De Broglie Hypothesis

In 1924, Prince Louis de Broglie made a striking proposal based purely on an argument from symmetry in nature. If radiation -- normally thought of as a wave -- had been shown (through the photoelectric effect and Compton scattering) to also possess particle-like properties, then perhaps MATTER -- normally thought of as made of particles -- might, symmetrically, also possess wave-like properties. This was a bold guess with no direct experimental support at the time it was proposed; the confirming evidence (the Davisson-Germer experiment) came a few years later and is described in the next section.

De Broglie built his hypothesis by directly extending the photon relations. For a photon, momentum and energy are related by $p=E/c$ (valid for any massless particle travelling at speed c, from Einstein's special relativity); combined with the Einstein relation $E=h\nu$ and $c=\nu\lambda$, this gives

De Broglie proposed that this SAME relation should connect the wave properties (frequency $\nu$, wavelength $\lambda$) and particle properties (energy E, momentum p) of ANY moving material particle, not just a photon. For a particle of mass m moving with velocity v (so $p=mv$), the associated MATTER WAVE has

The wavelength given by Eq. (14.6) is called the DE BROGLIE WAVELENGTH, and Eq. (14.6) itself the DE BROGLIE RELATION. It immediately shows an inverse relationship between momentum and wavelength: the LARGER a particle's momentum, the SHORTER its de Broglie wavelength.

For a particle of mass m and kinetic energy $E_K=\frac{1}{2}mv^2$ (so $v=\sqrt{2E_K/m}$), substituting into $\lambda=h/(mv)$ gives the useful energy form

For a CHARGED particle of charge q, accelerated from rest through a potential difference V, the work done $qV$ equals the kinetic energy gained, so $E_K=qV$, giving

This relation applies to any charged particle -- electron, proton, or a charged ion -- with m being that particle's own mass; it stops being valid once V becomes so large that the particle's speed approaches the speed of light, where relativistic corrections (not covered in this course) become necessary. …