Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Advances in High Energy Physics

Quark-Antiquark Effective Potential in Symplectic Quantum Mechanics

R. R. Luz, G. X. A. Petronilo, Ademir Eugenio de Santana, Caroline Costa, R. G. G. Amorim, Rendisley Aristóteles dos Santos Paiva

Published Feb 7, 2022• 4 citations
Publisher

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.

DOI
10.1155/2022/3409776

Related papers

Quark-Antiquark Effective Potential in Symplectic Quantum Mechanics

R. R. Luz, Caroline Silva Rocha Costa, G. X. A. Petronilo +3 more

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.

arXiv:2110.12223•Oct 23, 2021•0 citations

Fractional Effective Quark-Antiquark Interaction in Symplectic Quantum Mechanics

Mohamed Abu-Shady, R. R. Luz, G. X. A. Petronilo +2 more

We investigate within the formalism of Symplectic Quantum Mechanics a two-dimensional non-relativistic strong interacting system that represents the bound heavy quark-antiquark state, where it was considered a linear potential in the context of generalized fractional derivatives. For this purpose, it was solved the Schrödinger equation in phase space with the linear potential. The solution (ground state) is obtained, analyzed through the Wigner function comparing with the original solution, the Airy function for the meson $c\overline{c}$. The identified eigenfunctions are connected to the Wigner function via the Weyl product and the Galilei group representation theory in phase space. In some ways, compared to the wave function, the Wigner function makes it simpler to see how the meson system is non-classical.

arXiv:2209.12083•Sep 24, 2022•1 citation

The Landau problem and nonclassicality

G. X. A. Petronilo, S. C. Ulhoa, K. V. S. Araújo +3 more

Exploring the concept of the extended Galilei group [Formula: see text]. Representations for field theories in a symplectic manifold have been derived in association with the method of the Wigner function. The representation is written in the light-cone of a de Sitter space–time in five dimensions. A Hilbert space is constructed, endowed with a symplectic structure, which is used as a representation space for the Lie algebra of [Formula: see text]. This representation gives rise to the spin-0 Schrödinger (Klein–Gordon-like) equation for the wave functions in phase space, such that the dependent variables have the content of position and linear momentum. This is a particular example of a conformal theory, such that the wave functions are associated with the Wigner function through the Moyal product. We construct the Pauli–Schrödinger (Dirac-like) equation in phase space in its explicitly covariant form. In addition, we analyze the gauge symmetry for spin-1/2 particles in phase space and show how implement the minimal coupling in this case. We applied to the problem of an electron in an external field, and we recovered the nonrelativistic Landau levels. Finally, we study the parameter of negativity associated with the nonclassicality of the system.

arXiv:2009.07171•International Journal of Modern Physics A•Sep 3, 2020•4 citations

The Half Transform Ansatz: Quarkonium Dynamics in Quantum Phase Space

Gabriel Nowaskie

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.

arXiv:2303.16356•Mar 28, 2023•0 citations

SOLUTIONS FOR THE LANDAU PROBLEM USING SYMPLECTIC REPRESENTATIONS OF THE GALILEI GROUP

R. G. G. Amorim, M.C.B. Fernandes, F. C. Khanna +2 more

Symplectic unitary representations for the Galilei group are studied. The formalism is based on the noncommutative structure of the star-product, and using group theory approach as a guide, a consistent physical theory in phase space is constructed. The state of a quantum mechanics system is described by a quasi-probability amplitude that is in association with the Wigner function. As a result, the Schrödinger and Pauli–Schrödinger equations are derived in phase space. As an application, the Landau problem in phase space is studied. This shows how this method of quantum mechanics in phase space is to be brought to the realm of spatial noncommutative theories.

International Journal of Modern Physics A•Mar 10, 2013•7 citations

Analytical solution for quantum quartic oscillator in phase space

A. X. Martins, T. M. Rocha Filho, R. G. G. Amorim +3 more

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.

International Journal of Modern Physics A•Jul 9, 2020•1 citation
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.
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 cc‾c\overline{c}cc 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.
We investigate within the formalism of Symplectic Quantum Mechanics a two-dimensional non-relativistic strong interacting system that represents the bound heavy quark-antiquark state, where it was considered a linear potential in the context of generalized fractional derivatives. For this purpose, it was solved the Schrödinger equation in phase space with the linear potential. The solution (ground state) is obtained, analyzed through the Wigner function comparing with the original solution, the Airy function for the meson cc‾c\overline{c}cc. The identified eigenfunctions are connected to the Wigner function via the Weyl product and the Galilei group representation theory in phase space. In some ways, compared to the wave function, the Wigner function makes it simpler to see how the meson system is non-classical.
Exploring the concept of the extended Galilei group [Formula: see text]. Representations for field theories in a symplectic manifold have been derived in association with the method of the Wigner function. The representation is written in the light-cone of a de Sitter space–time in five dimensions. A Hilbert space is constructed, endowed with a symplectic structure, which is used as a representation space for the Lie algebra of [Formula: see text]. This representation gives rise to the spin-0 Schrödinger (Klein–Gordon-like) equation for the wave functions in phase space, such that the dependent variables have the content of position and linear momentum. This is a particular example of a conformal theory, such that the wave functions are associated with the Wigner function through the Moyal product. We construct the Pauli–Schrödinger (Dirac-like) equation in phase space in its explicitly covariant form. In addition, we analyze the gauge symmetry for spin-1/2 particles in phase space and show how implement the minimal coupling in this case. We applied to the problem of an electron in an external field, and we recovered the nonrelativistic Landau levels. Finally, we study the parameter of negativity associated with the nonclassicality of the system.
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.
Symplectic unitary representations for the Galilei group are studied. The formalism is based on the noncommutative structure of the star-product, and using group theory approach as a guide, a consistent physical theory in phase space is constructed. The state of a quantum mechanics system is described by a quasi-probability amplitude that is in association with the Wigner function. As a result, the Schrödinger and Pauli–Schrödinger equations are derived in phase space. As an application, the Landau problem in phase space is studied. This shows how this method of quantum mechanics in phase space is to be brought to the realm of spatial noncommutative theories.
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.