 # Two-Dimensional Oscillator in Magnetic Field

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The two-dimensional problem of a charged isotropic harmonic oscillator in a constant magnetic field can be solved exactly, as worked out in the Details. You can choose to display: (1) a contour plot of the solutions; (2) the radial distribution function in cylindrical coordinates; or (3) an energy-level diagram. You can select the oscillator frequency , the magnetic field and the quantum numbers and . In the contour plots, positive and negative regions are colored blue and yellow, respectively.

Contributed by: S. M. Blinder (March 2019)
Open content licensed under CC BY-NC-SA

## Snapshots   ## Details

For a charged particle (charge , mass ) in a magnetic field, the canonical form for the nonrelativistic Hamiltonian is given by ,

where is the vector potential. The magnetic field is given by . We consider an electron ( ) confined to the - plane, bound by an isotropic harmonic-oscillator potential and subjected to a constant magnetic field in the direction. This field can be represented by the vector potential , such that , , .

The Schrödinger equation, in Cartesian coordinates, can then be written .

Expanding the squares, we obtain .

Note now that , the component of angular momentum, and that , the Larmor frequency for an electron. It is convenient now to transform to cylindrical coordinates ( ), such that , which is an eigenfunction of with eigenvalues , . The radial function satisfies the equation ,

where . This has the form of the unperturbed two-dimensional oscillator and has the solutions (unnormalized, using atomic units ): , ,

where is an associated Laguerre polynomial. The corresponding energies are .

Using atomic units and expressing in teslas (T), . The energy, expanded in powers of the magnetic field, is then given by .

## Permanent Citation

S. M. Blinder

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