Landau-Level System

One-Sentence Description

The Landau-level system is the exactly solvable orbital motion of a charged particle in a uniform magnetic field, with harmonic-oscillator energy spacing and a large guiding-center degeneracy.

Physical Setup

Take a spinless particle of mass $m$ and charge $q$ moving in the plane perpendicular to a uniform magnetic field

$$\mathbf B=B\,\hat{\mathbf z}.$$

The vector potential $\mathbf A$ is gauge-dependent, but the magnetic field, energy spacing, and degeneracy density are physical. In three dimensions, free motion parallel to $\mathbf B$ adds a continuous kinetic-energy term.

Hilbert Space

For the ideal two-dimensional orbital model,

$$\mathcal H=L^2(\mathbb R^2).$$

Finite samples require boundary conditions. Those choices affect edge states and finite-size counting but not the bulk spacing of ideal Landau levels.

Hamiltonian

The orbital Hamiltonian is

$$\hat H = \frac{1}{2m} \left(\hat{\mathbf p}-q\mathbf A(\hat{\mathbf r})\right)^2, \qquad \nabla\times\mathbf A=\mathbf B.$$

Define

$$\omega_c=\frac{\lvert q\rvert B}{m}, \qquad \ell_B=\sqrt{\frac{\hbar}{\lvert q\rvert B}}.$$

Here $\omega_c$ is the cyclotron frequency and $\ell_B$ is the magnetic length.

Parameters

Symbol	Meaning
$m$	particle mass
$q$	charge, including sign
$B$	magnitude of the uniform magnetic field
$\omega_c$	cyclotron frequency $\lvert q\rvert B/m$
$\ell_B$	magnetic length $\sqrt{\hbar/(\lvert q\rvert B)}$
$A$	sample area when finite-area degeneracy is discussed

Solvability

The kinetic momenta obey a harmonic-oscillator algebra. The remaining guiding-center degrees of freedom commute with the Hamiltonian and label the degeneracy inside each Landau level.

Spectrum and Eigenstates

For the two-dimensional spinless orbital problem,

$$E_n = \hbar\omega_c\left(n+\frac12\right), \qquad n=0,1,2,\ldots$$

Each level is highly degenerate. In a large region of area $A$, the bulk orbital state count per spinless Landau level is approximately

$$N_\phi = \frac{A}{2\pi\ell_B^2} = \frac{\lvert q\rvert BA}{2\pi\hbar}.$$

This is the number of magnetic flux quanta through the sample, up to boundary conventions.

Key Observables

• Landau-level index $n$ labels cyclotron excitation.
• Guiding-center coordinates label the degenerate subspace.
• The magnetic length controls orbital size.
• Degeneracy density controls filling factors and quantum Hall applications.

What It Teaches

Landau levels teach magnetic quantization, gauge-dependent bases, gauge-invariant spectra, macroscopic degeneracy, and the bridge from single-particle quantum mechanics to quantum Hall physics.

Canonical Links

• Landau Levels
• Landau Gauge and Symmetric Gauge
• Degeneracy of Landau Levels
• Landau-Level Formula
• Charged Particle in a Magnetic Field Hamiltonian

Variants

• inclusion of spin and Zeeman splitting;
• three-dimensional Landau levels with free motion along the field;
• Dirac Landau levels in graphene-like models;
• finite samples with edges;
• many-electron Landau levels in quantum Hall systems.

Common Mistakes

• Treating the gauge label of one basis state as a gauge-invariant position observable.
• Forgetting that degeneracy counting needs an area and boundary convention.
• Adding a factor of two for spin without checking Zeeman splitting or spin polarization.
• Taking the $B\to0$ limit one energy level at a time instead of recovering a dense continuum.

Quick Checks

1. What is the bulk degeneracy density of one spinless Landau level?

Solution

Divide $N_\phi=A/(2\pi\ell_B^2)$ by the area $A$. The density is $1/(2\pi\ell_B^2)=\lvert q\rvert B/(2\pi\hbar)$.

2. Why is a displayed Landau-gauge wavefunction not itself a gauge-invariant object?

Solution

Changing gauge changes the wavefunction by a position-dependent phase and may also change the convenient basis inside a degenerate Landau level. Energies, degeneracy density, and physical densities of complete filled levels are gauge-invariant; a single basis label is not.

References

• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. E. Prange and S. M. Girvin, eds., The Quantum Hall Effect, 2nd ed., Springer, 1990.