A new approximation scheme to the centrifugal term is proposed to obtain the $l\neq 0$ solutions of the Schrödinger equation with the Manning-Rosen potential. We also find the corresponding normalized wave functions in terms of the Jacobi polynomials. To show the accuracy of the new approximation scheme, we calculate the energy eigenvalues numerically for arbitrary quantum numbers $n$ and $l$ with two different values of the potential parameter $α.$ The bound state energies of various states for a few $% HCl,$ $CH,$ $LiH$ and $CO$ diatomic molecules are also calculated. The numerical results are in good agreement with those obtained by using program based on a numerical integration procedure. Our solution can be also reduced to the s-wave ($l=0$) case and to the Hulthén potential case.
The solutions of trigonometric Scarf potential, PT/non-PT-symmetric and non-Hermitian q-deformed hyperbolic Scarf and Manning-Rosen potentials are obtained by solving the Schrodinger equation. The Nikiforov-Uvarov method is used to obtain the real energy spectra and corresponding eigenfunctions.
The Schrodinger equation for the Manning-Rosen potential with the centrifugal term is solved approximately to obtain bound states energies. Additionally, the corresponding wave functions are expressed by the Jacobi polynomials. The Nikiforov-Uvarov (NU) method is used in the calculations. To show the accuracy of our results, we calculate the eigenvalues numerically for arbitrary quantum numbers n and l with two different values of the potential parameter α. It is shown that the results are in good agreement with the those obtained by other methods for short potential range, small l and α. This…
The Schrodinger equation for the Manning-Rosen potential with the centrifugal term is solved approximately to obtain bound states energies. Additionally, the corresponding wave functions are expressed by the Jacobi polynomials. The Nikiforov-Uvarov (${\rm NU}$) method is used in the calculations. To show the accuracy of our results, we calculate the eigenvalues numerically for arbitrary quantum numbers $n$ and $l$ with two different values of the potential parameter $α.$ It is shown that the results are in good agreement with the those obtained by other methods for short potential range, small…
An alternative treatment is proposed for the calculations carried out within the frame of Nikiforov-Uvarov method, which removes a drawback in the original theory and by pass some difficulties in solving the Schrodinger equation. The present procedure is illustrated with the example of orthogonal polynomials. The relativistic extension of the formalism is discussed.
Exact bound state solutions of the Schrodinger equation for generalized noncentral potential are examined by means of the Nikiforov-Uvarov method. The wavefunctions and the corresponding energy eigenvalues of the system are obtained analytically. The results examined for the potential are compatible with those obtained by the other methods.
The one-dimensional Dirac equation is solved for the PT-symmetric generalized Hulthen potential. The Nikiforov-Uvarov method which is based on solving the second-order linear differential equations by reduction to a generalized equation of hypergeometric type is used to obtain exact energy eigenvalues and corresponding eigenfunctions.
The Nikiforov-Uvarov method is employed to calculate the the Schrodinger equation with a rotation Morse potential. The bound state energy eigenvalues and the corresponding eigenfunction are obtained. All of these calculation present an effective and clear method under a Pekeris approximation to solve a rotation Morse model. Meanwhile the results got here are in a good agreement with ones before.
The one-dimensional Klein-Gordon (KG) equation has been solved for the PT-symmetric generalized Woods-Saxon (WS) potential. The Nikiforov-Uvarov(NU} method which is based on solving the second-order linear differential equations by reduction to a generalized equation of hypergeometric type is used to obtain exact energy eigenvalues and corresponding eigenfunctions. We have also investigated the positive and negative exact bound states of the s-states for different types of complex generalized WS potentials.
The one-dimensional spinless Salpeter equation has been solved for the PT-symmetric generalized Hulthén potential. The Nikiforov-Uvarov {NU) method which is based on solving the second-order linear differential equations by reduction to a generalized equation of hypergeometric type is used to obtain exact energy eigenvalues and corresponding eigenfunctions. We have investigated the positive and negative exact bound states of the s-states for different types of complex generalized Hulthen potentials.
The general solutions of Schrodinger equation for non central potential are obtained by using Nikiforov Uvarov method. The Schrodinger equation with general non central potential is separated into radial and angular parts and energy eigenvalues and eigenfunctions for these potentials are derived analytically. Non central potential is reduced to Coulomb and Hartmann potential by making special selections, and the obtained solutions are compared with the solutions of Coulomb and Hartmann ring shaped potentials given in literature.
The exact solutions of the one-dimensional Klein-Gordon equation for the Rosen-Morse type potential with equal scalar and vector potentials are presented. First we briefly review Nikiforov-Uvarov mathematical method. Using this method, wave-functions and corresponding exact energy equation are obtained for the s-wave bound state. It has been shown that the results for Rosen-Morse type potentials reduce to standard Rosen-Morse well and Eckart potentials in the special case. The PT-symmetry for these potentials is also considered.
Using the Nikiforov-Uvarov method, the bound state energy eigenvalues and eigenfunctions of the PT-/non-PT-symmetric and non-Hermitian generalized Woods-Saxon (WS) potential with the real and complex-valued energy levels are obtained in terms of the Jacobi polynomials. According to the PT-symmetric quantum mechanics, we exactly solved the time-independent Schrodinger equation with same potential for the s-states and also for any l-state as well. It is shown that the results are in good agreement with the ones obtained before.
We present analytically the exact energy bound-states solutions of the Schrodinger equation in D-dimensions for a recently proposed modified Kratzer potential plus ring-shaped potential by means of the conventional Nikiforov-Uvarov method. We give a clear recipe of how to obtain an explicit solution to the wave functions in terms of orthogonal polynomials. The results obtained in this work are more general and true for any dimension which can be reduced to the standard forms in three-dimensions given by other works.
Approximate analytical bound-state solutions of the Dirac particle in the field of both attractive and repulsive RM potentials including Coulomb-like tensor (CLT) potential are obtained for arbitrary spin-orbit quantum number The Pekeris approximation is used to deal with the spin-orbit coupling terms In the presence of exact spin and pseudospin (p-spin) symmetries, the energy eigenvalues and the corresponding normalized two-component wave functions are found by using the parametric generalization of the Nikiforov-Uvarov (NU) method. The numerical results show that the CLT interaction removes …
We present analytically the exact energy bound-states solutions of the Schrodinger equation in D-dimensions for an alternative (often used) pseudo-Coulomb potential-plus- ring-shaped potential of the form $V(r)=-% \frac{a}{r}+\frac{b}{r^{2}}+\frac{β\cos ^{2}θ}{r^{2}\sin ^{2}θ}+c$ by means of the conventional Nikiforov-Uvarov method. We give a clear recipe of how to obtain an explicit solution to the radial and angular parts of the wave functions in terms of orthogonal polynomials. The total energy of the system is different from the pseudo-Coulomb potential because of the contribution of the a…
The energy levels of the Schrödinger equation under the Eckart-Hellmann potential (EHP) energy function are studied by the Nikiforov-Uvarov-Functional Analysis (NUFA) method. We obtained the analytic solution of the energy spectra and the wave function in closed form with the help of Greene-Aldrich approximation. The numerical bound states energy for various screening parameters at different quantum states and vibrational energies of EHP for CuLi, TiH, VH, and TiC diatomic molecules were computed. Four particular cases of this potential were achieved. To test the accuracy of our results, we co…
In this work, we used a tool of conventional Nikiforov-Uvarov method to determine bound state solution of Schrodinger equation with quantum interaction potential called Hulthen-Yukawa inversely quadratic potential (HYIQP). We obtained the energy eigen values and the total wave function . We employed Hellmann-Feynmann Theorem (HFT) to compute expectation values for four different diatomic molecules: Hydrogen molecule (H2), Lithium hydride molecule (LiH), Hydrogen Chloride molecule (HCl) and Carbon(II)Oxide molecule. The resulting energy equation reduces to three well known potentials which are:…
The extended Cornell potential which the harmonic oscillator potential is included in the original Cornell potential. The Dirac equation is solved by reducing the Dirac equation to the form of Schrodinger equation. The Nikiforov-Uvarov method is applied to obtain the energy eigenvalues and corresponding wave functions. The obtained results are important to calculate many characteristics of fermion relativistic particles.
The creation and annihilation operators of a two-term diatomic molecular potential are studied and it is observed that they satisfy the commutation relations of a SU(1,1) algebra. To study the Lie algebraic realization of the present potential, the normalized eigenfunctions and eigenvalues are computed by using the Nikiforov-Uvarov method.
The approximate analytical bound state solution of the Schrödinger equation for the Manning-Rosen potential is carried out by taking a new approximation scheme to the orbital centrifugal term. The Nikiforov-Uvarov method is used in the calculations. We obtain analytic forms for the energy eigenvalues and the corresponding normalized wave functions in terms of the Jacobi polynomials or hypergeometric functions for different screening parameters 1/b. The rotational-vibrational energy states for a few diatomic molecules are calculated for arbitrary quantum numbers n and l with different values of…
In this paper, we present the exact solution of one dimensional Schrödinger equation for Wood-Saxon plus Rosen-Morse plus symmetrical double well potential via Nikiforov-Uvarov mathematical method. The eigenvalues and eigenfunctions of this potential are obtained. The energy equations and the corresponding wave function for special cases of this potential are briefly discussed. The PT-symmetry and Hermiticity for this potential are also considered.
A nonpolynomial one-dimensional quantum potential in the form of an isotonic oscillator (harmonic oscillator with a centripetal barrier) is studied. We provide the non-relativistic bound state energy spectrum E_{n} and the wave functions ψ_{n}(x) in terms of the associated Laguerre polynomials in the framework of the Nikiforov-Uvarov method. Under the spin and pseudospin symmetric limits, the analytic eigenvalues and the corresponding two-component upper- and lower-spinors of the Dirac particle are obtained, in closed form.
In this work, the non-relativistic wave equation via the Schrödinger wave equation under the influence of the Aharonov-Bohm flux field Subject to physical potentials of various kinds is investigated. These potentials are modified Coulomb potential, modified harmonic oscillator potential, the Kratzer-Feus potential, and the Mie-type potential which have wide applications in different branches of physics and chemistry. We solve the Schrodinger wave equation using the Nikiforov-Uvarov (NU) method and obtain the energy profiles and the wave function of the non-relativistic particle, and analyze th…
For applications to quasi-exactly solvable Schrödinger equations in quantum mechanics, we establish the general conditions that have to be satisfied by the coefficients of a second-order differential equation with at most $k+1$ singular points in order that this equation has particular solutions which are $n$th-degree polynomials. In a first approach, we extend the Nikiforov-Uvarov method, which was devised to deal with hypergeometric-type equations (i.e., for $k=2$), and show that the conditions involve $k-2$ integration constants. In a second approach, we consider the functional Bethe ansatz…
Using the Nikiforov Uvarov method, an application of the relativistic Duffin Kemmer Petiau equation in the presence of a deformed Hulthen potential is presented for spin zero particles. We derived the first order coupled differential radial equations which enable the energy eigenvalues as well as the full wavefunctions to be evaluated by using of the Nikiforov Uvarov method that can be written in terms of the hypergeometric polynomials.
The Schrodinger equation with the PT-symmetric Hulthen potential is solved exactly by taking into account effect of the centrifugal barrier for any l-state. Eigenfunctions are obtained in terms of the Jacobi polynomials. The Nikiforov-Uvarov method is used in the computations. Our numerical results are in good agreement with the ones obtained before. Keywords: Energy Eigenvalues and Eigenfunctions; Hulthen potential; PT-symmetry; Nikiforov-Uvarov Method.
The Schrödinger equation in $D$-dimensions for the Manning-Rosen potential with the centrifugal term is solved approximately to obtain bound states eigensolutions (eigenvalues and eigenfunctions). The Nikiforov-Uvarov(NU) method is used in the calculations. We present numerical calculations of energy eigenvalues to two- and four-dimensional systems for arbitrary quantum numbers $n$ and $l$ with three different values of the potential parameter $α.$ It is shown that because of the interdimensional degeneracy of eigenvalues, we can also reproduce eigenvalues of a upper/lower dimensional sytem fr…
Two physically important potentials (Manning-Rosen and Pöschl-Teller) are considered for the ro-vibrational energy in diatomic molecules. An improved new approximation is invoked for the centrifugal term, which is then used for their solution within the Nikiforov-Uvarov framework. This employs a recently proposed scheme, which combines the two widely used Greene-Aldrich and Pekeris-type approximations. Thus, approximate analytical expressions are derived for eigenvalues and eigenfunctions. The energies are examined with respect to two approximation parameters, $λ$ and $ν$. The original approxi…
Exact analytical spectra of an effective field-modified radial Schrödinger equation by the Nikiforov--Uvarov method
College of Computing and Mathematical Sciences, Department of Applied Mathematics and Sciences, Khalifa University of Science and Technology（哈利法科学技术大学计算与数学科学学院应用数学与科学系）；Center for Cyber-Physical Systems (C2PS), Khalifa University of Science and Technology（哈利法科学技术大学网络物理系统中心）
The effective mass Klein-Gordon equation in one dimension for the Woods-Saxon potential is solved by using the Nikiforov-Uvarov method. Energy eigenvalues and the corresponding eigenfunctions are computed. Results are also given for the constant mass case.
A general form of the effective mass Schrodinger equation is solved exactly for Hulthen potential. Nikiforov-Uvarov method is used to obtain energy eigenvalues and the corresponding wave functions. A free parameter is used in the transformation of the wave function.
Using the Nikiforov Uvarov method, we obtained the eigenvalues and eigenfunctions of the Woods Saxon potential with the negative energy levels based on the mathematical approach. According to the PT Symmetric quantum mechanics, we exactly solved the time independent Shcrodinger equation for the same potential. Results are obtained for the s states.
The trigonometric Pöschl-Teller (PT) potential describes the diatomic molecular vibration. By using the Nikiforov-Uvarov (NU) method, we have obtained the exact analytical s-wave solutions of the radial Schrödinger equation (SE) for the trigonometric PT potential. The energy eigenvalues and corresponding eigenfunctions are calculated in closed forms. Some numerical results are presented too.
We investigate the approximate solutions of the Dirac equation with the position-dependent mass particle in the Eckart potential field including the Coulomb tensor interaction by using the parametric Nikiforov-Uvarov method. Taking an appropriate approximation to deal with the centrifugal term, the Dirac energy states and the corresponding normalized two-spinor components of the wave function are obtained in closed form. Some special cases of our solution are investigated. Furthermore, we present the correct solutions obtained via the asymptotic iteration method which are in agreement with the…
In this paper, the Schrodinger equation for s-wave and arbitrary angular momenta with the Modified Mobuis Square potential is investigated respectively. The eigenfunctions as well as energy eigenvalues are obtained in an exact analytical manner via the Nikiforov Uvarov method using two approximations scheme. Some special cases of this potentials are also studied.
Exact solution of Schrodinger equation for the Mie potential is obtained for an arbitrary angular momentum. The energy eigenvalues and the corresponding wavefunctions are calculated by the use of the Nikiforov-Uvarov method. Wavefunctions are expressed in terms of Jacobi polynomials. The bound states are calculated numerically for some values of l and $n$ with n<5. They are applied to several diatomic molecules.
In this paper, we solve analytically the Schrodinger equation for s-wave and arbitrary angular momenta with the Hua potential is investigated respectively. The wave function as well as energy equation are obtained in an exact analytical manner via the Nikiforov Uvarov method using two approximations scheme. Some special cases of this potentials are also studied.
With the aid of an improved short-range approximation, the bound state energies and the scattering phase shifts for a Hulthén- type potential plus Yukawa potential are calculated within the framework of Nikiforov-Uvarov and standard methods, respectively. Our numerical calculations show that the presence of Yukawa potential has a significant influence on bound state energies and the wave number-dependent phase shifts for any arbitrary values of the screening parameter α and angular momentum l.
In this paper, we solve the D-dimensional Schrödinger equation with hyperbolic Poschl-Teller potential plus a generalized ring-shaped potential. After the separation of variable in the hyperspherical coordinate. We used Nikiforov-Uvarov (NU) method to solve the resulting radial equation and obtain explicitly the energy level and the corresponding wave function in closed form. The solutions to the angular part are solved using the NU approach as well.
The radial part of the Klein-Gordon equation for the generalized Woods-Saxon potential is solved by using the Nikiforov-Uvarov method in the case of spatially dependent mass within the new approximation scheme to the centrifugal potential term. The energy eigenvalues and corresponding normalized eigenfunctions are computed. The solutions in the case of constant mass are also studied to check out the consistency of our new approximation scheme.
The one-dimensional Klein-Gordon equation is solved for the PT-symmetric generalized Hulthen potential in the scalar coupling scheme. The relativistic bound-state energy spectrum and the corresponding wave functions are obtained by using the Nikiforov-Uvarov method which is based on solving the second-order linear differential equations by reduction to a generalized equation of hypergeometric type.
The Dirac-Morse problem are investigated within the framework of an approximation to the term proportional to $1/r^2$ in the view of the position-dependent mass formalism. The energy eigenvalues and corresponding wave functions are obtained by using the parametric generalization of the Nikiforov-Uvarov method for any $κ$-value. It is also studied the approximate energy eigenvalues, and corresponding wave functions in the case of the constant-mass for pseudospin, and spin cases, respectively.
Focusing on an improved approximation scheme, we present how to treat the centrifugal and the Coulombic behavior terms and then to obtain the bound state solutions of the Klein-Gordon (KG) equation with the Manning-Rosen plus a Class of Yukawa potentials. By means of the Nikiforov-Uvarov (NU) and supersymmetric quantum mechanics (SUSYQM) methods, we present the energy spectrum for any $\ell$-state and the corresponding radial wave functions in terms of the hypergeometric functions. From both methods we obtain the same results. Several special cases for the potentials which are useful for other…
Effective mass Schrodinger equation is solved exactly for a given potential. Nikiforov-Uvarov method is used to obtain energy eigenvalues and the corresponding wave functions. A free parameter is used in the transformation of the wave function. The effective mass Schrodinger equation is also solved for the Morse potential transforming to the constant mass Schrödinger equation for a potential. One can also get solution of the effective mass Schrodinger equation starting from the constant mass Schrodinger equation.
We present an approximate analytic solution of the Klein-Gordon equation in the presence of equal scalar and vector generalized deformed hyperbolic potential functions by means of parameteric generalization of the Nikiforov-Uvarov method. We obtain the approximate bound state rotational-vibrational (ro-vibrational) energy levels and the corresponding normalized wave functions expressed in terms of the Jacobi polynomial for a spin-zero particle in a closed form. Special cases are studied including the non-relativistic solutions obtained by appropriate choice of parameters and also the s-wave so…
The Hellmann potential is a superposition potential that consists of an attractive Coulomb potential and a Yukawa potential. By using the generalized parametric Nikiforov-Uvarov (NU) method, we have studied the approximate analytical solutions of the Dirac equation with the Hellmann potential including a Coulomb-like tensor potential for arbitrary spin-orbit quantum number k under the presence of exact spin and pseudo-spin (p-spin) symmetries. We show that tensor interaction removes degeneracies between spin and pseudospin doublets. As particular cases, we found the energy levels of non-relati…
The approximate analytical solutions of the Dirac equations with the reflectionless-type and Rosen-Morse potentials including the spin-orbit centrifugal (pseudo-centrifugal) term are obtained. Under the conditions of spin and pseudospin (pspin) symmetry concept, we obtain the bound state energy spectra and the corresponding two-component upper- and lower-spinors of the two Dirac particles by means of the Nikiforov-Uvarov (NU) method in closed form. The special cases of the s-wave κ=\pm1 (l=l=0) Dirac equation and the non-relativistic limit of Dirac equation are briefly studied.
It is shown that the application of the Nikiforov-Uvarov method by Ikhdair for solving the Dirac equation with the radial Rosen-Morse potential plus the spin-orbit centrifugal term is inadequate because the required conditions are not satisfied. The energy spectra given are incorrect and the wave functions are not physically acceptable. We clarify the problem and prove that the spinor wave functions are expressed in terms of the generalized hypergeometric functions. The energy eigenvalues for the bound states are given by the solution of a transcendental equation involving the hypergeometric f…
We give the approximate analytic solutions of the Dirac equations for the Rosen-Morse potential including the spin-orbit centrifugal term. In the framework of the spin and pseudospin symmetry concept, we obtain the analytic bound state energy spectra and corresponding two-component upper- and lower-spinors of the two Dirac particles, in closed form, by means of the Nikiforov-Uvarov method. The special cases of the s-wave kappa=1,-1 (l=l bar=0) Rosen-Morse potential, the Eckart-type potential, the PT-symmetric Rosen-Morse potential and non-relativistic limits are briefly studied.
Analytical solutions of the N-dimensional Schrödinger equation for the newly proposed Varshni-Hulthén potential are obtained within the framework of Nikiforov-Uvarov method by using Greene-Aldrich approximation scheme to the centrifugal barrier. The numerical energy eigenvalues and the corresponding normalized eigenfunctions are obtained in terms of Jacobi polynomials. Special cases of the potential are equally studied and their numerical energy eigenvalues are in agreement with those obtained previously with other methods. However, the behavior of the energy for the ground state and several e…
The Yukawa potential is often used to compute bound-state normalizations and energy levels of neutral atoms. By using the generalized parametric Nikiforov-Uvarov (NU) method, we have obtained the approximate analytical solutions of the radial Schrödinger equation (SE) for the Yukawa potential. The energy eigenvalues and corresponding eigenfunctions are calculated in closed forms. Some numerical results are presented and showed that these results are in good agreement with those are obtained previously by other methods. Also, we found the energy levels of the familiar pure Coulomb potential ene…
We obtain the bound-state solutions of the radial Schrödinger equation (SE) with the shifted Deng-Fan (sDF) oscillator potential in the frame of the Nikiforov-Uvarov (NU) method and employing Pekeris-type approximation to deal with the centrifugal term. The analytical expressions for the energy eigenvalues and the corresponding wave functions are obtained in closed form for arbitrary -state. The ro-vibrational energy levels for a few diatomic molecules are also calculated. They are found to be in good agreement with those ones previously obtained by the Morse potential.
