# The de Broglie wavelength

It is not only water waves that can interfere, all waves can – also sound waves and light. For light, this was proven by Thomas Young in his famous double-slit experiment, in which he essentially observed the same thing as we did for water waves. However, light is a bit special as it can also behave like a particle. The higher the energy of the photon (blue) the faster the electron gets. This works only out if the photon has momentum which can be transferred to the electron.

When you illuminate a metal plate with light of the right colour, electrons are emitted – they are kicked out of the metal. For this to happen, the light must transfer momentum to the electrons. A prerequisite for this is that light itself has momentum and consequently mass. For this seminal idea to explain the so-called photoelectric effect Einstein was awarded the Nobel Prize in physics in 1921 .

The de Broglie wavelength The physicist Louis de Broglie

Soon afterwards the physicist Louis de Broglie had the ingenious idea that if each wave (for instance light) can be attributed a mass, this must also work the other way round: each massive particle can be attributed a wave-like property . The respective de Broglie wavelength λdB depends on h, the mass m of the particle and their speed v.

λdB = h / mv

This tells us that each particle has a wavelength. The implications of this statement are really groundbreaking. It implies that not only small particles like electrons but also atoms, molecules, and even we have a wavelength. As Plank’s constant h is very small

h = 0.000000000000000000000000000000000626 J·s,

so is our wavelength. An apple (m = 250 g) falling from a 3 m high tree has a de Broglie wavelength of 3.5×10-34 m when it hits the ground. This is as tiny as h itself (although the units are different, of course) and much, much smaller than the wavelength of green light of λ = 0.00000053 m. For molecules the situation is more favourable: the de Broglie wavelength of C60 (a football made of 60 carbon atoms) moving a 90 km/h is 2.2×10-11 m. This is 24’000 times smaller than the wavelength of green light but way larger than λdB of the apple. Molecular interference

The question is how do we detect such a small wavelength? This is done by looking for interference. Only if the particle behaves like a wave, we should be able to see the characteristic pattern. The first experiment observing diffraction of large molecules diffracted the carbon football C60 at a mechanical grating with a period of 1×10-7 m.  The pattern consists of regions where many molecules arrive and regions where very little molecule hit the detector. This is exactly what you can observe in water tank experiments: There are regions where the waves cancel each other (smooth surface) and areas where they amplify each other (large waves). In the molecular diffraction pattern on the left, there are regions where more molecules arrive (red arrows) and areas where less arrive (blue arrows). Also in water tanks, you can create situations, where the water level changes strongly in some places while it does not in others.

However, there is a fundamental requirement to see interference: the molecules have to be delocalized. How we can make molecules forget where they are is discussed here.

 A. Einstein, Über einen die Erzeugung und Verwandlung des Liches betreffenden heuristischen Gesichtspunkt, Ann. Phys. (Berlin) 322, 132 (1905)
 L. de Broglie, Waves and Quanta, Nature 112, 540 (1923)
 M. Arndt et al., Wave-particle duality of C60 molecules, Nature 401, 680 (1999)