الفهرس 
1 IntroductionOnly 14 pages are availabe for public view
 |  | from | 241 |  |  |
| | | from | 241 | | |
Abstract
One of the most significant developments in recent decades has been the solution of wave equations in relativistic and non-relativistic quantum mechanics. Therefore, many authors have dedicated their time and energy to studying the bound state solutions of relativistic and non-relativistic wave equations in order to study many difficulties and phenomena in the various domains of physics and science in general, particularly quantum chemistry and particle physics. As a consequence, we seek to investigate some solutions to relativistic and non-relativistic wave equations and their applications to diatomic molecules and hadrons in this thesis.
The thesis is divided into seven chapters, with English and Arabic summaries. A list of published papers from this thesis was also included, as well as a list of references used in the thesis.
Chapter (1):
In this chapter, we introduce essential outlines of the following points: quantum chemistry, the interplay between quantum chemistry and particle physics, types of wave equations, the standard model, quarks, hadrons, quarkonia, types and properties of diatomic molecules, besides the history of fractional derivatives and their applications. Furthermore, the radial Schrödinger equation in D-dimensions and the Dirac equation for radial potentials are derived. Finally, we present a review of the literature on the applicability of relativistic and non-relativistic models in hadrons and molecules.
Chapter (2):
In this chapter, the bound state solutions of the Dirac equation under the condition of spin symmetry have been obtained using the generalized Cornell potential model, which blends the Cornell potential plus a combination of the harmonic and inversely quadratic potentials. Both the relativistic and non-relativistic energy eigenvalues for the generalized Cornell potential model have been derived within the scope of the Nikiforov-Uvarov method. The energies spectra of the Kratzer potential and the modified Kratzer potential have been obtained as particular cases of the generalized Cornell potential model. The current finding have been applied to some diatomic molecules as well as heavy and heavy-light mesons. The ro-vibrational energies of some diatomic molecules have been
calculated using the Kratzer and modified Kratzer potentials, and the results are entirely consistent with those reported in the literature. Moreover, the spin-averaged mass spectra of heavy and heavy-light mesons have been predicted. It is evident that our predictions are improved with the previous studies and closely match the experimental results.
The following paper is extracted from this chapter titled as: “Bound state solutions of the Dirac equation for the generalized Cornell potential model.” Published in International Journal of Modern Physics A, (2021), 36(29), 2150195.
Chapter (3):
In this chapter, the relativistic and non-relativistic solutions of the Dirac equation with spin symmetry for the generalized Cornell potential model (GCPM) in the presence of external magnetic and Aharanov–Bohm (AB) flux fields are studied using the wave function ansatz method. For various vibrational and magnetic quantum numbers, the relativistic energy eigenvalues and their related eigenfunctions are determined. Through adjustments to the GCPM parameters, we extract the relativistic and non-relativistic energy eigenvalues both in the presence and absence of external fields for a range of potential models, such as the Kratzer, modified Kratzer, Cornell, harmonic oscillator, pseudoharmonic, anharmonic, and Killingbeck potentials. Furthermore, the non-relativistic energy spectra of the Kratzer and modified Kratzer potentials are evaluated with and without external magnetic and AB flux fields for a variety of diatomic molecules. We found that the non-relativistic energy spectrum grows and degeneracy vanishes in the presence of external magnetic and AB flux fields. Moreover, the energy spectrum is more affected by the AB flow field than by the magnetic field.
The following paper is extracted from this chapter titled as: “The influence of magnetic and Aharanov–Bohm fields on energy spectra of diatomic molecules in the framework of the Dirac equation with the generalized interaction potential.” Published in International Journal of Quantum Chemistry, (2022), 123(4), e27031.
Chapter (4):
In this chapter, the generalized fractional Nikiforov-Uvarov method is used to solve the D-dimensional radial Schrödinger equation with the Deng-Fan
potential. The analytical expressions of energy eigenvalues and corresponding eigenfunctions for the Deng-Fan potential are produced. The impact of the fractional parameter on the energy levels of different diatomic molecules is illustrated both numerically and graphically for the Deng-Fan and its shifted potentials. We observed that an increase in the fractional parameter leads to a good improvement in the energy eigenvalues. Additionally, the energy spectra of various diatomic molecules are computed in three dimensions and higher dimensions. Noteworthy is the fact that the energy spectrum rises with the number of dimensions. The graphic representation of the energy levels of the Deng-Fan and its shifted potentials with the reduced mass, screening parameter, equilibrium bond length, rotational and vibrational quantum numbers has been created for the I2 molecule. The energy levels of some diatomic molecules are estimated using the Deng-Fan and its shifted potentials for the classical case in comparison with earlier studies to validate our findings.
The following paper is extracted from this chapter titled as: “The generalized fractional NU method for the diatomic molecules in the Deng–Fan model.” Published in The European Physical Journal D, (2022), 76(9), 159.
Chapter (5):
In this chapter, new solutions to the D-dimensional Schrödinger equation have been discussed utilizing the improved Tietz potential within the context of the generalized fractional derivative. The analytic solutions of the energy spectra and wave functions have been formulated in terms of the fractional parameters in D-dimensions by the utilization of the generalized fractional Nikiforov-Uvarov technique. The potential energy curves for the several diatomic molecules (DMs) have been produced using the improved Tietz potential with the help of molecular constants. The vibrational energy levels for several DMs have been predicted in both the fractional and classical forms to confirm the effectiveness of the methodology employed in this work. It has been shown that, in comparison to earlier research, the predicted values for the vibrational energy spectra far more closely match the observed data. Furthermore, it is found that the vibrational energies of various DMs computed in the presence of fractional parameters agree with the experimental Rydberg-Klein-Rees (RKR) data more accurately than the classical case calculations. The average absolute deviation (AAD) and mean absolute percentage deviation (MAPD) have been estimated as goodness of fit indicators. The improved Tietz potential is an excellent model for fitting the RKR
data for each of the chosen diatomic molecules based on the estimated AAD and MAPD values.
The following paper is extracted from this chapter titled as: “On prediction of the fractional vibrational energies for diatomic molecules with the improved Tietz potential.” Published in Molecular Physics, (2022), 120(24), e2140720.
Chapter (6):
In this chapter, we have obtained new solutions to the D-dimensional Schrödinger equation through the improved Rosen-Morse potential (IRMP) within the framework of the generalized fractional derivative. The generalized fractional Nikiforov-Uvarov approach has been applied to construct the analytical formulations of the energy eigenvalues and wave functions in terms of the fractional parameters in D-dimensions by using the Pekeris-type approximation to the centrifugal term. With the help of molecular parameters, the improved Rosen-Morse potential is utilized to reproduce potential energy curves for a set of diatomic molecules (DMs) which have numerous uses in different fields of physics. To highlight the efficiency of the method used in this work, the pure vibrational energy spectra of the selected diatomic molecules are determined using both the fractional and ordinary forms. In comparison to previous studies, we showed that our calculated vibrational energies correspond significantly more closely to the observed Rydberg-Klein-Rees data (RKR). Furthermore, it is found that the vibrational energy spectra of various DMs computed in the presence of fractional parameters superior to those computed in the absence of fractional parameters for fitting the observed RKR data. As goodness of fit indicators, both the mean absolute percentage deviation (MAPD) and the average absolute deviation (AAD) are used. According to the estimated AAD and MAPD outcomes, the IRMP is a suitable model for fitting the RKR data for all of the diatomic molecules under examination.
The following paper is extracted from this chapter titled as: “A precise estimation for vibrational energies of diatomic molecules using the improved Rosen-Morse potential.” Published in Scientific Reports, (2023), 13(1), 11578.
Chapter (7):
This chapter summarizes the thesis and offers some suggestions for future works