The two-dimensional isotropic harmonic oscillator is defined by the Hamiltonian

,

in units where the mass

*, *the angular frequency

,

* *and Planck's constant

* * equal one. Its energy levels are

, with

.

The degeneracy of the level associated with the energy

is also

*. *This remarkable degeneracy is due to the presence of three constants of the motion,

,

which generate an SU(2)

* *algebra, just like three angular-momentum operators. The wavefunctions associated with an

level may be taken to be simultaneous eigenfunctions of

* *and one of the

* *operators. The wavefunctions as well as the Wigner functions may accordingly be labeled by {

*, *} with

.

* *Eigenfunctions common to

and

correspond to separation of the

and

variables. The case of

and

is completely analogous, but in a new coordinate system obtained by rotating the original one through an angle of

*.* In both cases the wavefunctions and the Wigner functions are products of the corresponding one-dimensional quantities.

In this Demonstration we consider the less familiar common eigenstates of

* *and

that correspond to a separation of variables in polar coordinates

. The wavefunctions are

, with

.

The corresponding Wigner functions have the form

.

They depend on the phase-space variables

,

, and

*, *where

* *and

* *are the lengths of the

*-* and

*- *vectors, and

is the angle between them. The variables enter through the functions

and

.

The Demonstration shows the dependence of the Wigner function on the quantum numbers

* *and

by means of contour plots. To see the dependence, move the

,

, and

* *sliders. The contour values vary between

and

and become visible as you move the pointer across the contour lines.