https://doi.org/10.1140/epjp/s13360-025-06630-4
Regular Article
Dirac equation solution with generalized tanh-shaped hyperbolic potential: application to charmonium and bottomonium mass spectra
1
Institute for Physical Problems, Baku State University, AZ-1148, Baku, Azerbaijan
2
Center for Theoretical Physics, Khazar University, AZ-1096, Baku, Azerbaijan
3
Department of Theoretical Physics, Baku State University, AZ-1148, Baku, Azerbaijan
4
Department of Chemistry and Physics, Lankaran State University, AZ-4200, Lankaran, Azerbaijan
5
Department of Physics, University of Bayreuth, 95447, Bayreuth, Germany
6
Bavarian Center for Battery Technologies, University of Bayreuth, 95448, Bayreuth, Germany
a
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Received:
15
March
2025
Accepted:
7
July
2025
Published online:
26
August
2025
In this study, we present an analytical solution of the Dirac equation in a generalized tanh-shape hyperbolic potential, which allows us to unify various well-known quantum potentials under a single theoretical framework. This versatile potential model is used to compute the mass spectra for charmonium and bottomonium, with excellent agreement with experimental measurements, and does better than some potential models in predicting the several orbital states. Our results not only validate the GTHP as a powerful tool for describing heavy quarkonium systems but also suggest its broader applicability in exploring quantum systems where similar potentials are effective. This work is a stepping stone for new research into fermionic systems with complex interactions, by jointly providing insights into foundational aspects of quantum mechanics as well as applications in particle physics.
Supplementary Information The online version contains supplementary material available at https://doi.org/10.1140/epjp/s13360-025-06630-4.
© The Author(s) 2025
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