arXiv subjects

V. M. Red'kov

Publications and source records attributed to V. M. Red'kov.

At least 55 records · Page 3Linked to original sources

arXiv2011

On simulating a medium with special reflecting properties by Lobachevsky geometry (One exactly solvable electromagnetic problem)

Lobachewsky geometry simulates a medium with special constitutive relations. The situation is specified in quasi-cartesian coordinates (x,y,z). Exact solutions of the Maxwell equations in complex 3-vector form, extended to curved space models within the tetrad formalism, have been found in Lobachevsky space. The problem reduces to a second order differential equation which can be associated with an 1-dimensional Schrodinger problem for a particle in external potential field U(z) = U_{0} e^{2z}. In quantum mechanics, curved geometry acts as an effective potential barrier with reflection coefficient R=1; in electrodynamic context results similar to quantum-mechanical ones arise: the Lobachevsky geometry simulates a medium that effectively acts as an ideal mirror. Penetration of the electromagnetic field into the effective medium, depends on the parameters of an electromagnetic wave, ω, k_{1}^{2} + k_{2}^{2}, and the curvature radius ρ.

math-ph↗

arXiv2011

Parabolic Coordinates and the Hydrogen Atom in Spaces H_{3} and S_{3}

The Coulomb problem for Schrödinger equation is examined, in spaces of constant curvature, Lobachevsky H_{3} and Riemann S_{3} models, on the base of generalized parabolic coordinates. In contrast to the hyperbolic case, in spherical space S_{3} such parabolic coordinates turn to be complex-valued, with additional constraint on them. The technique of the use of such real and complex coordinates in two space models within the method of separation of variables in Schrödinger equation with Kepler potential is developed in detail; the energy spectra and corresponding wave functions for bound states have been constructed in explicit form for both spaces; connections with Runge-Lenz operators in both curved space models are described.

quant-ph↗

arXiv2011

On the wave functions and energy spectrum for a spin 1 particle in external Coulomb field

Quantum-mechanical system -- spin 1 particle in external Coulomb field is studied on the base of the matrix Duffin-Kemmer-Petiau formalism with the use of the tetrad technique. Separation of the variables is performed with the help of Wigner functions. With the help of parity operator, the radial 10-equation system is divided into two subsystem of 4 and 6 equations that correspond to parity P=(-1)^{j+1} and P= (-1)^{j} respectively. The system of 4 equation is reduced to a second order differential equation which coincides with that arising in the case of a scalar particle in Coulomb potential. It is shown that the 6-equation system reduces to two different second order differential equations for a "main" function. One main equation reduces to a confluent Heun equation and provides us with energy spectrum. Another main equation is a more complex one, and any solutions for it are not constructed.

math-ph↗

arXiv2011

On a Dirac particle in an uniform magnetic field in 3-dimensional spaces of constant curvature

There are constructed exact solutions of the quantum-mechanical Dirac equation for a spin S=1/2 particle in Riemannian space of constant negative curvature, hyperbolic Lobachevsky space, in presence of an external magnetic field, analogue of the homogeneous magnetic field in the Minkowski space. A generalized formula for energy levels, describing quantization of the transversal motion of the particle in magnetic field has been obtained. The same problem is solved for spin 1/2 particle in the space of constant positive curvature, spherical Riemann space. A generalized formula for energy levels, describing quantization of the transversal and along the magnetic field motions of the particle on the background of the Riemann space geometry, is obtained.

math-ph↗

arXiv2010

Exact spectrum for quantum oscillator in spaces of constant curvature from WKB-quantization

Quantum-mechanical WKB-method is elaborated for the known quantum oscillator problem in curved 3-spaces models Euclid, Riemann, and Lobachevsky E_{3}, H_{3}, S_{3} in the framework of the complex variable function theory. Generalized generally covariant Schrödinger equation is considered. In all three space models, exact energy levels are found with the help of constructing special formal WKB-sieries.

math-ph↗

arXiv2010

Motion Caused by Magnetic Field in Lobachevsky Space

We study motion of a relativistic particle in the 3-dimensional Lobachevsky space in the presence of an external magnetic field which is analogous to a constant uniform magnetic field in the Euclidean space. Three integrals of motion are found and equations of motion are solved exactly in the special cylindrical coordinates. Motion on surface of the cylinder of constant radius is considered in detail.

math-ph↗

arXiv2010

Transitivity in the theory of the Lorentz group and the Stokes -- Mueller formalism in optics

Group-theoretical analysis of arbitrary polarization devices is performed, based on the theory of the Lorentz group. In effective "non-relativistic" Mueller case, described by 3-dimensional orthogonal matrices, results of the one polarization measurement determine group theoretical parameters within the accuracy of an arbitrary numerical variable. There are derived formulas, defining Muller parameter of the non-relativistic Mueller device uniquely and in explicit form by by the results of two independent polarization measurements. Analysis is extended to Lorentzian optical devices, described by 4-dimensional Mueller matrices.

math-ph↗

arXiv2010

On exact solutions of the Dirac equation in a homogeneous magnetic field in the Riemann spherical space

There are constructed exact solutions of the quantum-mechanical Dirac equation for a spin S=1/2 particle in the space of constant positive curvature, spherical Riemann space, in presence of an external magnetic field, analogue of the homogeneous magnetic field in the Minkowski space. A generalized formula for energy levels, describing quantization of the motion of the particle in magnetic field on the background of the Riemann space geometry, is obtained.

math-ph↗

arXiv2010

On exact solutions of the Dirac equation in a homogeneous magnetic field in the Lobachevsky space

math-ph↗

arXiv2010

Exact solutions for quantum mechanical particle with spin 1 in the external homogeneous magnetic field

With the use of the general covariant matrix 10-dimensional Petiau -- Duffin -- Kemmer formalism in cylindrical coordinates and tetrad there are constructed exact solutions of the quantum-mechanical equation for a particle with spin 1 in presence of an external homogeneous magnetic field. There are separated three linearly independent types of solutions; in each case the formula for energy levels has been found.

math-ph↗

arXiv2010

Schrödinger particle in magnetic and electric fields in Lobachevsky and Riemann spaces

Schrödinger equation in Lobachevsky and Riemann 4-spaces has been solved in the presence of external magnetic field that is an analog of a uniform magnetic field in the flat space. Generalized Landau levels have been found, modified by the presence of the space curvature. In Lobachevsky4-model the energy spectrum contains discrete and continuous parts, the number of bound states is finite; in Riemann 4-model all energy spectrum is discrete. Generalized Landau levels are determined by three parameters, the magnitude of the magnetic field $B$, the curvature radius $ρ$ and the magnetic quantum number $m$. It has been shown that in presence of an additional external electric field the energy spectrum in the Riemann model can be also obtained analytically.

hep-th↗

arXiv2010

Shapiro's plane waves in spaces of constant curvature and separation of variables in real and complex coordinates

The aim of the article to clarify the status of Shapiro plane wave solutions of the Schrödinger's equation in the frames of the well-known general method of separation of variables. To solve this task, we use the well-known cylindrical coordinates in Riemann and Lobachevsky spaces, naturally related with Euler angle-parameters. Conclusion may be drawn: the general method of separation of variables embraces the all plane wave solutions; the plane waves in Lobachevsky and Riemann space consist of a small part of the whole set of basis wave functions of Schrödinger equation. In space of constant positive curvature $S_{3}$, a complex analog of horospherical coordinates of Lobachevsky space $H_{3}$ is introduced. To parameterize real space $S_{3}$, two complex coordinates $(r,z)$ must obey additional restriction in the form of the equation $r^{2} = e^{z-z^{*}} - e^{2z} $. The metrical tensor of space $S_{3}$ is expressed in terms of $(r,z)$ with additional constraint, or through pairs of conjugate variables $(r,r^{*})$ or $(z,z^{*})$; correspondingly exist three different representations for Schrödinger Hamiltonian. Shapiro plane waves are determined and explored as solutions of Schrödinger equation in complex horosperical coordinates of $S_{3}$. In particular, two oppositely directed plane waves may be presented as exponentials in conjugated coordinates. $Ψ_{-}= e^{-αz}$ and $Ψ_{+}= e^{-αz^{*}}$. Solutions constructed are single-valued, finite, and continuous functions in spherical space and correspond to discrete energy levels.

math-ph↗

arXiv2010

Electromagnetic Waves in the De Sitter Space

5-Dimensional wave equation for a massive particle of spin 1 in the background of de Sitter space-time model is solved in static coordinates. The spherical 5-dimensional vectors $A_{a}, a= 1,...,5$ of three types, $j,j+1, j-1$ are constructed. In massless case they give electromagnetic wave solutions, obeying the Lorentz condition. 5-form of equations in massless case is used to produce recipe to build electromagnetic wave solutions of the types $Π, E,M$; the first is trivial and can be removed by a gauge ransformation. The recipe is specified to produce spherical $Π, E, M$ solutions in static coordinates.

hep-th↗

arXiv2010

Classical Particle in Presence of Magnetic Field, Hyperbolic Lobachevsky and Spherical Riemann Models

Motion of a classical particle in 3-dimensional Lobachevsky and Riemann spaces is studied in the presence of an external magnetic field which is analogous to a constant uniform magnetic field in Euclidean space. In both cases three integrals of motions are constructed and equations of motion are solved exactly in the special cylindrical coordinates on the base of the method of separation of variables. In Lobachevsky space there exist trajectories of two types, finite and infinite in radial variable, in Riemann space all motions are finite and periodical. The invariance of the uniform magnetic field in tensor description and gauge invariance of corresponding 4-potential description is demonstrated explicitly. The role of the symmetry is clarified in classification of all possible solutions, based on the geometric symmetry group, SO(3,1) and SO(4) respectively.

math-ph↗

arXiv2009

Maxwell Equations in Complex Form of Majorana - Oppenheimer, Solutions with Cylindric Symmetry in Riemann S_{3} and Lobachevsky H_{3} Spaces

Complex formalism of Riemann - Silberstein - Majorana - Oppenheimer in Maxwell electrodynamics is extended to the case of arbitrary pseudo-Riemannian space - time in accordance with the tetrad recipe of Tetrode - Weyl - Fock - Ivanenko. In this approach, the Maxwell equations are solved exactly on the background of static cosmological Einstein model, parameterized by special cylindrical coordinates and realized as a Riemann space of constant positive curvature. A discrete frequency spectrum for electromagnetic modes depending on the curvature radius of space and three parameters is found, and corresponding basis electromagnetic solutions have been constructed explicitly. In the case of elliptical model a part of the constructed solutions should be rejected by continuity considerations. Similar treatment is given for Maxwell equations in hyperbolic Lobachevsky model, the complete basis of electromagnetic solutions in corresponding cylindrical coordinates has been constructed as well, no quantization of frequencies of electromagnetic modes arises.

math-ph↗

arXiv2009

Maxwell equations in matrix form, squaring procedure, separating the variables, and structure of electromagnetic solutions

The Riemann -- Silberstein -- Majorana -- Oppenheimer approach to the Maxwell electrodynamics in vacuum is investigated within the matrix formalism. The matrix form of electrodynamics includes three real 4 \times 4 matrices. Within the squaring procedure we construct four formal solutions of the Maxwell equations on the base of scalar Klein -- Fock -- Gordon solutions. The problem of separating physical electromagnetic waves in the linear space λ_{0}Ψ^{0}+λ_{1}Ψ^{1}+λ_{2}Ψ^{2}+ lambda_{3}Ψ^{3} is investigated, several particular cases, plane waves and cylindrical waves, are considered in detail.

math-ph↗

arXiv2009

Maxwell Equations in Complex Form, Spherical Waves in Spaces of Constant Curvature of Lobachevsky and Riemann

Complex formalism of Riemann - Silberstein - Majorana - Oppenheimer in Maxwell electrodynamics is extended to the case of arbitrary pseudo-Riemannian space - time in accordance with the tetrad recipe of Tetrode - Weyl - Fock - Ivanenko. In this approach, the Maxwell equations are solved exactly on the background of simplest static cosmological models, spaces of constant curvature of Riemann and Lobachevsky parameterized by spherical coordinates. Separation of variables is realized in the basis of Schrödinger -- Pauli type, description of angular dependence in electromagnetic complex 3-vectors is given in terms of Wigner D-functions. In the case of compact Riemann model a discrete frequency spectrum for electromagnetic modes depending on the curvature radius of space and three discrete parameters is found. In the case of hyperbolic Lobachevsky model no discrete spectrum for frequencies of electromagnetic modes arises.

math-ph↗

arXiv2009

Maxwell equations in Duffin - Kemmer tetrad form, spherical waves in Riemann space S_3

The Duffin-Kemmer form of massless vector field (Maxwell field) is extended to the case of arbitrary pseudo-Riemannian space-time in accordance with the tetrad recipe of Tetrode-Weyl-Fock-Ivanenko. In this approach, the Maxwell equations are solved exactly on the background of simplest static cosmological model, space of constant curvature of Riemann parameterized by spherical coordinates. Separation of variables is realized in the basis of Schrödinger- Pauli type, description of angular dependence in electromagnetic filed functions is given in terms of Wigner D-functions. A discrete frequency spectrum for electromagnetic modes depending on the curvature radius of space and three discrete parameters is found. 4-potentials for spherical electro- magnetic waves of magnetic and electric type have been constructed.

math-ph↗