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arXiv:quant-ph/0703070

Non-relativistic Schroedinger theory on q-deformed quantum spaces I, Mathematical framework and equations of motion

Hartmut Wachter

Published Mar 8, 2007• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0703070

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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.
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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.
The aim of Part II of this paper is to try to describe wave functions on q-deformed versions of position and momentum space. This task is done within the framework developed in Part I of the paper. In order to make Part II self-contained the most important results of Part I are reviewed. Then it is shown that q-deformed exponentials and q-deformed delta functions play the role of momentum and position eigenfunctions, respectively. Their completeness and orthonormality relations are derived. For both bases of eigenfunctions matrix elements of position and momentum operators are calculated. A q-deformed version of the spectral decomposition of multiplication operators is discussed and q-analogs of Heaviside functions are proposed. Interpreting the results from the point of view provided by the concept of quasipoints gives the formalism a physical meaning. The definition of expectation values and the calculation of probability densities are explained in detail. Finally, it is outlined how the considerations so far carry over to antisymmetrized spaces.
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.
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.