Bound State and Ro-Vibrational Energies Eigenvalues of Selected Diatomic Molecules with a Class of Inversely Quadratic Yukawa Plus Hulthén Potential Model

Keywords: Schrödinger equation, Nikiforov-Uvarov method, Сlass of inversely quadratic plus Hulthén potential, Diatomic molecules, Bound state


 The Nikiforov-Uvarov approach is used in this study to solve the Schrödinger equation utilizing a class of inversely quadratic Yukawa plus Hulthén potential model with an approximation to the centrifugal term. The normalized wave function and energy eigenvalue equation were obtained. The numerical bound state for a few diatomic molecules (N2, O2, NO, and CO) for various rotational and vibrational quantum numbers was calculated using the energy equation and the related spectroscopic data. Our results show that, with no divergence between the s-wave and l-wave, the energy eigenvalues are very sensitive to the potential and diatomic molecule properties, suggesting that the approximation approach is appropriate for this set of potentials. The results are consistent with earlier studies in the literature, and we also found four special cases of this potential. 


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E.P. Inyang, E.O. Obisung, J. Amajama, E.S William, and I.B. Okon, “The Effect of Topological Defect on the Mass Spectra of Heavy and Heavy-Light Quarkonia,” Eurasian Physical Technical Journal, 19(4), 78-87 (2022).

E.S. William, E.P. Inyang, I.O. Akpan, J.A. Obu, A.N. Nwachukwu, and E.P. Inyang, “Ro-vibrational energies and expectation values of selected diatomic molecules via Varshni plus modified Kratzer potential model,” Indian Journal of Physics, 96, 34613476 (2022).

E.P. Inyang, E.P. Inyang, I.O. Akpan, J.E. Ntibi, and E.S. William, Masses and thermodynamic properties of a Quarkonium system. Canadian Journal of Physics. 99(11), 982-990 (2021).

I.B. Okon, C.A. Onate, R. Horchani, O.O. Popoola, E. Omugbe, E.S. William, U.S. Okorie, et al., “Thermomagnetic properties and its effects on Fisher entropy with Schioberg plus Manning Rosen potential (SPMRP) using Nikiforov Uvarov functional analysis (NUFA) and supersymmetric quantum mechanics (SUSYQM) methods,” Scientifc Reports, 13, 8193 (2023).

F. Ayedun, E.P. Inyang, E.A. Ibanga, and K.M. Lawal, “Analytical Solutions to The Schrödinger Equation with Collective Potential Models: Application to Quantum Information Theory,” East Eur. J. Phys. 4, 87-98 (2022).

E.S. William, S.C. Onye, A.N. Ikot, A.N. Nwachukwu, E.P. Inyang, I.B. Okon, I.O. Akpan and B. I. Ita, “Magnetic susceptibility and Magnetocaloric effect of Frost-Musulin potential subjected to Magnetic and Aharonov-Bohm(Flux)for CO and NO diatomic molecules“, Journal of Theoretical and Applied Physics, 17, 1-12 (2023).

C. Berkdermir, A. Berkdemir, and R. Sever, “Polynomial solutions of the Schrodinger equation for the generalized Woods-Saxon potential,” Phys. Rev. C, 72, 027001 (2008).

M. Abu-Shady, and E.P. Inyang, “The Fractional Schrödinger Equation with The Generalized Woods-Saxon Potential,” East European Journal of Physics, 1. 63-68 (2023).

S.M. Ikhdair, “The bound state solutions of the Manning-Rosen potential including an improved approximation to the orbital centrifugal term,” Phys. Scr. 83, 015010 (2011).

J. Lu, “Approximate spin and pseudospin solutions of the Dirac equation,” Physica Scripta, 72, 349 (2005).

R.L. Greene, and C. Aldrich, “Variational wave functions for a screened Coulomb potential,” Phys. Rev. A, 14, 2363 (1976).

C.S. Jia, T. Chen, and 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, 1621-1626 (2009).

E.L. Hill, “The Theory of Vector Spherical Harmonics,” Am. J. Phys. 22, 211-214 (1954).

C.L. Pekeris, “The Rotation-Vibration Coupling in Diatomic Molecules,” Phys. Rev. 45, 98 (1934).

B.H. Yazarloo, H. Hassanabadi, and S. Zarrinkamar, “Oscillator strengths based on the Mobius square potential under Schrodinger equation,” Eur. Phys. J. Plus, 127, 51 (2012).

S.H. Dong, W.C. Qiang, G.H. Sun, and V.B. Bezerra, “Analytical approximations to the l-wave solutions of the Schrödinger equation with the Eckart potential,” J. Phys. A, 40, 10535 (2007).

S.K. Nikiforov, and V.B. Uvarov, Special functions of Mathematical Physics, (Birkhauser, Basel, 1988).

E.S. William, E.P. Inyang, and E.A. Thompson, “Arbitrary ℓ -solutions of the Schrödinger equation interacting with Hulthén-Hellmann potential model,” Rev. Mex. Fis. 66, 730 (2020).

I.O. Akpan, E.P. Inyang, E.P. Inyang, and E.S. William, “Approximate solutions of the Schrödinger equation with Hulthen-Hellmann Potentials for a Quarkonium system,” Rev. Mex. Fis. 67, 482-490 (2021).

E.P. Inyang, E.O. Obisung, E.S. William, and I.B. Okon, “Non-Relativistic study of mass spectra and thermal properties of a quarkonium system with Eckart-Hellmann potential,” East European Journal of Physics, 3, 104-114 (2022).

E.P. Inyang, F.O. Faithpraise, J. Amajama, E.S. William, E.O. Obisung, and J.E. Ntibi, “Theoretical Investigation of Meson Spectrum using Exact Quantization Rule Technique,” East European Journal of Physics, 1, 53-62 (2023).

A.N. Ikot, U.S. Okorie, P.O. Amadi, C.O. Edet, G.J. Rampho, and R. Sever, “The Nikiforov-Uvarov Functional Analysis (NUFA) Method: A new approach for solving exponential-Type potentials,” Few-Body System, 62, 9 (2021).

C.O. Edet, S. Mahmoud, E.P. Inyang, N. Ali, S.A. Aljunid, R. Endut, A.N. Ikot, and M. Asjad, “Non-Relativistic Treatment of the 2D Electron System Interacting via Varshni-Shukla Potential Using the Asymptoptic Iteration Method,” Mathematics, 10, 2824 (2022).

A.N. Ikot, L.F. Obagboye, U.S. Okorie, E.P. Inyang, P.O. Amadi, and A. Abdel-Aty, Solutions of Schrodinger equation with generalized Cornell potential (GCP) and its applications to diatomic molecular systems in D-dimensions using Extended Nikiforov–Uvarov (ENU) formalism,” The European Physical Journal Plus, 137, 1370 (2022).

E.O. Omugbe, E. Osafile, E.P. Inyang, and A. Jahanshir, “Bound state solutions of the hyper-radial Klein-Gordon equation under the Deng-Fan potential by WKB and SWKB methods,” Physica Scripta, 96, 125408 (2021).

E.S. William, E.P. Inyang, J.E. Ntibi, J.A. Obu, and E.P. Inyang, “Solutions of the Non-relativistic Equation Interacting with the Varshni-Hellmann potential model with some selected Diatomic molecules,” Jordan Journal of Physics, 15, 179-193 (2022).

M. Abu-Shady, and E.P. Inyang, “Heavy-Light Meson masses in the Framework of Trigonometric Rosen-Morse Potential using the Generalized Fractional Derivative,” East Eur. J. Phys. 4, 80-87 (2022).

B.I. Ita, “Solutions of the Schrödinger equation with inversely quadratic Hellmann plus Mie-type potential using Nikiforov-Uvarov method,” International Journal of Recent Advances in Physics. 2(4), 25 (2013).

B.I. Ita, and A.I. Ikeuba, “Solutions to the Schrödinger Equation with Inversely Quadratic Yukawa Plus Inversely Quadratic Hellmann Potential Using Nikiforov-Uvarov Method,” Phys. Journal of Atomic and Molecular Physics, 582610 (2013).

E.P. Inyang, E.S. William, J.O. Obu, B.I. Ita, E.P. Inyang, and I.O. Akpan, “Energy spectra and expectation values of selected diatomic molecules through the solutions of Klein-Gordon equation with Eckart-Hellmann potential model,” Molecular Physics, 119(23), e1956615 (2021).

B.I. Ita, C.O. Ehi-Eromosele, A. Edobor-Osoh, and A.I. Ikeuba, “Solutions of the Schrödinger equation with inversely quadratic Hellmann plus inversely quadratic potential using Nikiforov-Uvarov method,” AIP Conf. Proc. 1629, 360 (2014).

K.J. Oyewumi, and E.A. Bangudu, “Isotropic harmonic oscillator plus inverse quadratic potential in N-dimensional spaces,” Arab. J. Sci. Eng. 28, 173-182 (2003).

R.H. Parmar, K.R. Purohit, and A.K. Rai, “Approximaate analytical solution of the extended Hulthen-Yukawa with inverse square and Coulombic term plus ring shape potential,” AIP Conf. Proc. 2220, 140071 (2020).

B.I. Ita, N. Nzeata-Ibe, T.O. Magu, and L. Hitler, “Bound-State Solutions of the Schrödinger Equation with Woods–Saxon Plus Attractive Inversely Quadratic Potential via Parametric Nikiforov-Uvarov Method,” Manila Journal of Science, 11, 58-67 (2018).

A. Maireche, “New Exact Non-Relativistic Energy Eigen Values for Modified Inversely Quadratic Hellmann Plus Inversely Quadratic Potential,” J. Nanosci. Curr. Res. 2, 1000115 (2017).

L. Hulthen, “Über die eigenlosunger der Schrödinger-Gleichung des deuterons,” Ark. Mat. Astron. Fys. A, 28, 5 (1942).

E.P. Inyang, E.S. William, E. Omugbe, and F. Ayedun, “The study of H2 and N2 Diatomic Molecules in Arbitrary Dimensions with Collective Potential Model,” Bulgarian Journal of Physics, 50, 1-15 (2023).

I.B. Okon, O. Popoola, and E.E. Ituen, “Bound state solution to Schrödinger equation with Hulthen plus exponential Coulombic potential with centrifugal potential barrier using parametric Nikiforov Uvarov method,” Intl. J. Rec. adv. Phys. 5, 5101 (2016).

E.P. Inyang, J. Ntibi, E.A. Ibanga, F. Ayedun, E.P. Inyang, and E.S. William, “Thermal Properties, Mass Spectra and Root Mean Square Radii of Heavy Quarkonium System with Class of Inversely Quadratic Yukawa Potential,” AIP Conference Proceedings 2679, 030003 (2023).

E.S. William, I.B. Okon, O.O. Ekerenam, I.O. Akpan, B.I. Ita, E.P. Inyang, I.P. Etim, and I.F. Umoh, “Analyzing the effects of magnetic and Aharonov-Bohm (AB) flux fields on the energy spectra and thermal properties of N2, NO, CO, and H2 diatomic molecules,” International Journal of Quantum Chemistry, (2022).

O. Bayrak, G. Kocak, and I. Boztosun, “Any l-state solutions of the Hulth´en potential by the asymptotic iteration method, J. Phys. A, 39, 11521 (2006).

I.B. Okon, and O. Popoola, “Bound-State solution of Schrodinger equation with Hulthen plus generalized exponential Coulomb potential using Nikiforov-Uvarov method,” Intl. J. Rec. Adv. Phys. 4(3), 1-12 (2015).

K.J. Oyewumi, and O.J. Oluwadare, “The scattering phase shifts of the Hulthen-type potential plus Yukawa potential,” Eur. Phys. J. Plus, 131, 295 (2016).

W.C. Qiang, Y. Gao, and R. Zhou, “Arbitrary l-state approximate solutions of the Hulthen potential through the exact quantization rule,” Cen. Eur. Phys. J. Phys. 6, 356 (2008).

S.M. Ikhdair, “An improved approximation scheme for the centrifugal term and the Hulth´en potential, The Eur. Phys. J. A, 39, 307 (2009).

L. Hitler, B.I. Ita, P.A. Isa, N. Nzeata-Ibe, I. Joseph, O. Ivan, and T.O. Magu, “Wkb Solutions for Inversely Quadratic Yukawa plus Inversely Quadratic Hellmann Potential,” World Journal of Applied Physics, 2, 4 (2017).

E.P. Inyang, E.S. William, J.E. Ntibi, J.A. Obu, P.C. Iwuji, and E.P. Inyang, “Approximate solutions of the Schrödinger equation with Hulthén plus screened Kratzer Potential using the Nikiforov-Uvarov – functional analysis (NUFA) method: an application to diatomic molecules,” Can. J. Phys. 100(10), 473 (2022).

E.P. Inyang, I.B. Okon, F.O. Faithpraise, E.S. William, P.O. Okoi, and E.A. Ibanga, “Quantum mechanical treatment of Shannon entropy measure and energy spectra of selected diatomic molecules with the modified Kratzer plus generalized inverse quadratic Yukawa potential model,” Journal of Theoretical and Applied Physics, 17(4), (2023).

E.P. Inyang, F. Ayedun, E.A. Ibanga, K.M. Lawal, I.B. Okon, E.S. William, O. Ekwevugbe, et al., “Analytical Solutions of the N-Dimensional Schrödinger equation with modified screened Kratzer plus Inversely Quadratic Yukawa potential and Thermodynamic Properties of selected Diatomic Molecules,” Results in Physics, 43, 106075 (2022).

E.P. Inyang, E.S. William, J.E. Ntibi, J.A. Obu, P.C. Iwuji, and E.P. Inyang, “Approximate solutions of the Schrodinger equation with Hulthen plus screened Kratzer potential using the Nikiforov-Uvarov-Functional analysis method: An Application to diatomic molecules,” Canadian Journal of Physics, 100(10), 463-473 (2022).

K.J. Oyewumi, and K.D. Sen, “Exact solutions of the Schrödinger equation for the pseudoharmonic potential: an application to some diatomic molecules,” J. Math. Chem. 50, 1039-1059 (2012).

How to Cite
Faithpraise, F. O., & Inyang, E. P. (2023). Bound State and Ro-Vibrational Energies Eigenvalues of Selected Diatomic Molecules with a Class of Inversely Quadratic Yukawa Plus Hulthén Potential Model. East European Journal of Physics, (3), 158-166.