 Compton scattering

Classic theory predicts that the output frequency must be the same as the input one since the electromagnetic wave makes the electron vibrate with same frequency as itself, which then irradiates further waves.

But the output waves are longer because photons are discrete and energy is proportional to frequency:

The formula is exactly that of two relativistic billiard balls colliding.

Therefore this is evidence that photons exist and have momentum.

Compton scattering is a quantum mechanical phenomenon that describes the elastic scattering of X-rays or gamma rays off charged particles, most commonly electrons. This effect is significant because it demonstrates the particle-like behavior of photons, the quantum particles of light. The process occurs when a photon collides with a free or loosely bound electron. During the collision, energy and momentum are conserved, leading to an increase in the wavelength of the scattered photon (which corresponds to a decrease in its energy).