In 1905, Einstein made use of the idea of light quanta in order to explain the photoelectric effect and later applied it to the emission as well as the absorption of radiation by atoms. The light quantum hypothesis states not only that the energy of monochromatic radiation of frequency ν is made up of integral multiples of the quantum hν, but also that the momentum is made up of integral multiples of the quantum h/λ, where λ is the wavelength of the radiation (ν and λ are related by the equation νλ = c). This hypothesis contrasts sharply with the classical picture in which the energy and momentum are regarded as continuously variable. The existence of discrete light quanta, or photons, is not immediately evident on a macroscopic scale, however. Due to the smallness of Planck's constant, even in a weak electromagnetic field there is an enormous number of photons, provided the frequency is not too high. For example, black-body radiation at a temperature of 300 K (room temperature) contains about 5.5 × 108 photons/cm3, most of which correspond to frequencies in the infrared part of the electromagnetic spectrum. At a temperature of 6000 K (roughly that at the surface of the sun), the bulk of the radiation has frequencies in the visible spectrum, and there are about 4.4 × 1012 photons/cm3. (The total number of photons in black-body radiation is proportional to the cube of the absolute temperature.)

Individual photons manifest themselves only through their interaction with atomic systems. According to Einstein's treatment of the absorption and emission of radiation, for example, an atom in a stationary state can make a transition to a lower or a higher energy level accompanied by the creation or annihilation, respectively, of a photon. If the atomic energies are Er and Es, where Er > Es, then the energy hν of the photon must equal the difference Er – Es. This is called Bohr's frequency condition and is equivalent to the law of conservation of energy applied to the complete system of atom and radiation; any energy lost or gained by the atom is given up to or abstracted from the radiation field in the form of photons. It should be noted that the number of photons in the radiation field need not be constant—photons can be created or annihilated through the interaction of the field with atoms.

The scattering of X-rays by free electrons also furnishes direct evidence for the corpuscular properties of radiation. In 1922, Compton discovered that when X-rays of wavelength λ are incident on a graphite target, the scattered X-rays have intensity peaks at two wavelengths, λ and λ′, where λ′ > λ. The shift in wavelength given by Δλ = λ′ − λ is a function of the angle of scattering (i.e., the angle between the direction of the incident and scattered X-rays) but is independent of wavelength and the target material. The X-rays with unchanged wavelength λ were understood to have been elastically scattered by atoms, which suffer no appreciable recoil, and they could readily be accounted for on the basis of classical electrodynamics. The scattered X-rays with shifted wavelength λ′, however, required a new interpretation. If it is assumed that the incident X-rays consist of photons, then these may collide with essentially free electrons in the target. In this case a photon gives up some of its energy hν to an electron and is scattered with a lower frequency ν′ and a longer wavelength λ′, where ν′λ′ = c.

The wavelength shift Δλ can be calculated as a function of scattering angle by using the laws of conservation of energy and momentum. By treating the problem relativistically and taking the electron to be at rest initially (see Fig. 3), one can easily show that Δλ depends on the scattering angle θ.