Matter Waves and Wave-Particle Duality

If a photon is a particle, it should be able to collide with things and transfer momentum. In 1923 Arthur Compton fired X-rays at a solid target and measured the X-rays that scattered off its electrons. The scattered X-rays had a longer wavelength than the incoming ones, and the shift grew with the scattering angle. A wave scattering off a charge should come out with its wavelength unchanged. A photon behaving like a billiard ball, though, gives some of its energy and momentum to the electron it hits, so it leaves with less energy and therefore a longer wavelength. This Compton effect was strong evidence that photons carry momentum exactly as $p=h/\lambda$ predicts.

In 1924 Louis de Broglie proposed that the relationship $p=h/\lambda$ works both ways: any moving particle with momentum $p=mv$ has an associated wavelength. For everyday objects this wavelength is absurdly small, because $h$ is so small and their masses are so large. For an electron, with its tiny mass, it comes out comparable to the spacing between atoms in a crystal — which suggests exactly how to test the idea.

Diffraction is only noticeable when a wave meets an opening or obstacle comparable in size to its wavelength. The regularly spaced atoms of a crystal act like a diffraction grating for waves with wavelengths around a tenth of a nanometre. In 1927 Clinton Davisson and Lester Germer directed a beam of electrons at a nickel crystal and found strong reflections only at particular angles — a diffraction pattern — and G. P. Thomson saw diffraction rings when electrons passed through thin metal foils. The measured wavelengths matched $\lambda =h/(mv)$. Electron diffraction is the key evidence for the wave nature of matter, and electron microscopes exploit it: electron wavelengths are far shorter than those of visible light, so they can resolve far finer detail.

A hot, low-pressure gas does not glow with a continuous rainbow. It emits only a few sharp wavelengths, a line spectrum unique to each element. Niels Bohr's 1913 model explained hydrogen's lines by allowing its electron only certain energy levels. An electron that drops from a higher level to a lower one emits a single photon carrying exactly the energy difference; an electron can jump up only by absorbing a photon with exactly the right energy. Matter waves later gave a reason for the allowed levels: an electron's wave must fit around the atom as a standing wave, a whole number of wavelengths, much like the harmonics on a guitar string.