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Doklady of the National Academy of Sciences of Belarus

КОГЕРЕНТНЫЕ СОСТОЯНИЯ НА ОРИСФЕРЕ ТРЕХМЕРНОГО ПРОСТРАНСТВА ЛОБАЧЕВСКОГО

Y. A. Kurochkin, І. Ю. Рибак, Д. В. Шёлковый

Published Jun 10, 2016• 0 citations
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Abstract

The separation of variables in the Laplace–Beltrami operator (Hamiltonian for a free quantum mechanical particle) of three-dimensional Lobachevsky space in the horospherical and quasi Cartesian coordinates provides an opportunity to introduce standard (conventional) coherent states. We consider problems, (oscillator, quantum mechanical particle in the uniform magnetic field) for which introduced coherent states are the adequate method of investigation.

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Coherent states on horospheric three-dimensional Lobachevsky space

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In the paper it is shown that due to separation of variables in the Laplace-Beltrami operator (Hamiltonian of a free quantum particle) in horospheric and quasi-Cartesian coordinates of three dimensional Lobachevsky space, it is possible to introduce standard (“conventional” according to Perelomov [Generalized Coherent States and Their Applications (Springer-Verlag, 1986), p. 320]) coherent states. Some problems (oscillator on horosphere, charged particle in analogy of constant uniform magnetic field) where coherent states are suitable for treating were considered.

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Description of a free quantum-mechanical particle in the Lobachevsky space based on the integral equation

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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.

Proceedings of the National Academy of Sciences of Belarus Physics and Mathematics Series•Oct 7, 2019•0 citations

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М. Н. Сергеенко

The wave equation in quantum mechanics and its general solution in the phase space are obtained.

ФИЗИКА ЭЛЕМЕНТАРНЫХ ЧАСТИЦ И АТОМНОГО ЯДРА•Dec 9, 2024•0 citations
The separation of variables in the Laplace–Beltrami operator (Hamiltonian for a free quantum mechanical particle) of three-dimensional Lobachevsky space in the horospherical and quasi Cartesian coordinates provides an opportunity to introduce standard (conventional) coherent states. We consider problems, (oscillator, quantum mechanical particle in the uniform magnetic field) for which introduced coherent states are the adequate method of investigation.
In the paper it is shown that due to separation of variables in the Laplace-Beltrami operator (Hamiltonian of a free quantum particle) in horospheric and quasi-Cartesian coordinates of three dimensional Lobachevsky space, it is possible to introduce standard (“conventional” according to Perelomov [Generalized Coherent States and Their Applications (Springer-Verlag, 1986), p. 320]) coherent states. Some problems (oscillator on horosphere, charged particle in analogy of constant uniform magnetic field) where coherent states are suitable for treating were considered.
В монографии изложен тетрадный метод обобщения уравнений для частиц различных спинов, учитывающий неевклидовую геометрию пространства-времени. В пространствах постоянной кривизны Лобачевского и Римана найдены точные решения Шредингера Дирака во внешнем магнитном поле. На основе матричного формализма Даффина-Кеммера-Петье в пространстве Минковского найдены точные решения релятивистского уравнения для частицы со спином 1 во внешнем магнитном поле, выполнен анализ этой задачи также в нерелятивистском приближении.
In this study, we construct the coherent states for a particle in the Smorodinsky-Winternitz potentials, which are the generalizations of the two-dimensional harmonic oscillator problem and the Kepler-Coulomb problem. In the first case we find the nonspreading wave packets by transforming the system into four oscillators in Cartesian, and also polar, coordinates. In the second case, the coherent states are constructed in Cartesian coordinates by transforming the system into three nonisotropic harmonic oscillators. All of these states evolve in physical-time. In the third case, the system is transformed into four oscillators and the parametric-time coherent states are constructed in two coordinate frames. In the fourth case, the system is transformed into two oscillators with the reflection symmetry and the parametrictime coherent states are constructed in two coordinate frames.
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.