We present a new approximation scheme for the centrifugal term to solve the Schrodinger equation with the Hulthen potential for any arbitrary l state by means of a mathematical Nikiforov-Uvarov (NU) method. We obtain the bound state energy eigenvalues and the normalized corresponding eigenfunctions expressed in terms of the Jacobi polynomials or hypergeometric functions for a particle exposed to this potential field. Our numerical results of the energy eigenvalues are found to be in high agreement with those results obtained by using the program based on a numerical integration procedure. The s-wave (l=0) analytic solution for the binding energies and eigenfunctions of a particle are also calculated. The physical meaning of the approximate analytical solution is discussed. The present approximation scheme is systematic and accurate.
The bound state energies and wave functions for a particle exposed to the Hulthén potential field in the D-dimensional space are obtained within the improved quantization rule for any arbitrary l state. The present approximation scheme used to deal with the centrifugal term in the effective Hulthén potential is systematic and accurate. The solutions for the three-dimensional (D=3) case and the s-wave (l=0) case are briefly discussed. Keywords: Hulthén potential, improved quantization rule, approximation schemes. 03.65.Ge, 12.39.Jh
The bound state (energy spectrum and two-spinor wave functions) solutions of the Dirac equation with the Hulthen potential for all angular momenta based on the spin and pseudospin symmetry are obtained. The parametric generalization of the Nikiforov-Uvarov method is used in the calculations. The orbital dependency (spin-orbit and pseudospin-orbit dependent coupling too singular 1/r^{2}) of the Dirac equation are included to the solution by introducing a more accurate approximation scheme to deal with the centrifugal (pseudo-centrifugal) term. The approximation is also made for the less singula…
The radial part of the effective mass Klein-Gordon equation for the Hulthen potential is solved by making an approximation to the centrifugal potential. The Nikiforov-Uvarov method is used in the calculations. Energy spectra and the corresponding eigenfunctions are computed. Results are also given for the case of constant mass.
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.
Approximate bound state solutions of the Dirac equation with the Hulthén plus a new generalized ring-shaped (RS) potential are obtained for any arbitrary -state. The energy eigenvalue equation and the corresponding two-component wave function are calculated by solving the radial and angular wave equations within a recently introduced shortcut of Nikiforov-Uvarov (NU) method. The solutions of the radial and polar angular parts of the wave function are given in terms of the Jacobi polynomials. We use an exponential approximation in terms of the Hulthen potential parameters to deal with the stron…
The problem of a Dirac particle moving in a deformed Hulthen potential is solved in the framework of the path integral formalism. With the help of the Biedenharn transformation, the construction of a closed form for the Green's function of the second-order Dirac equation is done by using a proper approximation to the centrifugal term and the Green's function of the linear Dirac equation is calculated. The energy spectrum for the bound states is obtained from the poles of the Green's function. A Dirac particle in the standard Hulthen potential for q=1 and a Dirac hydrogen-like ion when q = 1 an…
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 role of the Hulthen potential on the spin and pseudospin symmetry solutions is investigated systematically by solving the Dirac equation with attractive scalar S(r) and repulsive vector V(r) potentials. The spin and pseudospin symmetry along with orbital dependency (pseudospin-orbit and spin-orbit dependent couplings) of the Dirac equation are included to the solution by introducing the Hulthen-square approximation. This effective approach is based on forming the spin and pseudo-centrifugal kinetic energy term from the square of the Hulthen potential. The analytical solutions of the Dirac …
We present a new approximation scheme for the centrifugal term to obtain a quasi-exact analytical bound state solutions within the framework of the position-dependent effective mass radial Klein-Gordon equation with the scalar and vector Hulthén potentials in any arbitrary $D$ dimension and orbital angular momentum quantum numbers $l.$ The Nikiforov-Uvarov (NU) method is used in the calculations. The relativistic real energy levels and corresponding eigenfunctions for the bound states with different screening parameters have been given in a closed form. It is found that the solutions in the ca…
We solve the Klein-Gordon equation in any $D$-dimension for the scalar and vector general Hulthén-type potentials with any $l$ by using an approximation scheme for the centrifugal potential. Nikiforov-Uvarov method is used in the calculations. We obtain the bound state energy eigenvalues and the corresponding eigenfunctions of spin-zero particles in terms of Jacobi polynomials. The eigenfunctions are physical and the energy eigenvalues are in good agreement with those results obtained by other methods for D=1 and 3 dimensions. Our results are valid for $q=1$ value when $l\neq 0$ and for any $q…
An approximate solution of the position-dependent mass Dirac equation with the Hulthen potential is obtained in $D$-dimensions within frame work of an exponential approximation of the centrifugal term. The relativistic energy spectrum is worked out using direct transformation method; the two-component spinor wavefunctions are obtained in terms of the Jacobi polynomials. Dependence of the energy levels on some parameters is discussed. The results obtained are in good agreement with previous works.
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…
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.
The approximate analytic bound state solutions of the Klein-Gordon equation with equal scalar and vector exponential-type potentials including the centrifugal potential term are obtained for any arbitrary orbital angular momentum number l and dimensional space D. The relativistic/non-relativistic energy spectrum equation and the corresponding unnormalized radial wave functions, in terms of the Jacobi polynomials P_{n}^{(α,β)}(z), where α>-1, β>-1 and z\in[-1,+1] or the generalized hypergeometric functions _{2}F_{1}(a,b;c;z), are found. The Nikiforov-Uvarov (NU) method is used in the solution. …
Approximate analytical bound state solutions of the radial Schrödinger equation are studied for a two-term diatomic molecular potential in terms of the hypergeometric functions for the cases where $q\geq1$ and $q=0$. The energy eigenvalues and the corresponding normalized wave functions of the Manning-Rosen potential, the 'standard' Hulthén potential and the generalized Morse potential are briefly studied as special cases. It is observed that our analytical results are the same with the ones obtained before.
An alternate formalism is developed to determine the energy eigenvalues of quantum mechanical systems, confined within a rigid impenetrable spherical box of radius $r_0$, in the framework of Wentzel-Kramers-Brillouin (WKB) approximation. Instead of considering the Langer correction for the centrifugal term, the approach adopted here is that of Hainz and Grabert : The centrifugal term is expanded perturbatively (in powers of $\hbar$), decomposing it into 2 terms -- the classical centrifugal potential and a quantum correction. Hainz and Grabert found that this method reproduced the exact energie…
We examine the bound state solutions of the Dirac equation under the spin and pseudospin symmetries for a new suggested combined potential, Hulten plus a class of Yukawa potential including a Coulomb-like tensor interaction. An improved scheme is employed to deal with the centrifugal (pseudo-centrifugal) 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. Modifiable of our results into some particular potential cases, u…
The generalized pseudospectral method is employed to calculate the bound states of Hulthén and Yukawa potentials in quantum mechanics, with special emphases on higher excited states and stronger couplings. Accurate energy eigenvalues, expectation values and radial probability densities are obtained through a nonuniform and optimal spatial discretization of the radial Schrödinger equation. Results accurate up to thirteen to fourteen significant figures are reported for all the 55 eigenstates of both these potentials with $n\leq$10 for arbitrary values of the screening parameters covering a wide…
In this paper, the bound state solution of the modified Klein-Fock-Gordon equation is obtained for the Hulthén plus ring-shaped lake potential by using the developed scheme to overcome the centrifugal part. The energy eigenvalues and corresponding radial and azimuthal wave functions are defined for any $l\neq0$ angular momentum case on the conditions that scalar potential is whether equal and nonequal to vector potential, the bound state solutions of the Klein-Fock-Gordon equation of the Hulthén plus ring-shaped like potential are obtained by Nikiforov-Uvarov (NU) and supersymmetric quantum me…
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…
Analytical solutions are presented for eigenvalues, eigenfunctions of {\color{red} D-dimensional Schrodinger equation having Eckart potential} within Nikiforov-Uvarov method. This uses a new, improved approximation for centrifugal term, from a combination of Greene-Aldrich and Pekeris approximations. Solutions are obtained in terms of hypergeometric functions. It facilitates an accurate representation in entire domain. Its validity is illustrated for energies in an arbitrary $\ell \neq 0$ quantum state. Results are compared for a chosen set of potential parameters in different dimensions. In s…
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…
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…
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 a…
The approximated energy eigenvalues and the corresponding eigenfunctions of the spherical Woods-Saxon effective potential in $D$ dimensions are obtained within the new improved quantization rule for all $l$-states. The Pekeris approximation is used to deal with the centrifugal term in the effective Woods-Saxon potential. The inter-dimensional degeneracies for various orbital quantum number $l$ and dimensional space $D$ are studied. The solutions for the Hulthén potential, the three-dimensional (D=3), the $% s$-wave ($l=0$) and the cases are briefly discussed.
By using an improved approximation scheme to deal with the centrifugal (pseudo-centrifugal) term, we solve the Dirac equation for the generalized Morse potential with arbitrary spin-orbit quantum number κ. In the presence of spin and pseudospin symmetry, the analytic bound state energy eigenvalues and the associated upper- and lower-spinor components of two Dirac particles are found by using the basic concepts of the Nikiforov-Uvarov method. We study the special cases when κ=\pm1 (l=l=0, s-wave), the non-relativistic limit and the limit when α becomes zero (Kratzer potential model). The presen…
A new approximation scheme to the centrifugal term is proposed to obtain the $l\neq 0$ bound-state solutions of the Schrödinger equation for an exponential-type potential in the framework of the hypergeometric method. The corresponding normalized wave functions are also found in terms of the Jacobi polynomials. To show the accuracy of the new proposed approximation scheme, we calculate the energy eigenvalues numerically for arbitrary quantum numbers $n$ and $l$ with two different values of the potential parameter $σ_{\text{0}}.$ Our numerical results are of high accuracy like the other numeric…
In this work, the analytical solutions of the $D$-dimensional Schrödinger equation are studied in great detail for the Wood-Saxon potential by taking advantage of the Pekeris approximation. Within a novel improved scheme to surmount centrifugal term, the energy eigenvalues and corresponding radial wave functions are found for any angular momentum case within the context of the Nikiforov-Uvarov (NU) and Supersymmetric quantum mechanics (SUSYQM) methods. In this way, based on these methods, the same expressions are obtained for the energy eigenvalues, and the expression of radial wave functions …
Accurate solution of the Schrödinger equation with Deng-Fan potential is presented by means of Nikiforov-Uvarov method. A modified Pekeris-type approximation is proposed for the centrifugal term, from a linear combination of the $r \to 0$ and $r \to r_e$ limits. It can potentially offer a series of approximations (depending on an adjustable parameter $λ$). The existing approximations in the literature can then be recovered in certain special cases. Its efficiency and feasibility is demonstrated by a critical comparison of eigenvalues produced at various $λ$'s for four molecules, \emph{viz.}, H…
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…
