Abstract
In this work, we proposed ultra generalized exponential–hyperbolic potential (UGEHP) and derived various well known exponential–hyperbolic type potentials by setting parameters in UGEHP and using approximation suggested by Greene–Aldrich. The bound state solutions of the multi (D)-dimensional Schrödinger equation for UGEHP have been presented using the parametric Nikiforov–Uvarov method. The approximate analytical bound state energy eigenvalues and the corresponding un-normalized eigenfunctions expressed in terms of hypergeometric functions were obtained. We also investigated the rotational vibrational (RV) partition function from the eigenvalue of UGEHP. By the setting parameters, we obtained eigenvalue spectrum and RV partition function for the screened cosine Kratzer potential, screened Kratzer potential, attractive radial potential, quadratic exponential-type potential, Manning Rosen with class of Yukawa potential and Yukawa potential, class of Yukawa potential, mixed class of Yukawa potential, quantum interaction potential or Hulthën–Yukawa inversely quadratic potential, Hulthën plus inversely quadratic exponential Mie-type potential and Hulthën plus exponential Coulombic potential with centrifugal potential barrier. We studied the behavior of energy eigenvalues and RV partition function for the UGEHP. We computed and tabulated numerical results for \(CO, NO, I_2\), HCl and LiH diatomic molecules and compared with numerical results available in literature for same diatomic molecules.
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References
S. Dong, G.-H. Sun, S.H. Dong, Arbitrary l-wave solutions of the Schröodinger equation for the screen Coulomb potential. Int. J. Mod. Phys. E 22(6), 1350036 (2013)
A. Ghoshal, Y.K. Ho, Ground states of helium in exponential-cosine-screened Coulomb potentials. J. Phys. B 42, 075002 (2009)
A. Soylu, Plasma screening effects on the energies of hydrogen atom. Phys. Plasmas 19, 072701 (2012)
A.N. Ikot, W. Azogor, U.S. Okorie, F.E. Bazuaye, M.C. Onjeaju, C.A. Onate, E.O. Chukwuoch, Exact and Poisson summation thermodynamic properties for diatomic molecules with shifted Tietz potential. Indian J. Phys. (2019). https://doi.org/10.1007/s12648-019-01375-0
B.I. Ita, P. Ekuri, Bound state solutions of Schrödinger equation for a more general exponential screened Coulomb potential via Nikiforov–Uvarov method. Ecleticalquimica 35, 103–107 (2010)
S.M. Ikhdair, R. Sever, Bound states of a more general exponential screened Coulomb potential. J. Math. Chem. 41, 343–353 (2006)
B.I. Ita et al., Bound state solutions of Schrödinger equation for a more general exponential screened Coulomb potential plus Yukawa (MGESCY) potential using Nikiforov–Uvarov method. J. Quantum Inf. Sci. 8, 24–45 (2018)
C.C. Gerry, B.T. James, A large-N phase integral approximation for Coulomb type system using SO (2, 1) coherent states. J. Phys. A 19 (1986)
A. Soylu, O. Bayrak, I. Boztosun, Exact solutions of the Klein–Gordon equation with equal scalar and vector Rosen–Morse type potential. Chin. Phys. Lett. 25, 2754–2757 (2008)
B.I. Ita, A.I. Ikeuba, Solutions to the Schrödinger equation with inversely quadratic Yukawa plus inversely quadratic Hellmann potential using Nikiforov–Uvarov method. J. Atomic Mol. Phys. Article ID 582610 (2013)
A.K. Roy, Critical parameters and spherical confinment of H atom in screened Coulomb potential. Int. J. Quantum Chem. 116, 953–960 (2016)
C.A. Onate, J.O. Ojonubah, Eigensolutions of the Schrödinger equation with a class of Yukawa potential via Supersymmetric approach. J. Theor. Appl. Phys. 10, 21–26 (2016)
A.N. Ikot, H.P. Obon, T.M. Abbey, J.D. Olisa, Approximate analytical solutions of the Klein–Gordon equation with an exponential-type potential. Sae Mulli New Phys. 65, 825–836 (2015)
K.R. Purohit, R.H. Parmar, A.K. Rai, Eigensolution and various properties of the screened cosine Kratzer potential in D dimensions via relativistic and non-relativistic treatment. Eur. Phys. J. Plus. 135, 286 (2020)
A.N. Ikot, U.S. Okorie, R. Sever, G.J. Rampho, Eigensolution, expectation values and thermodynamic properties of the screened Kratzer potential. Eur. Phys. J. Plus. 134, 386 (2019)
I.B. Okon, O. Popoola, C.N. Isonguyo, A.D. Antia, Solutions of Schrödinger and Klein-Gordon equations with Hulthen plus inversely quadratic exponential Mie-Type potential. Phys. Sci. Int. J. 19(2), 1–27 (2018)
U.A. Deta, A. Suparmi, C. Cari, Approximate solution of Schrödinger equation in D-Dimensions for scarf hyperbolic plus non-central Poschl–Teller potential using Nikiforov–Uvarov method. J. Phys. Conf. Ser. 539, 012018 (2014)
A.N. Ikot, E.J. Ibanga, O.A. Awoga, L.E. Akpabio, A.D. Antia, Solutions of Schrödinger equation with generalized inverted hyperbolic potential. J. Mod. Phys. 3, 1849–1855 (2011)
A.D. Antia, Analytical solutions of Schrödinger equation with generalized hyperbolic potential using Nikiforov–Uvarov method. Afr. Rev. Phys. 6, 0026 (2011)
A.D. Antia, Solutions of nonrelativistic Schrödinger equation with scarf II plus Rosen–Morse II potential via ansaltz method. Am. J. Phys. Chem. 4(5), 38 (2015)
A.S. Halberg, Quasi-exact solvability of a hyperbolic intermolecular potential induced by an effectivemass step. Int. J. Math. Math. Sci. Article ID 358198 (2011)
S.M. Ikhdair, Bound state of the Klein–Gordon for exponential-type potential in D-dimensions. J. Quantum Inf. Sci. 1, 73–86 (2011)
F.-K. Wen, Z.-Y. Yang, C. Liu, W.-L. Yang, Y.-Z. Zhang, Exact polynomial solutions of Schrödinger equation with various hyperbolic potentials. Commun. Theor. Phys. 62(2), 153 (2014)
A. Onate, J.O. Ojonubah, Relativistic and nonrelativistic solutions of the generalized Pöschl–Teller and hyperbolical potentials with some thermodynamic properties. Int. J. Mod. Phys. E 24(03), 1550020 (2015)
T. Das, Analytical approximate bound state solution of Schrödinger equation in D-dimensions with a new mixed class of potential for arbitrary ‘ state via asymptotic iteration method. Chin. J. Phys. 24(5), 850–858 (2016)
R.L. Greene, C. Aldrich, Variational wave functions for a screened Coulomb potential. Phys. Rev. A 14, 2363 (1976)
C.O. Edet, U.S. Okorie, A.T. Ngiangia, A.N. Ikot, Bound state solutions of the Schrödinger equation for the modified Kratzer potential plus screened Coulomb potential. Indian J. Phys. (2019). https://doi.org/10.1007/s12648-019-01477-9
A. Kratzer, Z. Phys. 3, 289 (1920)
C.A. Onate, O. Adebimpe, A.F. Lukman, I.J. Adama, E.O. Davids, Approximate eigensolutions of the attractive potential via parametric Nikiforov-Uvarov method. Heliyon 4, e00977 (2018)
A.N. Ikot, O.A. Awoga, H. Hassanabadi, E. Maghsoodi, Analytical approximate solution of Schrödinger equation in D Dimensions with quadratic exponential-type potential for arbitrary state. Commun. Theor. Phys.61(4) (2014)
A. Arda, R. Sever, C. Tezcan, Approximate pseudospin and spin solutions of the Dirac equation for a class of exponential potentials. Chin. J. Phys. 48, 27 (2010)
X. Zou, L.Z. Yi, C.S. Jia, Bound states of the Dirac equation with vector and scalar Eckart potentials. Phys. Lett. A 346, 54 (2005)
M. Eshghi, M. Hamzavi, Spin symmetry in Dirac-attractive radial problem and tensor potential. Commun. Theor. Phys. 57, 355–360 (2012)
B.I. Ita, H. Louis, T.O. Magu, N.A. Nzeata-Ibe, Bound state solutions of the Schrödinger equation with Manning-Rosen plus a class of Yukawa potential using Pekeris-like approximation of the Coulomb term and parametric Nikiforov–Uvarov method. World Sci. News 70, 367–385 (2017)
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. 45, 323–333 (2018)
I.B. Okon, O. Popoola, C.N. Isonguyo, Approximate solutions of Schrödinger equation with some diatomic molecular interactions using Nikiforov–Uvarov method. Advances in High Energy Physics 9671816 (2017)
A. Murat, The energy eigenvalues of the exponential cosine screened Coulomb potential with magnetic field. Bitlis Eren Univ. J. Sci. Technol. 3(2), 32–38 (2013)
S.M. Ikhdair, R. Sever, Bound state energies for the exponential cosine screened Coulomb potential. Z. Phys. D 28, 1 (1993)
S.M. Ikhdair, R. Sever, Bound energy for the exponential-cosine-screened Coulomb potential. J. Math. Chem. 41, 329–341 (2007)
M.K. Bahar, An alternative approach to solutions of the MGECSC potential in presence of external electricfield. Advances in High Energy Physics. 807417 (2015)
I.B. Okon, O. Popoola, E.E. Ituen, Bound state solution to Schrödinger equation with Hulthen plus exponential Coulombic potential with centrifugal potential barrier using parametric Nikiforov-Uvaarov method. Int. J. Rec. Adv. Phys. (IJRAP) 5(2), 1–15 (2016)
S.H. Dong, W.C. Qiang, G.H. Sun, V.R. Bezerra, Analytical approximations to the \(\ell \) wave solutions of the Schrödinger equation with the Eckart potential. J. Phys. A Math. Theor. 40, 10535–10540 (2007)
K.J. Oyewunmi, B.J. Falaye, C.A. Onate, O.J. Oluwadare, W.A. Yahya, Thermodynamic properties and the approximate solutions of the (1993) equation with the shiftedDeng Fan potential model. Mol. Phys. 112(1), 127–141 (2014)
C.A. Onate, C.A. Ojonubah, Relativistic and nonrelativistic solutions of the generalized Pöschl–Teller and hyperbolical potentials with some thermodynamic properties. Int. J. Mod. Phys. E. 24 (2015)
U.S. Okorie, A.N. Ikot, M.C. Onyeaju, E.O. Chukwuocha, Bound state solutions of Schrödinger equation with modified Mobius square potential (MMSP) and its thermodynamic properties. J. Mol. Mod. 24, 289 (2018)
A.N. Ikot, Thermodynamical properties of diatomic molecule with general molecular potential. Pramana. J. Phys. 90, 22 (2018). https://doi.org/10.1007/s12043-1007-151-0
A.N. Ikot, U.S. Okorie, R. Sever, G.J. Rampho, Eigensolution, expectation values and thermodynamical properties of the screened Kratezar potential. Eur. Phys. J. Plus 134, 386 (2019)
A.D. Antia, I.E. Essien, E.B. Umoren, C.C. Eze, Approximate solutions of the nonrelativistic Schrödinger equation with inversely quadratic Yukawa plus Mobius square potential via parametric Nikiforov–Uvarov method. In: Advances in Physics Theories and Applications. vol. 44, pp. 1–13 (2015)
A.D. Antia, I.O. Akpan, A.O. Akankpo, Relativistic treatment of spinless particles subject to modified Scarf II potential. Int. J. High Energy Phys. 2(4), 50–55 (2015)
H. Hassanabadi, S. Zarrinkamar, A.A. Rajabi, Exact solutions of D-dimensional Schrödinger equation for energy-dependent potential by Nikiforov–Uvarov method. Commun. Theor. Phys. 55(4), 541–544 (2011)
W.C. Qiang, S.H. Dong, Analytical approximations to the solutions of the Manning–Rosen potential with centrifugal term. Phys. Lett. A 368, 13–17 (2007)
A.N. Ikot, Z.E. Maghsoodi, S. Zarrinkamar, H. Hassanabadi, Solutions of Dirac equation in the presence o fmodified Tietz and modified Pöschl–Teller potentials plus a Coulomb-like tensor interaction using SUSYQM. Few-Body Syst. 54(11), 2027–2040 (2013)
F. Cooper, A. Khare, U. Sukhatme, Supersymmetry and quantum mechanics. Phys. Rep. 251, 267–365 (1995)
R.H. Parmar, Construction of solvable non-central potential using vector superpotential: a new approach. Indian J. Phys. 93(9), 1163–1170 (2019)
S.M. Ikhdair, R. Sever, Exact solutionsof the modified Kratzer potential plus ring shape potential in the D dimensional Schrödinger equation by the Nikiforov–Uvarov method. Int. J. Mod. Phys. C 19(2), 221–235 (2008)
A.D. Antia, E.E. Ituen, H.P. Obong, C.N. Isonguyo, Analytical solutions of the modified coulomb potential using the factorization method. Int. J. Rec. Adv. Phys. 4(1), 55–65 (2015)
O. Bayrak, I. Boztosun, H. Ciftci, Exact analytical solutions to the Kratzer potential by the asymptotic iteration method. Int. J. Quantum Chem. 107, 540 (2007)
W.C. Qiang, S.H. Dong, Proper quantization rule. EPL (Eur. Lett.) 89, 10003 (2010)
Z.Q. Ma, B.W. Xu, Quantum correction in exact quantization rules. Eur. Phys. Lett. 69, 685–691 (2005)
A.F. Nikiforov, V.B. Uvarov, Special Functions of Mathematical Physics (Birkhauser, Basel, 1988)
H. Karayer, D. Demirhan, F. Büyükkilic, A particular solution of Heun equation for Hulthen and Woods-Saxon potentials. Ann. Phys. 526, 11–12 (2014)
R.H. Parmar, Generalized improved non-central potential and solution of Schrödinger equation with extended ring-shaped potential via Nikiforov–Uvarov method. Eur. Phys. J. Plus 134, 86 (2019)
C. Tezcan, U. Baskent, R. Sever, A general approach for the exact solution of the Schrödinger equation. Int. J. Theor. Phys. 48(2), 337–350 (2009)
J.F. Wang, X.L. Peng, L.H. Zhang, C.W. Wang, C.S. Jia, Entropy of gaseous boron monobromide. Chem. Phys. Lett. 686, 131 (2017)
M. Toutounji, A new approach to the exact and approximate anharmonic vibrational partition function of diatomic and polyatomic molecules utilizing Morse and Rosen–Morse oscillators. Int. J. Quant. Chem 111, 1885 (2011)
A.N. Ikot, W. Azogor, U.S. Okorie, F. E. Bazuaye, M. C. Onjeaju, C. A. Onate, E. O. Chukwuocha, Exact and Poisson summation thermodynamic properties for diatomic molecules with shifted Tietz potential. Indian J. Phys. https://doi.org/10.1007/s12648-019-01375-0 (2019)
M.L. Strekalov, An accurate closed-form expression for the partition function of Morse oscillators. Chem. Phys. Lett. 439, 209 (2007)
M.L. Strekalov, On the partition function of Morse oscillators. Chem. Phys. Lett. 393, 192 (2004)
P.M. Morse, H. Feshbash, Methods of Theoretical Physics (McGraw-Hill, New York, 1953)
X.Q. Song, C.W. Wang, C.S. Jia, Thermodynamic properties for the sodium dimer. Chem. Phys. Lett. 673, 50 (2017)
O. Ebomwonyi, C.A. Onate, S.A. Ekong, M.C. Onyeaj, Thermodynamic Properties for the Carbon monoxide molecule under the influence of the Coulomb-Hulthen-Pöschl- Teller potential. J. Sci. Technol. Res. 1(1), 122–136 (2019)
S.M. Ikhdair, R. Sever, Approximate analytical solutions of the generalized Woods-Saxon potentials including the spin-orbit coupling term and spin symmetry. J. Math. Chem. 45, 1137 (2009)
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Parmar, R.H. Solution of the Ultra Generalized Exponential–Hyperbolic Potential in Multi-dimensional Space. Few-Body Syst 61, 39 (2020). https://doi.org/10.1007/s00601-020-01572-2
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DOI: https://doi.org/10.1007/s00601-020-01572-2