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arXiv:quant-ph/0307195

Nonrelativistic system of interacting particles in the model of the noncommutative operators of coordinates and momenta of different particles

M. V. Kuzmenko

Published Jul 28, 2003• 0 citations
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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 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.

DOI
10.48550/arxiv.quant-ph/0307195

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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 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.
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.
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 quantum mechanical model for a system of interacting bodies, taking into account noncommutativity of the coordinate and impulse operators for different particles and also the correlation equalities, is considered. The noncommutativity of the operators is here the result of the action of interparticle forces and represents natural generalization of the conventional commutation relation between the coordinate and impulse operators for a single particle. The efficiency of the model is evidenced by the specific calculations of well-known systems of atomic physics.
A quantum-mechanical model for systems of interacting bodies that takes into account noncommutativity of the operators of coordinates and momenta of different particles and correlations of the uncertainties of these quantities is considered. Here, the noncommutativity of these operators is due to the effect of interparticle forces and arises as a natural generalization of the usual commutation relation between the coordinate and momentum operators for a single particle. The efficiency of the model is proven by specific calculations for systems known in atomic and nuclear physics.