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arXiv:2010.08826

Nonrelativistic one-particle problem on $q$-deformed Euclidean space

Hartmut Wachter

Published Oct 17, 2020• 0 citations
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Abstract

We consider time-dependent Schrödinger equations for a free nonrelativistic particle on the three-dimensional $q$-deformed Euclidean space. We determine plane wave solutions to these Schrödinger equations and show that they form a complete orthonormal system. We derive $q$-deformed expressions for propagators of a nonrelativistic particle. Considerations about expectation values for position or momentum of a nonrelativistic particle conclude our studies.

DOI
10.48550/arxiv.2010.08826

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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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Conservation laws for a qqq-deformed nonrelativistic particle
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.
Scattering of a particle on the qqq-deformed Euclidean space
We develop a formalism for the scattering of a particle on the qqq-deformed Euclidean space. We write down qqq-versions of the Lippmann-Schwinger equation. Their iterative solutions for a weak scattering potential lead us to qqq-versions of the Born series. With the expressions for the wave functions of the scattered particle, we can write down S-matrix elements. We show that these S-matrix elements satisfy unitarity conditions. Considerations about the interaction picture for a quantum system in the qqq-deformed Euclidean space and a discussion of a qqq-version of time-dependent perturbation theory conclude our studies.
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