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

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

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…

$so(2,1)$ scattering via Euclidean Connection for the Natanzon Hypergeometric Potentials

The $so(2,1)$ Euclidean Connection formalism is used to calculate the $S$ matrix for the Hypergeometric Natanzon Potentials ($HNP$).

A note on scattering in deformed space with minimal length

We consider the elastic scattering in deformed space with minimal length. We give the basic relation for the elastic scattering in deformed space. We also investigate the partial wave method in deformed space. It is shown that the relations for the scattering amplitude and cross-section formally coincides with ordinary ones.

Scattering and Bound States of a Deformed Quantum Mechanics

We construct the exact position representation of a deformed quantum mechanics which exhibits an intrinsic maximum momentum and use it to study problems such as a particle in a box and scattering from a step potential, among others. In particular, we show that unlike usual quantum mechanics, the present deformed case delays the formation of bound states in a finite potential well. In the process we also highlight some limitations and pit-falls of low-momentum or perturbative treatments and thus resolve two puzzles occurring in the literature.

Non-relativistic Schroedinger theory on q-deformed quantum spaces II, The free non-relativistic particle and its interactions

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 moment…

Wigner时间延迟与Hartman效应：沿变形黎曼流形的量子运动

Wigner time delay and Hartman effect in quantum motion along deformed Riemannian manifolds

本研究分析变形一维黎曼流形中的弹性量子散射，发现局部曲率变化导致共振态和Wigner时间延迟，低能下与经典时间延迟不符，中高能下两者一致，并探讨了Hartman效应。

Conservation laws for a $q$-deformed nonrelativistic particle

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.

$q$-Deformed Relativistic Fermion Scattering

In this article, after introducing a kind of q-deformation in quantum mechanics, first, q-deformed form of Dirac equation in relativistic quantum mechanics is derived. Then three important scat erring problem in physics are studied. All results have satisfied what we had expected before. Furthermore, effects of all parameters in the problems on the reflection and transmission coefficients are calculated and shown graphically.

Generalized MICZ-Kepler system, duality, polynomial and deformed oscillator algebras

We present the quadratic algebra of the generalized MICZ-Kepler system in three-dimensional Euclidean space $E_{3}$ and its dual the four dimensional singular oscillator in four-dimensional Euclidean space $E_{4}$. We present their realization in terms of a deformed oscillator algebra using the Daskaloyannis construction. The structure constants are in these cases function not only of the Hamiltonian but also of other integrals commuting with all generators of the quadratic algebra. We also present a new algebraic derivation of the energy spectrum of the MICZ-Kepler system on the three sphere …

Quantum oscillator and Kepler-Coulomb problems in curved spaces: deformed shape invariance, point canonical transformations, and rational extensions

The quantum oscillator and Kepler-Coulomb problems in $d$-dimensional spaces with constant curvature are analyzed from several viewpoints. In a deformed supersymmetric framework, the corresponding nonlinear potentials are shown to exhibit a deformed shape invariance property. By using the point canonical transformation method, the two deformed Schrödinger equations are mapped onto conventional ones corresponding to some shape-invariant potentials, whose rational extensions are well known. The inverse point canonical transformations then provide some rational extensions of the oscillator and Ke…