Approximate solutions of Schrodinger equation with combination of Kratzer and modified Eckart potential

Document Type : Original Article

Authors

1 Department of Physics, Federal University of Technology, Owerri, Nigeria

2 Department of Physics/Electronics, Federal Polytechnic, Nekede, Nigeria

3 Department of Physics, Abia State University, Uturu, Nigeria

Abstract

We solve the Schrodinger equation for the combination of Kratzer and Modified Eckart potential by using an approximation to the centrifugal term. Analytical expressions of the energy and the corresponding eigenfunctions have been obtained by using the parametric Nikiforov-Uvarov method. Special cases of the potential are discussed. In addition, numerical results of the energy are computed. Furthermore, it is reported that the energy E and the quantum numbers n and l are inversely proportional to each other.

Keywords


[1] H.I. Ahmadov, E.A. Dadashov, N.Sh. Huseynova, V.H. Badalov, Generalized tanh-shaped hyperbolic potential: bound state solution of Schrödinger equation, Eur. Phys. J. Plus 136 (2) 2021, 1-12.
[2] H.I. Ahmadov, M.V. Qocayeva, N.Sh. Huseynova, The bound state solutions of the D-dimensional Schrödinger equation for the Hulthén potential within SUSY quantum mechanics, Int. J. Mod. Phys E, 26 (5) 2017, 1750028.
[3] H.I. Ahmadov, C. Aydin, N.SH. Huseynova, O. Uzun, Analytical solutions of the Schrödinger equation with the manning–rosen potential plus a ring-shaped-like potential, Int. J. Mod. Phys. E 22 (10) 2013, 1350072.
[4] A.I. Ahmadov, M. Demirci, S.M. Aslanova, M.F. Mustamin, Arbitrary -state solutions of the Klein-Gordon equation with the Manning-Rosen plus a Class of Yukawa potentials, Phys. Lett. A, 384 (12) 2020, 126372.
[5] A.I. Ahmadov, S.M. Aslanova, M.Sh. Orujova, S.V. Badalov, S.H. Dong, Approximate bound state solutions of the Klein-Gordon equation with the linear combination of Hulthén and Yukawa potentials, Phys. Lett. A, 383 (24) 2019, 3010-317.
[7] D. Agboola, Schrödinger Equation with Hulthén Potential Plus Ring-Shaped Potential, Comm. Theor. Phys. 55(6) 2011, 972.
[8] A.I. Ahmadov, V.H. Badalov, H.I .Ahmadov, Analytical solutions of the Schroedinger equation with the Woods–Saxon potential for arbitrary l state, Int. J. Mod. Phys E, 18 (3) 2009, 631-641.
[9] H.I. Ahmadov, Sh.I. Jafarzade, M.V. Qocayeva, Analytical solutions of the Schrödinger equation for the Hulthén potential within SUSY quantum mechanics, Int. J. Mod. Phys. A 30 (32) 2015, 1550193.
[10] P. Boonserm, M. Visser, Quasi-normal frequencies: key analytical results, 2011, arxiv:1005.4483v3.
[11] J. Cai, P. Cai, A. Inomata, Path-integral treatment of the Hulthén potential, Phys. Rev. A 34 (6) 1986, 4621.
[12] F. Cooper, A. Kahare, U. Sukhatme, Supersymmetry and Quantum Mechanics, Phys. Rept. 251(5-6) 1995, 267-385.
[13] A. Cimas, M. Aschi, C. Barrientos, V.M. Rayon, J.A. Sardo, A. Largo, Computational study on the kinetics of the reaction of N(4S) with CH 2F, Chem. Phys. Lett. 374 (5) 2003, 594-600.
[14] S.H. Dong, Factorization Method in Quantum Mechanics, Springer, Netherlands, 2007, ISBN-13 978-1-4020-5795-3 (HB) ISBN-13 978-1-4020-5796-0 (e-book).
[15] S.H. Dong, W.C. Qiang, G.H. Sun, V.B. Bezerra, Analytical approximations to the l-wave solutions of the Schrödinger equation with the Eckart potential, J. Phys. A: Math. Theor. 40 (34) 2007, 10535
[16] S.H. Dong, XY. GU, Arbitrary l state solutions of the Schrödinger equation with the Deng-Fan molecular potential, J. Phys. Conf. Series 96(1) 2008, 012109.
[17] M. Eshghi, H. Mehraban, S.M. Ikhdair, The relativistic bound states of a non-central potential, Pram. J. Phys. 88(4) 2017, 1-10.
[18] C.O. Edet, U. S. Okorie, A. T. Ngiangia, A. N. Ikot, Bound state solutions of the Schrodinger equation for the modified Kratzer potential plus screened Coulomb potential, Ind. J. Phys. 94(4) 2019, 425-433.
[19] C.O. Edet, K. O. Okorie, H. Louis, N. A. Nzeata-Ibe, Any l-state solutions of the Schrodinger equation interacting with Hellmann–Kratzer potential model, Ind. J. Phys. 94 (2) 2019, 243-251.
[20] B.J. Falaye, S.M. Ikhdair, M. Hamzavi, Formula Method for Bound State Problems, Few-Body Syst., 56(1) 2015, 63-78.
[21]  B.J. Falaye, Any ℓ-state solutions of the Eckart potential via asymptotic iteration method, Cent. Eur. J. Phys. 10(4) 2012, 960-965.
[22] B.J. Falaye, Any ℓ-state solutions of the Eckart potential via asymptotic iteration method, Cent. Eur. J. Phys. 10(4) 2012, 960-965.
[23] J. Gao, M.C. Zhang, Analytical Solutions to the D-Dimensional Schrödinger Equation with the Eckart Potential, Chin. Phys. Lett. 33(1) 2016, 010303.
[24] R.L. Greene, C. Aldrich, Variational Wave Functions for a Screened Coulomb Potential, Phys. Rev. A 14(6) 1976, 2363-2366.
[25] H. Hassanabadi, B. H.Yazarloo, S. Zarrinkamar, Bound states and the oscillator strengths for the Klein-Gordon equation under Möbius square potential, Turk. J. Phys. 37(2) 2013, 268-274.
[26] M. Hamzavi, A.A. Rajabi, H. Hassanabadi, The rotation–vibration spectrum of diatomic molecules with the Tietz–Hua rotating oscillator and approximation scheme to the centrifugal term, Mol. Phys. 110(7) 2012, 389-393.
[27] E.P. Inyang, I.O. Akpan, J.E. Ntibi, E.S. William, Analytical Solutions of the Schrödinger Equation with Class of Yukawa Potential for a Quarkonium System Via Series Expansion Method, Eur. J. Appl. Phys. 2(6) 2020.
 [28] B.I. Ita, H. Louis, O.U. Akakuru, N.A. Nzeata-Ibe, A.I. Ikeuba, T.O. Magu, P.I. Amos, C.O. Edet, Approximate Solution to the Schrödinger Equation with Manning-Rosen plus a Class of Yukawa Potential via WKBJ Approximation Method, Bulg. J. Phys. 45 2018, 323.
[29] E.P. Inyang, E.S. William, J.A. Obu. Eigensolutions of the N-dimensional Schrödinger equation ìnteracting with Varshni-Hulthen potential model, Rev. Mexi. Fisi. 67 2021, 193.
[30] AN. Ikot, B.H. Yazarloo, S. Zarrinkamar, H. Hassanabadi, Symmetry limits of (D+1)-dimensional Dirac equation with Möbius square potential, Eur. Phys. J. Plus, 129 (79) 2014.
[31] L.D. Landau, E.M. Lifshitz, Quantum Mechanics-Non-Relativistic Theory, Pergamon, Oxford, 1977.
[32] M. G. Miranda, G.H. Sun, S.H. Dong, The solution of the second pöschlteller like potential by nikiforov-uvarov method, Intl. J. Mod. Phys. E 19(1) 2010, 123-129.
[33] A.F. Nikiforov, V.B. Uvarov, Special Functions of Mathematical Physics, Birkhauser, Basel, 1988.
[35] C.A. Onate, M.C. Onyeaju, A.N. Ikot, J.O. Ojonubah, Analytical solutions of the Klein–Gordon equation with a combined potential, Chin. J. Phys. 54 (5) 2016, 820-829.
[36] C.P. Onyenegecha, C.A. Onate, O.K. Echendu, A.A. Ibe, H. Hassanabadi, Solutions of Schrodinger equation for the modified Mobius square plus Kratzer potential, Eur. Phys. J. Plus 135 (3) 2020, 1-9.
[37] C.P. Onyenegecha, A.I. Opara, I.J. Njoku, S.C. Udensi, U.M. Ukewuihe, C.J. Okereke, A. Omame, Analytical solutions of D-dimensional Klein–Gordon equation with modified Mobius squared potential, Res. Phys. 25, 2021, 104144
[38] W.C. Qiang, S. H. Dong, Analytical Approximation to the Solutions of the Manning-Rosen Potential with Centrifugal Term, Phys. Letts. A 368 (1-2) 2007, 13-17.
[39] L.I. Schiff, Quantum Mechanics, 3rd edition, McGraw-Hill, New York, 1955, 306.
[40] A. Suparmi, C. Cari, S. Faniandari, Eigen solutions of the Schrodinger equation with variable mass under the influence of the linear combination of modified Woods-Saxon and Eckart potentials in toroidal coordinate, Mol. Phys. 118 (3) 2020, 1-9.
[41] F. Taskin, G. Kocak, Approximate solutions of Schrödinger equation for Eckart potential with centrifugal term, Chin. Phys. B, 19(9) 2010, 090314.
[42] C. Tezcan, R. Sever, A General Approach for the Exact Solution of the Schrödinger Equation, Int. J. Theor. Phys. 48 2009, 337-350.
[43] F. Taskin, G. Kocak, Approximate solutions of Schrödinger equation for Eckart potential with centrifugal term, Chin. Phys. B, 19(9) 2010, 090314.
[44] J.J. Weiss, Mechanism of Proton Transfer in Acid-Base Reactions, J. Chem. Phys. 41(4) 1964, 1120-1124.
[45] B.H. Yazarloo, H. Hassanabadi, S. Zarrinkamar, Oscillator strengths based on the Möbius square potential under Schrödinger equation, European Physical Journal Plus, 127(5) 2012, 1-11.
[46] S. Zarrinkamar, H. Panahi, M. Rezaci, M. Baradaran, Dirac Equation for Scalar, Vector and Tensor Generalized Cornell Interaction. Few-Body Systems, 57(2) 2016, 109-120.
Volume 3, Issue 2
May 2022
Pages 48-61
  • Receive Date: 08 December 2021
  • Revise Date: 04 February 2022
  • Accept Date: 08 February 2022
  • First Publish Date: 01 May 2022