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Effective Mass and Pseudoscalar Interaction in the Dirac Equation with Woods–Saxon Potential

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Abstract

We consider the one-dimensional Dirac equation with the Woods–Saxon potential in the Framework of position dependent mass and pseudoscalar interaction. By imposing appropriate constraints on the mass function and the pseudoscalar term new exact solvable models are obtained. A detailed study of the scattering and bound-states problems for these models is presented. Meanwhile, we work out the exact expressions for the transmission and reflection probabilities of scattered states and obtain the exact equation for the energy eigenvalues associated to bound states. In particular, transmission resonance at zero-momentum is observed for supercritical states.

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References

  1. Bastard, G.: Wave Mechanics Applied to Semiconductor Hetero-Structures, EDP Sciences. Les Editions de Physique, Les Ulis, France (1992)

  2. Von Roos O.: Position-dependent effective masses in semiconductor theory I. Phys. Rev. B 27, 7547 (1983)

    Article  ADS  Google Scholar 

  3. Von Roos O., Mavromatis H.: Position-dependent effective masses in semiconductor theory II. Phys. Rev. B 31, 2294 (1985)

    Article  ADS  Google Scholar 

  4. Morrow R.A., Brownstein K.R.: Model effective-mass Hamiltonians for abrupt heterojunctions and the associated wave-function-matching conditions. Phys. Rev. B 30, 678 (1984)

    Article  ADS  Google Scholar 

  5. Morrow R.A.: Establishment of an effective-mass Hamiltonian for abrupt heterojunctions. Phys. Rev. B 35, 8074 (1987)

    Article  ADS  Google Scholar 

  6. Gora T., Williams F.: Theory of electronic states and transport in graded mixed semiconductors. Phys. Rev. 177, 1179 (1969)

    Article  ADS  Google Scholar 

  7. Hamdouni Y.: Motion of position-dependent effective mass as a damping–antidamping process: application to the Fermi gas and to the Morse potential. J. Phys. A Math. Gen. 44, 385301 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Mustafa O.: Radial power-law position-dependent mass: cylindrical coordinates, separability and spectral signatures. J. Phys. A Math. Gen. 44, 355303 (2011)

    Article  MATH  Google Scholar 

  9. Morris J.R.: New scenarios for classical and quantum mechanical systems with position-dependent mass. Quantum Stud. Math. Found. 2, 359 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Thomsen J., Einevoll G.T., Hemmer P.C.: Operator ordering in effective-mass theory. Phys. Rev. B 39, 12783 (1989)

    Article  ADS  Google Scholar 

  11. Einevoll G.T., Hemmer P.C., Thomsen J.: Operator ordering in effective-mass theory for heterostructures. I. Comparison with exact results for superlattices, quantum wells, and localized potentials. Phys. Rev. B 42, 3485 (1990)

    Article  ADS  Google Scholar 

  12. Einvoll G.T.: Operator ordering in effective-mass theory for heterostructures. II. Strained systems. Phys. Rev. B 42, 3497 (1990)

    Article  ADS  Google Scholar 

  13. Levy-Leblond J.-M.: Position-dependent effective mass and Galilean invariance. Phys. Rev. A 52, 1845 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  14. Mustafa O., Habib Mazharimousavi S.: Ordering ambiguity revisited via position dependent mass pseudo-momentum operators. Int. J. Theor. Phys. 46, 1786 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Tkachuk V.M., Voznyak O.: Effective Hamiltonian with position-dependent mass and ordering problem. Eur. Phys. J. Plus 130, 161 (2015)

    Article  Google Scholar 

  16. Alhaidari A.D.: Relativistic extension of the complex scaling method. Phys. Rev. A 75, 042707 (2007)

    Article  ADS  Google Scholar 

  17. Novoselov K.S., Geim A.K., Morozov S.V., Jiang D., Zhang Y., Dubonos S.V., Grigorieva I.V., Firsov A.A.: Electric field effect in atomically thin carbon films. Science 306, 666 (2004)

    Article  ADS  Google Scholar 

  18. Novoselov K.S., Geim A.K., Morozov S.V., Jiang D., Katsnelson M.I., Grigorieva I.V., Dubonos S.V., Firsov A.A.: Two-dimensional gas of massless Dirac fermions in graphene. Nature 438, 197 (2005)

    Article  ADS  Google Scholar 

  19. Castro Neto A.H., Guinea F., Peres N.M.R., Novoselov K.S., Geim A.K.: The electronic properties of graphene. Rev. Mod. Phys. 81, 109 (2009)

    Article  ADS  Google Scholar 

  20. Alhaidari A.D.: Solution of the Dirac equation with position-dependent mass in the Coulomb field. Phys. Lett. A 322, 72 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Peng X.-L., Liu J.-Y., Jia C.-S.: Approximation solution of the Dirac equation with position-dependent mass for the generalized Hulthén potential. Phys. Lett. A 352, 478 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. De Souza Dutra A., Jia C.-S.: Classes of exact Klein–Gordon equations with spatially dependent masses: regularizing the one-dimensional inversely linear potential. Phys. Lett. A 352, 484 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Alhaidari A.D., Bahlouli H., Al-Hasan A., Abdelmonem M.S.: Relativistic scattering with a spatially dependent effective mass in the Dirac equation. Phys. Rev. A 75, 062711 (2007)

    Article  ADS  Google Scholar 

  24. Vakarchuk I.O.: The Kepler problem in Dirac theory for a particle with position-dependent mass. J. Phys. A Math. Gen. 38, 4727 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Ikhdair S.M., Sever R.: Approximate bound state solutions of Dirac equation with Hulthén potential including Coulomb-like tensor potential. Appl. Math. Comput. 216, 911 (2010)

    MathSciNet  MATH  Google Scholar 

  26. Ikhdair S.M., Falaye B.J.: Bound states of spatially dependent mass Dirac equation with the Eckart potential including Coulomb tensor interaction. Eur. Phys. J. Plus 129, 1 (2013)

    Article  Google Scholar 

  27. Egrifes H., Sever R.: Bound-State Solutions of the Klein-Gordon Equation for the Generalized PT-Symmetric Hulthén Potential. Int. J. Theor. Phys. 46, 935 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Jia C.-S., Chen T., Cui L.-G.: Approximate analytical solutions of the Dirac equation with the generalized Pöschl–Teller potential including the pseudo-centrifugal term. Phys. Lett. A 373, 1621 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Jia C.-S., De Souza Dutra A.: Extension of PT-symmetric quantum mechanics to the Dirac theory with position-dependent mass. Ann. Phys. 323, 566 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Panella O., Biondini S., Arda A.: New exact solution of the one-dimensional Dirac equation for the Woods–Saxon potential within the effective mass case. J. Phys. A Math. Gen. 43, 325302 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Aydoğdu O., Arda A., Sever R.: Effective-mass Dirac equation for Woods-Saxon potential: scattering, bound states, and resonances. J. Math. Phys. 53, 042106 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Sinha A., Roy P.: (1+1)-dimensional Dirac equation with non-hermitian interaction. Mod. Phys. Lett. A 20, 2377 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Jia C.-S., De Souza Dutra A.: Position-dependent effective mass Dirac equations with PT-symmetric and non-PT-symmetric potentials. J. Phys. A Math. Gen. 39, 11877 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Mustafa O., Mazharimousavi S.H.: Comment on ‘Position-dependent effective mass Dirac equations with PT-symmetric and non-PT-symmetric potentials’. J. Phys. A Math. Gen. 40, 863 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Jia C.-S., Liu J.-Y., Wang P.-Q., Che C.-S.: Trapping neutral fermions with a PT-symmetric trigonometric potential in the presence of position-dependent mass. Phys. Lett. A 369, 274 (2007)

    Article  ADS  Google Scholar 

  36. Mustafa O., Mazharimousavi S.H.: (1+1)-Dirac particle with position-dependent mass in complexified Lorentz scalar interactions: effectively PT-symmetric. Int. J. Theor. Phys. 47, 1112 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Jia C.-S., Wang P.-Q., Liu J.-Y., He S.: Relativistic confinement of neutral fermions with partially exactly solvable and exactly solvable PT-symmetric potentials in the presence of position-dependent mass. Int. J. Theor. Phys. 47, 2513 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. Luis Castro B.: On the Dirac equation with PT-symmetric potentials in the presence of position-dependent mass. Phys. Lett. A 375, 2510 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Kennedy P.: The Woods–Saxon potential in the Dirac equation. J. Phys. A 35, 689 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Senn P.: Threshold anomalies in one-dimensional scattering. Am. J. Phys. 56, 916 (1988)

    Article  ADS  Google Scholar 

  41. Sassoli de Bianchi M.: Levinson’s theorem, zero-energy resonances, and time delay in one-dimensional scattering systems. J. Math. Phys. 35, 2719 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Dombey N., Kennedy P., Calogeracos A.: Supercriticality and transmission resonances in the Dirac equation. Phys. Rev. Lett. 85, 1787 (2000)

    Article  ADS  Google Scholar 

  43. Diaz-Torres A., Scheid W.: Two center shell model with Woods–Saxon potentials: adiabatic and diabatic states in fusion. Nucl. Phys. A 757, 373 (2005)

    Article  ADS  Google Scholar 

  44. Hagino, K., Dasgupta, M., Gontchar, I.I, Hinda, D.J., Morton, C.R., Newton, J.O.: Surface diffuseness anomaly in heavy-ion fusion potentials. In: Kubono, S. et al. (eds.) Proceedings of the Fourth Italy-Japan Symposium on Heavy-Ion Physics, Tokyo, Japan. World Scientific, Singapore, pp. 87–98 (2002)

  45. Molique H., Dudek J.: Fock-space diagonalization of the state-dependent pairing Hamiltonian with the Woods-Saxon mean field. Phys. Rev. C 56, 1795 (1997)

    Article  ADS  Google Scholar 

  46. Costa L.S., Prudente F.V., Acioli P.H., Soares Neto J.J., Vianna J.D.M.: A study of confined quantum systems using the Woods-Saxon potential. J. Phys. B At. Mol. Opt. Phys. 32, 2461 (1999)

    Article  ADS  Google Scholar 

  47. Gradshteyn I.S., Ryzhik I.M.: Table of Integrals Series and Products. Academic Press, New York (2007)

    MATH  Google Scholar 

  48. Dosch H.G., Jensen J.H.D., Müller V.F.: Einige Bemerkungen zum Kleinschen Paradoxon. Phys. Nor. 5, 151 (1971)

    Google Scholar 

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Authors and Affiliations

  1. Physics Department, College of Science and Arts at ArRass, Qassim University, PO Box 53, ArRass, 51921, Saudi Arabia

    Yassine Chargui

  2. Unité de Recherche de Physique Nucléaire et des Hautes Energies, Faculté des Sciences de Tunis, Université de Tunis El Manar, 2092, Tunis, Tunisia

    Yassine Chargui

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  1. Yassine Chargui
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Correspondence to Yassine Chargui.

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Chargui, Y. Effective Mass and Pseudoscalar Interaction in the Dirac Equation with Woods–Saxon Potential. Few-Body Syst 57, 289–306 (2016). https://doi.org/10.1007/s00601-016-1060-x

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