Landau Levels in a Magnetic Field

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

This Demonstration considers the quantum-mechanical system of a free electron in a constant magnetic field, with definite values of the linear and angular momentum in the direction of the field. The wavefunction is plotted in a plane normal to the magnetic field. The corresponding energies are the equally spaced Landau levels, similar to the energies of a harmonic oscillator. These results find application in the theory of the quantum Hall effects.


You can select a 3D plot of the wavefunction, a plot of the radial function or an energy-level diagram. The first slider varies the magnetic field strength . You can then select and , the radial and angular quantum numbers, respectively.


Contributed by: S. M. Blinder (December 2018)
Open content licensed under CC BY-NC-SA


The nonrelativistic Hamiltonian for an electron in a magnetic field , where is vector potential, is given by


where and are the mass and charge of the electron, respectively. We also make use of the Coulomb gauge condition .

For a constant field in the direction, , it is convenient to work in cylindrical coordinates, . With a convenient choice of gauge, the vector potential can be represented by



This gives


The Schrödinger equation for is given by


The equation is separable in cylindrical coordinates, and we can write


for definite values of the angular and linear momenta. We consider only angular momentum anticlockwise about the axis. We set and consider only motion in a plane perpendicular to the magnetic field. Introducing atomic units , the radial equation reduces to


The solution with the correct boundary conditions as is given by


where is an associated Laguerre polynomial. The corresponding energy eigenvalues are


These are the well-known Landau levels, which are equivalent to the levels of a two-dimensional harmonic oscillator with


Recall that

is the cyclotron frequency for an electron in a magnetic field.


[1] L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-relativistic Theory, 2nd ed., Oxford: Pergamon Press, 1965, pp. 424ff.

[2] D. ter Haar (ed. and tr.), Problems in Quantum Mechanics, 3rd ed., London: Pion, 1975 pp. 38, 254ff.


Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.