``Furry approximation,'' a2/(J + ½)2 ≪ 1, a = Ze2/4πh/c. The Klein‐Gordon Green's function in this approximation is shown to be at the same time the exact Green's function for the Klein‐Gordon equation without the potential squared term. An alternate and very simple derivation of the approximate Green's function for the iterated Dirac equation is given using perturbation theory. From this Green's function, an approximate Coulomb Green's function in closed form for the Dirac equation itself is constructed. Certain known results for Coulomb wavefunctions with modified plane‐wave behavior at large distances are rederived using the foregoing methods and results.