Landau Levels in a Magnetic Field

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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.

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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.

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


Details

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.

References

[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.


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