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E. M. Ovsiyuk

Publications and source records attributed to E. M. Ovsiyuk.

At least 37 records · Page 2Linked to original sources
arXiv2012

On exact solutions for quantum particles with spin S= 0, 1/2, 1 and de Sitter event horizon

Exact wave solutions for particles with spin 0, 1/2 and 1 in the static coordinates of the de Sitter space-time model are examined in detail. Firstly, for a scalar particle, two pairs of linearly independent solutions are specified explicitly: running and standing waves. A known algorithm for calculation of the reflection coefficient $R_{εj}$ on the background of the de Sitter space-time model is analyzed. It is shown that the determination of R_{εj} requires an additional constrain on quantum numbers ερ/ \hbar c >> j, where ρis a curvature radius. When taken into account of this condition, the R_{εj} vanishes identically. It is claimed that the calculation of the reflection coefficient R_{εj} is not required at all because there is no barrier in an effective potential curve on the background of the de Sitter space-time. The same conclusion holds for arbitrary particles with higher spins, it is demonstrated explicitly with the help of exact solutions for electromagnetic and Dirac fields.

hep-th↗
arXiv2011

Particle with spin 1 in a magnetic field on the hyperbolic plane H_{2}

There are constructed exact solutions of the quantum-mechanical equation for a spin S=1 particle in 2-dimensional Riemannian space of constant negative curvature, hyperbolic plane, 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 vector particle in magnetic field on the 2-dimensional space H_{2} has been found, nonrelativistic and relativistic equations have been solved.

math-ph↗
arXiv2011

Spin 1/2 Particle in the Field of the Dirac String on the background of Anti de Sitter Space-Time

The Dirac monopole string is specified for anti de Sitter cosmological model. Dirac equation for spin 1/2 particle in presence of this monopole has been examined on the background of anti de Sitter space-time in static coordinates. Instead of spinor monopole harmonics, the technique of Wigner D-functions is used. After separation of the variables radial equations have been solved exactly in terms of hypergeometric functions. The complete set of spinor wave solutions has been constructed, the most attention is given to treating the states of minimal values for total moment quantum number j_{min}. At all values of j, the energy spectrum is discrete.

math-ph↗
arXiv2011

Quantum Kepler problem for spin 1/2 particle in spaces on constant curvature. I. Pauli theory

Transition to a nonrelativistic Pauli equation in Riemann space of constant positive curvature for a Dirac particle in presence of the Coulomb field is performed in the system of radial equations, exact solutions are constructed in terms of Hein functions, the energy spectrum is derived. The same is done for the Kepler quantum problem in hyperbolic Lobachevsky space, solutions are constructed in terms of Hein functions, the energy spectrum is obtained.

math-ph↗
arXiv2011

Coulomb problem for a Dirac particle in flat Minkowski space and the Heun functions, extension to curved models

It is shown that there exist several ways to treat the quantum-mechanical Coulomb problem for a Dirac particle in flat Minkowski space with the help of the Heun differential equation, Fuchs's equation with four singular points. When extending the problem to curved spaces of constant curvature, Lobachevsky H_{3} and Rimann S_{3}, there arise 2-nd order differential equations of the Fuchs type with 6 singular points, a method to get relevant equations with five singular points has been elaborated.

math-ph↗
arXiv2011

Particle with spin 1 in a magnetic field on the spherical plane S_{2}

There are constructed exact solutions of the quantum-mechanical equation for a spin S=1 particle in 2-dimensional Riemannian space of constant positive curvature, spherical plane, 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 vector particle in magnetic field on the 2-dimensional space S_{2} has been found, nonrelativistic and relativistic equations have been solved.

math-ph↗
arXiv2011

Spin 1/2 particle in the field of the Dirac string on the background of de Sitter space-time

The Dirac monopole string is specified for de Sitter cosmological model. Dirac equation for spin 1/2 particle in presence of this monopole has been examined on the background of de Sitter space-time in static coordinates. Instead of spinor monopole harmonics, the technique of Wigner D-functions is used. After separation of the variables, detailed analysis of the radial equations is performed; four types of solutions, singular, regular, in- and out- running waves, are constructed in terms of hypergeometric functions. The complete set of spinor wave solutions Ψ_{ε, j,m, λ}(t,r, θ, ϕ) has been constructed, special attention is given to treating the states of minimal values of the total angular moment j_{\min}.

quant-ph↗
arXiv2011

Particle with spin 2 and anomalous magnetic moment in external electromagnetic and gravitational fields

Tensor 50-component form of the first order relativistic wave equation for a particle with spin 2 and anomalous magnetic moment is extended to the case of an arbitrary curved space-time geometry. An additional parameter considered in the presence of only electromagnetic field as related to anomalous magnetic moment, turns to determine additional interaction terms with external geometrical background through Ricci R_{kl} and Riemann R_{klmn} tensors.

math-ph↗
arXiv2011

Exact solutions for a quantum-mechanical particle with spin 1 and additional intrinsic characteristics in a homogeneous magnetic field

With the use of the general covariant matrix 10-dimensional Petiau-Duffin-Kemmer formalism in cylindrical coordinates exact solutions of the quantum-mechanical equation for a particle with spin 1 in the presence of an external homogeneous magnetic field are constructed. Three linearly independent types of solutions are separated; in each case the formula for the energy levels has been found.Within similar technique for the quantum-mechanical equation for a particle with spin 1 and additional intrinsic electromagnetic characteristics - polarizability, exact solutions are found in the presence of an external homogeneous magnetic field.

quant-ph↗
arXiv2011

Spherical waves of spin 1 particle in anti de Sitter space-time

Three possible techniques to deal with a vector particle in the anti de Sitter cosmological model are viewed: Duffin-Kemmer-Petiau matrix formalism based on the general tetrad recipe, group theory 5-dimensional approach based on the symmetry group SO(3.2), anda tetrad form of Maxwell equations in complex Riemann-Silberstein-Majorana-Oppenheimer representation. In the first part, a spin 1 massive field is considered in static coordinates of the anti de Sitter space-time in tetrad-based approach. The complete set of spherical wave solutions with quantum numbers (ε, j,m,l) is constructed; angular dependence in wave functions is described with the help of Wigner functions. The energy quantization rule has been found. Transition to massless case of electromagnetic field is specified, and electromagnetic solutions in Lorentz gauge have been constructed. In the second part, the problem of a particle with spin 1 is considered on the base of 5-dimensional wave equation specified in the same static coordinates. In the third part, a rarely used approach, based on tetrad form of Maxwell equations in complex representation is examined in the anti de Sitter model.

math-ph↗
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

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↗