  In this paper, we study within the structure of Symplectic Quantum Mechanics
a bi-dimensional non-relativistic strong interaction system which represent the
bound state of heavy quark-antiquark, where we consider a Cornell potential
which consists of Coulomb-type plus linear potentials. First, we solve the
Schr\"odinger equation in the phase space with the linear potential. The
solution (ground state) is obtained and analyzed by means of the Wigner
function related to Airy function for the $c\overline{c}$ meson. In the second
case, to treat the Schr\"odinger-like equation in the phase space, a procedure
based on the Bohlin transformation is presented and applied to the Cornell
potential. In this case, the system is separated into two parts, one analogous
to the oscillator and the other we treat using perturbation method. Then, we
quantized the Hamiltonian with the aid of stars operators in the phase space
representation so that we can determine through the algebraic method the
eigenfunctions of the undisturbed Hamiltonian (oscillator solution), and the
other part of the Hamiltonian was the perturbation method. The eigenfunctions
found (undisturbed plus disturbed) are associated with the Wigner function via
Weyl product using the representation theory of Galilei group in the phase
space. The Wigner function is analyzed, the non-classicality of ground state
and first excited state is studied by the non-classicality indicator or
negativity parameter of the Wigner function for this system. In some aspects,
we observe that the Wigner function offers an easier way to visualize the
non-classic nature of meson system than the wavefunction does.
