Landau Levels: Why a Magnetic Field Quantizes Motion

The classical cyclotron orbit

A charged particle of charge q and mass m moving in a uniform magnetic field B, with its velocity perpendicular to the field, feels the Lorentz force qv×B and traces out a perfect circle — uniform cyclotron motion at the cyclotron frequency ω_c = qB/m, a rotation rate that is famously independent of both the particle's speed and the radius of its orbit. Classically, this leaves a continuous family of allowed orbits: a faster particle simply traces a bigger circle at the same period, and there is nothing to forbid any particular radius or energy.

Lev Landau's 1930 quantization

Solve the Schrödinger equation for a charged particle confined to two dimensions and immersed in a perpendicular magnetic field, and the problem reduces — through a change of variables that is the key technical trick of the whole calculation — to exactly the same mathematics as a quantum harmonic oscillator. The energy is quantized in precisely the same evenly-spaced ladder that the harmonic oscillator has, with the cyclotron frequency taking the place of the oscillator's natural frequency. Lev Landau published this result in 1930, originally motivated by trying to explain the surprisingly weak diamagnetism of a free electron gas.

E_n = (n + 1/2) · ħω_c          n = 0, 1, 2, 3, …

ω_c = qB / m                    cyclotron frequency
degeneracy per unit area ≈ qB / h    grows linearly with B

Massive degeneracy — and why it matters

Unlike an ordinary quantum harmonic oscillator, which has exactly one state per energy level, each Landau level is enormously degenerate: for a two-dimensional sample of area A in field B, there are roughly qBA/h independent quantum states all sharing precisely the same energy E_n. This huge degeneracy is a direct consequence of a classical fact hiding in the quantum problem — a cyclotron orbit's center can sit anywhere in the plane without changing the orbit's energy at all, since only its radius (set by the quantum number n) matters. It is exactly this degeneracy per Landau level that sets the spacing between quantum Hall plateaus.

From de Haas-van Alphen to the quantum Hall effect

As the magnetic field is swept, successive Landau levels sweep past the material's Fermi energy one by one, and every time a level crosses it, bulk measurable properties oscillate periodically in 1/B. Magnetization shows this as the de Haas-van Alphen effect, discovered in 1930, and resistivity shows the closely related Shubnikov-de Haas effect — both remain the standard experimental technique for mapping out the detailed shape of a metal's Fermi surface. At low temperature and strong field, this same Landau-level structure, combined with disorder-induced localization, is exactly what produces the flat, precisely quantized plateaus of the quantum Hall effect.

Landau levels beyond ordinary metals

In graphene, electrons behave as massless Dirac fermions rather than ordinary massive electrons, and their Landau levels follow a distinctly different, characteristic square-root spacing, E_n ∝ ±√(n·B), instead of the evenly-spaced n·ω_c formula that applies to ordinary electrons. This unusual spacing is a direct experimental fingerprint of relativistic-like Landau quantization, and observing it was one of the first clear confirmations that graphene's electrons genuinely obey a Dirac-like equation, shortly after graphene was first isolated in 2004–2005.

Frequently asked questions

Try it live

Everything above runs in your browser — open Landau Levels and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.