Conformable Schrodinger Equation with Pseudoharmonic Potential and Its Thermodynamic Properties
Abstract
In this work, the Nikiforov-Uvarov (NU) method is used to obtain the exact solution of the conformable radial Schrödinger equation (SE) for the pseudoharmonic potential. We derive both the energy states and the corresponding wave functions, and the results are compared with those in the existing literature for the case of the traditional derivative (α=1). Additionally, the obtained energy states are used to calculate the conformable partition function in the classical limit. Thermodynamic properties, including the conformable Helmholtz free energy, conformable mean energy, conformable entropy, and conformable specific heat capacity, are calculated and analyzed for N2 and CO molecules.
Downloads
References
C.P. Onyenegecha, C.A. Onate, O.K. Echendu, A.A. Ibe, and H. Hassanabadi, ”Solutions of Schrodinger equation for the modified Mobius square plus Kratzer potential,” The European Physical Journal Plus, 135, 289 (2020). https://doi.org/10.1140/epjp/s13360-020-00304-z
A.N. Ikot, U.S. Okorie, I.B. Okon, L.F. Obagboye, A.I. Ahmadov, H.Y. Abdullah, K.W. Qadir, et al., ”Thermal properties of 2D Schrödinger equation with new Morse interacting potential,” The European Physical Journal D, 76, 208 (2022). https://doi.org/10.1140/epjd/s10053-022-00533-0
A. Arda, C. Tezcan, and R. Sever, ” Schrödinger Equation with a Non-Central Potential: Some Statistical Quantities,” The European Physical Journal Plus, Eur. Phys. J. Plus, 131, 323 (2016). https://doi.org/10.1140/epjp/i2016-16323-0
K.J. Oyewumi, B.J. Falaye, C.A. Onate, O.J. Oluwadare, and W.A. Yahya, ”Thermodynamic properties and the approximate solutions of the Schrödinger equation with the shifted Deng–Fan potential model,” Molecular Physics, 112(1), 127–141 (2013). https://doi.org/10.1080/00268976.2013.804960
I.B. Okon, O. Popoola, C.N. Isonguyo, and A.D. Antia, “Solutions of Schrödinger and Klein-Gordon Equations With Hulthen Plus Inversely Quadratic Exponential Mie-Type Potential,” Physical Science International Journal, 19(2), 1-27 (2018). https://doi.org/10.9734/PSIJ/2018/43610
F. Ahmed, ”Effects of Cosmic String on Non-Relativistic Quantum Particles with Potential and Thermodynamic Properties,” Int. J. Theor. Phys. 62, 142 (2023). https://doi.org/10.1007/s10773-023-05397-7
I.J. Njoku, C.P. Onyenegecha, C.J. Okereke, A.I. Opara, U.M. Ukewuihe, and F.U. Nwaneho, ”Approximate solutions of Schrodinger equation and thermodynamic properties with Hua potential,” Results in Physics, 24, 104208 (2021). https://doi.org/10.1016/j.rinp.2021.104208
H.A. Mardi, N. Nasaruddin, M. Ikhwan, N. Nurmaulidar, and M. Ramli, ”Soliton dynamics in optical fiber based on nonlinear Schrödinger equation,” Heliyon, 9(3), e14235 (2023). https://doi.org/10.1016/j.heliyon.2023.e14235
Y. Yulianto, and Z. Su’ud, ”The Analytical Solutions of the Schrodinger Equation for a Single Electron in the Nikiforov-Uvarov Framework,” Jurnal Fisika, 10(2), 1-10 (2020). https://doi.org/10.15294/jf.v10i2.25190
M. Al-Hawamdeh, A. Akour, E. Jaradat, O. Jaradat, M. Al-Hawamdeh, and A. Akour, ”Involving Nikiforov-Uvarov Method in Schrodinger Equation Obtaining Hartmann Potential,” East European Journal of Physics, (2), 117-123 (2023). https://doi.org/10.26565/2312-4334-2023-2-10
F. Yas¸uk, C. Berkdemir, and A. Berkdemir, ”Exact solutions of the Schrödinger equation with non-central potential by the Nikiforov–Uvarov method,” J. Phys. A: Math. Gen. 38, 6579 (2004). https://doi.org/10.1088/0305-4470/38/29/012
S.H. Dong, Factorization Method in Quantum Mechanics, (Springer, Berlin, 2007).
U.S. Okorie, E.E. Ibekwe, A.N. Ikot, M. C. Onyeaju, and 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
D. Altarawneh, A.A. Shukri, and A.N. Ikot, ”Solution of the Woods-Saxon potential and its application for study of thermodynamic properties,” Jordan Journal Of Physics, 17(4), 395–402 (2024). https://doi.org/10.47011/17.4.2
E.P. Inyang, E.P. Inyang, I.O. Akpan, J.E. Ntibi, and E.S. William, ”Analytical Solutions of the Schrödinger Equation with Class of Yukawa Potential for a Quarkonium System Via Series Expansion Method,” European Journal of Applied Physics, 2(6), (2020). https://doi.org/10.24018/ejphysics.2020.2.6.26
B. Hamil, B. L¨utf¨uo˘glu, and M. Merad, ”Dunkl-Schrodinger Equation in Higher Dimension,” (2024). https://doi.org/10.48550/arXiv.2409.12653
A.F. Nikiforov, and V.B. Uvarov, Special Functions of Mathematical Physics, (Birkhauser, Basle, 1988).
R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, ”A new definition of fractional derivative,” Journal of Computational and Applied Mathematics, 264. 65–70 (2014). https://doi.org/10.1016/j.cam.2014.01.002
K. Oldham, and J. Spanier, editors, The fractional calculus. Theory and applications of differentiation and integration to arbitrary order, (Dover Publications Inc., USA, 2006).
K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, (JohnWiley and Sons, New York, USA, 1993).
I. Podlubny, Fractional Differential Equations, (Academic Press, San Diego, California, USA, 1999).
R. Herrmann, Fractional Calculus, 2nd ed., (World Scientific, Singapore, 2014).
A. Atangana, and D. Baleanu, and A. Alsaedi, ”New properties of conformable derivative,” Open Mathematics, 13(1), 889–898 (2015). https://doi.org/10.1515/math-2015-0081
D. Zhao, and M. Luo, ”General conformable fractional derivative and its physical interpretation,” Calcolo, 54, 903-917 (2017). https://doi.org/10.1007/s10092-017-0213-8
M. Lazo, and D.F.M. Torres, ”Variational Calculus with Conformable Fractional Derivatives,” IEEE/CAA Journal of Automatica Sinica, 4(2), 340-352 (2016). https://doi.org/10.1109/JAS.2016.7510160
A. Kajouni, A. Chafiki, K. Hilal, and M. Oukessou, ”A New Conformable Fractional Derivative and Applications,” International Journal of Differential Equations, 2021, 624543 (2021). https://doi.org/10.1155/2021/6245435
H. Karayer, D. Demirhan and F. B¨uy¨ukkılıc¸, ”Conformable Fractional Nikiforov–Uvarov Method,” Communications in Theoretical Physics, 66(1), 12 (2016). https://doi.org/10.1088/0253-6102/66/1/012
K.J. Oyewumi, F.O. Akinpelu, and A.D. Agboola, ”Exactly Complete Solutions of the Pseudoharmonic Potential in N-Dimensions,” Int. J. Theor. Phys. 47, 1039-1057 (2008). https://doi.org/10.1007/s10773-007-9532-x
O. Olendski, ”One-dimensional pseudoharmonic oscillator: classical remarks and quantum-information theory,” Journal of Physics Communications, 7, 045002 (2023). https://doi.org/10.1088/2399-6528/acce20
J. Bao, and B.D. Shizgal, ”Pseudospectral method of solution of the Schrödinger equation for the Kratzer and pseudoharmonic potentials with nonclassical polynomials and applications to realistic diatom potentials,” Computational and Theoretical Chemistry, 1149, 49-56 (2019). https://doi.org/10.1016/j.comptc.2019.01.001
A. Ikot, U. Okorie, I. Okon, L. Obagboye, A. Ahmadov, H. Abdullah, K. Qadir, M.E. Udoh, and C.A. Onate, ”Thermal Properties of 2D Schrodinger equation with New Morse Interacting Potential,” The European Physical Journal D, 76, 208 (2022). https://doi.org/10.1140/epjd/s10053-022-00533-0
I.I. Goldman, and V.D. Krivchenkov, Problems in quantum mechanics, (Pergamon Press, New York, 1961).
R. Sever, C. Tezcan, M. Aktas¸, and Ozlem Yesiltas, ”Exact solution of Schrödinger equation for Pseudoharmonic potential,” J. Math. Chem. 43, 845–851 (2008). https://doi.org/10.1007/s10910-007-9233-y
C. Berkdemir, A. Berkdemir, and J. Han, ”Bound state solutions of the Schrödinger equation for modified Kratzer’s molecular potential, Chemical Physics Letters, 417, 326-329 (2006). https://doi.org/10.1016/j.cplett.2005.10.039
U.S. Okorie, A.N. Ikot, G.J. Rampho, P.O. Amadi, and H.Y. Abdullah, ”Analytical solutions of fractional Schrödinger equation and thermal properties of Morse potential for some diatomic molecules,” Modern Physics Letters A, 36(07), 2150041 (2021). https://doi.org/10.1142/S0217732321500413
M. Al-Masaeed, E. Rabei, and A. Al-Jamel, ”Analytical Solution of Conformable Schrodinger Wave Equation with Coulomb Potential,” Progr. Fract. Differ. Appl. 10(1), 137-153 (2024). http://dx.doi.org/10.18576/pfda/100113
E.E. Ibekwe, E.P. Inyang, J.B. Emah, A.O. Akpan, and O.J. Yawo, “Mass spectra and thermal properties of deformed Schrödinger Equation for pseudoharmonic potential,” Sri Lankan Journal of Physics, 23(2), 63-76 (2022). http://doi.org/10.4038/sljp.v23i2.8119
I. Haouam, ”On the conformable fractional derivative and its applications in physics,” J. Theor. Appl. phys. 18(6) (2024). https://doi.org/10.57647/j.jtap.2024.1806.74
A. Al-Jamel, M. Al-Masaeed, E.M. Rabei, and D. Baleanu, ”The effect of deformations of special relativity by conformable derivative,” Revista Mexicana de F´ısica, 68(5), 050705 (2022). https://doi.org/10.31349/revmexfis.68.050705
Copyright (c) 2025 Derar Altarawneh, Eqab Rabei
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
