Description of a free quantum-mechanical particle in the Lobachevsky space based on the integral equation
https://doi.org/10.29235/1561-2430-2019-55-3-319-324
Abstract
The quantum mechanical problem of the motion of a free particle in the three-dimensional Lobachevsky space is interpreted as space scattering. The quantum case is considered on the basis of the integral equation derived from the Schrödinger equation. The work continues the problem considered in [1] studied within the framework of classical mechanics and on the basis of solving the Schrödinger equation in quasi-Cartesian coordinates. The proposed article also uses a quasi-Cartesian coordinate system; however after the separation of variables, the integral equation is derived for the motion along the axis of symmetry horosphere axis coinciding with the z axis. The relationship between the scattering amplitude and the analytical functions is established. The iteration method and finite differences for solution of the integral equation are proposed.
Keywords
Lobachevsky space,
horosphere,
coordinates,
Schrödinger equation,
scattering,
scattering amplitude,
analytical presentation,
method,
iterations,
finite differences
Supporting agencies: The author is grateful to Doctor V. M. Redkov for useful comments and the participants of the seminar of the Center «Fundamental Interactions and Astrophysics» of the B. I. Stepanov Institute of Physics of the National Academy of Sciences of Belarus for valuable discussion of the work.
About the Author
Yu. A. Kurochkin
B. I. Stepanov Institute of Physics of the National Academy of Sciences of Belarus
Belarus
Belarus
Yurii A. Kurochkin – Dr. Sc. (Physics and Mathematics), Head of the Center of the Center «Fundamental Interactions and Astrophysics»
68-2, Nezavisimosti Ave., 220072, Minsk, Republic of Belarus
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