The 2D free particle in the phase space quantum mechanics

The Wigner eigenfunctions of a free quantum particle propagating on a plane are derived. Two possibilities are analysed. Firstly, the particle of given energy and angular momentum is discussed. In that case, a special choice of coordinates on the symplectic space $({\mathbb{R}}^4,\omega )$ suitable for the representation of eigenstates of the discussed particle is presented. Further, the Moyal ${\star }_{(\text{M})}$-product on the phase space is derived with the use of the Fedosov algorithm adapted to these coordinates on a flat phase space. Next, the eigenvalue equations for the Wigner eigenfunction are solved and the physically acceptable solutions are identified. Secondly, the particle with fixed components of the Cartesian momentum is considered. Finally, a relationship between the Wigner eigenfunction of the particle with the fixed components of the Cartesian momentum and the cross-Wigner functions of the particle with the given energy and angular momentum is found.