Relativistic Energy Eigenvalues Research Articles

Energy spectra with the Dirac equation of the q-deformed generalized Pöschl-Teller potential via the Feynman approach for .

The diatomic molecules of potassium is widely used in industrial chemicals and alternative energy. Besides that, is very useful for researching molecular interactions and energy states, especially in the context of quantum chemistry and spectroscopy. In the present work, a newly proposed diatomic potential model within relativistic and non-relativistic quantum mechanics has been considered, to obtain corresponding energy eigenvalues and related normalized eigenfunctions. The Dirac equation has been solved for an arbitrary spin-orbit quantum number using the path integral technique with the -deformed generalized Pöschl-Teller potential . By including a Pekeris-type approximation to handle the centrifugal factor, it was possible to obtain the spin and pseudospin-symmetric solution of the relativistic energy eigenvalues and wave equation. To assess the correctness of this work, Maple software was used to present some numerical findings for various values of and . With the constraint , it was shown that in the situation of pseudospin symmetry, only bound states exist with negative energy. In the non-relativistic limits, the non-relativistic ro-vibrational energy expression of the diatomic molecule is derived from the relativistic energy equation under spin symmetry. Under Varshni conditions, both vibrational and ro-vibrational energies of the molecule were computed and compared with the data. The average absolute percentage deviations from the data obtained for the potassium molecule are . This demonstrates that the model is a very consistent model to study and characterize diatomic molecules.

New Eigensolution of the Klein--Gordon and Shrödinger equations for improved modified Yukawa-Kratzer potential and its applications using Boop's Shift method and standard perturbation theory in the 3D-ERQM and 3D-ENRQM symmetries

The deformed Klein-Gordonequation has been solved in three-dimensional extended relativistic quantum mechanics (3D-ERQM) symmetries for the improved modified Yukawa-Kratzer potential (IMYKP) model under the influence of the deformation space-space symmetries. The new relativistic energy eigenvalues were calculated using the parametric Bopp’s shift method and standard perturbation theory in addition to the approximation scheme suggested by Greene and Aldrich for the inverse square terms. The new relativistic energy eigenvalues of (LiH, HCl, CO and H2) molecules under the IMYKP model it was shown to be sensitive to the atomic quantum numbers (j, l, s, m), mixed potential depths (V0, De, re), the screening parameter’s inverse α and noncommutativity parameters (Θ,τ ,χ). In addition, we analyzed the nonrelativistic energy values by applying the well-known transmission rules known in the literature. In addition, we studied many special cases useful to researchers in the framework of the new extended symmetries, such as the improved modified Kratzer potential, the improved generalized Kratzer potential, the improved Kratzer potential, the improved modified Kratzer plus screened Coulomb potential, the improved Hellmann potential, the improved Yukawa potential, and improved inversely square Yukawa potential. We noticed that these particular results are identical to our previous work and other known works in the literature. The study is further extended to calculate the mass spectra of mesons of charmonium (cc) and bottomonium (bb) within the framework of the IMYKP model in three-dimensional extended non-relativistic quantum mechanics (3D-ENRQM) symmetries.

Analysis of quantum systems subject to inverse trigonometry Scarf plus Coulomb potential in N-dimensions

We present approximate solutions of the N-dimensional Klein–Gordon equation in the presence of a superposition of inverse trigonometry Scarf potential and Coulomb potential through asymptotic iteration approach. With the purpose of dealing with the -state, we use Greene-Aldrich approximation which is valid for the small values of the screening parameter. The analytical expressions for the relativistic energy eigenvalues and the corresponding eigenfunctions are established in hyperspherical coordinates. The normalized hyper-radial wavefunction is expressed in terms of hypergeometric and Jacobi polynomials. We analyze the dependences of the relativistic energy levels on the dimensions. We also discuss the effect of the screening parameter on the relativistic energy spectrum. The expectation values of inverse position the square of inverse position kinetic energy and square of momentum in N-dimensional space are reported using the Hellmann-Feynmann theorem. Discussing the special cases of the relevant potential, we show the accuracy of our results.

Arbitrary (k, l) States-Solutions of the Dirac and Schrödinger Equations Interacting with Improved Spatially-Dependent Mass Coulomb Potential with an improved Coulomb-like tensor interaction Model for H-atoms from 3D-RNCS and 3D-NRNCS symmetries

The improved spatially dependent mass Coulomb potential with an improved Coulomb-like tensor interaction (ISDM(CP-CLTI)) model has been used to investigate the new bound state solutions of the deformed Dirac equation under the condition of spin symmetry and pseudospin symmetry in the 3D-relativistic noncommutative space (3D-RNCS) symmetries. The combination of spatially dependent mass Coulomb potential with Coulomb-like tensor interaction (ISDM(CP-CLTI)) and the terms (1/r^3 and 1/r^4 ) which are connected with the induced couplings (LΘ and L ̃Θ) produced from the topological defects of space-space gives ISDM(CP-CLTI) model. The new relativistic and non-relativistic energy eigenvalues for the hydrogen atoms (H-atoms) such as He+, Li+2, and Be3+ under the ISDM(CP-CLTI) model have been derived within the parameters of the parametric Bopp shift method and standard perturbation theory. The new values E_nc^sp(n,C,m_0,m_1,H,Θ,τ,χ,j,l,s,m) and E_nc^ps(n,C,m_0,m_1,H,Θ,τ,χ,j,l,l ̃,s ̃,m ̃) that we obtained appeared to be sensitive to the quantum atomic discrete quantum numbers (j,k,l,s,m,l ̃,s ̃,m ̃), the mixed potential depths (C/q,m_0,m_1,H), and noncommutativity parameters (NP) (Θ,σ,χ). Within the context of relativistic extended quantum mechanics, we have obtained certain significant special cases that, in our view, will be interesting to the specialized researcher. When we applied the three simultaneous limitations (Θ,σ,χ)→(0,0,0), NP we were able to retrieve the typical results of relativistic and non-relativistic cases in the literature. Our new model presented new physical features compared to other models known in the literature.

Arbitrary (k, l) States-Solutions of the Dirac and Schrödinger Equations Interacting with Improved Spatially-Dependent Mass Coulomb Potential with an improved Coulomb-like tensor interaction Model for H-atoms from 3D-RNCS and 3D-NRNCS symmetries

The deformed Dirac equation has been investigated, in the context of 3D-relativistic noncommutative space (3D-RNCS) symmetries, using the improved spatially dependent mass Coulomb potential with an improved Coulomb-like tensor interaction (ISDM(CP-CLTI)) model under the conditions of spin symmetry and pseudospin symmetry. The ISDM(CP-CLTI) model is the combining the spatially dependent mass Coulomb potential with the Coulomb-like tensor interaction (CP-CLTI) and the two central terms that are generated to the topological defects of spacespace. Within the confines of the parametric Bopp shift method and conventional perturbation theory, the new relativistic and non-relativistic energy eigenvalues for the hydrogen atoms ( H-atoms), such as He+,Li+2, and Be3+ under the ISDM(CP-CLTI) model have been derived. The novel values Encsp(n,C,m0,m1,H,Θ,τ,χ,j,l,s,m) and Encps(n,C,m0,m1,H,Θ,τ,χ,j,l,l̃,s̃,m̃) that we discovered examined to be dependent on the noncommutativity parameters (NP) (Θ,σ,χ), mixed potential depths C/q,m0,m1,H), and quantum atomic discrete quantum numbers (j,k,l,s,m,l̃,s̃,m̃). We have obtained several interesting special examples within the framework of relativistic extended quantum mechanics, which we believe will be of interest to the expert researcher. We were able to retrieve the typical results of relativistic and non-relativistic examples in the literature when we applied the three simultaneous constraints (Θ,σ,χ)→(0,0,0). Compared to previous models that are known from the literature, our new model had novel physical characteristics.

New relativistic and non-relativistic investigation of deformed Klein–Gordon and Schrödinger equations for new improved screened Kratzer potential in the framework of non-commutative space

The deformed Klein–Gordon equation has been solved in three-dimensional extended relativistic quantum mechanics (3D-ERQM) symmetries for the newly improved screened Kratzer potential (NISKP) model under the influence of the deformation space–space symmetries. The new relativistic energy eigenvalues were calculated using the parametric Bopp’s shift method and standard perturbation theory in addition to the approximation scheme suggested by Greene and Aldrich for the inverse square terms. The new relativistic energy eigenvalues of (LiH, HCl, CO and H2) molecules under the NISKP model were shown to be sensitive to the atomic quantum numbers ([Formula: see text]), mixed potential depths ([Formula: see text]), the screening parameter’s inverse [Formula: see text], control parameter [Formula: see text] and non-commutativity parameters ([Formula: see text], [Formula: see text], [Formula: see text]). In addition, we analyzed the non-relativistic energy values by applying the well-known transmission rules known in the literature. Furthermore, we studied many special cases useful to researchers in the framework of the new extended symmetries, such as the new screened cosine Kratzer potential, the new screened Kratzer potential (SKP), the new Kratzer potential, the new coshine Yukawa potential, the new coshine Yukawa potential, the new shifted improved SKP and the new shifted SKP. The study is further extended to calculate the mass spectra of mesons of the heavy quarkonium system such as charmonium [Formula: see text], bottomonium [Formula: see text], [Formula: see text] and light mesons [Formula: see text] and [Formula: see text] that have the quark and antiquark flavor within the framework of the NISKP model in 3D-ENRQM symmetries.

Effect of the non-commutativity of space on the improved Mobius square plus generalized Yukawa potentials of the Klein–Gordon and Schrödinger equations in 3D-RNCQS and 3D-NRNCQS symmetries

Under the influence of the deformation space-space symmetries, the improved Mobius square plus generalized Yukawa potentials (IMSGYPs) have been employed to solve the deformed Klien–Gordon equation in three-dimensional noncommutative relativistic quantum space (3D-RNCQS) symmetries. Combined with the approximation approach suggested by Greene and Aldrich, we also employ the parametric Bopp’s shift approach and standard perturbation theory to derive novel relativistic energy eigenvalues. The new relativistic energy eigenvalues of (N2, K2, NI, ScI, and RbH) diatomic molecules under the IMSGYPs were shown to be sensitive to the atomic quantum numbers ([Formula: see text]), the mixed potential depths ([Formula: see text]), the screening parameter’s inverse [Formula: see text] and non-commutativity parameters ([Formula: see text], [Formula: see text], [Formula: see text]). In addition, we analyzed the new non-relativistic energy values in three-dimensional noncommutative non-relativistic quantum space (3D-NRNCQS) symmetries, by applying the well-known mapping in the literature. Furthermore, we studied many special cases useful to researchers in the framework of the new extended symmetries, such as the newly generalized Mobius square potential, the newly generalized Yukawa potential, and the newly generalized Deng-Fan potential. The study is further extended to calculate the mass spectra of mesons of the heavy quarkonium system, such as [Formula: see text], bottomonium [Formula: see text], [Formula: see text] and light mesons [Formula: see text] and [Formula: see text], that have the quark and antiquark flavors within the framework of the IMSGYPs model in 3D-NRNCQS symmetries.

The influence of noncommutativity on the energy spectra of bosonic particles in the framework of the DKGE with improved spatially-dependent mass including mixed scalar-vector Coulomb potentials in the ERQM symmetries

The bound state solutions of the deformed Klien-Gordon equation (DKGE) have been determined in the extended relativistic quantum mechanics ERQM symmetries using the improved spatially-dependent mass Coulomb potential with mixed scalar-vector Coulomb potentials (ISDM-SVCPs) model. The spatially-dependent mass Coulomb potential, as well as a combination of ((1/(r³)) and (1/(r⁴))), are included in the ISDM-SVCPs model, which is coupled with the coupling LΘ, which explains the interaction of the physical features of the system with the topological deformations of space-time. The new relativistic energy eigenvalues for the ISDM-CP have been derived using the parametric Bopp's shift method and standard perturbation theory. Quantum numbers (j,l,s,m), mixed potential depths (q/s_{c},m₀,m₁), and noncommutativity parameters (Θ,τ,χ) seemed to affect the new values we obtained. Within the framework of relativistic extended quantum mechanics, we have addressed certain significant particular instances that we hope will be valuable to the specialized researcher. In DKGE symmetries, we've also looked at the improved pure scalar Coulomb-like potential. The formulation of total energy was also discovered in the context of extended symmetries, which unified the energies of bosonic particles and antiparticles into a single mathematical formula. When the three simultaneous limits (Θ,τ,χ) were applied, we recovered the normal results of relativistic in the literature (0,0,0).

Relativistic Symmetries of Bosonic Particles and Antiparticles in the Background of the Position-Dependent Mass for the Improved Deformed Hulthen Plus Deformed Type-Hyperbolic Potential in 3D-EQM Symmetries

The bound state solutions of the deformed Klien-Gordon equation have been determined in the three-dimensional extended relativistic quantum mechanics 3D-ERQM symmetries using position-dependent mass (PDM) with unequal scalar and vector potential for the improved Hulthén plus improved deformed type-hyperbolic potential (PDM-SVID(H-TP)) models. PDM with unequal scalar and vector potential for the Hulthén plus deformed type-hyperbolic potential (PDM-(SVH-DTP)) models, as well as a combination of radial terms, which are coupled with the coupling LΘ, which explains the interaction of the physical features of the system with the topological deformations of space-space. The new relativistic energy eigenvalues have been derived using the parametric Bopp shift method and standard perturbation theory which is sensitive to the atomic quantum numbers (j,l,s,m), mixed potential depths (V0S0V1S1), the rest, and perturbed mass (m0,m1), the screening parameter's inverse α, and noncommutativity parameters (Θ,τ,χ). Within the framework of 3D-ERQM symmetries, we have treated certain significant particular instances that we hope will be valuable to the specialized researcher. We have also treated the nonrelativistic limit and applied our obtained results to generate the mass spectra of heavy-light mesons (HLM) such as cc- and bb- under PDM-SE with improved deformed Hulthén plus improved hyperbolic potential (PDM‑ID(H-TP)) models. When the three simultaneous limits (Θ,τ,χ) were applied, we recovered the normal results of relativistic in the literature ( 0,0,0) for the PDM‑ID(H-TP)) models.

Deformed Dirac and Shrödinger Equations with Improved Mie-Type Potential for Diatomic Molecules and Fermionic Particles in the Framework of Extended Quantum Mechanics Symmetries

In this study, the bound-state solutions of the deformed Dirac equation (DDE) have been determined with the improved Mie-type potential including an improved Coulomb-like tensor potential (IMTPICLP) under the condition of the spin or pseudospin symmetry in the extended relativistic quantum mechanics (ERQM) symmetries. The IMTPICLP model includes a combination of the terms 1/r3 and 1/r4 which coupled with the couplings (LΘ and L̃︀Θ) between the physical properties of the system with the topological deformations of space-space. In the framework of the parametric Bopp’s shift method and standard perturbation theory, the new relativistic and nonrelativistic energy eigenvalues for the improved Mietype potential have been found. The new obtained values appeared sensitive to the quantum numbers (j, k, l,̃︀ l, s, s,̃︀ m, m̃︀ ), the mixed potential depths (A, B, C, α), and noncommutativity parameters (Θ, σ, χ). The new energy spectra of the improved Kratzer–Fues potential within an improved Coulomb-like tensor interaction and the improved modified Kratzer potential within the Coulomb-like tensor interaction have been derived as particular cases of IMTPICLP. We recovered the usual relativistic and nonrelativistic results from the literature by applying the three simultaneous limits (Θ, σ, χ) → (0, 0, 0). One can notice that our results are in close agreement with the recent studies.

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

AbstractIn this study, we present the relativistic and non‐relativistic solutions of the Dirac equation with the spin symmetry for the generalized Cornell potential (GCP) using the wave function ansatz method in the existence of external magnetic and Aharanov–Bohm (AB) flux fields. The relativistic energy eigenvalues and the corresponding eigenfunctions are found for varied vibrational and magnetic quantum numbers. By adapting the GCP parameters, we derive the relativistic and non‐relativistic energy eigenvalues with and without external fields for a set of potential models including the Killingbeck, harmonic oscillator, pseudoharmonic, anharmonic, Cornell, Coulomb, Kratzer, and modified Kratzer potentials. Additionally, the non‐relativistic energy spectra of the Kratzer and modified Kratzer potentials are reported with and without external magnetic and AB flux fields for several diatomic molecules (DMs). We discovered that in the existence of external magnetic and AB flux fields, the non‐relativistic energy spectrum increases and degeneracy disappears. Furthermore, the AB flux field has a stronger impact on the energy spectrum than the magnetic field. To substantiate our findings, we calculate the energy levels of the Kratzer and modified Kratzer potentials for diverse DMs and find that they are perfectly consistent with previous studies.

The influence of Aharonov–Casher effect on the generalized Dirac oscillator in the cosmic string space-time

In this paper, we investigate the influence of the Aharonov–Casher (A–C) effect on the generalized Dirac oscillator containing the Coulomb-type potential function related to a relativistic neutral particle having a permanent magnetic dipole moment interacting with the external electromagnetic fields in (1 + 2)-dimensional cosmic string space-time. The eigenfunctions and energy eigenvalues of such a Dirac oscillator are derived by using the Nikiforov–Uvarov method. We indicate that implementing the scenario gives the relativistic modified exact analytical solutions. In this way, we can see that the degeneracy of the relevant relativistic energy eigenvalues is broken by depending on the Coulomb strength parameter under the influence of the curvature effect and the A–C effect.

More accurate ro-vibrational energies for SiF +(X 1Σ+) molecule

The most appropriate potential energy function for the X 1Σ+ state of SiF+ molecule has been specified by comparing the vibrational energies obtained via special cases of the general molecular potential (GMP) which are Morse, improved Rosen-Morse, modified Rosen-Morse, improved Manning-Rosen and Tietz potentials with the vibrational energies obtained in the presence of improved generalized Pöschl-Teller (IGPT) potential and experimental data. It has been shown that the improved Rosen-Morse potential is better than the other well-known potential energy functions in fitting experimental energies of SiF +(X 1Σ+) molecule. By using relativistic rotational-vibrational energy eigenvalue relation for Rosen-Morse potential in improved form which is acquired by solving the Dirac equation under the GMP and Pekeris type approximation, the more accurate ro-vibrational energies of SiF +(X 1Σ+) molecule have been obtained. It has been demonstrated for SiF +(X 1Σ+) molecule that in order to procure more proper ro-vibrating energies for the SiF +(X 1Σ+) molecule, a Pekeris-type approach to the centrifugal term is better than the improved Greene-Aldrich in getting more accurate ro-vibrational energies.

Bound state solutions of the Dirac equation for the generalized Cornell potential model

In this study, the bound state solutions of the Dirac equation (DE) have been determined with the generalized Cornell potential model (GCPM) under the condition of spin symmetry. The GCPM includes the Cornell potential plus a combination of the harmonic and inversely quadratic potentials. In the framework of the Nikiforov–Uvarov (NU) method, the relativistic and nonrelativistic energy eigenvalues for the GCPM have been obtained. The energies spectra of the Kratzer potential (KP) and the modified Kratzer potential (MKP) have been derived as particular cases of the GCPM. The present results have been applied to some diatomic molecules (DMs) as well as heavy and heavy-light mesons. The energy eigenvalues of the KP and MKP have been computed for several DMs, and they are fully consistent with the results found in the literature. In addition, the energy eigenvalues of the GCPM have been employed for predicting the spin-averaged mass spectra of heavy and heavy-light mesons. One can note that our predictions are in close agreement with the experimental data as well as enhanced compared to the recent studies.

Analytical solutions of D-dimensional Klein–Gordon equation with modified Mobius squared potential

Approximate solutions of Klein–Gordon equation are obtained for the modified Mobius squared potential using the Nikiforov-Uvarov (NU) method. The relativistic energy eigenvalues and corresponding wave functions are obtained. It is further shown that in the non-relativistic limit, the energy eigenvalues reduces to that of Schrodinger equation. The behavior of CO, NO and HCl molecules are investigated subject to the modified Mobius squared potential. Numerical values of the energies of these diatomic molecules are also presented for arbitrary values of quantum numbers n and l. Finally, plots showing the variation of the energy against various potential parameters are presented for the selected diatomic molecules.

The Reduced Spinless Salpeter Wave Equation for Bound State Systems

In this work, the reduced spinless Salpeter equation is transformed into a relativistic second-order Schrodinger-like wave equation. The converted equation is so simple that one can solve it by using our proposed perturbation technique within the framework of the asymptotic iteration method. As an example, the Coulomb and Oscillator potential models are tested within this method. It is found that this method yields quite simple analytical solutions that will give rapidly the bound state energies with good accuracy compared with the available theoretical predictions. Furthermore, we predicted the relativistic energy eigenvalues of muonium, as well as, for the positronium bound states arising from the one-photon exchange in QED.

The solution of D+1-dimensional Dirac equation for diatomic molecules with the Morse potential

In this paper, the solutions of the D+1-dimensional Dirac equation in the presence of the Morse potential are investigated by using the supersymmetric quantum mechanics and shape invariance theory with spin symmetry. The bound state energy spectrum and the ground state wave function are calculated by supersymmetry approach. Also, it is shown that the energy spectrum and vibrational energy eigenvalue at the critical point can be obtained by a numerical method which is called the asymptotic iteration method. The $$X^2\Sigma ^+$$ state of the CP and $$X^1\Sigma ^+$$ state of the $$SiF^+$$ molecules are considered, and their relativistic energy eigenvalues are studied.

Analytical Bound State Solutions of the Klein-Fock-Gordon Equation for the Sum of Hulthén and Yukawa Potential within SUSY Quantum Mechanics

The relativistic wave equations determine the dynamics of quantum fields in the context of quantum field theory. One of the conventional tools for dealing with the relativistic bound state problem is the Klein-Fock-Gordon equation. In this work, using a developed scheme, we present how to surmount the centrifugal part and solve the modified Klein-Fock-Gordon equation for the linear combination of Hulthén and Yukawa potentials. In particular, we show that the relativistic energy eigenvalues and corresponding radial wave functions are obtained from supersymmetric quantum mechanics by applying the shape invariance concept. Here, both scalar potential conditions, which are whether equal and nonequal to vector potential, are considered in the calculation. The energy levels and corresponding normalized eigenfunctions are represented as a recursion relation regarding the Jacobi polynomials for arbitrary l states. Beyond that, a closed form of the normalization constant of the wave functions is found. Furthermore, we state that the energy eigenvalues are quite sensitive with potential parameters for the quantum states. The nonrelativistic and relativistic results obtained within SUSY QM overlap entirely with the results obtained by ordinary quantum mechanics, and it displays that the mathematical implementation of SUSY quantum mechanics is quite perfect.

Analytical bound state solutions of the Dirac equation with the Hulthén plus a class of Yukawa potential including a Coulomb-like tensor interaction

We examine the bound state solutions of the Dirac equation under the spin and pseudospin symmetries for a new suggested combined potential, Hulthén plus a class of Yukawa potential including a Coulomb-like tensor interaction. An improved scheme is employed to deal with the centrifugal (pseudocentrifugal) term. Using the Nikiforov–Uvarov and SUSYQM methods, we analytically develop the relativistic energy eigenvalues and associated Dirac spinor components of wave functions. We find that both methods give entirely the same results. Modification of our results to fit some particular potentials, which are useful in some other systems, is also discussed. We obtain complete agreement with the findings of previous works. The spin and pseudospin bound state energy spectra for various levels are presented in the absence as well as the presence of tensor coupling. Both energy spectrums are sensitive with regard to the quantum numbers $$\kappa $$ and n, as well as the parameter $$\delta $$ . We also notice that the degeneracies between Dirac spin and pseudospin-doublet eigenstate partners are completely removed by the tensor interaction. Finally, we present the parameter space of allowable bound state regions of potential strength $$V_0$$ with constants $$C_{S}$$ and $$C_{PS}$$ for both symmetry limits.

Klein–Gordon oscillator subject to a Cornell-type potential in the rotating cosmic string space-time and Aharonov–Bohm effect

Klein–Gordon oscillator in the background space-time generated by a rotating cosmic string subject to a Cornell-type scalar and Coulomb-type vector potentials including an internal magnetic flux is studied. We obtain the relativistic energy eigenvalues and the corresponding eigenfunctions and analyze a relativistic analogue of the Aharonov–Bohm effect for bound states.