Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:hep-th/0703263 · Physical Review A

Scattering problem in deformed space with minimal length

M. M. Stetsko, V. M. Tkachuk

Published Jul 16, 2007• 50 citations
arXivHTMLPDF

Abstract

We investigated the elastic scattering problem with deformed Heisenberg algebra leading to the existence of a minimal length. The continuity equations for the moving particle in deformed space were constructed. We obtained the Green's function for a free particle, the scattering amplitude, and the cross section in deformed space. We also calculated the scattering amplitudes and differential cross sections for the Yukawa and the Coulomb potentials in the Born approximation.

DOI
10.1103/physreva.76.012707

Related papers

Elastic Scattering Theory

Ana Marı́a Cetto, Luis de la Peña

Chapter 18 is entirely devoted to the quantum theory of scattering, which is normally covered in more detail in a course on quantum field theory. We introduce the main concepts, including scattering amplitude, differential and total cross sections, and form factors, and derive the main formulas for elastic scattering, in a nonrelativistic framework; this allows us to appreciate how the information obtained from scattering experiments is used to explore the intricacies of quantum particles that are otherwise inaccessible. The Born approximation is studied and applied to scattering by a periodic potential, a weak potential, and Rutherford scattering. The partial wave expansion is derived and applied to obtain the optical theorem. The chapter concludes with a complementary section on resonant scattering.

Cambridge University Press eBooks•Dec 18, 2025•0 citations

Quantum Theory of Scattering

Arjun Sharma

In this chapter we discuss the quantum theory of scattering of incident particles from the target where an incident beam has point particles, scattering is elastic, the scattering potential is time independent, there is no multiple scattering, and the interaction between an incident particle and a scattering center is described by the potential that depends only on relative positions of particles. Scattering cross-section and frame of reference, asymptotic expansion of wave function and formal expression for scattering amplitude, partial wave analysis, integral equation, and the Born approximation are covered in the chapter, in a broader sense. Free particle and asymptotic solutions, scattering amplitude and phase shift, optical theorem, scattering length, phase shift and its interpretation, scattering by a square well potential and by hard sphere potential, and interpretation of the phase shift and its expression are explained with the use of partial wave analysis. Scattering from bare and screened potentials within the Born approximation, validity of the Born approximation, and the transformation from the center of a mass coordinate system to a laboratory coordinate system are written towards the end of the chapter.

CRC Press eBooks•Jun 18, 2021•19 citations

Deformed Heisenberg algebra and its Hilbert space representations

Latévi M. Lawson, Ibrahim Nonkané, Kinvi Kangni

A deformation of Heisenberg algebra induces among other consequences a loss of Hermiticity of some operators that generate this algebra. Therefore, these operators are not Hermitian, nor is the Hamiltonian operator built from them. In the present paper, we propose a position deformation of Heisenberg algebra with both maximal length and minimal momentum uncertainties. By using a pseudo-similarity transformation to the non-Hermitian operators, we prove their Hermiticity with a suitable positive-definite pseudo-metric operator. We then construct Hilbert space representations associated with these pseudo-Hermitian operators. Finally, we study the eigenvalue problem of a free particle in this deformed space and we show that this deformation curved the quantum levels allowing particles to jump from one state to another with low energy transitions.

math-phCross-listedarXiv:2602.15801•Feb 18, 2026•0 citations

Deformed Heisenberg algebra and minimal length

T. Mas Lowski, Anatol Nowicki, V. M. Tkachuk

A one-dimensional deformed Heisenberg algebra [X, P ] = if (P ) is studied.We answer the question: For what function of deformation f (P ) there exists a nonzero minimal uncertainty in position (minimal length).We also find an explicit expression for the minimal length in the case of arbitrary function of deformation.

•Jan 1, 2013•25 citations

Deformed Heisenberg algebra and minimal length

T Masłowski, Anatol Nowicki, V M Tkachuk

A one-dimensional deformed Heisenberg algebra $[X,P]=if(P)$ is studied. We answer the question: For what function of deformation $f(P)$ there exists a nonzero minimal uncertainty in position (minimal length). We also find an explicit expression for the minimal length in the case of arbitrary function of deformation.

arXiv:1201.5545•Journal of Physics A Mathematical and Theoretical•Feb 3, 2012•27 citations

Two-body continuum states in noninteger geometry

E.R. Christensen, Eric Garrido, A. S. Jensen

Wave functions, phase shifts, and corresponding elastic cross sections are investigated for two short-range interacting particles in a deformed external oscillator field. For this we use the equivalent $d$ method employing a noninteger dimension $d$. Using a square-well potential, we derive analytic expressions for scattering lengths and phase shifts. In particular, we consider the dimension, ${d}_{E}$, for infinite scattering length, where the Efimov effect occurs by addition of a third particle. We give explicitly the equivalent continuum wave functions in $d$ and ordinary three-dimensional space, and show that the phase shifts are the same in both methods. Consequently the $d$ method can be used to obtain low-energy two-body elastic cross sections in an external field.

arXiv:2203.01133•Physical Review A•Mar 15, 2022•3 citations
We investigated the elastic scattering problem with deformed Heisenberg algebra leading to the existence of a minimal length. The continuity equations for the moving particle in deformed space were constructed. We obtained the Green's function for a free particle, the scattering amplitude, and the cross section in deformed space. We also calculated the scattering amplitudes and differential cross sections for the Yukawa and the Coulomb potentials in the Born approximation.
Chapter 18 is entirely devoted to the quantum theory of scattering, which is normally covered in more detail in a course on quantum field theory. We introduce the main concepts, including scattering amplitude, differential and total cross sections, and form factors, and derive the main formulas for elastic scattering, in a nonrelativistic framework; this allows us to appreciate how the information obtained from scattering experiments is used to explore the intricacies of quantum particles that are otherwise inaccessible. The Born approximation is studied and applied to scattering by a periodic potential, a weak potential, and Rutherford scattering. The partial wave expansion is derived and applied to obtain the optical theorem. The chapter concludes with a complementary section on resonant scattering.
In this chapter we discuss the quantum theory of scattering of incident particles from the target where an incident beam has point particles, scattering is elastic, the scattering potential is time independent, there is no multiple scattering, and the interaction between an incident particle and a scattering center is described by the potential that depends only on relative positions of particles. Scattering cross-section and frame of reference, asymptotic expansion of wave function and formal expression for scattering amplitude, partial wave analysis, integral equation, and the Born approximation are covered in the chapter, in a broader sense. Free particle and asymptotic solutions, scattering amplitude and phase shift, optical theorem, scattering length, phase shift and its interpretation, scattering by a square well potential and by hard sphere potential, and interpretation of the phase shift and its expression are explained with the use of partial wave analysis. Scattering from bare and screened potentials within the Born approximation, validity of the Born approximation, and the transformation from the center of a mass coordinate system to a laboratory coordinate system are written towards the end of the chapter.
A deformation of Heisenberg algebra induces among other consequences a loss of Hermiticity of some operators that generate this algebra. Therefore, these operators are not Hermitian, nor is the Hamiltonian operator built from them. In the present paper, we propose a position deformation of Heisenberg algebra with both maximal length and minimal momentum uncertainties. By using a pseudo-similarity transformation to the non-Hermitian operators, we prove their Hermiticity with a suitable positive-definite pseudo-metric operator. We then construct Hilbert space representations associated with these pseudo-Hermitian operators. Finally, we study the eigenvalue problem of a free particle in this deformed space and we show that this deformation curved the quantum levels allowing particles to jump from one state to another with low energy transitions.
A one-dimensional deformed Heisenberg algebra [X,P]=if(P)[X,P]=if(P)[X,P]=if(P) is studied. We answer the question: For what function of deformation f(P)f(P)f(P) there exists a nonzero minimal uncertainty in position (minimal length). We also find an explicit expression for the minimal length in the case of arbitrary function of deformation.
Wave functions, phase shifts, and corresponding elastic cross sections are investigated for two short-range interacting particles in a deformed external oscillator field. For this we use the equivalent ddd method employing a noninteger dimension ddd. Using a square-well potential, we derive analytic expressions for scattering lengths and phase shifts. In particular, we consider the dimension, dE{d}_{E}dE​, for infinite scattering length, where the Efimov effect occurs by addition of a third particle. We give explicitly the equivalent continuum wave functions in ddd and ordinary three-dimensional space, and show that the phase shifts are the same in both methods. Consequently the ddd method can be used to obtain low-energy two-body elastic cross sections in an external field.