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arXiv:2103.03356

Conservation laws for a $q$-deformed nonrelativistic particle

Hartmut Wachter

Published Feb 13, 2021• 0 citations
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Abstract

We derive $q$-versions of Green's theorem from the Leibniz rules of partial derivatives for the $q$-deformed Euclidean space. Using these results and the Schrödinger equations for a $q$-deformed nonrelativistic particle, we derive continuity equations for the probability density, the energy density, and the momentum density of a $q$-deformed nonrelativistic particle.

DOI
10.48550/arxiv.2103.03356

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We derive qqq-versions of Green's theorem from the Leibniz rules of partial derivatives for the qqq-deformed Euclidean space. Using these results and the Schrödinger equations for a qqq-deformed nonrelativistic particle, we derive continuity equations for the probability density, the energy density, and the momentum density of a qqq-deformed nonrelativistic particle.
Nonrelativistic one-particle problem on qqq-deformed Euclidean space
We consider time-dependent Schrödinger equations for a free nonrelativistic particle on the three-dimensional qqq-deformed Euclidean space. We determine plane wave solutions to these Schrödinger equations and show that they form a complete orthonormal system. We derive qqq-deformed expressions for propagators of a nonrelativistic particle. Considerations about expectation values for position or momentum of a nonrelativistic particle conclude our studies.
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Transparent closed forms of the radial SOq{\mathrm{SO}}_{\mathit{q}}SOq​(N) Schr\"odinger equation acting on the noncommutative quantum Euclidean space are written both for l=0 and l\ensuremath{\ne}0. Starting from the s-wave equation leads to the derivation of an alternative q analog of the radial Schr\"odinger equation in N space dimensions exhibiting a modified symmetry, now in terms of wave functions depending on the radial coordinate only. The harmonic oscillator and the Coulomb system are treated as concrete examples. These results open the way to the derivation of q-deformed (1/N)-energy formulas for arbitrary spherically symmetrical potentials. Comparisons with exact SOq{\mathrm{SO}}_{\mathit{q}}SOq​(N) results, as well as further interpretations, have also been made.