Wave particle duality of matter

Albert Einstein's explanation of the photoelectric effect had already established that light shows a dual nature: it behaves as a wave in phenomena like diffraction and interference, but as a stream of particle-like photons when it exchanges energy with matter (as in the photoelectric effect). Louis de Broglie's insight, in 1924, was to turn this idea around and apply it to matter: if light - normally thought of as a wave - can behave like a particle, then perhaps particles - normally thought of as point masses - can behave like waves. He proposed that every form of matter possesses this same dual, wave-and-particle character.

Deriving the de Broglie relation. De Broglie combined two already-known expressions for the energy of a photon, one written in terms of its wave character and the other in terms of its particle character:

• Planck's quantum hypothesis (wave character): $E=h\nu\qquad$ — eq. (2.6)
• Einstein's mass-energy relation (particle character): $E=mc^2\qquad$ — eq. (2.7)

Equating the two right-hand sides,

and using $\nu=c/\lambda$ (frequency times wavelength equals speed) gives

Equation (2.8) describes a photon, whose momentum is $mc$ even though a photon has zero rest mass. For an ordinary particle of matter with mass $m$ travelling at velocity $v$ (rather than at the speed of light), the same reasoning gives the general de Broglie wavelength:

This form is valid only for particles moving much slower than light. It says that any moving particle - not just a photon - has an associated wavelength, and conversely that a wave can carry the momentum-like properties normally associated with a particle. …