Eigensolutions of the N-dimensional Schrödinger equation interacting with Varshni-Hulthén potential model


  • E. P. Inyang University of Calabar
  • E. S. William University of Calabar
  • J. A. Obu University of Calabar




N-Dimensional Schrödinger equation, Nikiforov-Uvarov method, eigenvalues, eigenfunction, Varshni- Hulthén potential.


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 excited states is illustrated graphically.


O. Aysel Approximate bound state solutions of the Hellmann plus Kratzer potential in N-dimensional space, GU J. Sci. 33 (2020) 3 780-793. https://doi.org/10.35378/gujs.672684

W.A. Yahya and K.J. Oyewumi, Thermodynamic properties and approximate solutions of the l-state Poschi-Teller-type potential, J. of the Association of Arab Univ. for basic and applied sciences 21(2016) 1 53-58.http://doi.org/10.1016/j.jaubas.2015.04.001

K. J. Oyewumi, O. J. Oluwadare, The scattering phase shifts of the Hulth´en-type potential plus Yukawa potential, Eur. Phys. J. Plus, 131 (2016) 295. https://doi.org/10.1140/epjp/i2016-16295-y

H. Louis, B. I. Ita, and N. I. Nzeata, Approximate solution of the Schrödinger equation with Manning-Rosen plus Hellmann potential and its thermodynamic properties using the proper quantization rule. The European Physical Journal Plus, 134 (2019) 7. https://doi.org/10.1140/epjp/i2019-12835-3

J. Lu, Analytic Quantum Mechanics of Diatomic Molecules with Empirical Potentials, Phys. Scr. 72 (2005) 349. https://doi.org/10.1238/Physica.Regular.072a00349

R. L. Greene, C. Aldrich, Variational wave functions for a screened Coulomb potential, Phys. Rev. A, 14 (1976) 6, 2363–2366. https://doi.org/10.1103/physreva.14.2363

C.S. Jia, T. Chen, L.-G. Cui, Approximate analytical solutions of the Dirac equation with the generalized Pöschl–Teller potential including the pseudo-centrifugal term, Phys. Lett. A 373, (2009) 1621. https://doi.org/10.1016/j.physleta.2009.03.006

E.H. Hill, The Theory of Vector Spherical Harmonics, Am. J. Phys. 22 (1954) 211. https://doi.org/10.1119/1.1933682

C. L. Pekeris. The Rotation-Vibration Coupling in Diatomic Molecules, Phys. Rev. 45 (1934) 98. https://doi.org/10.1103/PhysRev.45.98

B.H. Yazarloo, H. Hassanabadi, S. Zarinkarmar, Oscillator strengths based on the Möbius square potential under Schrödinger equation, Eur. Phys. J. Plus 127, (2012) 51. https://doi.org/10.1140/epjp/i2012-12051-9

J. E. Ntibi, E. P. Inyang, E. P. Inyang and E. S. William, Relativistic treatment of D-dimensional Klein-Gordon equation with Yukawa potential, Intl. J. Innov sci, engr. Tech. 7 (2020) 11. https//doi/10.13140/RG.2.2.32473.34406

K.J. Oyewumi, F.O. Akinpelu, and A.D. Agboola, Exactly Complete Solutions of the Pseudoharmonic Potential in N-Dimensions, Int J. Theor. Phys. 47 (2008) 1039. https://doi.org/10.1007/s10773-007-9532-x

B. Gönül and M. Koçak Explicit solutions for N -dimensional Schrödinger equations with position-dependent mass J. Maths Phys. 47, 102101 (2006). https://doi.org/10.1063/1.2354333

K.J. Oyewumi, Analytical solutions of the kratzer-fues potential in an arbitrary number of dimensions, Found. Phys. Lett. 18 (2005) 75. https://doi.org/10.1007/s10702-005-2481-9

S.M. Ikhdair and R. Sever, exact bound states of the d-dimensional Klein–Gordon equation with equal scalar and vector ring-shaped pseudoharmonic potential, Int. J. Mod. Phys. C 19 (2008) 221. https://doi.org/10.1142/S0129183108012923

Y.P. Varshni, Comparative Study of Potential Energy Functions for Diatomic Molecules, Rev. Mod. Phys. 29 (1957) 682. https://doi.org/10.1103/RevModPhys.29.664

C. O. Edet, P.O. Okoi, Any l-state solutions of the Schrodinger equation for q-deformed Hulthen plus generalized inverse quadratic Yukawa potential in arbitrary dimensions, Rev Mex Fis, 65 (2019) 333-344. https://doi.org/10.31349/RevMexFis.65.333

I. B. Okon, O. Popoola, E. E. Ituen, Bound state solution to Schrodinger equation with Hulthen plus exponential Coulombic potential with centrifugal potential barrier using parametric NikiforovUvarov method, Intl J. Rec. adv. Phys. ,5 (2016) 2. https://doi.org/10.14810/ijrap.2016.5101 1

O. Bayrak,and I Boztosun, Bound state solutions of the Hulthén potential by using the asymptotic iteration method. Physica Scripta, 76 (2007) 1, 92–96. https://doi.org/10.1088/0031-8949/76/1/016

Tazimi and A. Ghasempour, Bound State Solutions of Three-Dimensional Klein-Gordon Equation for Two Model Potentials by NU Method, N. Advances in High Energy Physics (2020) 2541837. https://doi.org/10.1155/2020/2541837

S. M. Ikhdair and J. Abu-Hasna, Quantization rule solution to the Hulthén potential in arbitrary dimension with a new approximate scheme for the centrifugal term. Physica Scripta, 83 J. (2011) 2, 025002. https://doi.org/10.1088/0031-8949/83/02/025002

U.S. Okorie, A. N. Ikot, P.O. Amadi, A. T. Ngiangia and E. E. Ibekwe Approximate solutions of the Schrodinger equation with energy- dependent screened Coulomb potential in D-dimensions. Ecletica Quimica J. 45 (2020) 4. https://doi.org/10.26850/1678-4618eqj.v45.4.2020.p40-56

C. O. Edet, P.O. Okoi, Any l-state solutions of the Schrodinger equation for q-deformed Hulthen plus generalized inverse quadratic Yukawa potential in arbitrary dimensions, Rev Mex Fis, 65 (2019) 333-344. https://doi.org/10.31349/RevMexFis.65.333

C. O. Edet, U. S. Okorie, A. T. Ngiangia and A. N. Ikot, Bound state solutions of the Schrodinger equation for the modified Kratzer potential plus screened Coulomb potential,l Indian J. Phys. 19 (2019) 01477. https://doi.org/10.1007/s12648-019-01477-9

C. O. Edet, K. O. Okorie, H. Louis, and N. A. Nzeata-Ibe, Any l-state solutions of the Schrodinger equation interacting with Hellmann–Kratzer potential model, Indian J. Phys. 19 (2019) 01467. https://doi.org/10.1007/s12648-019-01467-x

E. S. William, E. P. Inyang, E. A. Thompson, Arbitrary -solutions of the Schrödinger equation interacting with Hulthén-Hellmann potential model, Rev Mex Fis, 66 (2020) 6. https://doi.org/10.31349/RevMexFis.66.730

E. P. Inyang E. P. Inyang J. E. Ntibi, E. E. Ibekwe, and E. S. William, Approximate solutions of D-dimensional Klein-Gordon equation with Yukawa potential via Nikiforov-Uvarov method, Indian J. Phys. (2020) 00097R2. https://doi.org/10.13140/RG.2.2.32473.34406

Onate CA, Ojonubah JO. Relativistic and nonrelativistic solutions of the generalized Poschl-Teller and hyperbolical potentials with some thermodynamic properties, Intl J. Mod Phys E 24 (2015) 3 1550020. https://doi.org/10.1142/S0218301315500202

O. Ebomwonyi, C. A. Onate O. E. Odeyemi, Application of Formula Method for Bound State Problems in Schrödinger Equation. J. Appl. Sci. Environ Manage. 23(2019) 2. https://doi.org/10.4314/jasem.v23i2.19.

W.C. Qiang, Y. Gao, R.S. Zhou, Arbitrary l-state approximate solutions of the Hulthén potential through the exact quantization rule, Cent. Eur. J. Phys. 6 (2008) 356. https://doi.org/10.2478/s11534-008-0041-1

O. Bayrak, G. Kocak, I. Boztosun, Any l-state solutions of the Hulthén potential by the asymptotic iteration method, J. Phys. A: Math Gen. 39 (2006) 11521. https://doi.org/10.1088/0305-4470/39/37/012

S.M. Ikhdair, An improved approximation scheme for the centrifugal term and the Hulthén potential, Eur. Phys. J. A 39 (2009) 307. https://doi.org/10.1140/epja/i2008-10715-2

A. F. Nikiforov, V. B. Uvarov, Special Functions of Mathematical Physics, (Birkhauser, Bassel, 1988)




How to Cite

E. P. Inyang, E. S. William, and J. A. Obu, “Eigensolutions of the N-dimensional Schrödinger equation interacting with Varshni-Hulthén potential model”, Rev. Mex. Fís., vol. 67, no. 2 Mar-Apr, pp. 193–205, Jul. 2021.



04 Atomic and Molecular Physics