INTERPRETATION OF THE FREE MOTION OF PARTICLES IN THE LOBACHEVSKY SPACE IN THE TERM OF THE SCATTERING THEORY
Abstract
The problems of the motion of free particles in the three-dimensional Lobachevsky space are interpreted as scattering by space. The classical and quantum-mechanical cases are considered. A mechanical interpretation of parallel straight lines of the Lobachevsky space is given as the trajectories of non-interacting material points emitted from a point at infinity. Due to the properties of parallel lines in the Lobachevsky space, they can be considered as trajectories of particles scattered at an infinitely distant point. The concept of differential scattering cross sections in the horosphere element for the classical and quantum-mechanical problems is introduced. An analytical expression for the differential cross section in the quantum-mechanical problem is obtained. To derive this expression, we used the solutions of the Schrödinger equation in horospherical coordinates. It is noted that some part of a horosphere is a secant beam of parallel trajectories, can be considered as a model of a two-dimensional flat universe in the three-dimensional space with curvature – Lobachevsky space.
About the Author
Yu. A. Kurochkin
B. I. Stepanov Institute of Physics of the National Academy of Sciences of Belarus
Belarus
Belarus
D. Sc. (Physics and Mathematics), Head of the Center of Theoretical Physics
References
1. Hadamard J. Non Euclidean Geometry in the theory of the automorphic functions. Moskowa, Leningrad, state publishing house of technical literature, 1951. 132 p. (in Russian).
2. Olevskii M. N. Triorthogonal systems in spaces of constant curvature in which the equation Δ2u+λu=0Δ2u+λu=0 allows a complete separation of variables. Matematicheskii Sbornik = Sbornik: Mathematics, 1950, vol. 27, pp. 379–426 (in Russian).
3. Kurochkin Yu. A., Rybak I. Yu., Shelkovyi D. V. Coherent states on horospheric three-dimensional Lobachevsky space. Doklady Natsional’noi akademii nauk Belarusi = Proceedings of the National Academy of Sciences of Belarus, 2014, vol. 58, no 5, pp. 44–48 (in Russian).
4. Kurochkin Yu. A, Rybak I. Yu., Shoukavy D. V. Coherent states on horospheric three-dimensional Lobachevsky space. Journal of Mathematical Physics, 2016, vol. 57, no. 8, p. 082111. Doi: 10.1063/1.4960474
5. Kurochkin Y, Kulchitsky Y., Harkusha S., Russakovich N. Hadron as Coherent State on the Horosphere of the Lobachevsky Momentum Space. Physics of the Particles and Nuclei Letters, 2016, vol. 13, no. 3, pp. 285–288. Doi: 10.1134/ s1547477116030158
6. Ovsiyuk E. M. Exactly solved problems quantum mechanics and classical theory of the field in space with non-Euclidean geometry. Minsk, Republican Institute of Higher Education, 2013, 406 p. (in Russian).
7. Ovsiyuk E. M., Veco O. V. About modeling of the potential barrier in the Shcredinger theory by Lobachevsky space geometry. Vestnic Brestskogo universiteta. Seriya 4, Fizika. Matematika = Brest University Herald. Seriya 4 Physics. Mathamatics, 2011, no. 2, pp. 30–37 (in Russian).
8. Vilenkin N. Ya. Special functions and theory of the group representations. Мoskow, Nauka Publ., 1965. 588 p. (in Russian).
9. Migdal A. B. Quality methods in quantum theory. Мoskow, Nauka Publ., 1975, 335 p. (in Russian).
Review
Views: 652
Email this article 