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The differential cross section formula of the scattering of an electron on a nucleus are known to coincide when computed using either classical mechanics approach (Rutherford) or the Schroedinger equation. (These computations are presented in Landau-Lifshitz vol. 1 and 3 respectively).

I wonder whether it is possible to compute this cross section using the Dirac equation.

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    $\begingroup$ The exact Coulomb waves for the Dirac equation may be found in Rose's book, Relativistic Electron Theory archive.org/details/relativisticelec0000rose/mode/1up but they ain't pretty. $\endgroup$ Commented Nov 2 at 14:26
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    $\begingroup$ Chapter 3 of QED by Greiner & Reinhardt discusses the relativistic electron-proton scattering (first in an external Coulomb field approximation, then dealing with the effects of recoil and nuclear structure effects). It also has some remarks why higher-order cross-section calculations with free-particle Dirac states run into trouble with infinities and how using instead Dirac-Coulomb wave functions solves this problem. The non-relativistic limit of the leading-order part of the cross section gives back the Rutherford formula of course. $\endgroup$ Commented Nov 2 at 17:49

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