Quark-Antiquark Effective Potential in Symplectic Quantum Mechanics

Rendisley Paiva,Caroline Costa,Ademir de Santana,Ruilin Zhu,Ronni Amorim,Gustavo Petronilo,Renato Luz

doi:10.1155/2022/3409776

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.

Highlights

With the discovery of the J/Ψ meson in 1974 [1], the search to describe such heavy quark systems was based on the approach of potential models
We present a solution for the Schrödinger equation with Cornell potential in phase space and the associated Wigner function
We have studied heavy quarkonium bound states using the Schrödinger-like equation with an approach the Cornell potential in the Symplectic Quantum Mechanics framework

Summary

Introduction

With the discovery of the J/Ψ meson in 1974 [1], the search to describe such heavy quark systems was based on the approach of potential models. The system is characterized by a linear combination of the Coulomb and linear potentials The study of these potentials should take into account the two important features of the Quantum Chromodynamics (QCD), namely, asymptotic freedom and quark confinement [4,5,6,7]. Khoka et al [31, 33] utilized the analytical exact iterative method to solve N-dimensional Schrödinger equation with an extended Cornell potential [31] They found the energy and mass spectrum of heavy quarkonia [31].

Symplectic Quantum Mechanics

Quark Confinement and Schrödinger Equation in Phase Space

Cornell Potential and Schrödinger Equation in Phase Space

Concluding Remarks