Compton Scattering Equation

In his explanation of the Compton scattering experiment, Arthur Compton treated the x-ray photons as particles and applied conservation of energy and conservation of momentum to the collision of a photon with a stationary electron. Using the Planck relationship and the relativistic energy expression, conservation of energy takes the form

Conservation of momentum requires

where p=E/c is used for the photon momentum. Squaring this equation using the scalar product gives

Again using the Planck relationship and the relativistic energy expression:

The energy conservation expression above can be squared to give

These two forms can be equated to give

which can be rearranged to

and finally to the standard Compton formula: