Question

How to derive the Klein-Nishina formula from the Dirac equation?

Answer 1

Abstract
In 1928, Klein and Nishina investigated Compton scattering based on the Dirac equation just proposed in the same year, and derived the Klein–Nishina formula for the scattering cross section of a photon. At that time the Dirac equation had the following unsettled conceptual questions: the negative energy states, its four-component wave functions, and the spin states of an electron. Hence, during their investigation struggles, they encountered various difficulties. In this article, we describe their struggles to derive the formula using the “Sangokan Nishina Source Materials” retained in the the Nishina Memorial Foundation.
Keywords: Klein–Nishina formula, Dirac equation, semi-classical treatment, field quantization, negative energy states
1. Introduction
In Compton scattering, the scattering of an X-ray by an electron, the wavelength of the scattered X-ray varies with the scattering angle. In 1923, using the light quantum theory for X-rays, A.H. Compton1) explained the wavelength shift upon scattering by using the conservation principle of energy and momentum of the light quantum and electron system. This analysis showed that the Compton effect was one of the important experimental facts confirming the quantum theory of light.
In his analysis concerning the angular distribution of the intensity of the scattered wave, Compton used relativistic theory and the Doppler effect. In 1926, Breit2) discussed this problem using the correspondence principle in the old quantum theory, and obtained the same result as what Dirac3) and subsequently Gordon4) obtained independently, using quantum mechanics.
They deduced the energy and momentum conservation law on the light-quantum and electron system quantum mechanically, which Compton hypothesized, and derived a formula giving the angular distribution of the intensity of the scattered wave. In 1927, Klein5) discussed how the interaction between an electron and an electromagnetic field including Compton scattering should be treated quantum mechanically. As for Compton scattering, he obtained energy-momentum conservation for the Compton scattering, but did not show the intensity distribution of the scattered wave. In 1927, Gordon4) and Klein5) independently developed the so-called Klein–Gordon equation to describe the behavior of an electron to include relativistic effects in Compton scattering. The energy and momentum relation of an electron used by Dirac3) was equivalent to that of the Klein–Gordon equation.
Soon after Dirac presented his relativistic electron theory,6,7) O. Klein and Y. Nishina succeeded to derive the famous Klein–Nishina formula,8,9) calculating the intensity distribution of the scattered wave in the Compton scattering based on the Dirac equation. The Klein–Nishina formula has been firmly accepted and widely used, even now. When Klein and Nishina started to attack this problem, they intended in part to confirm the validity of the Dirac equation, as stated in Introduction of Ref. 8. Recalling that time, Bohr wrote to Nishina in 193410,11) that “the striking confirmation which this formula has obtained became soon the main support for the correctness of Dirac’s theory when it was apparently confronted with so many grave difficulties.” Ekspong discussed in Ref. 12, that the experiments conducted to confirm the Klein–Nishina formula were to suggest the existence of then unknown phenomena, namely the pair production and annihilation of positive and negative electrons.
The following arguments in this article are based on the Sangokan Nishina Source, preserved by the Nishina Memorial Foundation. In Section 2 through Section 5, we survey Klein–Nishina’s theory and consider the process they undertook when deriving their formula. In Section 6 and Section 7, we look back their efforts to solve the most difficult problem for them, namely how to set the final states of the electron after scattering, and consider why they adopted the semi-classical method in treating electromag