Quark-Antiquark Effective Potential in Symplectic Quantum Mechanics

Abstract

In this paper, we study within the structure of Symplectic Quantum Mechanics a bidimensional nonrelativistic 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 Schrödinger 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 c ¯ meson. In the second case, to treat the Schrödinger-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, and the nonclassicality of ground state and first excited state is studied by the nonclassicality 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 nonclassic nature of meson system than the wavefunction does phase space.

Fractional Effective Quark-Antiquark Interaction in Symplectic Quantum Mechanics

“…In this section, the symplectic Schrödinger equation is generalized to a fractional-space Schrödinger equation describing two particles interacting to each other by the linear part of the Cornell potential. Using the results of the previous section, the symplectic Schrödinger equation takes the form [33]…”

“…We have studied the symplectic Schrödinger-like equation in the presence of a linear potential using the formalism of generalized fractional derivatives for nonrelativistic heavy quarkonium bound state. For this purpose, we have investigated the behavior of the Wigner function for the ground-state c c meson considering the symplectic quantum mechanics and the generalized fractional derivative constructed in [33,34].…”

“…As a consequence, in order to obtain the heavy quarkonium masses, the energy eigenvalues and the associated wave functions are determined [2]. The eigen-solutions and an inverted polynomial potential were obtained by using the NU procedure [30][31][32][33].…”

“…However, the recent analysis of the Wigner function of such a system as c c mesons provides an interesting description of the confinement, by using the characteristics of the phase-space quantum mechanics. These achievements were carried out by considering the symplectic quantum mechanics, in which the Schrödinger equation is written in a phase-space representation [33]. Nevertheless, accounting for the physical richness of phase space, many aspects remain to be explored, such as the fractional structure of the symplectic Schrödinger describing a quark-antiquark system.…”