Quantum Mechanics of a Spin 1 Particle in the Magnetic Monopole Potential, in Spaces of Euclid and Lobachevsky: Non-Relativistic Approximation

  • E. M. Ovsiyuk I.P. Shamyakin Mozyr State Pedagogical University
  • O. V. Veko I.P. Shamyakin Mozyr State Pedagogical University
  • K. V. Kazmerchuk I.P. Shamyakin Mozyr State Pedagogical University
  • V. V. Kisel M. Tank Belarussian State Pedagogical University
  • V. M. Red’kov B.I. Stepanov Institute of Physics, Nat. Acad. of Sci. of Belarus
Keywords: magnetic monopole, the Duffin–Kemmer–Petiau equation, non-relativistic approximation, space of constant curvature, Coulomb field, oscillator potential

Abstract

A spin-1 particle is treated in the presence of a Dirac magnetic monopole in the non-relativistic approximation. After the separation of variables, the problem is reduced to the system of three interrelated equations, which can be disconnected with the use of a special linear transformation making the mixing matrix diagonal. As a result, there arise three separate differential equations which contain the roots of a cubic algebraic equation as parameters. The algorithm permits the extension to the case where external spherically symmetric fields are present. The cases of the Coulomb and oscillator potentials are treated in detail. The approach is generalized to the case of the Lobachevsky hyperbolic space. The exact solutions of the radial equation are constructed in terms of hypergeometric functions and Heun functions.

References

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https://doi.org/10.1007/BF02902362

V.M. Red'kov, Tetrad Formalism, Spherical Symmetry, and Schr¨odinger Basis (Belorusskaya Nauka, Minsk, 2011) (inRussian).

V.I. Strazhev and L.M. Tomil'chik, Electrodynamics with Magnetic Charge (Nauka i Tekhnika, Minsk, 1975) (in Russian).

D.A. Varshalovich, A.N. Moskalev, and V.K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1983)

V.M. Red'kov, Particle Fields in the Riemannian Space and the Lorentz Group (Belorusskaya Nauka, Minsk, 2009) (in Russian).

Published
2018-10-11
How to Cite
Ovsiyuk, E., Veko, O., Kazmerchuk, K., Kisel, V., & Red’kov, V. (2018). Quantum Mechanics of a Spin 1 Particle in the Magnetic Monopole Potential, in Spaces of Euclid and Lobachevsky: Non-Relativistic Approximation. Ukrainian Journal of Physics, 58(11), 1073. https://doi.org/10.15407/ujpe58.11.1073
Section
Archive