Advertisement

Symbolic-Numerical Calculations of High-|m| Rydberg States and Decay Rates in Strong Magnetic Fields

  • Alexander Gusev
  • Sergue Vinitsky
  • Ochbadrakh Chuluunbaatar
  • Vladimir Gerdt
  • Luong Le Hai
  • Vitaly Rostovtsev
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7442)

Abstract

Symbolic-numeric solving of the boundary value problem for the Schrödinger equation in cylindrical coordinates is given. This problem describes the impurity states of a quantum wire or a hydrogen-like atom in a strong homogeneous magnetic field. It is solved by applying the Kantorovich method that reduces the problem to the boundary-value problem for a set of ordinary differential equations with respect to the longitudinal variables. The effective potentials of these equations are given by integrals over the transverse variable. The integrands are products of the transverse basis functions depending on the longitudinal variable as a parameter and their first derivatives. To solve the problem at high magnetic quantum numbers |m| and study its solutions we present an algorithm implemented in Maple that allows to obtain analytic expressions for the effective potentials and for the transverse dipole moment matrix elements. The efficiency and accuracy of the derived algorithm and that of Kantorovich numerical scheme are confirmed by calculating eigenenergies and eigenfunctions, dipole moments and decay rates of low-excited Rydberg states at high |m|~200 of a hydrogen atom in the laboratory homogeneous magnetic field γ~2.35×10− 5(B~6T).

Keywords

Rydberg State Longitudinal Variable Magnetic Quantum Number Dipole Matrix Element Transverse Variable 
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.
    Chuluunbaatar, O., Gusev, A.A., Vinitsky, S.I., Abrashkevich, A.G.: KANTBP 2.0: New version of a program for computing energy levels, reaction matrix and radial wave functions in the coupled-channel hyperspherical adiabatic approach. Phys. Commun. 179, 685–693 (2008)zbMATHCrossRefGoogle Scholar
  2. 2.
    Gusev, A., Gerdt, V., Kaschiev, M., Rostovtsev, V., Samoylov, V., Tupikova, T., Vinitsky, S.: A Symbolic-Numerical Algorithm for Solving the Eigenvalue Problem for a Hydrogen Atom in Magnetic Field. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2006. LNCS, vol. 4194, pp. 205–218. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Chuluunbaatar, O., Gusev, A.A., Derbov, V.L., Kaschiev, M.S., Melnikov, L.A., Serov, V.V., Vinitsky, S.I.: Calculation of a hydrogen atom photoionization in a strong magnetic field by using the angular oblate spheroidal functions. J. Phys. A 40, 11485–11524 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Gusev, A.A., Derbov, V.L., Krassovitskiy, P.M., Vinitsky, S.I.: Channeling problem for charged particles produced by confining environment. Phys. At. Nucl. 72, 768–778 (2009)CrossRefGoogle Scholar
  5. 5.
    Chuluunbaatar, O., Gusev, A.A., Gerdt, V.P., Rostovtsev, V.A., Vinitsky, S.I., Abrashkevich, A.G., Kaschiev, M.S., Serov, V.V.: POTHMF: A program for computing potential curves and matrix elements of the coupled adiabatic radial equations for a hydrogen-like atom in a homogeneous magnetic field. Comput. Phys. Commun. 178, 301–330 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Gusev, A.A., Chuluunbaatar, O., Gerdt, V.P., Rostovtsev, V.A., Vinitsky, S.I., Derbov, V.L., Serov, V.V.: Symbolic-Numeric Algorithms for Computer Analysis of Spheroidal Quantum Dot Models. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2010. LNCS, vol. 6244, pp. 106–122. Springer, Heidelberg (2010); arXiv:1104.2292CrossRefGoogle Scholar
  7. 7.
    Vinitsky, S.I., Chuluunbaatar, O., Gerdt, V.P., Gusev, A.A., Rostovtsev, V.A.: Symbolic-Numerical Algorithms for Solving Parabolic Quantum Well Problem with Hydrogen-Like Impurity. In: Gerdt, V.P., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2009. LNCS, vol. 5743, pp. 334–349. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. 8.
    Chuluunbaatar, O., Gusev, A., Gerdt, V., Kaschiev, M., Rostovtsev, V., Samoylov, V., Tupikova, T., Vinitsky, S.: A Symbolic-Numerical Algorithm for Solving the Eigenvalue Problem for a Hydrogen Atom in the Magnetic Field: Cylindrical Coordinates. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2007. LNCS, vol. 4770, pp. 118–133. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Gusev, A.A., Vinitsky, S.I., Chuluunbaatar, O., Gerdt, V.P., Rostovtsev, V.A.: Symbolic-Numerical Algorithms to Solve the Quantum Tunneling Problem for a Coupled Pair of Ions. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2011. LNCS, vol. 6885, pp. 175–191. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Chuluunbaatar, O., Gusev, A.A., Vinitsky, S.I., Abrashkevich, A.G.: ODPEVP: A program for computing eigenvalues and eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm-Liouville problem. Comput. Phys. Commun. 180, 1358–1375 (2009)zbMATHCrossRefGoogle Scholar
  11. 11.
    Chuluunbaatar, O., Gusev, A.A., Vinitsky, S.I., Derbov, V.L., Melnikov, L.A., Serov, V.V.: Photoionization and recombination of a hydrogen atom in a magnetic field. Phys. Rev. A 77, 034702–1–034702–4 (2008)Google Scholar
  12. 12.
    Guest, J.R., Choi, J.-H., Raithel, G.: Decay rates of high-|m| Rydberg states in strong magnetic fields. Phys. Rev. A 68, 022509–1–022509–9 (2003)Google Scholar
  13. 13.
    Guest, J.R., Raithel, G.: High-|m| Rydberg states in strong magnetic fields. Phys. Rev. A 68, 052502–1–052502–9 (2003)Google Scholar
  14. 14.
    Abramovits, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Alexander Gusev
    • 1
  • Sergue Vinitsky
    • 1
  • Ochbadrakh Chuluunbaatar
    • 1
  • Vladimir Gerdt
    • 1
  • Luong Le Hai
    • 1
  • Vitaly Rostovtsev
    • 1
  1. 1.Joint Institute for Nuclear ResearchDubnaRussia

Personalised recommendations