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I understand that compton scattering is modeled as a collision between a photon and an electron, but why does this conclusively prove that light can act as a particle? Why couldn't the same conservation of momentum and energy equations have been applied assuming that light is a wave?

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The high energy photons are shown to have energy and momentum which is consistent with the Planck and de Broglie relations, which are quantum relationships that connect wave properties to particles.

The dynamics would be different if the interaction was wavelike: you see that in diffraction experiments, which follow the wave model of Bragg's law.

By the mid-1920s there was experimental evidence for particle behavior by photons, and of wave behavior by electrons. This is the origin of the wave-particle duality description.

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