One-dimensional generalised Coulomb problem

  • 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

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
Keywords: generalised Coulomb problem, curvature, space of constant curvature, Cayley – Klein geometries, factorisation method, one-dimensional space
How to Cite
LavrenovА. N., & Lavrenov, I. A. (2024). One-dimensional generalised Coulomb problem. Journal of the Belarusian State University. Physics, 1, 75-82. Retrieved from https://journals.bsu.by/index.php/physics/article/view/5882
Issue
No 1 (2024): Journal of the Belarusian State University. Physics
Section
Theoretical Physics

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