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 harmonicoscillator 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 twodimensional 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
.