Landau Levels

Landau levels are the discrete kinetic-energy levels of a charged particle moving in a uniform magnetic field. They are the canonical example of magnetic quantization: a continuous two-dimensional free-particle spectrum collapses into equally spaced levels, each with a large degeneracy.

This page treats a spinless nonrelativistic particle of mass $m$ and charge $q$ in a prescribed uniform magnetic field

$$\mathbf B=B\hat{\mathbf z}, \qquad B\gt0.$$

The scalar potential is set to zero. Spin, Zeeman splitting, disorder, interactions, finite-thickness effects, and physical edges are not included in the ideal model. A periodic box in the $y$ direction is used only to normalize the Landau-gauge states and count bulk orbitals; the bulk limit is taken before interpreting edge physics.

Required background. Minimal Coupling in Wave Mechanics supplies the gauge-covariant kinetic momentum and Hamiltonian used below.

Helpful background. Particle in a Uniform Magnetic Field separates parallel and transverse motion; the Quantum Harmonic Oscillator supplies the ladder; Dimensionless Variables and Scaling supplies scale checks.

Cyclotron frequency and magnetic length

The two natural scales are the cyclotron frequency

$$\omega_c = \frac{\lvert q\rvert B}{m}$$

and the magnetic length

$$\ell_B = \sqrt{ \frac{\hbar}{\lvert q\rvert B} }.$$

The frequency $\omega_c$ sets the level spacing. The length $\ell_B$ sets the transverse size of the oscillator wavefunctions and the area associated with one flux quantum. These scales are introduced more generally in Dimensionless Variables and Scaling.

Hamiltonian

Minimal coupling gives

$$\hat H = \frac{1}{2m} \left( \hat{\mathbf p}-q\mathbf A \right)^2.$$

Choose the Landau gauge

$$\mathbf A=Bx\,\hat{\mathbf y}.$$

Then

$$\nabla\times\mathbf A = B\hat{\mathbf z},$$

and the two-dimensional Hamiltonian is

$$\hat H = \frac{1}{2m} \left[ \hat p_x^2 + \left( \hat p_y-qBx \right)^2 \right].$$

For a three-dimensional particle, one simply adds the free longitudinal term $\hat p_z^2/(2m)$.

Landau-Gauge Solution

The Hamiltonian does not depend explicitly on $y$, so $\hat p_y$ commutes with $\hat H$. Use a simultaneous eigenfunction of $\hat p_y$:

$$\psi(x,y) = \frac{e^{ik_y y}}{\sqrt{L_y}}\varphi(x), \qquad \hat p_y\psi=\hbar k_y\psi.$$

Here $y$ is temporarily placed in a periodic box of length $L_y$, so $k_y=2\pi j/L_y$ with $j\in\mathbb Z$. We choose the oscillator functions to satisfy

$$\int_{-\infty}^{\infty}\!dx\,\lvert\varphi_n(x)\rvert^2=1.$$

Together with the factor $L_y^{-1/2}$, this gives unit norm on $\mathbb R\times[0,L_y)$. The box is a regulator for normalization and state counting, not a claim that the physical sample has no edges.

Substitution gives a one-dimensional equation for $\varphi(x)$:

$$\left[ -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + \frac{(qB)^2}{2m} \left( x-x_0 \right)^2 \right]\varphi = E\varphi,$$

where

$$x_0 = \frac{\hbar k_y}{qB}.$$

The sign of $x_0$ depends on the sign of $qB$, but the oscillator frequency is positive:

$$\omega_c = \frac{\lvert q\rvert B}{m}.$$

Thus the Landau-gauge problem is a harmonic oscillator in $x$, centered at $x_0$, with no energy dependence on $k_y$.

Energy Spectrum

The two-dimensional Landau-level energies are

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

The corresponding Landau-gauge wavefunctions have the form

$$\psi_{n,k_y}(x,y) = \frac{e^{ik_y y}}{\sqrt{L_y}} \varphi_n(x-x_0),$$

where $\varphi_n$ is the $n$th harmonic-oscillator wavefunction with width $\ell_B$.

For a three-dimensional particle with free motion along $\hat{\mathbf z}$,

$$E_{n,k_z} = \hbar\omega_c \left( n+\frac12 \right) + \frac{\hbar^2k_z^2}{2m}.$$

The discreteness is transverse to the magnetic field; the parallel motion remains free unless another potential or boundary condition quantizes it.

Dimensional and continuum checks

Both natural scales have the required dimensions:

$$[\omega_c]=T^{-1}, \qquad [\ell_B]=L, \qquad [\hbar\omega_c]=E.$$

The degeneracy density below also has dimensions of inverse area because $1/(2\pi\ell_B^2)=\lvert q\rvert B/h$. In the limit $B\to0$, the spacing $\hbar\omega_c$ tends to zero and the degeneracy per unit area of any one level tends to zero. The two-dimensional free-particle continuum is recovered only after summing over the increasingly dense ladder of Landau levels; taking $B\to0$ at fixed $n$ does not by itself reproduce a finite free-particle energy.

Degeneracy and Guiding Centers

The label $k_y$ changes the oscillator center $x_0$ but not the energy. In Landau gauge, this is how the degeneracy appears: many different guiding-center positions have the same cyclotron energy.

For a spinless two-dimensional particle in a large region of area $A$, the bulk degeneracy of each Landau level is

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

Equivalently, each orbital Landau level contains one state per flux quantum through the sample. Boundary conditions control the exact finite-size statement. The dedicated counting argument is in Degeneracy of Landau Levels.

Guiding-Center Intuition

Classically, a charged particle in a uniform magnetic field executes circular cyclotron motion. Quantum mechanically, the cyclotron motion is quantized into oscillator-like levels, while the center of the orbit remains a separate degree of freedom.

In Landau gauge, $k_y$ labels the guiding-center coordinate $x_0$. This label is gauge dependent; it is not simply a physical momentum in the ordinary free-particle sense. The physical content is that the magnetic field separates motion into:

• cyclotron motion, whose energy is quantized;
• guiding-center motion, which accounts for the degeneracy.

The kinetic momentum components do not commute,

$$[\hat\pi_x,\hat\pi_y] = i\hbar qB,$$

so the magnetic field creates an oscillator algebra from the transverse kinetic momenta. The guiding-center coordinates form a second noncommuting pair that labels states inside each Landau level.

Symmetric Gauge Overview

Another common choice is symmetric gauge:

$$\mathbf A = \frac12\mathbf B\times\mathbf r = \frac{B}{2} \left( -y\,\hat{\mathbf x} +x\,\hat{\mathbf y} \right).$$

This gauge preserves rotational symmetry about the $z$ axis, so angular-momentum labels become natural. The wavefunctions look different from the Landau-gauge wavefunctions, but the energies are identical:

$$E_n = \hbar\omega_c \left( n+\frac12 \right).$$

The difference between Landau gauge and symmetric gauge is descriptive, not physical. Gauge-invariant quantities such as energy, density profiles after summing over a complete degenerate subspace, flux counting, and current observables do not depend on the gauge used to compute them.

Common Mistakes

• Using $qB/m$ instead of $\lvert q\rvert B/m$ for the positive level spacing.
• Treating $k_y$ in Landau gauge as an ordinary gauge-invariant momentum.
• Forgetting that the ideal Landau levels here are spinless; electron spin adds Zeeman splitting.
• Counting degeneracy without specifying boundary conditions or sample area.
• Thinking the Landau-gauge and symmetric-gauge wavefunctions must look the same if they describe the same physics.
• Forgetting the free $k_z$ contribution in the three-dimensional problem.

Exercises

1. Starting from the Landau-gauge Hamiltonian, derive the oscillator center $x_0$ and frequency $\omega_c$.

Solution

With

$$\psi(x,y)=e^{ik_y y}\varphi(x),$$

the operator $\hat p_y$ acts as multiplication by $\hbar k_y$. Thus

$$\left( \hat p_y-qBx \right)^2 \to \left( \hbar k_y-qBx \right)^2.$$

Rewrite this as

$$(qB)^2 \left( x-\frac{\hbar k_y}{qB} \right)^2.$$

Therefore

$$x_0=\frac{\hbar k_y}{qB}.$$

The quadratic coefficient satisfies

$$\frac{(qB)^2}{2m} = \frac12m\omega_c^2,$$

so

$$\omega_c = \frac{\lvert q\rvert B}{m}.$$

2. Use guiding-center spacing to estimate the degeneracy of one Landau level in a rectangle of area $A=L_xL_y$.

Solution

Periodic boundary conditions along $y$ give

$$k_y=\frac{2\pi r}{L_y}.$$

Since $x_0=\hbar k_y/(qB)$, neighboring allowed $k_y$ values correspond to guiding-center spacing

$$\Delta x_0 = \frac{2\pi\hbar}{\lvert q\rvert BL_y} = \frac{2\pi\ell_B^2}{L_y}.$$

The number of centers that fit in width $L_x$ is approximately

$$N_\Phi = \frac{L_x}{\Delta x_0} = \frac{L_xL_y}{2\pi\ell_B^2} = \frac{A}{2\pi\ell_B^2}.$$

3. Add free motion along $z$ and find the three-dimensional spectrum.

Solution

For a uniform field along $\hat{\mathbf z}$, the Hamiltonian separates into transverse Landau motion plus free longitudinal motion:

$$\hat H = \hat H_\perp + \frac{\hat p_z^2}{2m}.$$

Using $\hat p_z e^{ik_z z}=\hbar k_z e^{ik_z z}$, the energy is

$$E_{n,k_z} = \hbar\omega_c \left( n+\frac12 \right) + \frac{\hbar^2k_z^2}{2m}.$$

Core Connections

• Minimal Coupling in Wave Mechanics supplies the Hamiltonian and gauge conventions.
• Landau Levels in Solids keeps this ideal oscillator result as an input and owns its controlled specialization to anisotropic material bands, internal splittings, valleys, nonparabolicity, and Dirac contrasts.

Further Connections

These pages connect the ideal Landau problem to its setup, symmetries, and applications:

• Particle in a Uniform Magnetic Field explains the physical setup, cyclotron frequency, kinetic-momentum algebra, and oscillator reduction.
• Magnetic Translations gives the symmetry explanation of the noncommuting guiding-center translations.
• Gauge Transformations: First Encounter explains why Landau and symmetric gauges describe the same magnetic field.
• Landau Gauge and Symmetric Gauge compares the rectangular and rotational bases used to represent each Landau level.
• Degeneracy of Landau Levels gives the canonical finite-area and flux-counting derivation.
• Charged Harmonic Oscillator in a Magnetic Field shows how parabolic confinement lifts the guiding-center degeneracy.
• Dimensionless Variables and Scaling defines $\ell_B$ and $\omega_c$ as natural magnetic scales.
• Quantum Harmonic Oscillator provides the oscillator spectrum used in the Landau-gauge derivation.
• Degeneracy in Separable Systems gives the earlier taxonomy of degeneracy before this magnetic-field example.
• Density of States: First Encounter gives the ordinary free-particle state-counting contrast to magnetic flux counting.
• Probability Current is needed to interpret currents in magnetic-field wavefunctions.
• Berry Phase is a later geometric phase topic that becomes important in magnetic bands and quantum Hall settings.
• Quantum Oscillations follows Landau-quantized orbits into Shubnikov–de Haas and de Haas–van Alphen acquisition, damping fits, Fermi-surface inference, and phase cautions.
• Integer Quantum Hall Effect uses Landau filling but adds disorder localization, chiral edges, quantized response, and metrology.
• Ginzburg–Landau Theory uses the lowest Landau level of the covariant order-parameter equation to derive the upper critical field.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• R. E. Prange and S. M. Girvin, eds., The Quantum Hall Effect, 2nd ed., Springer, 1990.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.