Approximate Bound State Solutions Research Articles

Approximate bound state solutions of the Dunkl-Schrödinger equation for a hyperbolic double-well interaction

We construct approximate solutions to the stationary, one-dimensional Schrödinger equation for a hyperbolic double-well potential within the Dunkl formalism. Our approximation is applied to an inverse quadratic term contributed by the Dunkl formalism in the effective potential. The solutions we obtain are given in terms of confluent Heun functions. We establish parity of these solutions, discuss their elementary cases, and present an example of a system admitting bound states.

[formula omitted]-dimensional Klein–Gordon equation in presence of deformed generalized Deng–Fan with Yukawa potential class: Approximate bound state solutions in relativistic and non-relativistic regimes

In this paper, we examine the bound state solutions of the D-dimensional Klein–Gordon equation for a deformed Deng–Fan potential in generalized form along with Yukawa potential class. The supersymmetric quantum mechanics method and Nikiforov–Uvarov method are employed, utilizing a proper correspondence to the centrifugal potential term. Remarkably, in both methods, we obtain the same analytical expressions for normalized wave functions and energy eigenvalues, expressed in closed form by hypergeometric functions and Jacobi polynomials, for all l and n quantum states. Additionally, we explore the thermodynamic quantities (internal energy, partition function, specific heat, entropy and free energy) in case of both relativistic and non-relativistic regimes for the aforementioned potential. Finally, we demonstrate energy variation against various potential parameters for a few diatomic molecules pictorially.

Approximate bound state solutions of the fractional Schr\"{o}dinger equation under the spin-spin-dependent Cornell potential

In this work, the approximate bound state solutions of the fractional Schr\"{o}dinger equation under a spin-spin-dependent Cornell potential are obtained via the convectional Nikiforov-Uvarov approach. The energy spectra are applied to obtain the mass spectra of the heavy mesons such as bottomonium, charmonium and bottom-charm. The masses for the singlet and triplet spin numbers increase as the quantum numbers increase. The fractional Schr\"{o}dinger equation improves the mass spectra compared to other theoretic masses. The bottomonium masses agree with the experimental results where percentage errors for fractional parameters of $\beta =1,\alpha =0.97$ and $\beta =1,\alpha =0.50$ were found to be 0.67\% and 0.49\% respectively. The respective percentage errors of 1.97\% and 1.62\% for fractional parameters of $\beta =1,\alpha =0.97$ and $\beta =1,\alpha =0.50$ were obtained for charmonium meson. The results indicate that the potential curves coupled with the fractional parameters account for the short-range gluon exchange between the quark-antiquark interactions and the linear confinement phenomena which is associated with the quantum chromo-dynamic and phenomenological potential models in particle and high-energy physics.

Energy spectrum and applications of Eckart plus Hellmann potential in hyperspherical coordinates

We investigate the approximate bound-state solutions of the N-dimensional Schrödinger equation with a combination of Eckart and Hellmann potentials via the asymptotic iteration approach. By the use of the Greene-Aldrich approximation, which is valid for small values of the screening parameter, we establish the N-dimensional energy spectrum and the N-dimensional radial wave function in approximate analytic form. In hyperspherical coordinates, the normalized radial wave function is expressed in terms of hypergeometric and Jacobi polynomials. To double-check the energy spectrum in hyperspherical coordinates, we utilize polynomial solutions in view of the asymptotic iteration approach. The expressions for the expectation values of inverse position, square of inverse position, kinetic energy, and square of momentum are derived in hyperspherical coordinates by using the Hellmann-Feynmann theorem. We also present the analytically calculated energy eigenvalues for Eckart plus Hellmann potential. We deduce special forms of the relevant potential, such as Eckart, Hellmann, and Hulthén potentials.

Solutions of Schrödinger equation based on Modified Exponential Screened Plus Yukawa Potential (MESPYP) within the frame of parametric Nikiforov-Uvarov method

In this paper, we deduced an approximate bound state solutions of Schrӧdinger equation based on Modified Exponential Screened Plus Yukawa Potential (MESPYP), with the help Greene–Aldrich Approximation to determine the centrifugal term. The analytical solution was obtained based on Nikiforov–Uvarov (NU) method. The bound state solutions of five diatomic molecules which are mercury hydride (HgH), zinc hydride (ZnH), cadmium hydride (CdH), hydrogen bromide (HBr) and hydrogen fluoride (HF) molecules were also calculated numerically. The estimated energy eigenvalues for these molecules were deduced using the resulting energy eigenequation and total unnormalized wavefunction is expressed in term of Jacobi polynomial. The numerical solution shows that the bound state solutions increase with increase in the quantum state.

Exact and approximate bound state solutions of the Schrödinger equation with a class of Kratzer-type potentials in the global monopole spacetime

This work investigates the motion of a non-relativistic charged particle in the spacetime of a global monopole. We introduce the Schrödinger equation to describe the particle’s motion with two interactions by considering the Kratzer and the screened modified Kratzer potential. The problem’s eigenfunctions and eigenvalues are obtained by deriving and solving the radial equation. The effective potential encompasses both the Kratzer and electrostatic self-interaction potential and leads to bound states solutions. The energy spectrum is investigated, particularly emphasizing its dependence on the system’s physical parameters. The screened modified Kratzer potential, and the screened self-interaction potential reveal an important role in influencing both the effective potential and the energy spectrum. Additionally, it also accommodates the existence of bound states. All these behaviors are illustrated with graphs and discussed in detail.

The Influence of Deformation Space-Space on High and Low Energy Spectra of Fermionic Particles and Spectra of Heavy Quarkonia with Improved Hulthén Plus Hyperbolic Exponential Inversely Quadratic Potential

In this work, the modified approximation to the centrifugal barrier term is applied to find approximate bound-state solutions of the deformed Dirac equation for the spin and pseudospin symmetries in a model with the improved hyperbolic Hulth´en and hyperbolic exponential inversely quadratic potentials (IHHEIQPs) using the parametric method of Bopp’s shift and the standard perturbation theory in the extended relativistic quantum mechanics (ERQM). Our results indicate that the new energy eigenvalues are highly sensitive to the potential parameters (ν1, A) and to the values of quantum atomicnumbers (j, k, l, m,̃︀ l, ̃︀ m, s, ̃︀s), range of the potential ν, and noncommutativity parameters (θ, σ, β). We found that the effect of a space-space deformation gives a correction in the energy spectrum, where the main energy term remains due to the effect of the hyperbolic Hulth´en and hyperbolic exponential inversely quadratic potentials known in the literature. The new nonrelativistic energies are obtained by applying the nonrelativistic limit to the relativistic spin-energy equation in the extended nonrelativistic quantum mechanics (ENRQM). The proposed potential model reduces to the improved Hulth´en and exponential inversely quadratic potentials as special cases in ERQM. The present results are applied for calculating the new mass spectra Mhiqpnc−nl of heavy mesons such as cc, bb, bc, bs, cs, and bq,q = (u, d) in ENRQM. It turns out that the values of masses come from the contribution of the mass spectra Mhiqpnl in NRQM, while the effect of a space-space deformation δMhiqpnc−nl is an infinitesimal correction as compared with Mhiqpnl. Our results seem to be significant and agree perfectly with the ones in the literature.

Bound state solutions and thermodynamic properties of modified exponential screened plus Yukawa potential

In this research paper, the approximate bound state solutions and thermodynamic properties of Schrӧdinger equation with modified exponential screened plus Yukawa potential (MESPYP) were obtained with the help Greene–Aldrich approximation to evaluate the centrifugal term. The Nikiforov–Uvarov (NU) method was used to obtain the analytical solutions. The numerical bound state solutions of five selected diatomic molecules, namely mercury hydride (HgH), zinc hydride (ZnH), cadmium hydride (CdH), hydrogen bromide (HBr) and hydrogen fluoride (HF) molecules were also obtained. We obtained the energy eigenvalues for these molecules using the resulting energy eigenequation and total unnormalized wave function expressed in terms of associated Jacobi polynomial. The resulting energy eigenequation was presented in a closed form and applied to study partition function (Z) and other thermodynamic properties of the system such as vibrational mean energy (U), vibrational specific heat capacity (C), vibrational entropy (S) and vibrational free energy (F). The numerical bound state solutions were obtained from the resulting energy eigenequation for some orbital angular quantum number. The results obtained from the thermodynamic properties are in excellent agreement with the existing literature. All numerical computations were carried out using spectroscopic constants of the selected diatomic molecules with the help of MATLAB 10.0 version. The numerical bound state solutions obtained increases with an increase in quantum state.

Approximate k-state solutions of the deformed Dirac equation in spatially dependent mass for the improved Eckart potential including the improved Yukawa tensor interaction in ERQM symmetries

In this work, we considered analytical approximate bound state solutions of the mass-dependent deformed Dirac equation with the improved vector Eckart potential including the Yukawa tensor interaction (IEPDMYI for short) in the context of extended relativistic quantum mechanics ERQM symmetries. The new relativistic energy spectra of the DDE for the IEPDMYI with spin symmetry and pseudospin symmetry are obtained. The solutions for arbitrary k-wave are obtained by using the basic concept of Bopp’s shift approach and standard perturbation by applying an approximation to the centrifugal-like term in the calculations. The new values that we get appeared sensitive to the discrete quantum numbers [Formula: see text], the mixed potential depths [Formula: see text], the range of the potential [Formula: see text] and noncommutativity parameters [Formula: see text]. We have highlighted three physical phenomena that automatically generate a result of the topological properties of noncommutativity, the first physical phenomena are the perturbative spin-orbit coupling, the second is the magnetic induction while the third corresponds to the rotational proper phenomena. We show that the corrections on the spectrum energy are smaller than the main energy in the usual cases of quantum field theory. In the ERQM symmetries, it is not possible to get the exact analytical solutions for [Formula: see text], only the approximate solutions are available. We have obtained the energy equation for several well-known potentials by choosing appropriate parameters in the improved Eckart potential model such as the improved Hulthén potential and the improved generalized Morse potential.

Spin and pseudospin solutions to Dirac equation and its thermodynamic properties using hyperbolic Hulthen plus hyperbolic exponential inversely quadratic potential

In this research article, the modified approximation to the centrifugal barrier term is applied to solve an approximate bound state solutions of Dirac equation for spin and pseudospin symmetries with hyperbolic Hulthen plus hyperbolic exponential inversely quadratic potential using parametric Nikiforov–Uvarov method. The energy eigen equation and the unnormalised wave function were presented in closed and compact form. The nonrelativistic energy equation was obtain by applying nonrelativistic limit to the relativistic spin energy eigen equation. Numerical bound state energies were obtained for both the spin symmetry, pseudospin symmetry and the non relativistic energy. The screen parameter in the potential affects the solutions of the spin symmetry and non-relativistic energy in the same manner but in a revised form for the pseudospin symmetry energy equation. In order to ascertain the accuracy of the work, the numerical results obtained was compared to research work of existing literature and the results were found to be in excellent agreement to the existing literature. The partition function and other thermodynamic properties were obtained using the compact form of the nonrelativistic energy equation. The proposed potential model reduces to Hulthen and exponential inversely quadratic potential as special cases. All numerical computations were carried out using Maple 10.0 version and Matlab 9.0 version softwares respectively.

Bound state solution and thermodynamic properties of the screened cosine Kratzer potential under influence of the magnetic field and Aharanov–Bohm flux field

In this article, approximate analytical bound state solutions of the Schrödinger equation in two-dimensional space for screened cosine Kratzer potential (SCKP) under the influence of the magnetic field and Aharanov–Bohm flux field have been investigated. We obtained energy eigenvalues and wave functions for SCKP with external fields (magnetic field and Aharanov–Bohm flux field) via parametric Nikiforov–Uvarov (pNU) method using the approximation method suggested by Greene–Aldrich for handling centrifugal barriers. We deduced energy eigenvalues for screened Kratzer potential (SKP) and Hellmann potential (HP) from the obtained energy spectrum of SCKP with an external field. We extended our results in D dimensions for Hellmann potential in the absence of external fields. Thermodynamic properties such as partition function Z(B→,ΦAB,β), mean energy U(B→,ΦAB,β), mean free energy F(B→,ΦAB,β), entropy S(B→,ΦAB,β), specific heat capacity Cs(B→,ΦAB,β), magnetization at finite temperature (B→,ΦAB,β) and magnetic susceptibility χm(B→,ΦAB,β) at finite temperature are presented. The obtained numerical resultsfor SKP and Hellmann potential are in very good agreementwith numerical results available in the literature with and without external fields respectively.

Approximate Bound State Solutions of the Hellmann Plus Kratzer Potential in N-dimensional Space

We have examined the approximate l_(N-1)-state solutions of the N-dimensional Schrödinger equation for a particle interacting with the Hellmann plus Kratzer potential. In hyperspherical coordinate system, we have constructed the bound state energy equation and the wavefunctions expressed by the hypergeometric function via the asymptotic iteration approach in detail. When considered the special cases of parameters in Hellmann plus Kratzer potential, this potential turns into several potential models. In this connection, the non-relativistic energy spectra for the modified Kratzer, Yukawa, Coulomb and Hellmann potentials in approximate analytic form have been obtained in hyperspherical coordinates. We have presented the numerical energy eigenvalues for the Hellmann, Yukawa and Coulomb potentials in N=3 dimensions. Our present results provide an appropriate test of the accuracy of asymptotic iteration formalism.

Eigensolution and various properties of the screened cosine Kratzer potential in D dimensions via relativistic and non-relativistic treatment

In this article, we proposed screened cosine Kratzer potential (SCKP). Using approximation suggested by Greene–Aldrich, the approximate bound-state solutions of the D-dimensional Klein–Gordon equation for SCKP have been obtained via the generalized Nikiforov–Uvarov method. The bound-state energy eigenvalues and the corresponding normalized wave functions expressed in terms of hypergeometric functions were obtained. From the proposed SCKP model, we recovered different potentials such as screened Kratzer potential, standard Kratzer potential, generalized cosine Yukawa potential, screened Coulomb potential, Coulomb potential, inversely quadratic Yukawa potential and its corresponding energy eigenvalues from obtained energy eigenvalues of the SCKP in both the relativistic and non-relativistic regime. We obtained rotational–vibrational energy for few heterogeneous (LiH, HCl, NO) and homogeneous (H $$_2$$ , I $$_2$$ , O $$_2$$ ) diatomic molecules in three dimensions. The numerical results obtained for LiH, HCl, NO and I $$_2$$ , O $$_2$$ diatomic molecules are in very good agreement with the results previously obtained by others. We obtained various properties of the SCKP such as thermodynamical properties (partition function, vibrational mean energy, vibrational mean free energy, vibrational specific heat capacity and vibrational entropy, information energy $$E(\gamma )$$ , Tsallis entropy T(q), Renyi entropy R(q), Shannon entropy $$ S(\gamma )$$ and Fisher information entropy $$I(\gamma )$$ ) using energy eigenvalues and eigenfunctions, the expressions for the expectation values of the square of inverse of position $$1/r^2$$ , inverse of position 1/r, kinetic energy T, and square of momentum $$p^2$$ via Hellmann–Feynman theorem. We also presented quarkonium mass spectroscopy using energy eigenvalues of the SCKP.

Non-Relativistic Bound State Solutions of Modified Quadratic Yukawa plus <i>q</i>-Deformed Eckart Potential

We have obtained approximate bound state solutions of Schrödinger wave equation with modified quadratic Yukawa plus q-deformed Eckart potential Using Parametric Nikiforov-Uvarov (NU) method. However, we obtained numerical energy eigenvalues and un-normalized wave function using confluent hypergeometric function (Jacobi polynomial). With some modifications, our potential reduces to a well-known potential such as Poschl-Teller and exponential inversely quadratic potential. Numerical bound state energies were carried out using a well-designed Matlab algorithm while the plots were obtained using origin software. The result obtained is in agreement with that of the existing literature.

Non-commutativity and non-inertial effects on the Dirac oscillator in a cosmic string space–time

We examine the non-inertial effects of a rotating frame on a Dirac oscillator in a cosmic string space-time with non-commutative geometry in phase space. We observe that the approximate bound-state solutions are related to the biconfluent Heun polynomials. The related energies cannot be obtained in a closed form for all the bound states. We find the energy of the fundamental state analytically by taking into account the hard-wall confining condition. We describe how the ground-state energy scales with the new non-commutative term as well as with the other physical parameters of the system.

Higher-dimensional fractional time-independent Schrödinger equation via fractional derivative with generalised pseudoharmonic potential

In this paper, we obtain approximate bound-state solutions of N-dimensional time-independent fractional Schrodinger equation for the generalised pseudoharmonic potential which has the form $$V(r^{\alpha })=a_1r^{2\alpha } + ({a_2}/{r^{2\alpha }})+a_3$$ . Here $$\alpha \;(0<\alpha <1)$$ acts like a fractional parameter for the space variable r. The entire study consists of the Jumarie-type fractional derivative and the elegance of Laplace transform. As a result, we can successfully express the approximate bound-state solution in terms of Mittag–Leffler function and fractionally defined confluent hypergeometric function. Our study may be treated as a generalisation of all previous works carried out on this topic when $$\alpha =1$$ and N arbitrary. We provide numerical result of energy eigenvalues and eigenfunctions for a typical diatomic molecule for different $$\alpha $$ close to unity. Finally, we try to correlate our work with a Cornell potential model which corresponds to $$\alpha = {1}$$ $$/$$ $${2}$$ with $$a_3=0$$ and predicts the approximate mass spectra of quarkonia.

Approximate bound state solutions of the Klein-Gordon equation with the linear combination of Hulthén and Yukawa potentials

Based on a developed scheme we show how to deal with the centrifugal term and the Coulombic behavior part and then to solve the Klein-Gordon (KG) equation for the linear combination of Hulthén and Yukawa potentials. Two cases, i.e., the scalar potential which is equal and unequal to vector potential, are considered for arbitrary l state. With the aid of the Nikiforov-Uvarov (NU) method and the traditional approach, we present the eigenvalues and the corresponding radial wave functions expressed by the Jacobi polynomials or hypergeometric functions and find that the results obtained by them are consistent. For given values of potential parameters V0,V0′,S0,S0′ and M=1, we notice that the energy levels E are sensitively relevant for the potential parameter δ and the energy levels E increase for δ>0.1 as quantum numbers nr and l increase. However, for δ∈(0,0.1) the energy levels E do not always increase with the quantum numbers nr and l. We find that the energy levels E are inversely proportional to quantum numbers nr and l when δ∈(0,0.05).

Energy spectra and the expectation values of diatomic molecules confined by the shifted Deng-Fan potential

The approximate bound state solutions of the Schrodinger equation with the shifted Deng-Fan potential was obtained via a proper quantization rule. The energy spectra for the homogenous diatomic molecules (H2, I2); the heterogeneous diatomic molecules (CO, HCl, LiH); the neutral transition metal hydrides (ScH, TiH, VH, CrH); the transition-metal lithide (CuLi); the transition-metal carbides (TiC, NiC); the transition metal nitrite (ScN) and the transition metal fluoride (ScF) were calculated. By applying the Hellmann-Feynman theorem, the expression for the expectation values of the square of inverse of position r-2, potential energy V, kinetic energy T and the square of momentum p2 were derived and the numerical values for diatomic molecules were presented. The result is reliable and very consistent with the available ones in the literature.

Time independent fractional Schrödinger equation for generalized Mie-type potential in higher dimension framed with Jumarie type fractional derivative

In this paper, we obtain approximate bound state solutions of an N-dimensional fractional time independent Schrödinger equation for a generalised Mie-type potential, namely, V(rα)=Ar2α+Brα+C. Here α(0 &amp;lt; α &amp;lt; 1) acts like a fractional parameter for the space variable r. When α = 1 the potential converts into the original form of Mie-type of potential that is generally studied in molecular and chemical physics. The entire study is composed with a Jumarie-type fractional derivative approach. The solution is expressed via the Mittag-Leffler function and fractionally defined confluent hypergeometric function. To ensure the validity of the present work, obtained results are verified with the previous studies for different potential parameter configurations, specially for α = 1. At the end, few numerical calculations for energy eigenvalue and bound state eigenfunctions are furnished for a typical diatomic molecule.

PT-/non-PT-symmetric and non-Hermitian Hellmann potential: approximate bound and scattering states with any ℓ-values

We investigated the approximate bound-state solutions of the Schrödinger equation for the PT-/non-PT-symmetric and non-Hermitian Hellmann potential. Exact energy eigenvalues and corresponding normalized wave functions were obtained. Numerical energy eigenvalues for the bound states were compared with ones obtained before. Scattering state solutions were also studied. Phase shifts of the potential are written in terms of the angular momentum quantum number ℓ.