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Journal of Physical Studies 21(4), Article 4002 [7 pages] (2017)
DOI: https://doi.org/10.30970/jps.21.4002 MANY-PARTICLE SYSTEM IN A ROTATIONALLY-INVARIANT SPACE WITH CANONICAL NONCOMMUTATIVITY OF COORDINATESKh. P. Gnatenko, V. M. Tkachuk
Department for Theoretical Physics, Ivan Franko National University of Lviv, |
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The rotationally-invariant noncommutative space with noncommutativity of coordinates of canonical type is studied. The problem of describing the motion of the center-of-mass of a composite system is studied in this space. We find that the motion of the center-of-mass is described with the help of the effective tensor of noncommutativity, which depends on the tensors of noncommutativity of individual particles and on their masses. We show that the motion of the center-of-mass is not independent of the relative motion in a rotationally-invariant noncommutative space. We propose a condition on the parameter in the tensor of coordinate noncommutativity on which noncommutative coordinates can be considered as kinematic variables; the coordinates of the center-of-mass and the coordinates of the relative motion commute; the effective tensor of noncommutativity does not depend on the composition of the system; the representations of the coordinates of the center-of-mass obtained on the basis of their definition and on the basis of the relations of noncommutative algebra are identical. According to this condition, the tensor of coordinate noncommutativity which corresponds to a particle is inversely proportional to its mass. In addition, the Hamiltonian of a system in a noncommutative space with preserved rotational symmetry is analyzed. We show that up to the second order in the $\Delta H$ given by (\ref{dd}) one can consider an effective Hamiltonian. This Hamiltonian is constructed by averaging the Hamiltonian of the system over the degrees of freedom of harmonic oscillators.
PACS number(s): 02.40.Gh, 04.20.Cv