Compton wavelength via dimensional analysis

Dimensional analysis is cool because it gets wonderful results out of literally nothing. In this post, we figure out the Compton wavelength of an electron by simply talking and doing little to zero maths.

The cool way of finding the Compton wavelength of an electron

1. Suppose we want to find the Compton wavelength of the electron.
2. We know that the Compton effect is seen when a photon scatters off an electron. Therefore the physical entities that play a role in Compton scattering are the electron and the photon. An electron has a fixed mass and charge. The photon has a fixed velocity. We argue, therefore, that the Compton wavelength, which is a fixed quantity, must depend on these other fixed quantities and nothing else.‘
3. We reduce the set of quantities that might make up the Compton wavelength by further noting that the Compton effect occurs because of relativistic energy momentum conservation in an inelastic scattering. Since the photon wavelength is determined purely by energy momentum conservation, the electric charge plays no role : it could have been anything and the results would have been the same. Also, we can forget about and because we will be working in natural units, which includes these two quantities by default. This is explained in detail in another blog post.
4. We conclude that the only quantity that might make up the Compton wavelength is the electron mass, which is 0.5 MeV.
5. We notice that has the units of length.
6. We conclude that the Compton wavelength must be in the scale of
7. 😎