Two-Dimensional Oscillator in Magnetic Field

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.

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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
.
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