Kratzer Potential Research Articles

RETRACTED ARTICLE: Non-relativistic analytical solutions of the Kratzer potential for a perturbed system in a magnetic field

Non-relativistic analytical solutions of a quantum mechanical system, perturbed by an external magnetic field, are obtained for the case of Kratzer potential. Such problem is a kind of Landau problem named “Kratzer-Landau problem” due to the presence of the magnetic field. Paying attention to hyperspherical coordinates, an analytical formula for the Landau levels is achieved dealing with Schrodinger equation by using the perturbation method in the frame of the asymptotic iteration method. The analytical formula obtained is also tested by numerically cross-checking the energy eigenvalues with those of the literature in two dimensions (polar coordinates). Moreover, effects of the parameters, such as magnetic field strength and azimuthal angle (in three dimensions), on the eigenvalues are presented.

Eigensolution, expectation values and thermodynamic properties of the screened Kratzer potential

Within the framework of non-relativistic quantum mechanics via the Nikiforov-Uvarov (NU) method, we obtained the energy eigenvalues and the corresponding normalized eigenfunctions of a newly proposed screened Kratzer potential for lithium hydride (LiH) and hydrogen chloride (HCl) diatomic molecules. With the help of the Hellman-Feynman theorem, the expressions for the expectation values of the square of inverse of position, $ r^{-2}$ , inverse of position, $ r^{-1}$ , kinetic energy, T , and square of momentum, $ p^2$ , and their respective numerical values for the selected diatomic molecules are presented. The vibrational partition function and other thermodynamic functions are also obtained in closed form for the diatomic molecules, using the energy eigenvalues. The results obtained clearly agree with the previously obtained results in the literature.

Solution of the Schrödinger equation with inversely quadratic Yukawa potential (IQYP) plus Kratzer-Fues (KFP) potential using the WKB quantum mechanical formalism

The main objective of this research work is theoretical investigate the bound state solutions of the non-relativistic Schrödinger equation with a mixed potential composed of the Inversely Quadratic Yukawa/Attractive Coulomb potential plus a Modified Kratzer potential (IQYCKFP) by utilizing the Wentzel-Kramers-Brillouin (WKB) quantum theoretical formalism. The energy eigenvalues and its associated wave functions have successfully been obtained sequel to certain diatomic molecules includes; HCL, HBr, LiH.

The effect of deformation-dependent mass term on the Bohr model in the presence of rigidity parameter for Kratzer potential

In this work, we have solved the Bohr Hamiltonian with a deformation-dependent mass term in the presence of the free parameter $ \chi$ by using the Nikiforov-Uvarov (NU) method. We have studied such Hamiltonian by employing the generalized Kratzer potential for the $ \beta$ part of the total potential. While in $ \gamma$ -angular parts, we have used both axially symmetric prolate deformed nuclei and $ \gamma$ -unstable nuclei. The energy eigenvalues and the corresponding wave functions have been calculated to investigate the competition between $ \gamma$ -stable and $ \gamma$ -rigid ( $ \gamma$ -unstable and $ \gamma$ -rigid) collective conditions. Also, the energy spectra and transition rates have been determined for some nuclei and compared with experimental data. Our results show good agreement with experimental data.

Any l-state solutions of the Schrodinger equation interacting with Hellmann–Kratzer potential model

The approximate analytical solutions of the radial Schrodinger equation have been obtained with a newly proposed potential called Hellmann–Kratzer potential. The potential is a superposition of Hellmann potential and modified Kratzer potential. The Hellmann–Kratzer potential actually comprises of three different potentials which include Yukawa potential, Coulomb potential and Kratzer potential. The aim of combining these potentials is to have a wide application. The energy eigenvalue and the corresponding wave function are calculated in a closed and compact form using the Nikiforov–Uvarov method. The energy equation for some potentials such as Kratzer, Hellmann, Yukawa and Coulomb potentials has also been obtained by varying some potential parameters. Our results excellently agree with the already existing literature. Some numerical results have been computed. We have plotted the behaviour of the energy eigenvalues with different potential parameters and also reported on the numerical result.

Bound state solutions of the Schrodinger equation for the modified Kratzer potential plus screened Coulomb potential

We obtained an approximate solution of the Schrodinger equation for the modified Kratzer potential plus screened Coulomb potential model, within the framework of Nikiforov–Uvarov method. The bound state energy eigenvalues for N2, CO, NO, and CH diatomic molecules were computed for various vibrational and rotational quantum numbers. Special cases were considered when the potential parameters were altered, resulting into modified Kratzer potential, screened Coulomb potential, and standard Coulomb potential, respectively. Their energy eigenvalues expressions and numerical computations agreed with the already existing literatures.

Exact solutions of Schrödinger and Pauli equations for a charged particle on a sphere and interacting with non-central potentials

We calculate exact solutions of the Schrödinger equation for a particle constrained to move along a spherical surface and interacting with non-central potentials, namely, (i) Makarov, (ii) ring-shaped pseudo-harmonic oscillatory, and (iii) Kratzer potentials. We also study exact solutions of the Pauli equation in the same geometrical setting for a charged particle in the presence of a uniform magnetic field. In this case, the two-component spinor can adhere to the surface only if the magnetic field intensity has certain special values. The solutions of Schrödinger equations allow us to obtain exact Pauli spinors and their corresponding energy eigenvalues for the same non-central potentials.

Odd deformed nuclei with $\gamma$ γ -instability

The properties of deformed odd nuclei away from the E(5/4) critical point have been studied by using an adjustable Kratzer potential for the even-even core coupled to a fermion moving in a single j = 3/2 shell. This choice preserves the $\gamma$ -instability of the whole system and gives precise analytic predictions for spectra and transitions rates, that are compared with available spectroscopic information in iridium isotopes.

D-dimensional Klein-Gordon equation with spatially q-deformed of radial momentum for Kratzer potential analyzed by using hypergeometric method

Hypergeometric method can be used to solve D-dimensional Klein-Gordon equation with q-deformed of radial momentum for Kratzer potential. By substituting q-deformed of radial momentum variable, the D-dimensional and the potential parameters to obtain the second order of D-dimensional Klein-Gordon equation which is used to determine the energy and wave function equations. The analytical result of energy can be founded with D-dimension and q-deformed of radial momentum variations. As a result, the rise of D-dimension values cause the rise of energy values, while the rice of q deformation values causes the decline of energy values.

Pseudospectral method of solution of the Schrödinger equation for the Kratzer and pseudoharmonic potentials with nonclassical polynomials and applications to realistic diatom potentials

The pseudoharmonic and Kratzer potentials have been extensively employed by numerous workers to model the vibrational states of diatomic molecules. These potentials belong to SUperSYmmetric (SUSY) quantum mechanics and the eigenvalues of the Schrödinger equation are known. The energy eigenvalues of the Schrödinger equation for the pseudoharmonic and Kratzer potentials have been determined with different numerical methods as a benchmark of the numerical schemes. We employ a pseudospectral method based on a quadrature grid defined with a nonclassical polynomial basis set. This basis set is defined orthogonal with respect to the square of ground state wavefunction as the weight function. A discrete matrix representation of the Hamiltonian of dimension N is constructed and vibrational energies are calculated with the numerical diagonalization of this matrix. This pseudospectral method is employed to calculate the vibrational energy levels of H2, CO and NO modelled with the pseudoharmonic and Kratzer potentials in comparison with the more realistic Morse potential. The vibrational energy eigenstates for H2 with a realistic quantum mechanical potential and an approximate Morse potential are also calculated. The oversimplified pseudoharmonic and Kratzer potentials are useful for benchmarking numerical methods. However, they represent potential energy curves that can deviate drastically from more exact potentials such as the Morse potential.

Bound-state solutions of the Dirac equation for the Kratzer potential with pseudoscalar-Coulomb term

We present exact analytical solutions of the Dirac equation in (1 + 1) dimensions for the generalized Kratzer potential by taking the pseudoscalar interaction term as an attractive Coulomb potential. We study the problem for a particular (spin) symmetry of the Dirac Hamiltonian. After a qualitative analyse, we study the results for some special cases, such as the Dirac-Coulomb problem in the presence of pseudoscalar interaction, and the “pure” Coulomb problem by discussing some points about pseudospin and spin symmetries in one dimension. We also plot some figures representing the dependence of the energy on the quantum number, and potential parameters.

Information entropy and complexity measure in generalized Kratzer potential

Shannon entropy (S), Fisher information (I) and a measure equivalent to Fisher-Shannon complexity (CIS) of a ro-vibrational state of diatomic molecules (O2, O2+, NO, NO+) with generalized Kratzer potential is analyzed. Exact analytical expression of Ir is derived for the arbitrary state, whereas the same could be done for Ip with {n,ℓ,m=0} state. It is found that shifting from neutral to the cationic system, Ir increases while Sr decreases, consistent with the interpretation of a localization in the probability distribution. Additionally, this study reveals that CIS increases with the number of nodes in a system.

The Klein–Gordon equation with the Kratzer potential in the noncommutative space

The Klein–Gordon equation is considered for the Kratzer potential in the spherical polar coordinate in laboratory frame in noncommutative space. The energy shift due to noncommutativity is obtained via the perturbation theory. After rather cumbersome algebra, we found the eigenfunctions and eigenvalues of the system for a noncommutative phase space.

Bound state solutions to the Schrödinger equation for some diatomic molecules

The bound state solutions to the radial Schrodinger equation are obtained in three-dimensional space using the series expansion method within the framework of a general interaction potential. The energy eigenvalues of the pseudoharmonic and Kratzer potentials are given as special cases. The obtained analytical results are applied to several diatomic molecules, i.e. $$\mathrm {N}_2, \mathrm {CO}, \mathrm {NO}$$ and $$\mathrm {CH}$$ . In order to check the accuracy of the present method, a comparison is made with similar results obtained in the literature by using other techniques.

Electric quadrupole transitions of deformed nuclei via Davidov-Chaban Hamiltonian within the Kratzer potential

In this work, we investigate energy spectra, corresponding wave functions and the electric quadrupole probability transition rate for even-even nuclei using the collective model and the Davydov-Chaban Hamiltonian. Within the Kratzer potential for the $\beta$ part and $\gamma$ fixed at $30^{\circ}$ , we compare our results with the experimental ones to show our model’s accuracy.

Excited states of neutral donor bound excitons in GaN

We investigate the excited states of a neutral donor bound exciton (D0X) in bulk GaN by means of high-resolution, polychromatic photoluminescence excitation (PLE) spectroscopy. The optically most prominent donor in our sample is silicon accompanied by only a minor contribution of oxygen—the key for an unambiguous assignment of excited states. Consequently, we can observe a multitude of Si0X-related excitation channels with linewidths down to 200 μeV. Two groups of excitation channels are identified, belonging either to rotational-vibrational or electronic excited states of the hole in the Si0X complex. Such identification is achieved by modeling the excited states based on the equations of motion for a Kratzer potential, taking into account the particularly large anisotropy of effective hole masses in GaN. Furthermore, several ground- and excited states of the exciton-polaritons and the dominant bound exciton are observed in the photoluminescence (PL) and PLE spectra, facilitating an estimate of the associated complex binding energies. Our data clearly show that great care must be taken if only PL spectra of D0X centers in GaN are analyzed. Every PL feature we observe at higher emission energies with regard to the Si0X ground state corresponds to an excited state. Hence, any unambiguous peak identification renders PLE spectra highly valuable, as important spectral features are obscured in common PL spectra. Here, GaN represents a particular case among the wide-bandgap, wurtzite semiconductors, as comparably low localization energies for common D0X centers are usually paired with large emission linewidths and the prominent optical signature of exciton-polaritons, making the sole analysis of PL spectra a challenging task.

Coherent states for Kratzer-type potentials

The coherent states for a particle in Kratzer type potentials are constructed by solving Feynman’s path integral. The action of the generalized Kratzer potential is transformed into two harmonic oscillators by Levi-Civita transformation to derive the parametric time coherent states. Green’s function, energy eigenvalues, and the corresponding wave functions for this potential are calculated. The evaluated results are reduced to the modified Kratzer potentials and Kratzer-Fues oscillators which are special cases of the generalized Kratzer potential.

Perturbed Coulomb Potentials in the Klein–Gordon Equation: Quasi-Exact Solution

Using the Lie algebraic approach, we present the quasi-exact solutions of the relativistic Klein–Gordon equation for perturbed Coulomb potentials namely the Cornell potential, the Kratzer potential and the Killingbeck potential. We calculate the general exact expressions for the energies, corresponding wave functions and the allowed values of the parameters of the potential within the representation space of sl(2) Lie algebra. In addition, we show that the considered equations can be transformed into the Heun’s differential equations and then we reproduce the results using the associated special functions. Also, we study the special case of the Coulomb potential and show that in the non-relativistic limit, the solution of the Klein–Gordon equation converges to that of Schrodinger equation.

Screw dislocation and external fields effects on the Kratzer pseudodot

In this research, we investigate the influence of the screw dislocation on the energy levels of nonrelativistic electrons on a two-dimensional solid taking into account the deformed Kratzer potential under an external magnetic and the Aharonov–Bohm (AB) fields. We obtain the bound state energy eigenvalues equation and the corresponding wave functions by using the Frobenius method. We also calculate the direct interband light absorption coefficient and threshold frequency of absorption in a quantum pseudodot. Some of the numerical results are given.

Some aspects of an induced electric dipole moment in rotating and non-rotating frames.

Quantum effects on a neutral particle (atom or molecule) with an induced electric dipole moment are investigated when it is subject to the Kratzer potential and a scalar potential proportional to the radial distance. In addition, this neutral is placed in a region with electric and magnetic fields. This system is analysed in both non-rotating and rotating reference frames. Then, it is shown that bound state solutions to the Schrödinger equation can be achieved and, in the search for polynomial solutions to the radial wave function, a restriction on the values of the cyclotron frequency is analysed in both reference frames.