A charged particle moving through a uniform magnetic field is bent into a circular cyclotron orbit. Quantum mechanics turns these orbits into a discrete ladder of energies — the Landau levels — equally spaced by the cyclotron energy. This simulation links the classical circular motion on the left with the quantised energy ladder and probability rings on the right.
r = mv/(qB); increasing the field B tightens the orbit.E_n = (n+½)ℏω_c, with cyclotron frequency ω_c = qB/m.E_n = (n + ½)·ℏ·ω_c · ω_c = qB/m · r = mv/(qB) · degeneracy ∝ B
Landau levels, derived by Lev Landau in 1930, are the foundation of the quantum Hall effect — a discovery so precise that it now helps define the international standard for electrical resistance.
What are Landau levels? Landau levels are the quantised energy states of a charged particle moving in a uniform magnetic field. Instead of a continuous range of energies, only discrete values E_n = (n+½)ℏω_c are allowed.
What is the cyclotron frequency? The cyclotron frequency ω_c = qB/m is the angular frequency at which a charged particle of charge q and mass m circles in a magnetic field B. It sets the spacing between Landau levels.
Why is the lowest energy not zero? Because of the +½ zero-point term. Even in the ground state (n=0) the energy is ½ℏω_c, a purely quantum effect analogous to the harmonic oscillator's zero-point energy.
Classically the radius r = mv/(qB) shrinks as B increases. A stronger field forces the particle into tighter circles, which is why increasing B compresses the orbits in the simulation.
Each Landau level can hold many electrons. The degeneracy (number of states) per unit area is proportional to B: stronger fields pack more states into each level.
Landau levels are evenly spaced by ℏω_c. Because ω_c ∝ B, raising the magnetic field widens the gaps between consecutive levels in the energy ladder.
The quantum problem of a charged particle in a magnetic field maps mathematically onto a 1D quantum harmonic oscillator, giving the same evenly-spaced (n+½)ℏω ladder.
They underlie the quantum Hall effect, de Haas–van Alphen oscillations, and the electronic properties of two-dimensional materials such as graphene in strong magnetic fields.
They represent the quantum probability density of finding the electron. Higher Landau levels have more radial nodes and a larger characteristic radius, shown as concentric rings.
It is a faithful qualitative model: the classical orbit radius follows r = mv/(qB) and the quantised ladder follows E_n = (n+½)ℏω_c, with units scaled for clear on-screen visualisation.
Landau levels are discrete energy levels of a charged particle (typically an electron) confined to a plane and placed in a perpendicular magnetic field. They arise from the quantisation of the circular cyclotron orbits that electrons undergo in a magnetic field, and were first described by Lev Landau in 1930. The energy levels are equally spaced: Eₙ = ℏωc(n + 1/2), where ωc = eB/m is the cyclotron frequency, B is the magnetic field strength, and n = 0, 1, 2, ... is the Landau level index.
Each Landau level is highly degenerate — it contains a large number of states proportional to the magnetic flux through the sample (eB/(h) states per unit area per spin). As the magnetic field increases, fewer Landau levels are needed to accommodate all electrons, and each time the highest occupied level empties into a lower one, the density of states at the Fermi level oscillates. These quantum oscillations give rise to the de Haas–van Alphen effect (oscillations in magnetisation) and the Shubnikov–de Haas effect (oscillations in electrical resistance).
Landau quantisation is the foundation of the Integer Quantum Hall Effect (IQHE), discovered by Klaus von Klitzing in 1980 (Nobel Prize 1985). When the Fermi energy lies in the gap between Landau levels, the Hall conductance is quantised to integer multiples of e²/h with extraordinary precision — a quantum metrological standard now used to define the ohm. The Fractional Quantum Hall Effect (FQHE), observed in 1982 by Tsui, Störmer, and Gossard (Nobel Prize 1998), arises from strong electron-electron interactions within a Landau level, producing quasiparticles with fractional charge and exotic statistics.
Landau levels are quantised energy states of electrons in a 2D plane subjected to a perpendicular magnetic field. Classically, electrons undergo circular cyclotron orbits; quantum mechanics quantises the orbital energy into discrete levels Eₙ = ℏωc(n + 1/2). Between levels are energy gaps where no states exist — the system behaves like a 1D harmonic oscillator in the direction perpendicular to its motion.
When a 2D electron system in a strong perpendicular magnetic field is cooled to low temperatures, the Hall conductance (the ratio of current to transverse voltage) is quantised to integer multiples of e²/h (approximately 3.87×10⁻⁵ siemens). This quantisation is topologically protected, extraordinarily precise, and independent of sample impurities — a consequence of filled Landau levels and the topology of electronic band structure.
The von Klitzing constant RK = h/e² ≈ 25,813 ohms is a fundamental constant depending only on Planck's constant and the electron charge. The quantised Hall resistance can be measured with a precision of 10⁻¹⁰ and is independent of temperature, material, and sample geometry, making it an ideal primary resistance standard. Since 2019, the International System of Units (SI) defines the ohm in terms of h and e using this effect.
The FQHE occurs at very high magnetic fields when a Landau level is partially filled. Strong Coulomb repulsion between electrons within the level creates a correlated ground state (Laughlin state) with a gap to excitations. Quasiparticle excitations carry fractional charge (1/3 of the electron charge for the ν=1/3 state) and obey fractional (anyonic) statistics, behaviour impossible for free electrons and with potential applications in topological quantum computing.
As the magnetic field is swept, Landau levels pass through the Fermi energy successively. Each time a level empties, there is a discontinuity in the number of electrons at the Fermi level, causing oscillations in magnetisation (de Haas-van Alphen) and resistance (Shubnikov-de Haas) with period 1/B. The frequency of these oscillations is proportional to the area of the Fermi surface cross-section perpendicular to B, allowing its detailed mapping.