Why are Landau levels equally spaced like an oscillator?

Minimal coupling in a magnetic field maps the kinetic energy in the plane to a harmonic oscillator with frequency ω_c = qB/m. Hence E_n = ħ ω_c (n+½), linear in B. This is not an external spring; the ‘spring constant’ is set by B itself.

What is special about n_B = qB/h?

Each Landau level can hold one state per flux quantum Φ₀ = h/q through the sample area, giving degeneracy density qB/h. Filling an integer number of levels yields the integer quantum Hall plateaus. The simulator highlights that counting; it does not compute Hall resistivity.

Does the classical orbit radius equal the quantum r_n?

r_n = ℓ_B √(2n+1) matches the rms cyclotron radius associated with level n in the symmetric gauge, useful for intuition. A full quantum state is a degenerate manifold of guiding-center locations, not a single classical circle — the animation is pedagogical.

Landau Levels in a Magnetic Field

Charged particle in a uniform B-field: equally-spaced Landau ladder E_n = ℏω_c(n+½) with cyclotron frequency ω_c = qB/m, magnetic length ℓ_B = √(ℏ/qB) and orbit radius r_n = ℓ_B√(2n+1). Animated cyclotron orbit + linear-in-B fan diagram; the n_B = qB/h degeneracy underlies the quantum Hall effect.

About this model

A charged particle in a uniform magnetic field B occupies equally spaced Landau levels E_n = ħ ω_c (n+½) with cyclotron frequency ω_c = qB/m. The magnetic length ℓ_B = √(ħ/qB) sets the orbit scale, and the classical cyclotron radius for level n is r_n = ℓ_B √(2n+1). The simulator animates a cyclotron orbit and shows a fan diagram of energies linear in B; the areal degeneracy density n_B = qB/h underlies the integer quantum Hall effect. The model is a single free charge in a uniform perpendicular B with no disorder, no electric field, and no spin or Zeeman splitting unless you treat them separately. You vary B and level index n to see level spacing and orbit size change.

Who it's for: Condensed-matter and quantum mechanics courses on Landau quantization and the quantum Hall effect.

Key terms

Landau levels
Cyclotron frequency
Magnetic length
Quantum Hall effect
Degeneracy density
Fan diagram

How it works

Landau levels of a 2-D charged particle in a perpendicular magnetic field B: the equally-spaced ladder E_n = ℏω_c(n+½), the magnetic length ℓ_B = √(ℏ/qB), the cyclotron radius r_n, and the linear-in-B fan diagram — the building block of the integer quantum Hall effect.

Frequently asked questions

Why are Landau levels equally spaced like an oscillator?: Minimal coupling in a magnetic field maps the kinetic energy in the plane to a harmonic oscillator with frequency ω_c = qB/m. Hence E_n = ħ ω_c (n+½), linear in B. This is not an external spring; the ‘spring constant’ is set by B itself.
What is special about n_B = qB/h?: Each Landau level can hold one state per flux quantum Φ₀ = h/q through the sample area, giving degeneracy density qB/h. Filling an integer number of levels yields the integer quantum Hall plateaus. The simulator highlights that counting; it does not compute Hall resistivity.
Does the classical orbit radius equal the quantum r_n?: r_n = ℓ_B √(2n+1) matches the rms cyclotron radius associated with level n in the symmetric gauge, useful for intuition. A full quantum state is a degenerate manifold of guiding-center locations, not a single classical circle — the animation is pedagogical.