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arXiv:quant-ph/0009036

Two-particle model with noncommuting operators of coordinates and momenta

M. V. Kuzmenko

Published Sep 8, 2000• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0009036

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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 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.
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 system of two equations is found that has solutions which coincide with the solutions of the Klein‐Gordon equation in the rest frame. This system includes the Schrödinger equation for a free neutral spinless particle. Using the Schrödinger equation as an additional condition for solving the Klein‐Gordon equation in the rest frame leads to two Helmholtz equations. Helmholtz equations can be solved by specifying a particle model and boundary conditions. One of the Helmholtz equations leads to discreteness of the rest masses of relativistic particles.
The Klein-Gordon equation describes the wave-like behavior of spinless particles since it is Lorentz invariant. While it seemed initially ripe for explaining the electronic structure of the hydrogen atom, the lack of a unconditional positive probability density really limited its applications. Yet, it is intimately connected with fermions. Any solution to the Dirac equation is automatically a solution to the Klein-Gordon equation. What is even more surprising, the Klein-Gordon equation for a free particle turns into the Schrödinger equation in the non-relativistic limit. In this work we show that these problems disappear when we use the relativistic mass instead of the rest mass. While the Klein-Gordon equation losses its Lorentz invariance because of this transformation, it gains most of the features present the Schrödinger equation, including the unconditional positivity of probability density, while keeping most of its relativistic characteristics intact, including the matter-wave dispersion relation. What is even more surprising, the non-relativistic, quasi-static limit of the Klein-Gordon equation with relativistic mass is simply the Schrödinger equation under all possible conditions. So, it can be argued that this Klein-Gordon equation is a sort of relativistic Schrödinger equation.