The search for fractional order in heavy quarkonia spectra

2019 ◽  
Vol 34 (10) ◽  
pp. 1950054 ◽  
Author(s):  
Ahmed Al-Jamel
Keyword(s):  
Fractional Order ◽  
Mass Spectra ◽  
Dimensional Space ◽  
Heavy Quarkonia ◽  
Eigenvalues And Eigenfunctions ◽  
Energy Eigenvalues ◽  
Dimension Formula ◽  
Fractional Models ◽  
Uvarov Method ◽  
Exact Energy

Using the concept of conformable fractional derivative, we study the properties of fractional [Formula: see text]-dimensional Schrödinger equation for the potential [Formula: see text]. The extended Nikiforov–Uvarov method is generalized to the fractional domain and then employed to obtain the analytic exact energy eigenvalues and eigenfunctions and their dependence on the fractional order [Formula: see text] and the dimension [Formula: see text]. To test its applicability, we apply the method on heavy quarkonia systems, and reproduce their mass spectra and fractional radial probabilities at different values of [Formula: see text] and [Formula: see text]. Comparing the mass spectra with the experimental data, we discuss to what extent fractional models can account for some features in the description of heavy quarkonia at certain dimensional space.

scholarly journals EIGENVALUES AND EIGENFUNCTIONS OF WOODS–SAXON POTENTIAL IN PT-SYMMETRIC QUANTUM MECHANICS

2006 ◽  
Vol 21 (27) ◽  
pp. 2087-2097 ◽  
Author(s):  
AYŞE BERKDEMIR ◽  
CÜNEYT BERKDEMIR ◽  
RAMAZAN SEVER
Keyword(s):  
Quantum Mechanics ◽  
Real Spectrum ◽  
Saxon Potential ◽  
Eigenvalues And Eigenfunctions ◽  
Discrete Energy ◽  
Energy Eigenvalues ◽  
Symmetric Quantum ◽  
Symmetric Version ◽  
Uvarov Method ◽  
S States

Using the Nikiforov–Uvarov method which is based on solving the second-order differential equations, we firstly analyzed the energy spectra and eigenfunctions of the Woods–Saxon potential. In the framework of the PT-symmetric quantum mechanics, we secondly solved the time-independent Schrödinger equation for the PT and non-PT-symmetric version of the potential. It is shown that the discrete energy eigenvalues of the non-PT-symmetric potential consist of the real and imaginary parts, but the PT-symmetric one has a real spectrum. Results are obtained for s-states only.

scholarly journals Heavy quarkonia properties from a hard-wall confinement potential model with conformal symmetry perturbing effects

2019 ◽  
Vol 34 (37) ◽  
pp. 1950307 ◽  
Author(s):  
Ahmed Al-Jamel
Keyword(s):  
Mass Spectra ◽  
Potential Model ◽  
Conformal Symmetry ◽  
Heavy Quarkonia ◽  
Heavy Flavor ◽  
Hard Wall ◽  
Theoretical Approaches ◽  
Radial Schrödinger Equation ◽  
Uvarov Method ◽  
Good Agreement

Heavy [Formula: see text] and [Formula: see text] quarkonia are considered as systems confined within a hard-wall potential shaped after a linear combination of a cotangent — with a square co-secant function. Wave functions and energy spectra are then obtained in closed forms in solving by the Nikiforov–Uvarov method the associated radial Schrödinger equation in the presence of a centrifugal term. The interest in this potential is that in one parametrization, it can account for a conformal symmetry of the strong interaction, and in another for its perturbation, a reason for which we here employ it to study status of conformal symmetry in the heavy flavor sector. The resulting predictions on heavy quarkonia mass spectra and root mean square radii are compared with the available experimental data, as well as with predictions by other theoretical approaches. We observe that a relatively small conformal symmetry perturbing term in the potential suffices to achieve good agreement with data.

GALILEAN COVARIANT DIRAC EQUATION WITH A WOODS–SAXON POTENTIAL

2013 ◽  
Vol 22 (12) ◽  
pp. 1350092 ◽  
Author(s):  
A. A. OTHMAN ◽  
M. DE MONTIGNY ◽  
F. C. KHANNA
Keyword(s):  
Dirac Equation ◽  
Dimensional Manifold ◽  
Saxon Potential ◽  
Eigenvalues And Eigenfunctions ◽  
Covariant Approach ◽  
Energy Eigenvalues ◽  
Energy Eigenvalues And Eigenfunctions ◽  
Galilean Space ◽  
Uvarov Method ◽  
Pekeris Approximation

We derive and solve the Galilean covariant Dirac equation, also called "Lévy-Leblond equation", for spin-½ particles in a Woods–Saxon potential. We obtain this wave equation with a Galilean covariant approach, which is based on a (4+1)-dimensional manifold with light-cone coordinates followed by a reduction to the (3+1)-dimensional Galilean space-time. We apply the Pekeris approximation and exploit the Nikiforov–Uvarov method to find the energy eigenvalues and eigenfunctions.

scholarly journals Binding Energies and Dissociation Temperatures of Heavy Quarkonia at Finite Temperature and Chemical Potential in the N-Dimensional Space

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
M. Abu-Shady ◽  
T. A. Abdel-Karim ◽  
E. M. Khokha
Keyword(s):  
Finite Temperature ◽  
Chemical Potential ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Binding Energies ◽  
Heavy Quarkonia ◽  
Energy Eigenvalues ◽  
Cornell Potential ◽  
Radial Schrödinger Equation ◽  
Three Dimensional Space

The N-dimensional radial Schrödinger equation has been solved using the analytical exact iteration method (AEIM), in which the Cornell potential is generalized to finite temperature and chemical potential. The energy eigenvalues have been calculated in the N-dimensional space for any state. The present results have been applied for studying quarkonium properties such as charmonium and bottomonium masses at finite temperature and quark chemical potential. The binding energies and the mass spectra of heavy quarkonia are studied in the N-dimensional space. The dissociation temperatures for different states of heavy quarkonia are calculated in the three-dimensional space. The influence of dimensionality number (N) has been discussed on the dissociation temperatures. In addition, the energy eigenvalues are only valid for nonzero temperature at any value of quark chemical potential. A comparison is studied with other recent works. We conclude that the AEIM succeeds in predicting the heavy quarkonium at finite temperature and quark chemical potential in comparison with recent works.

scholarly journals Study on the Applicability of Varshni Potential to Predict the Mass Spectra of the Quark-Antiquark Systems in a Non-relativistic Framework

2021 ◽  
Vol 14 (4) ◽  
pp. 339-347
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Mass Spectra ◽  
Laguerre Polynomials ◽  
Heavy Meson ◽  
Energy Eigenvalues ◽  
Relativistic Regime ◽  
Uvarov Method ◽  
Relativistic Framework ◽  
Good Agreement

Abstract: In this work, we obtain the Schrödinger equation solutions for the Varshni potential using the Nikiforov-Uvarov method. The energy eigenvalues are obtained in non-relativistic regime. The corresponding eigenfunction is obtained in terms of Laguerre polynomials. We applied the present results to calculate heavy-meson masses of charmonium cc ¯ and bottomonium bb ¯. The mass spectra for charmonium and bottomonium multiplets have been predicted numerically. The results are in good agreement with experimental data and the works of other researchers. Keywords: Schrödinger equation, Varshni potential, Nikiforov-Uvarov method, Heavy meson. PACs: 14.20.Lq; 03.65.-w; 14.40.Pq; 11.80.Fv.

scholarly journals Quarkonium masses in a hot QCD medium using conformable fractional of the Nikiforov–Uvarov method

2019 ◽  
Vol 34 (31) ◽  
pp. 1950201
Author(s):  
M. Abu-Shady
Keyword(s):  
Order Parameter ◽  
Dimensional Space ◽  
Binding Energies ◽  
Heavy Quarkonium ◽  
Energy Eigenvalues ◽  
Debye Mass ◽  
Higher Dimensional ◽  
Radial Schrödinger Equation ◽  
Uvarov Method ◽  
Nu Method

By using the conformable fractional of the Nikiforov–Uvarov (CF–NU) method, the radial Schrödinger equation is analytically solved. The energy eigenvalues and corresponding functions are obtained, in which the dependent temperature potential is employed. The effect of fraction-order parameter is studied on the heavy-quarkonium masses such as charmonium and bottomonium in a hot QCD medium in the 3D and the higher-dimensional space. This paper discusses the flavor dependence of their binding energies and explores the nature of dissociation by employing the perturbative, nonperturbative, and the lattice-parametrized form of the Debye masses in the medium-modified potential. A comparison is studied with recent works. We conclude that the fractional-order plays an important role in a hot QCD medium in the 3D with consideration of a form of the Debye mass.

Saturation in heavy quarkonia spectra with energy-dependent confining potential in N-dimensional space

2018 ◽  
Vol 33 (32) ◽  
pp. 1850185 ◽  
Author(s):  
Ahmed Al-Jamel
Keyword(s):  
Mass Spectra ◽  
Dimensional Space ◽  
Heavy Quarkonia ◽  
Energy Spectra ◽  
Quadratic Term ◽  
Typical Application ◽  
Confining Potential ◽  
Quantum Numbers ◽  
Energy Dependent ◽  
Large Quantum

We study the energy spectra of Schrödinger Hamiltonian in N-dimensional space with Coulomb plus inverse quadratic term [Formula: see text] and energy-dependent confining potential of the form [Formula: see text]. It is found that the energy spectra exhibit saturation effect, where these spectra are suppressed to finite values at large quantum numbers n and/or [Formula: see text] at any dimensional space N. As a typical application, we obtained a compact mass formula for the spectra of heavy quarkonia ([Formula: see text], [Formula: see text]). It is found that the model is capable of reproducing the mass spectra of these systems with observed saturation governed by a factor [Formula: see text]. The comparison of this work predictions with available experimental data as well as other relevant theoretical works is also presented. The eigenfunctions are expressed in terms of the bi-confluent Heun functions.

Exact solutions of the D-dimensional Schrödinger equation for a ring-shaped pseudoharmonic potential

Open Physics ◽  
2008 ◽  
Vol 6 (3) ◽  
Author(s):  
Sameer Ikhdair ◽  
Ramazan Sever
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Jacobi Polynomials ◽  
Wave Functions ◽  
Three Dimensions ◽  
Eigenvalues And Eigenfunctions ◽  
Energy Eigenvalues ◽  
Pseudoharmonic Potential ◽  
Uvarov Method

AbstractA new non-central potential, consisting of a pseudoharmonic potential plus another recently proposed ring-shaped potential, is solved. It has the form $$ V(r,\theta ) = \tfrac{1} {8}\kappa r_e^2 \left( {\tfrac{r} {{r_e }} - \tfrac{{r_e }} {r}} \right)^2 + \tfrac{{\beta cos^2 \theta }} {{r^2 sin^2 \theta }} $$. The energy eigenvalues and eigenfunctions of the bound-states for the Schrödinger equation in D-dimensions for this potential are obtained analytically by using the Nikiforov-Uvarov method. The radial and angular parts of the wave functions are obtained in terms of orthogonal Laguerre and Jacobi polynomials. We also find that the energy of the particle and the wave functions reduce to the energy and the wave functions of the bound-states in three dimensions.

REAL SPECTRUM OF NON-HERMITIAN HAMILTONIANS FOR HULTHÉN POTENTIAL

2008 ◽  
Vol 23 (25) ◽  
pp. 2077-2084 ◽  
Author(s):  
SANJIB MEYUR ◽  
S. DEBNATH
Keyword(s):  
State Energy ◽  
Jacobi Polynomials ◽  
Bound State ◽  
Real Spectrum ◽  
Eigenvalues And Eigenfunctions ◽  
Hulthén Potential ◽  
Exact Bound ◽  
Energy Eigenvalues ◽  
Hulthen Potential ◽  
Uvarov Method

The non-Hermitian Hamiltonians of the type [Formula: see text] is solved for the generalized Hulthén potential in terms of Jacobi polynomials by using Nikiforov–Uvarov method. The exact bound-state energy eigenvalues and eigenfunctions are presented.

scholarly journals Variable-Order Fractional Models for Wall-Bounded Turbulent Flows

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 782
Author(s):  
Fangying Song ◽  
George Em Karniadakis
Keyword(s):  
Fractional Derivative ◽  
Turbulent Flows ◽  
Fractional Order ◽  
Variable Order ◽  
Universal Curve ◽  
Reynolds Stresses ◽  
Continuous Change ◽  
Mean Velocity ◽  
Symmetric Variable ◽  
Fractional Models

Modeling of wall-bounded turbulent flows is still an open problem in classical physics, with relatively slow progress in the last few decades beyond the log law, which only describes the intermediate region in wall-bounded turbulence, i.e., 30–50 y+ to 0.1–0.2 R+ in a pipe of radius R. Here, we propose a fundamentally new approach based on fractional calculus to model the entire mean velocity profile from the wall to the centerline of the pipe. Specifically, we represent the Reynolds stresses with a non-local fractional derivative of variable-order that decays with the distance from the wall. Surprisingly, we find that this variable fractional order has a universal form for all Reynolds numbers and for three different flow types, i.e., channel flow, Couette flow, and pipe flow. We first use existing databases from direct numerical simulations (DNSs) to lean the variable-order function and subsequently we test it against other DNS data and experimental measurements, including the Princeton superpipe experiments. Taken together, our findings reveal the continuous change in rate of turbulent diffusion from the wall as well as the strong nonlocality of turbulent interactions that intensify away from the wall. Moreover, we propose alternative formulations, including a divergence variable fractional (two-sided) model for turbulent flows. The total shear stress is represented by a two-sided symmetric variable fractional derivative. The numerical results show that this formulation can lead to smooth fractional-order profiles in the whole domain. This new model improves the one-sided model, which is considered in the half domain (wall to centerline) only. We use a finite difference method for solving the inverse problem, but we also introduce the fractional physics-informed neural network (fPINN) for solving the inverse and forward problems much more efficiently. In addition to the aforementioned fully-developed flows, we model turbulent boundary layers and discuss how the streamwise variation affects the universal curve.

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