Quasiclassical Green function in an external field and small-angle scattering

TL;DR

This paper derives quasiclassical Green functions for Dirac and Klein-Gordon equations in external electric fields, including first corrections, and demonstrates their broader applicability compared to eikonal approximation through scattering amplitude calculations.

Contribution

It introduces a refined quasiclassical Green function approach that accounts for first corrections and applies it to small-angle scattering and photon scattering in external fields.

Findings

Green functions differ from eikonal approximation and are more widely applicable.

First correction to scattering amplitude is proportional to scattering angle.

Real part of forward photon scattering amplitude in screened Coulomb potential is obtained.

Abstract

The quasiclassical Green functions of the Dirac and Klein-Gordon equations in the external electric field are obtained with the first correction taken into account. The relevant potential is assumed to be localized, while its spherical symmetry is not required. Using these Green functions, the corresponding wave functions are found in the approximation similar to the Furry-Sommerfeld-Maue approximation. It is shown that the quasiclassical Green function does not coincide with the Green function obtained in the eikonal approximation and has a wider region of applicability. It is illustrated by the calculation of the small-angle scattering amplitude for a charged particle and the forward photon scattering amplitude. For charged particles, the first correction to the scattering amplitude in the non-spherically symmetric potential is found. This correction is proportional to the scattering angle. The real part of the amplitude of forward photon scattering in a screened Coulomb potential is obtained.