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

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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.

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References

  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. Heun, K., Zur Theorie der Riemannschen Functionen zweiter Ordnung mit vier Verzweigungspunkten, Math. Ann., 1889, vol. 33, pp. 161–179.

    Article  MATH  MathSciNet  Google Scholar 

  3. Heun’s Differential Equation, A. Ronveaux, A. and Arscott, F., Eds., Oxford: Oxford Univ. Press, 1995.

    Google Scholar 

  4. Slavyanov, S.Ju. and Lay, W., Special Functions. A Unified Theory Based on Singularities, Oxford, 2000.

    MATH  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. 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. 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. 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.

    MATH  MathSciNet  Google Scholar 

  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.

    MATH  Google Scholar 

  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. 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. 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. Zaitsev, V.F. and Polyanin, A.D., Spravochnik po obyknovennym differentsial’nym uravneniyam (Handbook of Ordinary Differential Equations), Moscow: Fizmatlit, 2011.

    Google Scholar 

  17. Landau, L.D. and Lifshitz, E.M., Kvantovaya mekhanika. Nerelyativistskaya teotiya (Quantum Mechanics: Non-Relativistic Theory), Moscow: Nauka, 1975.

    Google Scholar 

  18. http://reference.wolfram.com/mathematica/ref/DifferentialRoot.html

  19. Birkhoff, G. and Rota, G.C., Ordinary Differential Equations, New York: Wiley, 1989.

    Google Scholar 

  20. http://reference.wolfram.com/mathematica/

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Authors and Affiliations

  1. Stepanov Institute of Physics, National Academy of Sciences of Belarus, pr. Nezavisimosti 68, Minsk, 220072, Belarus

    V. M. Red’kov & A. V. Chichurin

  2. John Paul II Catholic University of Lublin, Al. Racławickie 14, Lublin, 20-950, Poland

    V. M. Red’kov & A. V. Chichurin

Authors
  1. V. M. Red’kov
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  2. A. V. Chichurin
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Correspondence to V. M. Red’kov.

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Original Russian Text © V.M. Red’kov, A.V. Chichurin, 2014, published in Programmirovanie, 2014, Vol. 40, No. 2.

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Red’kov, V.M., Chichurin, A.V. A symbolic-numerical method for solving the differential equation describing the states of polarizable particle in Coulomb potential. Program Comput Soft 40, 86–92 (2014). https://doi.org/10.1134/S0361768814020091

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