DERIVATION. “To show that his model was correct, Compton used it to derive the expression for the Compton shift. In his derivation, he assumed that both photon and electron are relativistic particles and that the collision obeys two commonsense principles: (1) the conservation of linear momentum and (2) the conservation of total relativistic energy.”
https://opentextbc.ca/universityphysicsv3openstax/chapter/the-compton-effect/#:~:text=The%20Compton%20effect%20is%20the,wavelength%20of%20the%20incident%20radiation