The isotropic three-dimensional harmonic oscillator is described by the Schrödinger equation
, in units such that
. The wavefunction is separable in Cartesian coordinates, giving a product of three one-dimensional oscillators with total energies
. More interesting is the solution separable in spherical polar coordinates:
, with the radial function
. Here,
is an associated Laguerre polynomial,
, a spherical harmonic and
, a normalization constant. The energy levels are then given by
, being
-fold degenerate. For a given angular momentum quantum number
, the possible values of
are
. The conventional code is used to label angular momentum states, with
representing
.
This Demonstration shows contour plots in the
plane for the lower-energy eigenfunctions with
to
. For
, the eigenfunctions are complex. In all cases, the real parts of
are drawn. The wavefunctions are positive in the blue regions and negative in the white regions. The radial functions are also plotted, as well as an energy-level diagram, with each dash representing the degenerate set of
eigenstates for a given
.
Browse all topics
