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Solutions of Klein–Gordon and Dirac Equations for Non-pure Dipole Potential in 2D Systems

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Abstract

We study the quantum relativistic wave equations (Klein–Gordon and Dirac) for the non-pure dipole potential \(V(r)=-Ze/r+D\cos \theta /r^{2}\), in the case of two-dimensional systems. We consider either spin symmetry or anti-spin symmetry cases in our computations. We give the analytical expressions of the eigenfunctions, compute the exact values of the energies and study their dependence according to the dipole moment D. Our study generalizes the energies of the Kratzer potential as well as the magnetic quantum number m, which is replaced with the Mathieu characteristic values obtained during the resolution of the angular equations. For each magnetic quantum number, we demonstrate the existence of a critical value for the dipole moment, beyond which the corresponding bound state can no longer exist. We find that the critical value is null when \(m=0\); this means that these s-states cannot exist for this system and this is in agreement with non-relativistic studies.

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Acknowledgements

This work was done with funding from the DGRSDT of the Ministry of Higher Education and Scientific Research in Algeria as part of the PRFU B00L02UN070120190003. The authors would like to thank the reviewer who greatly contributed to improving the quality of this work with his remarks and comments.

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Authors and Affiliations

  1. Laboratory of Photonic Physics and Nano-Materials (LPPNNM), Department of Matter Sciences, University of Biskra, Biskra, Algeria

    Mustafa Moumni, Mokhtar Falek & Mebarek Heddar

  2. Laboratory of Radiation and their Interactions with Matter (PRIMALAB), Department of Physics, University of Batna1, Batna, Algeria

    Mustafa Moumni

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  1. Mustafa Moumni
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Correspondence to Mustafa Moumni.

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Moumni, M., Falek, M. & Heddar, M. Solutions of Klein–Gordon and Dirac Equations for Non-pure Dipole Potential in 2D Systems. Few-Body Syst 61, 47 (2020). https://doi.org/10.1007/s00601-020-01580-2

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