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Journal of Nonlinear Mathematical Physics

A Model of Control of an Equation for Two-Electron Interaction

В. И. Соколов

Published Jan 1, 1997• 0 citations
Publisher

Abstract

Quantum Schrödinger equation, describing dynamical spin-interaction of two electrons with external magnetic field, is considered as an object for cybernetic research. Indeed, because of having a possibility to change external magnetic field, we can influence the interaction of particles. The algorithm of investigation of the algebraic structure of the spin-system is given. A specification of this algorithm is essentially connected with a control aspect. The spin-system is decomposed into six subsystems by us-ing the found algebraic structure. Five from them are one-dimensional noninteract subsystems. Every from the latter has his own one-dimensional control. 1 Description of the system Let us consider a Schrödinger equation for the spin-state X (t) of two spin-interacting electrons in an external inhomogeneous magnetic field [1] ih̄ d dt

DOI
10.2991/jnmp.1997.4.1-2.24

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Quantum Schrödinger equation, describing dynamical spin-interaction of two electrons with external magnetic field, is considered as an object for cybernetic research. Indeed, because of having a possibility to change external magnetic field, we can influence the interaction of particles. The algorithm of investigation of the algebraic structure of the spin-system is given. A specification of this algorithm is essentially connected with a control aspect. The spin-system is decomposed into six subsystems by us-ing the found algebraic structure. Five from them are one-dimensional noninteract subsystems. Every from the latter has his own one-dimensional control. 1 Description of the system Let us consider a Schrödinger equation for the spin-state X (t) of two spin-interacting electrons in an external inhomogeneous magnetic field [1] ih̄ d dt
Quantum control of elementary particle at nucleon had already been investigated for a couple of decades at the frontier area both in quantum physics and system control. At the nucleus, a kind of control for particles motion had been considered in the form of Klein-Gordon-Schrödinger equation, which is a system describing the Yukawa interaction between nucleon (e.g. proton, neutron) and meson. The aim of this paper is trying to find an external force to act at one of the particles to change the particle from its ground state to reach its exciting state.
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An exact evaluation of memory functions to describe dynamic localization and excitation transfer in a small quantum system is provided to clarify the connection between two recently reported phenomena, one in quantum optics and the other in condensed matter physics. The possibility of quantum control on the basis of the calculation, with relevance to systems as widely different as electrons in a crystal, atoms interacting with light, spins in crossed magnetic fields, and Bose condensates falling in a gravitation field, is briefly mentioned. The connection to the interacting electron−phonon system and to nonlinear Schroedinger equations is pointed out.
Abstract The density-matrix-renormalization-group method is used to study the influence of external magnetic field on the stability of the excitonic phase induced by local hybridization in the one-dimensional spin-1/2 Falicov-Kimball model. It is shown that a fine tuning of external magnetic field through the quantum critical point is able to induce significant changes (continuous as well as discontinuous) in the excitonic ⟨ d + f ⟩ expectation average providing new set of physical properties and technological applications, like possibilities of faster switching ferroelectrics and controlling their optical properties with magnetic fields. Moreover, effects of some other interactions (which may be present in real d-f materials) on the stability of excitonic phase are examined and it is shown that the Hubbard interaction in the d-electron subsystem, the interband d-f Coulomb interaction, as well as the exchange d-f interaction stabilize significantly excitonic correlations below the quantum critical point, while the anisotropic spin-dependent interaction of the Ising type between f and d electrons as well as the f-electron hopping suppress the excitonic correlations for magnetic fields below as well as above the quantum critical point.
Based on the Schrodinger equation, exact expressions for the non-relativistic particle energy in the local external field and the external field potential are derived as inhomogeneous density functionals. On this basis, it is shown that, when considering more than two noninteracting electrons, the energy of such a system cannot be an inhomogeneous density functional. The result is extended for the system of interacting electrons. This means that the Hohenberg-Kohn lemma which assert that in the ground state to each inhomogeneous density corresponds only one potential of the external field cannot be a justification of the existence of the universal density functional in the general case. At the same time, statements of the density functional theory remain valid when considering any number of noninteracting ground-state bosons due to the Bose condensation effect.