The typical wavelength of an electron

Electrons in circuits and in atoms are typically moving through voltage differences on the order of 1 V, so that a typical energy is (e)(1 V), which is on the order of 10^-19 J. What is the wavelength of an electron with this amount of kinetic energy?

This energy is nonrelativistic, since it is much less than mc². Momentum and energy are therefore related by the nonrelativistic equation KE = p²/2m. Solving for p and substituting in to the equation for the wavelength, we find

This is on the same order of magnitude as the size of an atom, which is no accident: as we will discuss in the next chapter in more detail, an electron in an atom can be interpreted as a standing wave. The smallness of the wavelength of a typical electron also helps to explain why the wave nature of electrons wasn't discovered until a hundred years after the wave nature of light. To scale the usual wave-optics devices such as diffraction gratings down to the size needed to work with electrons at ordinary energies, we need to make them so small that their parts are comparable in size to individual atoms. This is essentially what Davisson and Germer did with their nickel crystal.