Wigner Function of Two-Dimensional Isotropic Harmonic Oscillator

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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.


Contributed by: Jens Peder Dahl and Wolfgang P. Schleich (July 2009)
After work by: Jens Peder Dahl and Wolfgang P. Schleich
Open content licensed under CC BY-NC-SA




[1] J. P. Dahl and W. P. Schleich, "State Operator, Constants of the Motion, and Wigner Functions: The Two-Dimensional Isotropic Harmonic Oscillator," Phys. Rev. A 79, 024101,2009.

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