The nonrelativistic Hamiltonian for an electron in a magnetic field
is vector potential, is given by
are the mass and charge of the electron, respectively. We also make use of the Coulomb gauge condition
For a constant field in the
, 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
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.