A closed expression is obtained for the n-dimensional Wigner oscillator function, and a diagrammatic technique is developed for finding the explicit dependence of the function on Euler angles at any dimension n.
We derive explicit expressions for the Wigner function of wave functions in D dimensions which depend on the hyperradius---that is, of s waves. They are based either on the position or the momentum representation of the s wave. The corresponding Wigner function depends on three variables: the absolute value of the D-dimensional position and momentum vectors and the angle between them. We illustrate these expressions by calculating and discussing the Wigner functions of an elementary s wave and the energy eigenfunction of a free particle.
Euler angles often are used to describe the orientation of a secondary system rotated relative to a primary system by generating two auxiliary systems and the secondary with three rotations. We use basis vectors to show that the auxiliary systems of the y-convention Euler angles (used in quantum mechanics) are the cylindrical and spherical systems associated with the primary Cartesian system. We then apply Euler angle geometry to obtain an improved expression (and understanding) of the Wigner D-function addition theorem and related geometrical issues. We introduce helicity vectors and express vector spherical harmonics as combinations of Wigner D-functions to show that the D-function addition theorem is implicit within expansions of rotated vector spherical harmonics. As example applications of the Wigner D-function, we obtain an addition theorem for vector spherical harmonics and then simplify a dyadic Green function (related to Mie theory) describing inelastic light scattering from a laser-irradiated spherical droplet containing a fluorescing molecule.
We present two equivalent analytical expressions of the Wigner function (WF) of Ince–Gauss (IG) eigenstates of the Schrödinger equation in elliptic coordinates for the two-dimensional (2D) harmonic oscillator. To this end, we first derive closed-form expressions for the Weyl transforms for two different Hermite–Gauss (HG) states and two different helical Laguerre–Gauss (LG) states. Using these Weyl transforms, we also obtain the WF of the IG states through twofold finite summations and the WF of parity-definite LG states in a closed form. An alternative expression of the WF of the IG states in terms of the harmonic oscillator’s quadratic invariants is also presented.