In this work, we solve the nonrelativistic symplectic Schrödinger-type equation for a two-body system within the framework of symplectic quantum mechanics. By employing a Lie algebraic approach, we obtain an explicit solution for the wave function in phase space. Subsequently, we derive the corresponding Wigner function and analyze its behavior. As an application, we investigate the heavy quark-antiquark system, specifically the \(c\overline{c}\) meson, which interacts through a linear term of the Cornell potential model. The Wigner function is studied to describe the ground state of the meson. Furthermore, our results indicate that small variations in kinetic momentum significantly affect the maximum possible relative quark-antiquark separation q in \(\textrm{GeV}^{-1}\) . This suggests the existence of an upper limit for the Wigner function curves of the heavy quark-antiquark pair, dependent on the kinetic energy, as illustrated in our graphical analysis. These findings align with previous results in the literature. We also emphasize that the methodology adopted for the study of this equation is based on the theory of Lie groups for differential equations, and with application in the calculation of conservation laws using the Noether theorem.
