Advertisement

Programming and Computer Software

, Volume 40, Issue 2, pp 86–92 | Cite as

A symbolic-numerical method for solving the differential equation describing the states of polarizable particle in Coulomb potential

  • V. M. Red’kov
  • A. V. Chichurin
Article

Abstract

Methods for solving the differential equation describing the wave functions of a polarizable particle in the Coulomb potential are discussed. Relations between the coefficients under which the general solution of this equation can be found in analytical form are obtained. For the case of zero polarizability, the general solution to this equation in terms of special functions is obtained; for the first values of the parameter j, plots of the corresponding solutions are presented. For nonzero polarizability and certain specially chosen values of the energy level parameter, solutions possessing the required physical properties for the varying parameter j are constructed on fairly large intervals of the argument values using numerical methods and functional objects of the type DifferentialRoot. Instructions in Mathematica are presented that allow computer-aided analysis using numerical and analytical methods and visualization of the resulting solutions.

Keywords

General Solution Kepler Problem Coulomb Field Functional Object Heun Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Kisel, V., Krylov, G., Ovsiyuk, E., Amirfachrian, M., and Red’kov, V., Wave functions of a particle with polarizability in the Coulomb potential, http://arxiv.org/abs/1109.3302
  2. 2.
    Heun, K., Zur Theorie der Riemannschen Functionen zweiter Ordnung mit vier Verzweigungspunkten, Math. Ann., 1889, vol. 33, pp. 161–179.CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Heun’s Differential Equation, A. Ronveaux, A. and Arscott, F., Eds., Oxford: Oxford Univ. Press, 1995.Google Scholar
  4. 4.
    Slavyanov, S.Ju. and Lay, W., Special Functions. A Unified Theory Based on Singularities, Oxford, 2000.zbMATHGoogle Scholar
  5. 5.
    Ovsiyuk, E.M. and Red’kov, V.M., Coulomb problem for a Dirac particle in Minkowski space and the Heun functions, extension to curved models, http://arxiv.org/abs/1109.5452.
  6. 6.
    Bogush, A.A., Krylov, G.G., Ovsiyuk, E.M., and Red’kov, V.M., Maxwell equations in complex form of Majorana-Oppenheimer, solutions with cylindric symmetry in Riemann S 3 and Lobachevsky H 3 spaces, Ricerche di matematica, 2010, vol. 59, no. 1, pp. 59–96.CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Red’kov, V.M., Ovsiyuk, E.M., and Veko, O.V., Spin 1/2 particle in the field of the Dirac string on the background of de Sitter space-time, Uzhhorod University Scientific Herald, Ser. Physics, 2012, no. 32, pp. 131–140. http://arxiv.org/abs/1109.3009 Google Scholar
  8. 8.
    Ovsiyuk, E.M. and Red’kov, V.M., On simulating a medium with special reflecting properties by Lobachevsky geometry (one exactly solvable electromagnetic problem), http://arxiv.org/abs/1109.0126
  9. 9.
    Otchik, V.S. and Red’kov, V.M., Quantum mechanical Kepler problem in spaces of constant curvature, Preprint of the Institute of Physics, National Academy of Sciences of Belarus, Minsk, 1986, preprint no. 298.Google Scholar
  10. 10.
    Ovsiyuk, E.M., On solutions of Maxwell equations in the space-time of Schwarzschild black hole, NPCS, 2012. vol. 15, no. 1, pp. 81–91.Google Scholar
  11. 11.
    Ovsiyuk, E.M., Quantum Kepler problem for spin 1/2 particle in spaces of constant curvature. I. Pauli theory, NPCS, 2011, vol. 14, no. 1, pp. 14–26.zbMATHMathSciNetGoogle Scholar
  12. 12.
    Kisel, V.V., Ovsiyuk, E.M., and Red’kov, V.M., On the wave functions and energy spectrum for a spin 1 particle in external Coulomb field, NPCS, 2010, vol. 13, no. 4, pp. 352–367.zbMATHGoogle Scholar
  13. 13.
    Ovsiyuk, E., Veko, O., and Amirfakchrian, M., On Schröbinger equation with potential U = −αr −1 + βr + kr 2 and the bi-confluent Heun functions theory, NPCS, 2012, vol. 15, no. 2, pp. 163–170.Google Scholar
  14. 14.
    Shahverdyan, T.A., Mogilevtsev, D.S., Ishkhanyan, A.M., and Red’kov, V.M., Complete-return spectrum for a generalized Rosen-Zener two-state termcrossing model, NPCS, 2013, vol. 16, no. 1, pp. 86–92.Google Scholar
  15. 15.
    Ovsiyuk, E., Florea, O., Chichurin, A. and Red’kov, V., Electromagnetic field in Schwarzschild black hole background. Analytical treatment and numerical simulation, Computer Algebra Systems in Teaching and Research, 2013, vol. IV, no. 1, pp. 204–214.Google Scholar
  16. 16.
    Zaitsev, V.F. and Polyanin, A.D., Spravochnik po obyknovennym differentsial’nym uravneniyam (Handbook of Ordinary Differential Equations), Moscow: Fizmatlit, 2011.Google Scholar
  17. 17.
    Landau, L.D. and Lifshitz, E.M., Kvantovaya mekhanika. Nerelyativistskaya teotiya (Quantum Mechanics: Non-Relativistic Theory), Moscow: Nauka, 1975.Google Scholar
  18. 18.
  19. 19.
    Birkhoff, G. and Rota, G.C., Ordinary Differential Equations, New York: Wiley, 1989.Google Scholar
  20. 20.

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Stepanov Institute of PhysicsNational Academy of Sciences of BelarusMinskBelarus
  2. 2.John Paul II Catholic University of LublinLublinPoland

Personalised recommendations