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V. M. Tkachuk

Publications and source records attributed to V. M. Tkachuk.

At least 55 records · Page 3Linked to original sources
arXiv2017

Noncommutative phase space with rotational symmetry and hydrogen atom

We construct algebra with noncommutativity of coordinates and noncommutativity of momenta which is rotationally invariant and equivalent to noncommutative algebra of canonical type. Influence of noncommutativity on the energy levels of hydrogen atom is studied in rotationally invariant noncommutative phase space. We find corrections to the levels up to the second order in the parameters of noncommutativity and estimate the upper bounds of these parameters.

quant-ph↗
arXiv2017

Geometric characteristics of quantum evolution: curvature and torsion

We study characteristics of quantum evolution which can be called curvature and torsion. The curvature shows a deviation of the state vector in quantum evolution from the geodesic line. The torsion shows a deviation of state vector from the plane of evolution (a two-dimensional subspace) at a given time.

quant-ph↗
arXiv2016

Weak equivalence principle in noncommutative phase space and the parameters of noncommutativity

The weak equivalence principle is studied in a space with noncommutativity of coordinates and noncommutativity of momenta. We find conditions on the parameters of noncommutativity which give the possibility to recover the equivalence principle in noncommutative phase space. It is also shown that in the case when these conditions are satisfied the motion of the center-of-mass of a composite system in noncommutative phase space and the relative motion are independent, the kinetic energy of composite system has additivity property and is independent on the systems composition. So, we propose conditions on the parameters of noncommutativity which give the possibility to solve the list of problems in noncommutative phase space.

gr-qc↗
arXiv2016

Quantifying geometric measure of entanglement by mean value of spin and spin correlations for pure and mixed states

We quantify the geometric measure of entanglement in terms of mean values of observables of entangled system. For pure states we find the relation of geometric measure of entanglement with the mean value of spin one-half for the system composed of spin and arbitrary quantum system. The geometric measure of entanglement for mixed states of rank-2 is studied as well. We find the explicit expression for geometric entanglement and the relation of entanglement in this case with the values of spin correlations. These results allow to find experimentally the value of entanglement by measuring a value of the mean spin and the spin correlations for pure and mixed states, respectively. The obtained results are applied for calculation of entanglement during the evolution in spin chain with Ising interaction , two-spin Ising model in transverse fluctuating magnetic field, Schrödinger cat in fluctuating magnetic field.

quant-ph↗
arXiv2016

Two-particle system in noncommutative space with preserved rotational symmetry

We consider a system of two particles in noncommutative space which is rotationally invariant. It is shown that the coordinates of the center-of-mass position and the coordinates of relative motion satisfy noncommutative algebra with corresponding effective tensors of noncommutativity. The hydrogen atom is studied as a two-particle system. We find the corrections to the energy levels of the hydrogen atom up to the second order over the parameter of noncommutativity.

quant-ph↗
arXiv2016

Classical electrodynamics in a space with spin noncommutativity of coordinates

We propose a new relativistic Lorentz-invariant spin-noncommutative algebra. Using the Weyl ordering of noncommutative position operators, we build an analogue of the Moyal-Groenewald product for the proposed algebra. The Lagrange function of an electromagnetic field in the space with spin noncommutativity is constructed. In such a space electromagnetic field becomes non-abelian. A gauge transformation law of this field is also obtained. Exact nonlinear field equations of noncommutative electromagnetic field are derived from the least action principle. Within the perturbative approach we consider field of a point charge in a constant magnetic field and interaction of two plane waves. An exact solution of a plane wave propagation in a constant magnetic and electric fields is found.

hep-th↗
arXiv2016

One-dimensional Coulomb-like problem in general case of deformed space with minimal length

We present a definition of the two-sided inverse of position operator in general case of deformed Heisenberg algebra leading to minimal length. Energy spectrum and eigenfunctions in momentum space for 1D Coulomb-like potential in deformed space are found exactly. We analyse the energy spectrum for different partial cases of deformation function and find that correction due to the deformation highly depends on type of the deformation function.

quant-ph↗
arXiv2016

Effect of coordinate noncommutativity on the mass of a particle in a uniform field and the equivalence principle

We consider the motion of a particle in a uniform field in noncommutative space which is rotationally invariant. On the basis of exact calculations it is shown that there is an effect of coordinate noncommutativity on the mass of a particle. A particular case of motion of a particle in a uniform gravitational field is considered and the equivalence principle is studied. We propose the way to solve the problem of violation of the equivalence principle in the rotationally invariant noncommutative space.

quant-ph↗
arXiv2015

Geometry of a two-spin quantum state in evolution

We study the quantum evolution of a two-spin system described by the isotropic Heisenberg Hamiltonian in the external magnetic field. It is shown that this evolution happens on a two-parametric closed manifold. The Fubini-Study metric of this manifold is obtained. It is found that this is the metric of the torus. The entanglement of the states which belong to this manifold is investigated.

quant-ph↗
arXiv2015

Exactly solvable problems in the momentum space with a minimum uncertainty in position

A new approach in solution of simple quantum mechanical problems in deformed space with minimal length is presented. We propose the generalization of Schroëdinger equation in momentum representation on the case of deformed Heisenberg algebra with minimal length. Assuming that the kernel of potential energy operator do not change in the case of deformation, we obtain exact solution of eigenproblem of a particle in delta potential as well as double delta potential. Particle in Coulomb like potential is revisited and the problem of inversibility and hermicity of inverse coordinate operator is solved.

quant-ph↗
arXiv2015

Effective Hamiltonian with position dependent mass and ordering problem

We derive the effective low energy Hamiltonian for the tight-binding model with the hopping integral slowly varying along the chain. The effective Hamiltonian contains the kinetic energy with position dependent mass, which is inverse to the hopping integral, and effective potential energy. Changing of ordering in the kinetic energy leads to change of the effective potential energy and leaves the Hamiltonian the same one. Therefore, we can choose arbitrary von Roos ordering parameters in the kinetic energy without changing the Hamiltonian. Moreover, we propose a more general form for the kinetic energy than that of von Roos, which nevertheless together with the effective potential energy represent the same Hamiltonian.

quant-ph↗
arXiv2015

The quantum brachistochrone problem for an arbitrary spin in a magnetic field

We consider quantum brachistochrone evolution for a spin-$s$ system on rotational manifolds. Such manifolds are determined by the rotation of the eigenstates of the operator of projection of spin-$s$ on some direction. The Fubini-Study metrics of these manifolds are those of spheres with radii dependent on the value of the spin and on the value of the spin projection. The conditions for optimal evolution of the spin-$s$ system on rotational manifolds are obtained.

quant-ph↗
arXiv2014

Hydrogen atom in rotationally invariant noncommutative space

We consider the noncommutative algebra which is rotationally invariant. The hydrogen atom is studied in a rotationally invariant noncommutative space. We find the corrections to the energy levels of the hydrogen atom up to the second order in the parameter of noncommutativity. The upper bound of the parameter of noncommutativity is estimated on the basis of the experimental results for 1s-2s transition frequency.

quant-ph↗
arXiv2014

Preparation of quantum states of two spin-$\frac{1}{2}$ particles in the form of the Schmidt decomposition

We consider a system of two spins that are coupled via an isotropic Heisenberg Hamiltonian. For the first time, a two-step method for the preparation of an arbitrary quantum state of two qubits in the form of the Schmidt decomposition is proposed. The simplified version of this method is applied to the physical system of an atom having with a nuclear spin $1/2$ and one valence electron. As an example, the preparation of two-spin quantum states in the $^{31}$P system is considered.

quant-ph↗
arXiv2014

Quantum mechanical analogy and supersymmetry of electromagnetic wave modes in planar waveguides

We study an analogue of the equations describing TE and TM modes in a planar waveguide with an arbitrary continuous dependence of the electric permittivity and magnetic permeability on coordinates with the stationary Schrödinger equation. The effective potential energies involved in the Schrödinger equation for TE and TM modes are found. In general, the effective potential energies for TE and TM modes are different but in the limit of a weak dependence of the permittivity and permeability on coordinates they coincide. In the case when the product of a position-dependent permittivity and permeability is constant, it means that the refractive index is constant, we find that the TE and TM modes are described by the supersymmtric quantum mechanics.

quant-ph↗
arXiv2013

Relation of deformed nonlinear algebras with linear ones

The relation between nonlinear algebras and linear ones is established. For one-dimensional nonlinear deformed Heisenberg algebra with two operators we find the function of deformation for which this nonlinear algebra can be transformed to a linear one with three operators. We also establish the relation between Lie algebra of total angular momentum and corresponding nonlinear one. This relation gives a possibility to simplify and to solve the eigenvalue problem for the Hamiltonian in a nonlinear case using the reduction of this problem to the case of linear algebra. It is demonstrated on the example of harmonic oscillator.

math-ph↗
arXiv2013

Effect of the generalized uncertainty principle on Galilean and Lorentz transformations

Generalized Uncertainty Principle (GUP) was obtained in string theory and quantum gravity and suggested the existence of a fundamental minimal length which, as was established, can be obtained within the deformed Heisenberg algebra. We use the deformed commutation relations or in classical case (studied in this paper) the deformed Poisson brackets, which are invariant with respect to the translation in configurational space. We have found transformations relating coordinates and times of moving and rest frames of reference in the space with GUP in the first order over parameter of deformation. For the non-relativistic case we find the deformed Galilean transformation which is similar to the Lorentz one written for Euclidean space with signature $(+,+,+,+)$. The role of the speed of light here plays some velocity $u$ related to the parameter of deformation, which as we estimate is many order of magnitude larger than the speed of light $u\simeq 1.2 \times 10^{22} c$. The coordinates of the rest and moving frames of reference for relativistic particle in the space with GUP satisfy the Lorentz transformation with some effective speed of light. We estimate that the relative deviation of this effective speed of light $\tilde c$ from $c$ is ${(\tilde c-c)/ c}\simeq 3.5\times 10^{-45}$. The influence of GUP on the motion of particle and the Lorentz transformation in the first order over parameter of deformation is hidden in $1/c^2$ relativistic effects.

quant-ph↗