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arXiv:2110.12223

Quark-Antiquark Effective Potential in Symplectic Quantum Mechanics

R. R. Luz, Caroline Silva Rocha Costa, G. X. A. Petronilo, A. E. Santana, R. G. G. Amorim, R. A. S. Paiva

Published Oct 23, 2021• 0 citations
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Abstract

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 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 $c\overline{c}$ 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, 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.

DOI
10.48550/arxiv.2110.12223

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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.
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Since the groundwork published by Torres-Vega and Frederick, the Quantum Phase Space Representation (QPSR) has been explored as a method for solving a multitude of physical systems and describing phenomena. Most recently, Valentino A. Simpao has developed a method, the Heaviside Operational Ansatz, to solve the Time Dependent Schrodinger Equation (TDSE) in the QPSR, but there are still no general, direct methods to solve the Time Independent Schrodinger Equation in the QPSR. There is also no current formulation of quarkonium in phase space. In this paper, we describe the strong interactions of non-relativistic heavy quarks using the Cornell potential, and present a method, the Half-Transform Ansatz, to cast the Schrodinger Equation into a hyper-geometric form which can be solved for the phase space wave function and its energy eigenvalues using the Nikiforov-Uvarov method. This solution can be generalized for any two particle system with a scleronomic potential made up of polynomial and reciprocal terms. These results are compared to experimental results and other theoretical models. We also analyze the behavior of these wave functions, which suggest a correlation between radial momentum and the upper limit of existence in charm-anticharm mesons.
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Modified Schrodinger equation plays a fundamental role for describing the behavior of a particle at microscopic scale. In this context, we have studied the non-relativistic energy spectra of the modified quantum mechanical gravitational potential plus the harmonic oscillator (MQMGHO) potential using the generalized Bopp’s shift method and standard perturbation theory within the framework of non-commutative 3-dimensional phase space (NC: 3D-RSP) symmetries. Furthermore, we have shown that the modified Hamiltonian operator containing a perturbed Coulomb potential, which physically means the global Hamiltonian of MQMGHO potential, can be described as Hydrogenic atoms interacting with strong potential composed of quantum mechanical gravitational potential plus the harmonic oscillator (QMGHO) potential and an auxiliary part. The bound state energy eigenvalues, in terms of discreet atomic quantum numbers ( j , n, landm ), four infinitesimal parameters , , ,  induced by positionposition and phase-phase non-commutativity, in addition to the dimensional parameters of MQMGHO potential and the corresponding noncommutative Hamiltonian operator were obtained for Hydrogenic atoms. Moreover, we have shown that, the total complete degeneracy of energy
In this work, we address the quartic quantum oscillator in phase space using two approaches: computational and algebraic methods. In order to achieve such an aim, we built simplistic unitary representations for Galilei group, as a consequence the Schrödinger equation is derived in the phase space. In this context, the amplitudes of quasi-probability are associated with the Wigner function. In a computational way, we apply the techniques of Lie methods. As a result, we determine the solution of the quantum oscillator in the phase space and calculate the corresponding Wigner function. We also calculated the negativity parameter of the analyzed system.