Solutions of the Schrödinger equation with velocity-dependent potentials:a study on Kratzer, Mie and Hulthen potentials

The effect of isotropic velocity-dependent potentials on the bound state energy eigenvalues of the Kratzer, Mie and Hulthen potentials is obtained for any quantum states in the presence of constant form factor εr=γρ0. The corresponding energy eigenvalues and eigenfunctions are determined and tabulated for a set of finite quantum numbers n and l in the framework of the well-known Nikiforov-Uvarov method.


Introduction
In recent years, velocity-dependent potentials have held a significant position in quantum mechanics.Numerous successful studies have been conducted in this field for various physical systems.They are sometimes used in modeling the behavior of particles within the framework of quantum mechanics.The concept of the velocity-dependent potential was initially developed to analyze the scattering of mesons from complex nuclei in nuclear physics [1].In atomic physics, the velocity-dependent potential was utilized to illustrate the scattering of electrons from oxygen and neon atoms [2,3].Additionally, a nucleon-nucleus velocity-dependent optical model was introduced [4].Moreover, the solution of the Schrödinger equation using a velocity-dependent potential has been employed to describe particles with a position-dependent effective mass [5].In presence of positiondependent effective mass, the movement of electrons in semiconductor hetero-structures, such as quantum dots [6] and liquids [7] can be analyzed.And these systems can be transformed into well-known exactly solvable Schrödinger equations with constant mass through point canonical transformation [5][6][7].
The energy spectrum of the Kratzer potential [36] has been determined using similar methods as described in references [37,38].As an example within our study, the focus is on the Mie potential, and the corresponding bound-state energies for the non-relativistic Schrödinger wave equation have been calculated [39].
Until now, the energy eigenvalues of the Kratzer, Mie, and Hulthen potentials have been computed using various methods.However, no study has taken into account the effect of velocity-dependent potentials.This paper focuses on resolving the solution for the velocity-dependent versions of the Kratzer, Mie, and Hulthen potentials, considering a constant value for the form factor r 0 ( ) e gr = [40].The present work is organized as follows: section 2 provides a concise introduction to the NU method.The determination of the corresponding eigenfunctions and eigenvalues for Kratzer, Mie, and Hulthen potentials for any quantum states for a constant form factor are given in the subsequent section.The final section presents the conclusions and discusses notable results.

The method
Nikiforov-Uvarov (NU) is a method of solving the second order linear equations with special functions.The energy spectrums of many physical systems have been investigated by Nikiforov-Uvarov method [19,[41][42][43][44][45][46].The corresponding second order differential equation is considered as where s ( ) s and s ˜( ) s are polynomials, at most of second degree, and s ˜( ) t is a first-degree polynomial.To make the application of the NU method simpler and more direct, we introduce a more compact presentation of the idea.In order to do this, the equation (2.1) is rewritten as follows [22] With these equations the necessary parameters for the solution of the equation are derived.
(i) The relevant constant: 2.5 The essential polynomial functions: The energy equation: The wave function: are Jacobi polynomials with where N nk is a normalization constant.After some algebraic calculation, the wave function can be expressed as where c c 0, 0 The wave function can be reduced in form of the Laguerre function in the case of c 0,

The model
The velocity-dependent potential is a superposition of the V r eff ( ) and the isotropic velocity-dependent V r p , total ( )potentials which has form [1], + is the corresponding physical potential V r , ( )  is the Planck constant, and m is the mass of the particle.Substituting the potential form in the eigenvalue equation leads to for a particle with energy E moving under the isotropic velocity-dependent potential and which is the required Schrödinger equation including velocity-dependent potential function.Now, the final equation is amenable to apply the method with different potentials.

Kratzer potential
Kratzer potential [47] has been used extensively so far in order to describe the molecular structure and interactions.The graph of potential by choosing appropriate parameters is given in figure 1 and the potential form is defined as where the parameters A, B, and C are constants which are related with Kratzer potential.The SE with Kratzer potential takes the form, by using form factor r r , 0 By comparing with equation defined in method, the corresponding parameters are obtained as and the energy equations is Since the analytical solution of equation (3.6) could not be achieved by using NU method, numerical solutions of energy are obtained for some n and l values.The results obtained are given in the table 1 with fixed parameters A B 0. Besides, this energy figure gives pretty much close results compared to the non-velocity-dependent potential in the literature.

MIE potential
The Mie-type potential [48], which belongs to a class of multi-parameter exponential potentials, has applications in vibrational and rotational spectroscopy in physical sciences.This is because its interaction model includes both repulsive and attractive terms as seen in figure 3, catering to shortand large intermolecular distances for certain diatomic molecules [48] with where a is the positive constant which is strongly repulsive at shorter distances.Substituting this potential with r r 0 ( ) ( ) e gr gr = = into Schrödinger equation transforms to well-known form of second order differential equation Table 1.The energy eigenvalues of the Kratzer potential under By applying similar procedure, we get   The corresponding wave function is obtained easily as -+ gr gr gr gr gr gr Similar to the previous calculation, the numerical solutions of energy for n and l are given in table 2 with suitable parameters

Hulthen potential
The Hulthen potential [49] is very important exponential short range potential which behaves like a Coulomb potential for small values of r and decreases exponentially for large values of r.The Hulthen potential has been used in many branches of physic, such as atomic physic, nuclear physic, solid state physics and chemical physics.The graph of potential is presented in figure 5.The Hulthen potential is given by, where K and k are the strength and the range parameter of the potential respectively.Similarly, the Schrödinger equation is obtained as Table 2.The energy eigenvalues of the Mie potential under m a 0 0 To get the solution of eigenvalue equation, Pekeris approximation [50] is needed and is proposed as Finally, the transformed equation is reduced to desired form and are obtained with the energy equation for the Hulthen potential as The corresponding wave function becomes, The equation (3.16) cannot be solved analytically by using NU method.Therefore, the numerical solutions of energy of energy for n and l were calculated from energy equation with appropriate parameters  K 0.5,

Conclusions
In the case of isotropic velocity-dependent potentials Hulthen, Kratzer, and Mie potentials are determined for various quantum states using the Nikiforov-Uvarov (NU) method.By changing the parameters K , , , 0 g m r and , k the influence of the form factor on the corresponding potentials was analyzed and tabulated as in tables.
The energy spectrum is obtained numerically by using 0 l = in equations (2.7), (3.6), (3.10) and (3.16).It was shown how the change in these parameters alters the energy eigenfunction and energy eigenvalue of the Schrödinger wave equation.This study will shed light on research where solutions will be made in cases of variable form factor.
is plotted as shown in figure 4 for this values.Further, energy spectrum and wave function of Mie potential are compared to the literature and give the same results in case r 0. ( ) e = Looking at the energy change of Mie potential in figure, it shows a change similar to the energy change of Kratzer potential.

4 .
The graph of energy eigenvalues of the Mie potential for m a

as in table 2 .
Energy spectrums and wave function of Hulthen potential give the similar results compared to the literature for value of r 0 ( ) e = form factor.The graph of energy is plotted as a function of r in figure 6.The energy values of Hulthen potential showed a decrease negatively depending on the increase quantum number, at the same time, energy decreases negatively when the angular momentum increases as shown in table 3.Such as, E n = −0.00996143 in n = 1 and l 1 = 1, E n = −0.0113747 in n = 2 and l 2 = 2.The energy changes of the Hulthen potential is showed almost the same

Table 3 .
The energy eigenvalues of the Hulthen potential in