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arXiv:2208.05723

Scattering of a particle on the $q$-deformed Euclidean space

Wachter, Hartmut

Published Aug 11, 2022• 0 citations
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Abstract

We develop a formalism for the scattering of a particle on the $q$-deformed Euclidean space. We write down $q$-versions of the Lippmann-Schwinger equation. Their iterative solutions for a weak scattering potential lead us to $q$-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 $q$-deformed Euclidean space and a discussion of a $q$-version of time-dependent perturbation theory conclude our studies.

DOI
10.48550/arxiv.2208.05723

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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.
This is the third part of a paper about non-relativistic Schroedinger theory on q-deformed quantum spaces like the braided line or the three-dimensional q-deformed Euclidean space. Propagators for the free q-deformed particle are derived and their basic properties are discussed. A time-dependent formulation of scattering is proposed. In this respect, q-analogs of the Lippmann-Schwinger equation are given. Expressions for their iterative solutions are written down. It is shown how to calculate S-matrices and transition probabilities. Furthermore, attention is focused on the question what becomes of unitarity of S-matrices in a q-deformed setting. The examinations are concluded by a discussion of the interaction picture and its relation to scattering processes.
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.
The aim of these three papers (I, II, and III) is to develop a q-deformed version of non-relativistic Schroedinger theory. Paper I introduces the fundamental mathematical and physical concepts. The braided line and the three-dimensional q-deformed Euclidean space play the role of position space. For both cases the algebraic framework is extended by a time element. A short review of the elements of q-deformed analysis on the spaces under consideration is given. The time evolution operator is introduced in a consistent way and its basic properties are discussed. These reasonings are continued by proposing q-deformed analogs of the Schroedinger and the Heisenberg picture.
We summarize some basics about mathematical tools of analysis for the q-deformed Euclidean space. We use the new tools to examine q-deformed eigenfunctions of the momentum or position operator within the framework of the star product formalism. We show that these two systems of functions are complete and orthonormal. With the q-deformed momentum or position eigenfunctions, we calculate matrix elements of the momentum or position operator. Considerations about expectation values and probability densities conclude the studies.
This is the second part of a paper about a q-deformed analog of non-relativistic Schroedinger theory. It applies the general ideas of part I and tries to give a description of one-particle states on q-deformed quantum spaces like the braided line or the q-deformed Euclidean space in three dimensions. Hamiltonian operators for the free q-deformed particle in one as well as three dimensions are introduced. Plane waves as solutions to the corresponding Schroedinger equations are considered. Their completeness and orthonormality relations are written down. Expectation values of position and momentum observables are taken with respect to one-particle states and their time-dependence is discussed. A potential is added to the free-particle Hamiltonians and q-analogs of the Ehrenfest theorem are derived from the Heisenberg equations of motion. The conservation of probability is proved.