# 19.24 Electron waves

In post 19.23, I explained the idea that light could behave as a stream of particles, as well as being a wave. So, do particles (or any other objects) behave as waves? This question was explored by the French physicist Louis de Broglie in his doctoral thesis of 1924. (To find out how to pronounce his name, go to https://forvo.com/search/de%20Broglie/). The arrow in the picture above points to de Broglie, in the company of many other famous early twentieth century scientists (for more information on this photograph see post 19.3). The thesis was unusually short, so the authorities at the University of Paris supposed that it might be rubbish. As a result, they invited a very famous theoretical physicist (in the middle of the front row of the picture) to examine the work; de Broglie won the Nobel Prize for physics five years later.

To understand de Broglie’s proposal (sometimes called wave-particle duality), we need to go back to some ideas in earlier posts.

Let’s think about an object, of mass m, moving at the speed of light, c. According to Einstein’s theory of relativity, the mass of this object is converted into energy mc2 (post 17.41). In post 19.23, we saw that the energy of a photon (a particle moving at the speed of light) was hf; here h is Planck’s constant (post 19.19) and f is the frequency of the wave corresponding to a photon with this energy (post 19.23).

Equating these two expressions for the energy of a particle, moving at the speed of light, we get

mc2 = hf = hc/λ.

The final step comes from the relationship between the speed, frequency and wavelength, λ, of a wave (post 18.10). Rearranging this equation gives

mc = h/λ.

The left-hand side of this equation represents a mass multiplied by a speed, which is the definition of momentum, p (post 16.13), so we take

p = h/λ

as the definition of the momentum of a photon corresponding to light with a wavelength λ.

De Broglie’s idea was that an object with momentum p would be associated with a wavelength

λ = h/p.

What does this mean for an electron? Let’s think about an electron, with charge e (post 16.25), accelerated though a potential difference V; this electron has kinetic energy Ve (post 17.44). From the definition of kinetic energy (post 16.21), for an object of mass m and speed v,

Ve = mv2/2 = p2/2m

where p is its momentum (post 16.13). Rearranging this equation gives the momentum of the electron as

p = (2mVe)1/2.

So, the wavelength of the electron is

λ = h/p = h/(2mVe)1/2.

This prediction for the wavelength of an electron was tested experimentally by two independent research groups. One was led by G P Thomson, at the University of Aberdeen in Scotland, and the other consisted of Clinton Davisson and Lester Germer working at the Bell Telephone Laboratories in New Jersey, USA. They reasoned that electron waves would be deflected by the electric field of an atom (post 17.24). Since electrons accelerated through about 50 V are predicted to have a wavelength comparable to the distance between atoms in a crystal (post 16.37), they expected to observe electron diffraction, when electrons were accelerated towards crystal surfaces, and to be able to measure the wavelength of the electrons. As a result, they confirmed de Broglies’s idea; Davisson and Thomson shared the Nobel Prize for Physics in 1937.

Why don’t we notice that everyday moving objects have an associated wavelength? Think about a bullet of mass 0.03 kg moving at a speed of 1 000 m.s-1; its momentum is 30 kg.m.s-1. Using equation 1 and the value for h in post 19.19, the wavelength associated with the bullet is 2 × 10-35 m – very much shorter than for any experimentally available wave and immeasurable.

Once again (post 16.2) we need to consider quantum effects when dealing with particles like electrons. But, in the macroscopic world, they are negligible and we can describe the motion of everyday objects using Newton’s laws of motion.

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