JWKB Solution of the Modified Pöschl–Teller Double Ring-Shaped Coulomb Potential in an External Magnetic Field
Keywords:
Poschl–Teller, Double Ring-Shaped, Coulomb Potential, JWKB Approximation, Energy Eigenvalue, Magnetic FieldAbstract
In recent years, the analytical treatment of the Schrödinger equation for non-central potential has become increasingly significant due to its wide applicability in describing quantum systems. This study obtained the analytical bound-state solutions of the time-independent Schrödinger equation for a modified Pöschl–Teller double ring-shaped Coulomb (MPTDRSC) potential with the presence of an external magnetic field using the Jeffreys–Wentzel–Kramers–Brillouin (JWKB) semiclassical approximation. The proposed potential contains the Coulomb term, ring-shaped term, and magnetic term, along with two tunable reduced parameters, (x) and (y), that control the azimuthal and polar confinement geometry of the quantum system. Using the JWKB quantisation condition, the bound state expressions for the eigenvalues of the energy and the corresponding radial wave functions are derived analytically. The energy eigenvalue spectrum of the bound state of the system is obtained numerically for the principal quantum numbers n = 0–5 and angular momentum quantum numbers l = 0–4, and only negative energies are obtained, which proves the bound solution and stability of the system. A thorough parametric study also shows that the azimuthal ring parameter x is the major confinement modulator, with a range of x values giving rise to a potential variation of 467%, while the polar parameter y is the major confinement modulator with a potential variation of 267% over its investigated range. The magnetic field induction (B) has also a linear effect on the energy spectrum, inducing a Zeeman-like interaction, whereas the Coulomb strength parameter (beta) has a strictly linear additive effect on the effective potential, whereas the Coulomb strength parameter (beta) has a strictly linear additive effect on the effective potential.
References
Boztosun, I., & Karakoc, M. (2007). The asymptotic iteration method for the hydrogen atom in a magnetic field. Chinese Physics Letters, 24(12), 3028–3031. https://doi.org/10.1088/0256-307X/24/11/001
Champion, B., Hall, R. L., & Saad, N. (2008). Asymptotic iteration method for singular potentials. International Journal of Modern Physics A, 23(9), 1405–1430. https://doi.org/10.1142/S0217751X08039852
Chang-Yuan, C. (2013). Exact solutions of the Schrödinger equation with double ring-shaped oscillator potentials. Chinese Physics B, 22(10), 100302. https://doi.org/10.1088/1674-1056/22/10/100302
Dong, S.-H., & Gonzalez-Cisneros, A. (2008). Energy spectra of the hyperbolic and second Pöschl-Teller-like potentials solved by new exact quantization rule. Annals of Physics, 323(5), 1136–1149. https://doi.org/10.1016/j.aop.2007.12.007
Dong, S.-H., Qiang, W.-C., Sun, G.-H., & Bezerra, V. B. (2007). Analytical approximations to the Schrödinger equation for a second Pöschl-Teller-like potential with centrifugal term. Journal of Physics A: Mathematical and Theoretical, 40(34), 10535–10540. https://doi.org/10.1088/1751-8113/40/34/010
Eyube, E. S., Notani, P. P., & Dikko, A. B. (2023). Modelling of diatomic molecules with modified hyperbolical-type potential. European Physical Journal Plus, 138(3), 254. https://doi.org/10.1140/epjp/s13360-023-03859-5
Fernandez, F. M. (2004). Comment on "Supersymmetric approach to quasi-exactly solvable systems." Physical Review A, 70(5), 056101. https://doi.org/10.1103/PhysRevA.70.056101
Griffiths, D. J. (2005). Introduction to quantum mechanics (2nd ed.). Pearson Education.
Hassanabadi, H., Maghsoodi, E., Ikot, A. N., & Zarrinkamar, S. (2012). Dirac equation for generalized Pöschl-Teller scalar and vector potentials and a Coulomb tensor interaction by Nikiforov-Uvarov method. Journal of Mathematical Physics, 53(2), 022104. https://doi.org/10.1063/1.3683078
Ikhdair, S. M. (2012). Rotational and vibrational diatomic molecule in the Klein-Gordon equation with hyperbolic scalar and vector potentials. International Journal of Modern Physics C, 20(10), 1563–1582. https://doi.org/10.1142/S0129183109014606
Ikhdair, S. M., & Falaye, B. J. (2013). Approximate relativistic bound states of a particle in Pöschl–Teller field with shell-confined factor. Chemical Physics, 421, 84–95. https://doi.org/10.1016/j.chemphys.2013.05.021
Ita, B. I., Udoimuk, A. B., & Ehi-Eromosele, C. O. (2010). Arbitrary angular momentum solutions of the Schrödinger equation for the general exponential screened Coulomb potential. Journal of the Chemical Society of Nigeria, 35(1), 1–6.
Ngiania, A. T., & Nwabuzor, P. O. (2025). Thermal and quantum properties of nonlinear and inverted harmonic oscillators in magnetic fields: Applications to polyatomic molecules. Physical Science International Journal, 29(6), 128–159.
Onate, C. A., & Onyeaju, M. C. (2021). Approximate solution of the Schrödinger equation for the modified Pöschl-Teller potential and application. Results in Physics, 22, 103958. https://doi.org/10.1016/j.rinp.2021.103958
Ortakaya, S. (2012). Exact analytical solutions of the Schrödinger equation with a Coulomb-plus-linear potential. Few-Body Systems, 54(1–4), 151–159. https://doi.org/10.1007/s00601-012-0452-9
Oyewumi, K. J., & Akoshile, C. O. (2010). Bound-state solutions of the Dirac-Rosen-Morse potential with spin and pseudospin symmetry. European Physical Journal A, 45(3), 311–318. https://doi.org/10.1140/epja/i2010-11007-0
Qiang, W.-C., & Dong, S.-H. (2010). Analytical approximations to the solutions of the Manning-Rosen potential with centrifugal term. Physics Letters A, 368(1–2), 13–17. https://doi.org/10.1016/j.physleta.2007.03.057
Sur, S., & Debnath, S. (2020). Bound state solutions of the Klein-Gordon equation for the double ring-shaped Coulomb potential via the asymptotic iteration method. Journal of Physics: Conference Series, 1540, 012035. https://doi.org/10.1088/1742-6596/1540/1/012035
Yahya, W. A., & Oyewumi, K. J. (2018). Thermodynamic properties and approximate solutions of the ℓ-state Pöschl–Teller-type potential. Journal of the Association of Arab Universities for Basic and Applied Sciences, 21, 53–58. https://doi.org/10.1016/j.jaubas.2015.04.001
Yahya, W. A., Oyewumi, K. J., Akoshile, C. O., & Oluwadare, O. J. (2010). Scattering states and bound states of the relativistic Hulthén potential in D-dimension. Journal of Vectorial Relativity, 5(3), 1–12.
Yahya, W. A., Oyewumi, K. J., & Sen, K. D. (2014). Position and momentum information-theoretic measures of the pseudoharmonic potential. International Journal of Quantum Chemistry, 115(2), 1–10. https://doi.org/10.1002/qua.24864
