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A quantum theory of scattering of radiation by a stationary free electron based on photon conception and relativistic kinematics, applying the principles of conservation of energy and conservation of momentum was proposed by Compton to explain the scattering of X-rays and {\gamma}-rays by light elements. The relativistic differential cross-section for the Compton scattering of a photon by a stationary free electron was formulated by Klein and Nishina using Dirac's relativistic theory of electrons, and has been verified experimentally, when the binding energy of the electron is negligible compared to the incident photon energy. Here we show that the energy of scattered photons, and kinetic energy of recoiled electrons obtained from Compton's quantum theory of scattering of radiation, show a degree of matching (that increases with the increase of incident photon energy as quantified by chi-square test) with the differential Klein-Nishina electronic cross section per electron per unit solid angle for the scattering of an unpolarized photon by a stationary free electron when appropriate normalizations are invoked. There is a high degree of matching in a regime where the total electronic Klein-Nishina cross section for the Compton scattering on a free stationary electron scales as the inverse of the incident photon energy and the contribution of the electro-magnetic interaction to differential electronic cross section diminishes. Our results have significant implications to the foundations of quantum mechanics and to the understanding of the mechanisms of photon-electron interactions in the Compton scattering.

Signatures of the differential Klein‐Nishina electronic cross section in Compton's quantum theory of scattering of radiation