• Українська
  • English

< | >

Current issue   Ukr. J. Phys. 2017, Vol. 62, N 4, p.343-348

    Paper

Vasyuta V.M., Tkachuk V.M.

Ivan Franko National University of Lviv, Department for Theoretical Physics
(12, Dragomanov Str., Lviv 79005, Ukraine; e-mail: waswasiuta@gmail.com,
voltkachuk@gmail.com)

Inverse Square potential in a Space with Spin Noncommutativity of Coordinates

Section: General Problems of Theoretical Physics
Original Author's Text: Ukrainian

Abstract: An attractive inverse square potential has been considered in a space with the spin noncommutativity of coordinates. The corresponding effective potential energy, as well as the total energy, is shown to be bounded from below. Using the variational method, the upper limit of the ground-state energy, which turns out to be negative for a sufficiently large coupling constant, is found. As a result, it is proved that the inverse square potential creates stationary levels in the space concerned, unlike the case of commutative space, where a particle falls to the center.

Key words: inverse square potential, noncommutativity.

References:

  1. N. Seiberg, E. Witten. String theory and noncommutative geometry. J. High Energy Phys. 9909, 032 (1999).
    https://doi.org/10.1088/1126-6708/1999/09/032
  2. A. Connes, M. Douglas, A. Schwarz. Noncommutative geometry and matrix theory: Compactification on tori. J. High Energy Phys. 9802, 003 (1998).
  3. S. Doplicher, K. Fredenhagen, J.E. Roberts. The quantum structure of spacetime at the Planck scale and quantum fields. Commun. Math. Phys. 172, 187 (1995).
    https://doi.org/10.1007/BF02104515
  4. H.S. Snyder. Quantized space-time. Phys. Rev. 71, 38 (1947).
    https://doi.org/10.1103/PhysRev.71.38
  5. K.P. Gnatenko, V.M. Tkachuk. Hydrogen atom in rotationally invariant noncommutative space. Phys. Lett. A 378, 3509 (2014).
    https://doi.org/10.1016/j.physleta.2014.10.021
  6. K. Gnatenko, Y. Krynytskyi, V. Tkachuk. Perturbation of the ns levels of the hydrogen atom in rotationally invariant noncommutative space. Mod. Phys. Lett. A 30, 1550033 (2015).
    https://doi.org/10.1142/S0217732315500339
  7. H. Falomir, J. Gamboa, J. Lopez-Sarrion, F. Mendez, P.A.G. Pisani. Magnetic-dipole spin effects in noncommutative quantum mechanics. Phys. Lett. B 680, 384 (2009).
    https://doi.org/10.1016/j.physletb.2009.09.007
  8. V.M. Vasyuta, V.M. Tkachuk. Classical electrodynamics in a space with spin noncommutativity of coordinates. Phys. Lett. B 761, 462 (2016).
    https://doi.org/10.1016/j.physletb.2016.09.001
  9. M. Gomes, V.G. Kupriyanov, A.J. da Silva. Noncommutativity due to spin. Phys. Rev. D 81, 085024 (2010).
    https://doi.org/10.1103/PhysRevD.81.085024
  10. V.M. Vasyuta. Exact solution of harmonical oscillator in space with spin noncommutativity. J. Phys. Stud. 17, 3001 (2013).
  11. V. Efimov. Low-energy properties of three resonantly interacting particles. Sov. J. Nucl. Phys. 29, 546 (1979).
  12. L.V. Hau, M.M. Burns, J.A. Golovchenko. Bound states of guided matter waves: An atom and a charged wire. Phys. Rev. A 45, 6468 (1992).
    https://doi.org/10.1103/PhysRevA.45.6468
  13. J. Denschlag, J. Schmiedmayer. Scattering a neutral atom from a charged wire. Europhys. Lett. 38, 405 (1997).
    https://doi.org/10.1209/epl/i1997-00259-y
  14. J. Denschlag, G. Umshaus, J. Schmiedmayer. Probing a singular potential with cold atoms: A neutral atom and a charged wire. Phys. Rev. Lett. 81, 737 (1998).
    https://doi.org/10.1103/PhysRevLett.81.737
  15. V.M. Tkachuk. Binding of neutral atoms to ferromagnetic wire. Phys. Rev. A 60, 4715 (1999).
    https://doi.org/10.1103/PhysRevA.60.4715
  16. T.R. Govindarajan, V. Suneeta, S. Vaidya. Horizon states for AdS black holes. Nucl. Phys. B 583, 291 (2000).
    https://doi.org/10.1016/S0550-3213(00)00336-9
  17. D. Birmingham, K.S. Gupta, S. Sen. Near-horizon conformal structure of black holes. Phys. Lett. B 505, 191 (2001).
    https://doi.org/10.1016/S0370-2693(01)00354-9
  18. K.S. Gupta, S. Sen. Further evidence for the conformal structure of a Schwarzschild black hole in an algebraic approach. Phys. Lett. B 526, 121 (2002).
    https://doi.org/10.1016/S0370-2693(01)01501-5
  19. S.K. Chakrabarti, K.S. Gupta, S. Sen. Universal nearhorizon conformal structure and black hole entropy. Int. J. Mod. Phys A 23, 2547 (2008).
    https://doi.org/10.1142/S0217751X08040482
  20. M. Bawin. Electron-bound states in the field of dipolar molecules. Phys. Rev. A 70, 022505 (2004).
    https://doi.org/10.1103/PhysRevA.70.022505
  21. M. Bawin, S.A. Coon, B.R. Holstein. Anions and anomalies. Int. J. Mod. Phys. A 22, 4901 (2007).
    https://doi.org/10.1142/S0217751X07038268
  22. A. Alhaidari. Charged particle in the field of an electric quadrupole in two dimensions. J. Phys. A 40, 14843(2007).
    https://doi.org/10.1088/1751-8113/40/49/016
  23. P.R. Giri, K.S. Gupta, S. Meljanac, A. Samsarov. Electron capture and scaling anomaly in polar molecules. Phys. Lett. A 372, 2967 (2008).
    https://doi.org/10.1016/j.physleta.2008.01.008
  24. L.D. Landau and E.M. Lifshitz, Quantum Mechanics. Non-Relativistic Theory (Pergamon Press, 1981) [ISBN 10: 0080291406].
  25. V.M. Vasyuta, V.M. Tkachuk. Falling of a quantum particle in an inverse square attractive potential. Eur. Phys. J. D 70, 267 (2016).
    https://doi.org/10.1140/epjd/e2016-70463-3
  26. D. Bouaziz, M. Bawin. Regularization of the singular inverse square potential in quantum mechanics with a minimal length. Phys. Rev. A 76, 032112 (2007).
    https://doi.org/10.1103/PhysRevA.76.032112
  27. T.V. Fityo, I.O. Vakarchuk, V.M. Tkachuk. WKB approximation in deformed space with minimal length. J. Phys. A 39, 379 (2005).
    https://doi.org/10.1088/0305-4470/39/2/008
  28. A. Das, J. Gamboa, F. M’endez, F. Torres. Generalization of the Cooper pairing mechanism for spin-triplet in superconductors. Phys. Lett. A 375, 1756 (2011).
    https://doi.org/10.1016/j.physleta.2011.02.063
  29. A. Das, H. Falomir, M. Nieto, J. Gamboa, F. M’endez. Aharonov–Bohm effect in a class of noncommutative theories. Phys. Rev. D 84, 045002 (2011).
    https://doi.org/10.1103/PhysRevD.84.045002
  30. V.M. Vasyuta. Corrections to the energy levels of hydrogen atom in space with spin noncommutativity of coordinates. J. Phys. Stud. 18, 4001 (2014).
  31. V.B. Berestetskii, E.M. Lifshitz, and L.P. Pitaevskii, Relativistic Quantum Theory (Pergamon Press, 1982) [ISBN 10: 0750633719].
  32. I.S. Gradshtein, I M. Ryzhik. Table of Integrals, Series, and Products (Academic Press, 1980) [ISBN: 0122947606].