Abstract
Within the framework of non-relativistic quantum mechanics, the bound state approximate solution of the SE is solved for the Coshine Yukawa potential (CYP) using the Nikiforov–Uvarov (NU) method. By employing the Greene-Aldrich-type approximation scheme, we have obtained the explicit energy-eigenvalues and corresponding normalized eigen-functions in closed form for the newly proposed CYP for hydrogen-related diatomic molecules such as hydrogen dimer (H2), lithium hydride (LiH), scandium hydride (ScH) and hydrogen chloride (HCl). Our results show that the bound state energy is highly sensitive to the spectroscopic parameters of the diatomic molecules considered. The thermodynamic properties are also evaluated including the vibrational partition function, vibrational mean energy, vibrational mean free energy, vibrational entropy and vibrational specific heat capacity. Presented also are some numerical results which show an indication of similar correlation of energies, owing to their ion-ion coupling with regards to similar atomic radii existing among the diatomic molecules.
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References
W. Greiner, Relativistic Quantum Mechanics. Wave Equations II, 3rd edn. (Springer, Berlin, 2000)
S. Flügge, Practicle Quantum Mechanics (Springer, Berlin, 1994)
A.N. Ikot, U.S. Okorie, G. Osobonye, P.O. Amadi, C.O. Edet, M.J. Sithole, G.J. Rampho, R. Sever, Superstatistics of Schrödinger equation with pseudo-harmonic potential in external magnetic and Aharanov-Bohm fields. Heliyon. 6, e03738 (2020). https://doi.org/10.1016/j.heliyon.2020.e03738
S. Dong, G.H. Sun, B.J. Falaye, S.H. Dong, Semi-exact solutions to position-dependent mass Schrödinger problem with a class of hyperbolic potential V0tanh(ax). Phys. J. Plus Eur (2016). https://doi.org/10.1140/epjp/i2016-16176-5
C.O. Edet, P.O. Amadi, U.S. Okorie, A. Taş, A.N. Ikot, G. Rampho, Solutions of Schrödinger equation and thermal properties of generalized trigonometric Pöschl-Teller potential. Rev. Mex. Fis. 66, 824–839 (2020)
R. Horchani, H. Al-Aamri, N. Al-Kindi, A.N. Ikot, U.S. Okorie, G.J. Rampho, H. Jelassi, Energy spectra and magnetic properties of diatomic molecules in the presence of magnetic and AB fields with the inversely quadratic Yukawa potential. Eur. Phys. J. D. (2021). https://doi.org/10.1140/epjd/s10053-021-00038-2
E.E. Ibekwe, U.S. Okorie, J.B. Emah, E.P. Inyang, S.A. Ekong, Mass spectrum of heavy quarkonium for screened Kratzer potential (SKP) using series expansion method. Phys. J. Plus Eur (2021). https://doi.org/10.1140/epjp/s13360-021-01090-y
U.S. Okorie, A.N. Ikot, C.O. Edet, I.O. Akpan, R. Sever, G.J. Rampho, Solutions of the klein gordon equation with generalized hyperbolic potential in d-dimensions. J. Phys. Commun. (2019). https://doi.org/10.1088/2399-6528/ab42c6
E.E. Ibekwe, A.T. Ngiangia, U.S. Okorie, A.N. Ikot, H.Y. Abdullah, Bound state solution of radial schrodinger equation for the quark-antiquark interaction potential. Iran. J. Sci. Technol. Trans. A Sci. 44, 1191–1204 (2020). https://doi.org/10.1007/s40995-020-00913-4
M.R. Hadizadeh, A. Khaledi-nasab, Heavy tetraquarks in the diquark – antidiquark picture. Phys. Lett. B. 753, 8–12 (2016). https://doi.org/10.1016/j.physletb.2015.11.072
M.A. Shalchi, M.R. Hadizadeh, R-matrix calculations for few-quark bound states. Eur. Phys. J. C. (2016). https://doi.org/10.1140/epjc/s10052-016-4369-1
U.S. Okorie, E.E. Ibekwe, A.N. Ikot, M.C. Onyeaju, E.O. Chukwuocha, Thermodynamic properties of the modified yukawa potential. J. Korean Phys. Soc. 73, 1211–1218 (2018). https://doi.org/10.3938/jkps.73.1211
A.N. Ikot, U.S. Okorie, R. Sever, G.J. Rampho, Eigensolution, expectation values and thermodynamic properties of the screened Kratzer potential. Phys. J. Plus Eur (2019). https://doi.org/10.1140/epjp/i2019-12783-x
O. Ebomwonyi, C.A. Onate, S.A. Ekong, M.C. Onyeaju, Thermodynamic Properties for the Carbon Monoxide Molecule under the Influence of the Coulomb-Hulthen-Pöschl-Teller Potential. J. Sci. Technol. Res. 1, 122–136 (2019)
T. Sahraeian, M.R. Hadizadeh, Momentum space calculations of the binding energies of argon dimer. Int. J. Quantum Chem. (2019). https://doi.org/10.1002/qua.25807
C.O. Edet, P.O. Okoi, Any l -state solutions of the Schr odinger equation for q -deformed Hulthen plus generalized inverse quadratic Yukawa potential in arbitrary dimensions. Revista mexicana de física 65, 333–344 (2019)
P.O. Okoi, C.O. Edet, T.O. Magu, Relativistic treatment of the hellmann-generalized morse potential. Rev. Mex. Fis. 66, 1–13 (2020)
C.O. Edet, P.O. Okoi, S.O. Chima, Analytic solutions of the Schrödinger equation with non-central generalized inverse quadratic Yukawa potential. Bras. Ensino Fis Rev (2020). https://doi.org/10.1590/1806-9126-RBEF-2019-0083
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. Indian J. Phys. 94, 425–433 (2020). https://doi.org/10.1007/s12648-019-01477-9
A.N. Ikot, S. Zarrinkamar, B.H. Yazarloo, H. Hassanabadi, Relativistic symmetries of Deng - Fan and Eckart potentials with Coulomb-like and Yukawa-like tensor interactions. Chinese Phys. B. (2014). https://doi.org/10.1088/1674-1056/23/10/100306
D. Nath, A.K. Roy, dinger equation for Eckart Analytical solution of D dimensional Schr o potential with a new improved approximation in centrifugal term. Chem. Phys. Lett. 780, 138909 (2021). https://doi.org/10.1016/j.cplett.2021.138909
A.N. Ikot, B.H. Yazarloo, E. Maghsoodi, S. Zarrinkamar, H. Hassanabadi, Effects of tensors coupling to Dirac equation with shifted Hulthen potential via SUSYQM. J. Assoc. Arab Univ. Basic Appl. Sci. 18, 46–59 (2015). https://doi.org/10.1016/j.jaubas.2014.03.005
A. Arai, Exactly solvable supersymmetric quantum mechanics. J. Math. Anal. Appl. 158, 63–79 (1991). https://doi.org/10.1016/0022-247X(91)90267-4
C.N. Isonguyo, I.B. Okon, A.N. Ikot, H. Hassanabadi, Solution of klein gordon equation for some diatomic molecules with new generalized morse-like potential using SUSYQM. Bull. Korean Chem. Soc. 35, 3443–3446 (2014). https://doi.org/10.5012/bkcs.2014.35.12.3443
B.J. Falaye, Any ℓ-state solutions of the Eckart potential via asymptotic iteration method. Cent. Eur. J. Phys. 10, 960–965 (2012). https://doi.org/10.2478/s11534-012-0047-6
H. Ciftci, R.L. Hall, N. Saad, Perturbation theory in a framework of iteration methods. Phys. Lett. Sect A Gen. At. Solid State Phys. 340, 388–396 (2005). https://doi.org/10.1016/j.physleta.2005.04.030
K.J. Oyewumi, B.J. Falaye, C.A. Onate, O.J. Oluwadare, W.A. Yahya, Molecular physics : an international journal at the interface between chemistry and physics thermodynamic properties and the approximate solutions of the Schrödinger equation with the shifted Deng – Fan potential model. Molecular Physics (2013). https://doi.org/10.1080/00268976.2013.804960
M. Hamzavi, A.A. Rajabi, Solution of Dirac equation with Killingbeck potential by using wave function ansatz method under spin symmetry limit. Commun. Theor. Phys. 55, 35–37 (2011). https://doi.org/10.1088/0253-6102/55/1/07
B.J. Falaye, S.M. Ikhdair, M. Hamzavi, Formula method for bound state problems. Few-Body Syst. 56, 63–78 (2015). https://doi.org/10.1007/s00601-014-0937-9
Q. Wang, X. Xie, S. Li, Z. Zhang, X. Li, H. Yao, C. Chen, F. Cao, J. Sui, X. Liu, Q. Zhang, Enhanced thermoelectric performance in Ti(Fe Co, Ni)Sb pseudo-ternary Half-Heusler alloys. J. Mater. 7, 756–765 (2021). https://doi.org/10.1016/J.JMAT.2020.12.015
J.Y. Liu, G.D. Zhang, C.S. Jia, Calculation of the interaction potential energy curve and vibrational levels for the a3 Σu + state of Li 2 7 molecule. Phys. Lett. Sect A Gen. At. Solid State Phys. 377, 1444–1447 (2013). https://doi.org/10.1016/j.physleta.2013.04.019
M.C. Onyeaju, J.O.A. Idiodi, A.N. Ikot, M. Solaimani, H. Hassanabadi, Linear and nonlinear optical properties in spherical quantum dots: Manning-Rosen potential. J. Opt. 46, 254–264 (2017). https://doi.org/10.1007/s12596-016-0359-9
S.H. Dong, Factorization Method in Quantum Mechanics (Springer, Armsterdam, 2007)
C.S. Jia, Y. Jia, Relativistic rotation-vibrational energies for the Cs2 molecule. Eur. Phys. J. D. (2017). https://doi.org/10.1140/epjd/e2016-70415-y
C.S. Jia, X.L. Peng, S. He, Molecular spinless energies of the modified Rosen-Morse potential energy model. Bull. Korean Chem. Soc. 35, 2699–2703 (2014). https://doi.org/10.5012/bkcs.2014.35.9.2699
G. Chen, The exact solutions of the Schrödinger equation with the Morse potential via Laplace transforms. Phys. Lett. Sect A Gen. At. Solid State Phys. 326, 55–57 (2004). https://doi.org/10.1016/j.physleta.2004.04.029
S.M. Ikhdair, R. Sever, Exact quantization rule to the Kratzer-type potentials: an application to the diatomic molecules. J. Math. Chem. 45, 1137–1152 (2009). https://doi.org/10.1007/s10910-008-9438-8
S.M. Ikhdair, J. Abu-Hasna, Quantization rule solution to the Hulthén potential in arbitrary dimension with a new approximate scheme for the centrifugal term. Phys. Scr. (2011). https://doi.org/10.1088/0031-8949/83/02/025002
C. Grosche, Conditionally solvable path integral problems. J. Phys. A. Math. Gen. 28, 5889–5902 (1995). https://doi.org/10.1088/0305-4470/28/20/018
R.L. Greene, C. Aldrich, Variational wave functions for a screened Coulomb potential. Phys. Rev. A. 14, 2363–2366 (1976). https://doi.org/10.1103/PhysRevA.14.2363
A.N. Ikot, U.S. Okorie, G.J. Rampho, P.O. Amadi, C.O. Edet, I.O. Akpan, H.Y. Abdullah, R. Horchani, Klein–Gordon Equation and Nonrelativistic thermodynamic properties with improved screened Kratzer potential. J. Low Temp. Phys. 202, 269–289 (2021)
C.L. Pekeris, The rotation-vibration coupling in diatomic molecules. Phys. Rev. 45, 98–103 (1934)
B.I. Ita, H. Louis, O.U. Akakuru, T.O. Magu, I. Joseph, P. Tchoua, P.I. Amos, Bound state solutions of the schrödinger equation for the more general exponential screened coulomb potential plus Yukawa ( MGESCY ) potential using nikiforov-uvarov method. J. Quantum Inform. Sci. (2018). https://doi.org/10.4236/jqis.2018.81003
H.I. Ahmadov, E.A. Dadashov, N.S. Huseynova, V.H. Badalov, Generalized tanh-shaped hyperbolic potential: bound state solution of Schrödinger equation. Phys. J. Plus Eur. (2021). https://doi.org/10.1140/epjp/s13360-021-01202-8
P.M. Morse, Phys. Rev. Phys. Rev. 34, 57–64 (1929)
C. Berkdemir, J. Han, Any l-state solutions of the Morse potential through the Pekeris approximation and Nikiforov-Uvarov method. Chem. Phys. Lett. 409, 203–207 (2005). https://doi.org/10.1016/j.cplett.2005.05.021
R.F. Garcia Ruiz, R. Berger, J. Billowes, C.L. Binnersley, M.L. Bissell, A.A. Breier, A.J. Brinson, K. Chrysalidis, T.E. Cocolios, B.S. Cooper, K.T. Flanagan, T.F. Giesen, R.P. de Groote, S. Franchoo, F.P. Gustafsson, T.A. Isaev, Koszorús, G. Neyens, H.A.C.M.S.L.A.R.K.D.A.F.S.G.X.F. PerrettRickettsRotheSchweikhardVernonWendtWienholtzWilkinsYang, Spectroscopy of short-lived radioactive molecules. Nature 581, 396–400 (2020)
C. Eckart, The penetration of a potential barrier by electrons. Phys. Rev. 35, 1303–1309 (1930). https://doi.org/10.1103/PhysRev.35.1303
X. Zou, L.Z. Yi, C.S. Jia, Bound states of the Dirac equation with vector and scalar Eckart potentials. Phys. Lett. Sect A Gen. At. Solid State Phys. 346, 54–64 (2005). https://doi.org/10.1016/j.physleta.2005.07.075
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, 11521–11529 (2006). https://doi.org/10.1088/0305-4470/39/37/012
B.C. Lütfüoǧlu, A.N. Ikot, U.S. Okorie, A.T. Ngiangia, A statistical mechanical analysis on the bound state solution of an energy-dependent deformed Hulthén potential energy. Commun. Theor. Phys. 71, 1127–1138 (2019). https://doi.org/10.1088/0253-6102/71/9/1127
M.F. Manning, N. Rosen, Proceedings of the southeastern section of the american physical society. Phys. Rev. 44, 951–954 (1933). https://doi.org/10.1103/PhysRev.69.545
W.C. Qiang, S.H. Dong, Analytical approximations to the solutions of the Manning-Rosen potential with centrifugal term. Phys. Lett. Sect. A Gen. At. Solid State Phys. 368, 13–17 (2007). https://doi.org/10.1016/j.physleta.2007.03.057
W.C. Qiang, S.H. Dong, The Manning-Rosen potential studied by a new approximate scheme to the centrifugal term. Phys. Scr. (2009). https://doi.org/10.1088/0031-8949/79/04/045004
R.D. Woods, D.S. Saxon, Diffuse surface optical model for nucleon-nuclei scattering [19]. Phys. Rev. 95, 577–578 (1954). https://doi.org/10.1103/PhysRev.95.577
B.C. Lütfüoğlu, A.N. Ikot, E.O. Chukwocha, F.E. Bazuaye, Analytical solution of the Klein Gordon equation with a multi-parameter q-deformed Woods-Saxon type potential. Phys. J. Plus Eur (2018). https://doi.org/10.1140/epjp/i2018-12299-y
N. Rosen, P.M. Morse, On the vibrations of polyatomic molecules. Phys. Rev. 42, 210–217 (1932). https://doi.org/10.1103/PhysRev.42.210
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. Phys. Lett. Sect A Gen. At. Solid State Phys. 333, 212–217 (2004). https://doi.org/10.1016/j.physleta.2004.10.054
C.S. Jia, Y. Li, Y. Sun, J.Y. Liu, L.T. Sun, Bound states of the five-parameter exponential-type potential model. Phys. Lett. Sect A Gen. At. Solid State Phys. 311, 115–125 (2003). https://doi.org/10.1016/S0375-9601(03)00502-4
H. Egrifes, D. Demirhan, F. Büyükkilic, Exact solutions of the Schrodinger equation for the deformed hyperbolic potential well and the deformed four-parameter exponential type potential. Phys. Lett. Sect A Gen. At Solid State Phys. 275, 229–237 (2000). https://doi.org/10.1016/S0375-9601(00)00592-2
C.S. Jia, Y.F. Diao, M. Li, Q.B. Yang, L.T. Sun, R.Y. Huang, Mapping of the five-parameter exponential-type potential model into trigonometric-type potentials. J. Phys. A. Math. Gen. 37, 11275–11284 (2004). https://doi.org/10.1088/0305-4470/37/46/012
C.S. Jia, X.L. Zeng, L.T. Sun, PT symmetry and shape invariance for a potential well with a barrier. Phys. Lett. Sect A Gen. At. Solid State Phys. 294, 185–189 (2002). https://doi.org/10.1016/S0375-9601(01)00840-4
S.M. Ikhdair, R. Sever, Bound states of a more general exponential screened Coulomb potential. J. Math. Chem. 41, 343–353 (2007). https://doi.org/10.1007/s10910-007-9226-x
B.I. Ita, P. Ekuri, I.O. Isaac, A.O. James, Bound state solutions of schrÖdinger equation for a more general exponential screened coulomb potential via Nikiforovuvarov method. Eclet. Quim. 35, 103–107 (2010)
A.K. Roy, Critical parameters and spherical confinement of H atom in screened Coulomb potential. Int. J. Quantum Chem. 116, 953–960 (2016). https://doi.org/10.1002/qua.25108
U.S. Okorie, A.N. Ikot, E.O. Chukwuocha, G.J. Rampho, Thermodynamic properties of improved deformed exponential-type potential (IDEP) for some diatomic molecules. Results Phys (2020). https://doi.org/10.1016/j.rinp.2020.103078
G.T. Osobonye, M. Adekanmbi, A.N. Ikot, U.S. Okorie, Thermal properties of anharmonic Eckart potential model using Euler – MacLaurin formula. Pramana (2021). https://doi.org/10.1007/s12043-021-02122-z
C.A. Onate, M.C. Onyeaju, U.S. Okorie, A.N. Ikot, Thermodynamic functions for boron nitride with q-deformed exponential- type potential. Results Phys. 16, 102959 (2020). https://doi.org/10.1016/j.rinp.2020.102959
E.P. Inyang, E.P. Inyang, I.O. Akpan, J.E. Ntibi, E.S. William, Masses and thermodynamic properties of a quarkonium system. Can J. Phys. (2021). https://doi.org/10.1139/cjp-2020-0578
A.F. Nikiforov, V.B. Uvarov, Special Functions of Mathematical Physics: A Unified Introduction with Applications (Springer Basel AG). (1988)
M. MohammadiSabet, Solution of radial schrödinger equation with yukawa potential using bethe ansatz method. Acta Phys. Pol. A. 140, 97–102 (2021)
M. Hamzavi, M. Movahedi, K.E. Thylwe, A.A. Rajabi, Approximate analytical solution of the yukawa potential with arbitrary angular momenta. Chinese Phys. Lett. 29, 3–6 (2012). https://doi.org/10.1088/0256-307X/29/8/080302
O. Ebomwonyi, C.A. Onate, M.C. Onyeaju, A.N. Ikot, ScienceDirect Any [À states solutions of the Schr € odinger equation interacting with Hellmann-generalized Morse potential model. Karbala Int. J. Mod. Sci. 3, 59–68 (2017). https://doi.org/10.1016/j.kijoms.2017.03.001
S. Dong, W. Qiang, G. Sun, V.B. Bezerra, Analytical approximations to the l -wave solutions of the Schr odinger equation with the Eckart potential. J. Phys. A: Math. Theoretical (2007). https://doi.org/10.1088/1751-8113/40/34/010
U.S. Okorie, A.N. Ikot, P.O. Amadi, A.T. Ngiangia, E.E. Ibekwe, Approximate solutions of the Schrödinger equation with energy-dependent screened Coulomb potential in D - dimensions. Eclet. Quim. 45, 40–56 (2020)
M. Abramowitz, I.A. Stegun, Handbook of mathematical functions_ with formulas. Graphs, and Math. Tables-National Bureau of Standards. (1970). https://doi.org/10.1159/000452153
I.S. Gradshteyn, I.M.R., Table of integrals, series, and products: Seventh edition. (2007)
C. Berkdemir, Application of the Nikiforov-Uvarov Method in Quantum Mechanics. (2012)
D. Schiöberg, The energy eigenvalues of hyperbolical potential functions. Mol. Phys. An Int J. Interface Between Chem. Phys. 59, 1123–1137 (1986)
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Ekong, S.A., Okorie, U.S., Ikot, A.N. et al. Thermodynamic evaluation of Coshine Yukawa potential (CYP) for some diatomic molecule systems. Eur. Phys. J. Plus 138, 364 (2023). https://doi.org/10.1140/epjp/s13360-023-03982-7
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DOI: https://doi.org/10.1140/epjp/s13360-023-03982-7