Document Type : Original Article

**Author**

Applied Chemistry Department, Faculty of Gas and Petroleum, Yasouj UNIVERSITY

**Abstract**

Abstract

The Morse potential has important and significance rule to describe the diatomic molecule energy and structure. However there is no any analytical solution for Schrodinger equation with this potential without approximation, therefore other ways such as numerical, perturbation, variation and so on are taken to deal with this potential. In this work the the Nikiforov-Uvarov method is taken to obtain its energy eigenvalues and eigenfunctions. In the ground state the Schrodinger equation with this potential have exact solution but with arbitrary l-state the Morse potential with centrifugal term have no exact solution therefore it is solved analytically with use the Pekeris approximation. Here in this work we solved the Schrodinger in the space of D dimension and use the Nikiforove-Uvarov method which is based on solving the hyper geometric type second-order differential equations by means of the special orthogonal functions.

The Morse potential has important and significance rule to describe the diatomic molecule energy and structure. However there is no any analytical solution for Schrodinger equation with this potential without approximation, therefore other ways such as numerical, perturbation, variation and so on are taken to deal with this potential. In this work the the Nikiforov-Uvarov method is taken to obtain its energy eigenvalues and eigenfunctions. In the ground state the Schrodinger equation with this potential have exact solution but with arbitrary l-state the Morse potential with centrifugal term have no exact solution therefore it is solved analytically with use the Pekeris approximation. Here in this work we solved the Schrodinger in the space of D dimension and use the Nikiforove-Uvarov method which is based on solving the hyper geometric type second-order differential equations by means of the special orthogonal functions.

**Keywords**

[1] H. Ahmadov, S.M. Nagiyev, M. Qocayeva, K. Uzun, and V. Tarverdiyeva, Bound state solution of the Klein–Fock–Gordon

equation with the Hulthén plus a ring-shaped-like potential within SUSY quantum mechanics, International Journal of

Modern Physics A, 33(33) 2018, 1850203.

[2] H. Ahmadov, M. Qocayeva, and N. S. Huseynova, The bound state solutions of the D-dimensional Schrödinger equation

for the Hulthén potential within SUSY quantum mechanics, International Journal of Modern Physics E, 26(5) 2017,

1750028.

[3] A. Ahmadov, M. Naeem, M. Qocayeva, and V. Tarverdiyeva, Analytical bound-state solutions of the Schrödinger equation

for the Manning–Rosen plus Hulthén potential within SUSY quantum mechanics, International Journal of Modern Physics

A, 33(3) 2018, 1850021.

[4] C. Berkdemir, Application of the nikiforov-uvarov method in quantum mechanics, Theoretical Concepts of Quantum

Mechanics, 2012.

[5] C. Berkdemir, A. Berkdemir, and R. Sever, Systematical approach to the exact solution of the Dirac equation for a

deformed form of the Woods–Saxon potential, Journal of Physics A: Mathematical and General, 39(43) 2006, 13455.

[6] C. Berkdemir, A. Berkdemir, and J. Han, Bound state solutions of the Schrödinger equation for modified Kratzer’s

molecular potential, Chemical Physics Letters, 417(4-6) 2006, 326-329.

[7] V. Badalov, B. Baris, and K. Uzun, Bound states of the D-dimensional Schrödinger equation for the generalized Woods–

Saxon potential, Modern Physics Letters A, 34(14) 2019, 1950107.

[8] S.-H. Dong and J. Garcia-Ravelo, Exact solutions of the s-wave Schrödinger equation with Manning–Rosen potential,

Physica Scripta, 75(3) 2007, 307.

[9] B. Falaye and K. Oyewumi, Solutions of the Dirac equation with spin and pseudospin symmetry for trigonometric Scarf

potential in D-dimensions, arXiv preprint arXiv:1111.6501, 2011.

[10] B. Gönül and İ. Zorba, Supersymmetric solutions of non-central potentials, Physics Letters A, 269(2-3) 2000, 83-88.

[11] L.A. Girifalco, V.G. Weizer, Application of the Morse potential function to cubic metals, Physical Review, 114(3) 1959, 687.

equation with the Hulthén plus a ring-shaped-like potential within SUSY quantum mechanics, International Journal of

Modern Physics A, 33(33) 2018, 1850203.

[2] H. Ahmadov, M. Qocayeva, and N. S. Huseynova, The bound state solutions of the D-dimensional Schrödinger equation

for the Hulthén potential within SUSY quantum mechanics, International Journal of Modern Physics E, 26(5) 2017,

1750028.

[3] A. Ahmadov, M. Naeem, M. Qocayeva, and V. Tarverdiyeva, Analytical bound-state solutions of the Schrödinger equation

for the Manning–Rosen plus Hulthén potential within SUSY quantum mechanics, International Journal of Modern Physics

A, 33(3) 2018, 1850021.

[4] C. Berkdemir, Application of the nikiforov-uvarov method in quantum mechanics, Theoretical Concepts of Quantum

Mechanics, 2012.

[5] C. Berkdemir, A. Berkdemir, and R. Sever, Systematical approach to the exact solution of the Dirac equation for a

deformed form of the Woods–Saxon potential, Journal of Physics A: Mathematical and General, 39(43) 2006, 13455.

[6] C. Berkdemir, A. Berkdemir, and J. Han, Bound state solutions of the Schrödinger equation for modified Kratzer’s

molecular potential, Chemical Physics Letters, 417(4-6) 2006, 326-329.

[7] V. Badalov, B. Baris, and K. Uzun, Bound states of the D-dimensional Schrödinger equation for the generalized Woods–

Saxon potential, Modern Physics Letters A, 34(14) 2019, 1950107.

[8] S.-H. Dong and J. Garcia-Ravelo, Exact solutions of the s-wave Schrödinger equation with Manning–Rosen potential,

Physica Scripta, 75(3) 2007, 307.

[9] B. Falaye and K. Oyewumi, Solutions of the Dirac equation with spin and pseudospin symmetry for trigonometric Scarf

potential in D-dimensions, arXiv preprint arXiv:1111.6501, 2011.

[10] B. Gönül and İ. Zorba, Supersymmetric solutions of non-central potentials, Physics Letters A, 269(2-3) 2000, 83-88.

[11] L.A. Girifalco, V.G. Weizer, Application of the Morse potential function to cubic metals, Physical Review, 114(3) 1959, 687.

[12] B. Ita, H. Louis, T. Magu, N. Nzeata-Ibe, Bound State Solutions of the Klein Gordon Equation with Woods-Saxon Plus Attractive Inversely Quadratic Potential Via Parametric Nikiforov-Uvarov Method, World Scientific News, 74 2017, 280-287.

[13] C.-S. Jia, P. Guo, Y.-F. Diao, L.-Z. Yi, and X.-J. Xie, Solutions of Dirac equations with the Pöschl-Teller potential, The European Physical Journal A, 34(1) 2007, 41.

[14] R. Lincoln, K. Koliwad, and P. Ghate, Morse-potential evaluation of second-and third-order elastic constants of some cubic metals, Physical Review, 157(3) 1967, 463.

[15] S.M. Nagiyev, A. Ahmadov, Exact solution of the relativistic finite-difference equation for the Coulomb plus a ring-shaped-like potential, International Journal of Modern Physics A, 34(17) 2019,1950089.

[16] A.F. Nikiforov and V.B. Uvarov, Special functions of mathematical physics. Springer, 1988.

[17] M. Onyeaju, J. Idiodi, A. Ikot, M. Solaimani, H. Hassanabadi, Linear and nonlinear optical properties in spherical quantum dots: generalized Hulthén potential, Few-Body Systems, 57(9) 2016, 793-805.

[18] C. Pekeris, The rotation-vibration coupling in diatomic molecules, Physical Review, 45(2) 1934, 98.

[19] W.C. Qiang, S.H. Dong, Analytical approximations to the solutions of the Manning–Rosen potential with centrifugal term, Physics Letters A, 368(1-2) 2007, 13-17.

[20] G. Szegö, Orthogonal Polynomials, American Mathematical Society, New York, 1939.

[21] E. Yazdankish, Solving of the Schrodinger equation analytically with an approximated scheme of the Woods-Saxon potential by the systematical method of Nikiforov-Uvarov, International Journal of Modern Physics E, 29(6), 2020, 2050032.

[22] E. Yazdankish, Calculation of the energy eigenvalues of the Yukawa potential via variation principle, International Journal of Modern Physics E, 29(9) 2020, 2020.

[23] W. Yahya, K. Oyewumi, C. Akoshile, T. Ibrahim, Bound states of the relativistic dirac equation with equal scalar and vector Eckart potentials using the Nikiforov-Uvarov Method, Journal of Vectorial Relativity, 62410 2020, 2.

[24] L.Z. Yi, Y.F. Diao, J.-Y. Liu, C.-S. Jia, Bound states of the Klein–Gordon equation with vector and scalar Rosen–Morse-type potentials, Physics Letters A, 333(3-4) 2004, 212-217.

[13] C.-S. Jia, P. Guo, Y.-F. Diao, L.-Z. Yi, and X.-J. Xie, Solutions of Dirac equations with the Pöschl-Teller potential, The European Physical Journal A, 34(1) 2007, 41.

[14] R. Lincoln, K. Koliwad, and P. Ghate, Morse-potential evaluation of second-and third-order elastic constants of some cubic metals, Physical Review, 157(3) 1967, 463.

[15] S.M. Nagiyev, A. Ahmadov, Exact solution of the relativistic finite-difference equation for the Coulomb plus a ring-shaped-like potential, International Journal of Modern Physics A, 34(17) 2019,1950089.

[16] A.F. Nikiforov and V.B. Uvarov, Special functions of mathematical physics. Springer, 1988.

[17] M. Onyeaju, J. Idiodi, A. Ikot, M. Solaimani, H. Hassanabadi, Linear and nonlinear optical properties in spherical quantum dots: generalized Hulthén potential, Few-Body Systems, 57(9) 2016, 793-805.

[18] C. Pekeris, The rotation-vibration coupling in diatomic molecules, Physical Review, 45(2) 1934, 98.

[19] W.C. Qiang, S.H. Dong, Analytical approximations to the solutions of the Manning–Rosen potential with centrifugal term, Physics Letters A, 368(1-2) 2007, 13-17.

[20] G. Szegö, Orthogonal Polynomials, American Mathematical Society, New York, 1939.

[21] E. Yazdankish, Solving of the Schrodinger equation analytically with an approximated scheme of the Woods-Saxon potential by the systematical method of Nikiforov-Uvarov, International Journal of Modern Physics E, 29(6), 2020, 2050032.

[22] E. Yazdankish, Calculation of the energy eigenvalues of the Yukawa potential via variation principle, International Journal of Modern Physics E, 29(9) 2020, 2020.

[23] W. Yahya, K. Oyewumi, C. Akoshile, T. Ibrahim, Bound states of the relativistic dirac equation with equal scalar and vector Eckart potentials using the Nikiforov-Uvarov Method, Journal of Vectorial Relativity, 62410 2020, 2.

[24] L.Z. Yi, Y.F. Diao, J.-Y. Liu, C.-S. Jia, Bound states of the Klein–Gordon equation with vector and scalar Rosen–Morse-type potentials, Physics Letters A, 333(3-4) 2004, 212-217.

January 2021

Pages 61-70

**Receive Date:**08 January 2021**Revise Date:**17 January 2021**Accept Date:**18 January 2021**First Publish Date:**18 January 2021