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Nuclear Physics and Atomic Energy

Model of the non-commutative operators of coordinates and momenta of different particles

M. V. Kuzmenko

Published Dec 30, 2005• 0 citations
Publisher

Abstract

It is shown that the Schrödinger equation for a system of interacting particles whose Compton wavelengths are of the same order of magnitude as the system size has contradictions and is not strictly nonrelativistic, because it is based on the implicit assumption that the velocity of propagation of interactions is finite. In the framework of the model of the non-commutative operators of coordinates and momenta of different particles, the equation for wave function which has no above-mentioned drawbacks is deduced. The significant differences from solutions of the non-relativistic Schrödinger equation for large values of the interaction constant are found and the comparison of analogous results for hydrogenlike atoms with experimental data is carried out.

DOI
10.15407/jnpae2005.03.059

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It is shown that the Schrödinger equation for a system of interacting particles whose Compton wavelengths are of the same order of magnitude as the system size has contradictions and is not strictly nonrelativistic, because it is based on the implicit assumption that the velocity of propagation of interactions is finite. In the framework of the model of the non-commutative operators of coordinates and momenta of different particles, the equation for wave function which has no above-mentioned drawbacks is deduced. The significant differences from solutions of the non-relativistic Schrödinger equation for large values of the interaction constant are found and the comparison of analogous results for hydrogenlike atoms with experimental data is carried out.
It is shown that the Schrödinger equation for a system of interacting particles whose Compton wavelengths are of the same order of magnitude as the system size is contradictory and is not strictly nonrelativistic, because it is based on the implicit assumption that the velocity of propagation of interactions is finite. In the framework of the model of the noncommutative operators of coordinates and momenta of different particles, the equation for a wave function which has no above-mentioned drawbacks is deduced. The significant differences from solutions of the nonrelativistic Schrödinger equation for large values of the interaction constant are found, and the comparison of analogous results for hydrogenlike atoms with experimental data is carried out.
A nonrelativistic equation for the system of two interacting particles within the framework of a model with noncommuting operators of coordinates and momenta of different particles is proposed, and a self-consistent system of equations for the wave function of every quantum state is deduced. Solutions for the lowest states of a hydrogenlike atom are found, and the comparison with analogous solutions of the Klein-Gordon equation for the relativistic spinless problem is performed. In the case where the size of a two-particle system and the Compton wavelengths of particles forming it are of the same order, the essential differences with solutions of the Schrödinger nonrelativistic equation are revealed.
It is shown that the Schr\"odinger nonrelativistic equation of a system of interacting particles is not a rigorously nonrelativistic equation since it is based on the implicit assumption of finiteness of the interaction propagation velocity. For a system of interacting particles, a fully nonrelativistic nonlinear system of integro-differential equations is proposed. In the case where the size of the system of particles is of the same order as the Compton wavelength associated with particles, certain essential differences are shown to exist as compared with traditional consequences of the nonrelativistic Schr\"odinger equation.
A new operator approach to the solution of the non-relativistic Schrödinger equation for few-body quantum systems with pair-wise interaction is presented. The problem is solved on the basis of the sequential account of configurations (SAC) method proposed. The wave-function of a few-body system determined with allowance for the configuration interaction (CI-wavefunction) is expressed in terms of the one-configuration wavefunctions with use of the configuration mixing operator (±) introduced in the paper. The examination is carried out with an example of a few-electron atomic system. An analytical expression for the (±) -operator has been represented and an integral equation for the CI-wavefunction has been obtained. It is shown that: (I) the correlation interaction inside an isolated configuration does not destroy the independent particle approximation; (ii) in the framework of the present approach the self-action is excluded. The present approach has enabled to receive an analytical expression for the (±) -operator without applying the non-physical procedure of 'switching-on' the interconfiguration interaction.
A noncanonical quantum system, consisting of two nonrelativistic particles, interacting via a harmonic potential, is considered. The center-of-mass position and momentum operators obey the canonical commutation relations, whereas the internal variables are assumed to be the odd generators of the Lie superalgebra sl(1,2). This assumption implies a set of constraints in the phase space, which are explicitly written in the paper. All finite-dimensional irreducible representations of sl(1,2) are considered. Particular attention is paid to the physical representations, i.e., the representations corresponding to Hermitian position and momentum operators. The properties of the physical observables are investigated. In particular, the operators of the internal Hamiltonian, the relative distance, the internal momentum, and the orbital momentum commute with each other. The spectrum of these operators is finite. The distance between the constituents is preserved in time. It can take no more than three different values. For any non-negative integer or half-integer l there exists a representation, where the orbital momentum is l (in units of 2ℏ). The position of any one of the particles cannot be localized, since the operators of the coordinates do not commute with each other. The constituents are smeared with a certain probability within a finite surface, which moves with a constant velocity together with the center of mass.