Skip to main content
SpringerLink
Log in

Cusp Interaction in Minimal Length Quantum Mechanics

  • Published:
Few-Body Systems Aims and scope Submit manuscript

Abstract

The modified Schrödinger equation with a minimal length is considered under a Cusp potential which includes the exponential interaction. Next, exact analytical solutions of the problem are reported and thereby the scattering states as well as the corresponding transmission and reflection coefficients are reported.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Konishi K., Paffuti G., Provero P.: Minimum physical length and the generalized uncertainty principle in string theory. Phys. Lett. B 234, 276 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  2. Maggiore M.: A generalized uncertainty principle in quantum gravity. Phys. Lett. B 304, 65 (1993)

    Article  ADS  Google Scholar 

  3. Maggiore M.: Quantum groups, gravity, and the generalized uncertainty principle. Phys. Rev. D 49, 5182 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  4. Maggiore M.: The algebraic structure of the generalized uncertainty principle. Phys. Lett. B 319, 83 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  5. Garay L.J.: Quantum gravity and minimum length. Int. J. Mod. Phys. A 10, 145 (1995)

    Article  ADS  Google Scholar 

  6. Kempf A., Mangano G., Mann R.B.: Hilbert space representation of the minimal length uncertainty relation. Phys. Rev. D 52, 1108 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  7. Brau F.: Minimal length uncertainty relation and the hydrogen atom. J. Phys. A 32, 7691 (1999)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Scardigli F.: Generalized uncertainty principle in quantum gravity from micro-black hole gedanken experiment. Phys. Lett. B 452, 39 (1999)

    Article  ADS  Google Scholar 

  9. Ran Y., Xue L., Hu S., Su R.-K.: On the Coulomb-type potential of the one-dimensional Schrödinger equation. J. Phys. A 33, 9265 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Amelino-Camelia G.: Testable scenario for relativity with minimum length. Phys. Lett. B 510, 255 (2001)

    Article  ADS  MATH  Google Scholar 

  11. Chang L.N., Minic D., Okamura N., Takeuchi T.: Exact solution of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations. Phys. Rev. D 65, 125027 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  12. Scardigli F., Casadio R.: Generalized uncertainty principle, extra dimensions and holography. Class. Quantum Gravity 20, 3915 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Magueijo J., Smolin L.: String theories with deformed energy-momentum relations, and a possible nontachyonic bosonic string. Phys. Rev. D 71, 026010 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  14. Cortes J.L., Gamboa J.: Quantum uncertainty in doubly special relativity. Phys. Rev. D 71, 065015 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  15. Nozari K., Azizi T.: Some aspects of gravitational quantum mechanics. Gen. Relativ. Gravit. 38, 735 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. Nouicer K.: Pauli-Hamiltonian in the presence of minimal lengths. J. Math. Phys. 47, 122102 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  17. Battisti M.V., Montani G.: The Big-Bang singularity in the framework of a Generalized Uncertainty Principle. Phys. Lett. B 656, 96 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Slawny J.: Position and length operators in a theory with minimal length. J. Math. Phys. 48, 052108 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  19. Fityo T.V., Vakarchuk I.O., Tkachuk V.M.: The WKB approximation in the deformed space with the minimal length and minimal momentum. J. Phys. A Math. Theor. 41, 045305 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  20. Bambi C., Urban F.R.: Natural extension of the generalized uncertainty principle. Class. Quantum Gravity 25, 095006 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  21. Das S., Vagenas E.C.: Universality of Quantum Gravity Corrections. Phys. Rev. Lett. 101, 221301 (2008)

    Article  ADS  Google Scholar 

  22. Das S., Vagenas E.C.: Phenomenological implications of the generalized uncertainty principle. Can. J. Phys. 87, 233 (2009)

    Article  ADS  Google Scholar 

  23. Das S., Vagenas E.C., Ali A.F.: Discreteness of space from GUP II: Relativistic wave equations. Phys. Lett. B 690, 407 (2010)

    Article  ADS  Google Scholar 

  24. Hossain G.M., Husain V., Seahra S.S.: Background-independent quantization and the uncertainty principle. Class. Quantum Gravity 27, 165013 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  25. Bouaziz D., Ferkous N.: Hydrogen atom in momentum space with a minimal length. Phys. Rev. A 82, 022105 (2010)

    Article  ADS  Google Scholar 

  26. Ali A.F., Das S., Vagenas E.C.: Proposal for testing quantum gravity in the lab. Phys. Rev. D 84, 044013 (2011)

    Article  ADS  Google Scholar 

  27. Hassanabadi H., Zarrinkamar S., Maghsoodi E.: Scattering states of Woods-Saxon interaction in minimal length quantum mechanics. Phys. Lett. B 718, 678–682 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  28. Hassanabadi H., Zarrinkamar S., Rajabia A.A.: A simple efficient methodology for Dirac equation in minimal length quantum mechanics. Phys. Lett. B 718, 1111–1113 (2013)

    Article  ADS  Google Scholar 

  29. Jiang Y., Dong S.H., Antillón A., Lozada-Cassou M.: Low momentum scattering of the Dirac particle with an asymmetric cusp potential. Eur. Phys. J. C 45, 525 (2006)

    Article  ADS  Google Scholar 

  30. Villalba V.M., Greiner W.: Transmission resonances and supercritical states in a one-dimensional cusp potential. Phys. Rev. A 67, 052707 (2003)

    Article  ADS  Google Scholar 

  31. Villalba V.M., Rojas C.: Bound states of the Klein-Gordon equation in the presence of short range potentials. Int. J. Mod. Phys. A 21, 313–326 (2006)

    Article  ADS  Google Scholar 

  32. Adams F.C. et al.: Orbital Instabilities in a Triaxial Cusp Potential. Astrophys. J. 670, 1027 (2007)

    Article  ADS  Google Scholar 

  33. Arda A., Aydoğdu O., Sever R.: Scattering and bound state solutions of the asymmetric Hulthén potential. Phys. Scr. 84, 025004 (2011)

    Article  ADS  Google Scholar 

  34. Villalba V.M., Rojas C.: Scattering of a relativistic scalar particle by a cusp potential. Phys. Lett. A 362, 21 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Basic Sciences, Shahrood Branch, Islamic Azad University, Shahrood, Iran

    H. Hassanabadi & E. Maghsoodi

  2. Department of Basic Sciences, Garmsar Branch, Islamic Azad University, Garmsar, Iran

    S. Zarrinkamar

Authors
  1. H. Hassanabadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Zarrinkamar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. Maghsoodi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Maghsoodi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hassanabadi, H., Zarrinkamar, S. & Maghsoodi, E. Cusp Interaction in Minimal Length Quantum Mechanics. Few-Body Syst 55, 255–263 (2014). https://doi.org/10.1007/s00601-014-0875-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00601-014-0875-6

Keywords

Search

Navigation