Wigner function of the Ince–Gauss states of the two-dimensional quantum harmonic oscillator
Agustín Domínguez-Cruz, Julio C. Gutiérrez-Vega · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · 2025
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.