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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.

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      EUN,LOM,LRE4,work-cmr-id:398806,http://demonstrations.wolfram.com:http://demonstrations.wolfram.com/WignerFunctionOfTwoDimensionalIsotropicHarmonicOscillator/,ilox,learning resource exchange,LRE metadata application profile,LRE

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