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.
Highlights
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
Summary
The quantum mechanical problem of the motion of a free particle in the three-dimensional Lobachevsky space is interpreted as space scattering. 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 uses a quasi-Cartesian coordinate system; 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. Description of a free quantum-mechanical particle in the Lobachevsky space based on the integral equation.
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More From: Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series
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