 Solution

For a photon scattering through angle $\theta$ from a free electron initially at rest, conserve energy and momentum. Eliminating the final electron momentum by its relativistic energy relation givesor $E^{\prime }=E/[1+(E/m{c}^2)(1-\cos \theta )]$. The recoil electron gains kinetic energy $E-E^{\prime }$. The shift is zero forward, one electron Compton wavelength at $90^{\circ }$, and twice that wavelength backward.

Compton's X-ray scattering measurements compared scattered spectra at different angles and identified an angle-dependent shifted component. This supports photon momentum $p=E/c$ and inelastic recoil, beyond the unchanged-frequency prediction of classical elastic Thomson scattering. Bound electrons and coherent scattering can also produce an unshifted component; initial momentum and binding broaden or modify the ideal free-electron result. Treating an electron as initially free and stationary is therefore a kinematic approximation appropriate when the relevant photon transfer dominates binding scales.^[ref]