The Bethe-Maximon result.

We develop and illustrate a very simple eikonal technique in the framework of one-particle quantum mechanics for calculating radiation during scattering and apply it to derive the celebrated result of Bethe and Maximon for bremsstrahlung from a point coulomb target with a large charge Ze . This approach is a modification of that developed in a previous paper for calculating radiation during the collision of electron-positron beam pulses (beamstrahlung ). The differences in the physics of scattering from a localized target and from an extended target are clarified. A second reason for the present note is the simplicity of our method and its possible extension to other processes. Submitted to Physical Review D * Work supported by the Department of Energy, contract DE-AC03-76SF00515. I-. Introduction and Motivation In 1954, Bethe and Maximon’ calculated the differential cross sections for bremsstrahlung by high energy electrons and for electromagnetic lepton pair production in a coulomb field of arbitrary charge Ze without the use of the Born approximation. Their paper was a major tour de force of analysis utilizing appropriate approximations to exact continuum relativistic coulomb wave functions. The first treatment to lowest order in Zcu was of course given by Bethe and Heitler.2 Since this pioneering work, several useful approximations for treating high energy scattering processes have been developed. These lead to considerable simplification in the treatment of radiation during the scattering of electrically charged high energy particles. These include in particular the eikonal method 3-6 in potential scattering, operator methods for calculating radiation from an electron orbiting in a homogeneous purely magnetic field: and light-cone quantization techniques applied to QED in the infinite momentum limit.* In this paper we develop and illustrate a very simple eikonal technique in the framework of one particle quantum mechanics for calculating radiation during scattering. This method was developed in connection with a recent study9 of radiation during the collision of two beam pulses (‘beamstrahlung ‘), but it was found to be inappropriate (i.e., wrong) for that problem because of the extended nature of the target pulsesin the rest frame of one of the pulses, its length grows as E/m = 7. However for finite potential sources, this method leads directly and simply to the same expression for the amplitude as found by Bjorken, Kogut, and Soper, and confirms the results of Bethe and Maximon for bremsstrahlung during scattering in a pure coulomb field of arbitrary charge Ze . 2 . . .-2.-T-Matrix Consider the T-Matrix for the scattering of a particle with free Green’s function G in an external vector potential wV , where w is a scale parameter that will eventually be set equal to one, T = WV (1GwV)-’ . (2-l) Introducing the electromagnetic potential A for the radiation field in the standard gauge invariant manner leads to the substitution