A few years later, a French physics graduate student named Louis deBroglie proposed an astonishing idea in his graduate dissertation. Following on the ideas of Planck and Einstein, he was particularly struck by the dual nature of light; light behaves as a particle (photon) in the photoelectric effect and yet has wave properties of interference and diffraction. He proposed that a propagating wave is associated with the motion of all sorts of particle: photons, electrons, protons, and more! To work out the details of his idea, he referred to another of Einstein's theories - the special theory of relativity. (We will look at Einstein's theories of relativity further on in our study.)

The basic description of wave motion is the same for all types of waves, whether they be water, sound, or light. It is related to the motion of an object at constant speed, since waves travel at constant speed in an unvarying situation. This equation of wave motion says ln = c, where l is the wave's wavelength, n is its frequency, and c is its constant speed of travel. Equally, we can write this as n = c/l.

From Einstein's description of the photoelectric effect, we know that the energy of a photon of given frequency is E_photon = hn = hc/l, using our fact from wave motion. Einstein's special theory of relativity describes the relationship between energy and "particle" properties as E² = p²c² + m₀²c⁴, where p is the particle's momentum, c is the speed of light in empty space, and m₀ is the mass of the particle when it is not moving. Since a photon always moves at the speed of light, the special theory of relativity requires that its rest mass is zero, or it would have infinite energy. This makes the energy equation of a photon in special relativity come out to E_photon² = p²c².

When we put the energy descriptions from special relativity and the photoelectric effect together, we have E_photon = pc = hc/l. From this, we get the momentum of a photon (in its role as a particle) as p = h/l.