Landau Levels

A charged particle in a magnetic field has no choice but to go in circles. Classically it traces a cyclotron orbit at the frequency $\omega_c = eB/m$, tighter and faster the stronger the field. Quantize that circular motion and something striking happens to the energy: instead of the smooth continuum of a free particle, only a ladder of discrete values is allowed, the Landau levels $E_n = (n+\tfrac12)\hbar\omega_c$, evenly spaced rungs whose separation grows in direct proportion to $B$. Each rung is not a single state but an enormous bundle of them, with a degeneracy that also climbs with the field, so turning up $B$ both spreads the levels apart and packs more electrons onto each one. The scene shows the orbit shrinking toward the magnetic length $\ell_B = \sqrt{\hbar/eB}$ as the field rises, beside the energy ladder filled up to the Fermi level. The lower panel is where the consequences show up in real metals: the Landau fan, the energies $E_n$ plotted against $B$, a set of straight lines splaying out from the origin. As the field increases each line climbs and one by one they cross the Fermi energy and empty out, and that periodic depopulation is the microscopic origin of the de Haas-van Alphen and Shubnikov-de Haas oscillations, the tool experimentalists use to map the Fermi surfaces of metals.