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Exact Solution of Klein–Gordon and Dirac Equations with Snyder–de Sitter Algebra

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Abstract

In this paper, we present the exact solution of the \((1+1)\)-dimensional relativistic Klein–Gordon and Dirac equations with linear vector and scalar potentials in the framework of deformed Snyder–de Sitter model. We introduce some changes of variables, we show that a one-dimensional linear potential for the relativistic system in a space deformed can be equivalent to the trigonometric Rosen–Morse potential in a regular space. In both cases, we determine explicitly the energy eigenvalues and their corresponding eigenfunctions expressed in terms of Romonovski polynomials. The limiting cases are analyzed for \(\alpha _{1}\) and \(\alpha _{2} \rightarrow 0\) and are compared with those of literature.

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References

  1. R. Banerjee, S. Kulkarni, S. Samanta, Deformed symmetry in Snyder space and relativistic particle dynamics. J. High Energy Phys. 05, 077 (2006)

    Article  ADS  Google Scholar 

  2. S. Mignemi, Classical and quantum mechanics of the nonrelativistic Snyder model in curved space. Class. Quantum Gravity 29, 215019 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. S. Mignemi, R. Strajn, Quantum mechanics on a curved snyder space. Adv. High Energy Phys. 2016, 1328284 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Mignemi, Classical and quantum mechanics of the nonrelativistic Snyder model. Phys. Rev. D 84, 025021 (2011)

    Article  ADS  Google Scholar 

  5. C. Leiva, Harmonic oscillator in Snyder space: the classical case and the quantum case. Pramana J. Phys. 74, 172 (2010)

    Article  ADS  Google Scholar 

  6. C. Leiva, J. Saavedra, J.R. Villanueva, The Kepler problem in the Snyder space. Pramana J. Phys. 80, 945 (2013)

    Article  ADS  Google Scholar 

  7. M.M. Stetsko, Dirac oscillator and nonrelativistic Snyder–de Sitter algebra. J. Math. Phys. 56, 012101 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. M. Falek, M. Merad, T. Birkandan, Dufn–Kemmer–Petiau oscillator with Snyder–de Sitter algebra. J. Math. Phys. 58, 023501 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. S.N. Jena, M.R. Behera, S. Panda, Ground-state baryon masses in an equally mixed scalar-vector linear potential model. Phys. Rev. 55, 291 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  10. L. Dittmann, T. Heinzl, A. Wip, Effective theories of confinement. Nucl. Phys. Proc. Suppl. 108, 63 (2002)

    Article  ADS  MATH  Google Scholar 

  11. J.M. Overduin, P.S. Wesson, Kaluza–Klein gravity. Phys. Rep. 283, 303 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  12. Ø. Gron, Classical Kaluza–Klein description of the hydrogen atom. IL Nuovo Cimento B 91, 57 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  13. F. Dominguez-Adame, B. Mendez, Relativistic particles in orthogonal electric and magnetic fields with confining scalar potentials. Il Nuovo Cimento B 05, 489 (1992)

    Article  ADS  Google Scholar 

  14. S. Zarrinkamar, A. Rajabi, H. Hassanabadi, Dirac equation for the harmonic scalar and vector potentials and linear plus coulomb-like tensor potential; the SUSY approach. Ann. Phys. 325, 2522 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Y. Chargui, L. Chetouani, Exact solution of the one-dimensional Klein-Gordon equation with scalar and vector linear potentials in the presence of a minimal length. Chin. Phys. B 19, 020305 (2010)

    Article  ADS  Google Scholar 

  16. T.K. Jana, P. Roy, Exact solution of the Klein–Gordon equation in the presence of a minimal length. Phys. Lett. A 373, 1239 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. A. Tilbi, M. Merad, T. Boudjedaa, Particles of spin zero and 1/2 in electromagnetic field with confining scalar potential in modified heisenberg algebra. Few Body Syst. 56, 139 (2015)

    Article  ADS  Google Scholar 

  18. H.S. Snyder, Quantized space-time. Phys. Rev. 71, 38 (1947)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. C.B. Compean, M. Kirchbach, The trigonometric Rosen–Morse potential in the supersymmetric quantum mechanics and its exact solutions. J. Phys. A Math. Gen. 39, 550 (2005)

    MATH  Google Scholar 

  20. Alvaro P. Raposo, Hans J. Weber, David E. Alvarez-Castillo, Mariana Kirchbach, Romanovski polynomials in selected physics problems. J. Cent. Eur. Phys. 5, 254–273 (2007)

    Google Scholar 

  21. Y. Chargui, A. Trabelsi, L. Chetouani, Exact solution of the (1 + 1)-dimensional Dirac equation with vectorand scalar linear potentials in the presence of a minimal length. Phys. Lett. A 374, 533 (2010)

    ADS  MATH  Google Scholar 

  22. Su Ru-keng, Zhang Yuhong, Exact solutions of the Dirac equation with a linear scalar confining potential in a uniform electric field. J. Phys. A Math. Gen. 17, 854 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  23. R.N. Costa Filho, G. Alencar, B. Skagerstam, J.S. Andrade, Morse potential derived from first principles. J. EPL 101, 10009 (2013)

    Article  ADS  Google Scholar 

  24. D.E. Alvarez-Castillo, Exactly Solvable Potentials and Romanovski Polynomials in Quantum Mechanics, Master Thesis, March 2007. arXiv:0808.1642 [math-ph]

  25. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions Series, vol. 55 (Dover Publications Inc, New York, 1972), p. 37

    MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratoire (L.S.D.C), Département des Sciences de la Matière, Faculté des Sciences Exactes et Sciences de la Vie, Université de Oum El Bouaghi, 04000, Oum El Bouaghi, Algeria

    M. Merad & M. Hadj Moussa

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  1. M. Merad
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  2. M. Hadj Moussa
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Correspondence to M. Merad.

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Merad, M., Hadj Moussa, M. Exact Solution of Klein–Gordon and Dirac Equations with Snyder–de Sitter Algebra. Few-Body Syst 59, 5 (2018). https://doi.org/10.1007/s00601-017-1326-y

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  • DOI: https://doi.org/10.1007/s00601-017-1326-y

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