Landau Levels in a Magnetic Field
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.[more]
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.[less]
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
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
is the cyclotron frequency for an electron in a magnetic field.
 L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-relativistic Theory, 2nd ed., Oxford: Pergamon Press, 1965, pp. 424ff.
 D. ter Haar (ed. and tr.), Problems in Quantum Mechanics, 3rd ed., London: Pion, 1975 pp. 38, 254ff.