,

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.

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