One-dimensional generalised Coulomb problem

Authors

  • Аlexandre N. Lavrenov Belarusian State Pedagogical University named after Maxim Tank, 18 Savieckaja Street, Minsk 220030, Belarus
  • Ivan A. Lavrenov Octonion Technology, 25 Janki Kupaly Street, Minsk 220030, Belarus

Keywords:

generalised Coulomb problem, curvature, space of constant curvature, Cayley – Klein geometries, factorisation method, one-dimensional space

Abstract

The quantum-mechanical Coulomb problem, complicated in two directions, is considered in this article. The first generalisation is associated with the transition from Euclidean space to one-dimensional Cayley – Klein geometries, and the second one is connected with the addition of a singular term g/x2 to the Coulomb potential. It can be considered as a Calogerо – Sutherland potential, which is used to describe anyons, magnetic monopoles, dyons, etc. In addition to the methodological aspect, the problem under consideration will also be useful as a special case of the so-called coordinate-dependent mass model when describing nanostructures in quantum dots or on a plane, metamaterials and astronomical objects in strong magnetic fields. On the positive coordinate semiaxis, it turns into a certain generalisation of the model with the Kratzer potential, which is used to describe molecular energy and structure, interactions between molecules and non-bounded atoms. Using the factorisation method, the energy spectrum and wave functions of stationary states are found, having the curvature of space as a parameter. The formula for energy levels contains two terms. The first term gives the energy spectrum of the one-dimensional Coulomb problem, and the second term explicitly depends on the presence of curvature and is responsible for the spectrum of the particle on the circle S1(j). The coupling constant g of Calogero – Sutherland potential is non-linearly contained in both terms through a variable β0 (g2 = β0(β0 - 1)) representing an additive correction to the number of the energy level. In the special case of a purely Coulomb field, the results obtained coincide with the results published earlier.

Author Biographies

  • Аlexandre N. Lavrenov, Belarusian State Pedagogical University named after Maxim Tank, 18 Savieckaja Street, Minsk 220030, Belarus

    PhD (physics and mathematics), docent; associate professor at the department of informatics and methods of teaching informatics, faculty of physics and mathematics

  • Ivan A. Lavrenov, Octonion Technology, 25 Janki Kupaly Street, Minsk 220030, Belarus

    leading specialist

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Published

2024-01-18

Issue

No. 1 (2024): Journal of the Belarusian State University. Physics

Section

Theoretical Physics

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How to Cite

(1)
Lavrenov А. N. .; Lavrenov, I. A. . One-Dimensional Generalised Coulomb Problem. Журнал Белорусского государственного университета. Физика 2024, No. 1, 75-82.