  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.
