Quark-Antiquark Effective Potential in Symplectic Quantum Mechanics

In this paper, we explore a bi-dimensional nonrelativistic strong interaction system that represent the bound state of heavy quark-antiquark, where we consider a Cornell potential which it has Coulombian and linear terms. For this purpose, we solve the Schr\"odinger equation in the phase space with the linear potential. The eigenfunction found is associated with the Wigner function via Weyl product using the representation theory of Galilei group in the phase space. We find that the Wigner function offers an easier way to visualize the properties of $c\overline{c}$, $b\overline{b}$, $b\overline{c}$ mesons systems than the wavefunction does.