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:quant-ph/0607025

Deformed Heisenberg algebra with upper bound of momentum value

T V Fityo

Published Jul 4, 2006• 0 citations
arXivHTMLPDF

Abstract

We consider one dimensional deformed Heisenberg algebra leading to existence of minimal length for coordinate operator and minimal and maximal uncertainty of momentum operator. For this algebra an exactly solvable Hamiltonian is constructed.
DOI
10.48550/arxiv.quant-ph/0607025

Related papers

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

Algebraic Solution of the Harmonic Oscillator With Minimal Length Uncertainty Relations

Kay L. Gemba, Zvonimir Hlousek, Zoltán Papp

In quantum mechanics with minimal length uncertainty relations the Heisenberg-Weyl algebra of the one-dimensional harmonic oscillator is a deformed SU(1,1) algebra. The eigenvalues and eigenstates are constructed algebraically and they form the infinite-dimensional representation of the deformed SU(1,1) algebra. Our construction is independent of prior knowledge of the exact solution of the Schrödinger equation of the model. The approach can be generalized to the $D$-dimensional oscillator with non-commuting coordinates.

arXiv:0712.2078•Dec 13, 2007•2 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

Hilbert space representations for Hermitian position-deformed Heisenberg algebra and path integral formulation

Thomas Katsekpor, Latévi M. Lawson, Prince K. Osei +1 more

Position deformation of a Heisenberg algebra and Hilbert space representation of both maximal length and minimal momentum uncertainties may lead to loss of Hermiticity of some operators that generate this algebra. Consequently, the Hamiltonian operator constructed from these operators is also not Hermitian. In this paper, with an appropriate positive-definite Dyson map, we establish the Hermiticity of these operators by means of a similarity transformation. We then construct Hilbert space representations associated with these Hermitian operators that generate a Hermitian Heisenberg algebra. With the help of these representations, we establish the path integral formulation of any systems in this Hermitian algebra. Finally, using the path integral of a free particle as an example, we demonstrate that the Euclidean propagator, action, and kinetic energy of this system are constrained by the standard classical mechanics limits.

arXiv:2404.07082•Reviews in Mathematical Physics•Aug 25, 2025•0 citations

Minimal length scale in quantum mechanics from a deformation quantization perspective

Ziemowit Domański

Abstract We develop a deformation quantization approach to quantum mechanics that exhibits a nonzero minimal uncertainty in position through the modification of commutation relations for the operators of position and momentum. Deformation quantization is a method of quantizing a classical Hamiltonian system by suitably deforming the Poisson algebra of the system. The developed theory of deformation quantization is non-formal. An appropriate integral formula for the star-product is introduced, along with a suitable space of functions on which the star-product is well defined. A C*-algebra of observables and a space of states are constructed. Moreover, an operator representation in momentum space is presented. Finally, an example of states of maximal localization is given in terms of generalized Wigner functions.

Journal of Physics Conference Series•Dec 1, 2024•0 citations
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.
In quantum mechanics with minimal length uncertainty relations the Heisenberg-Weyl algebra of the one-dimensional harmonic oscillator is a deformed SU(1,1) algebra. The eigenvalues and eigenstates are constructed algebraically and they form the infinite-dimensional representation of the deformed SU(1,1) algebra. Our construction is independent of prior knowledge of the exact solution of the Schrödinger equation of the model. The approach can be generalized to the DDD-dimensional oscillator with non-commuting coordinates.
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.
Position deformation of a Heisenberg algebra and Hilbert space representation of both maximal length and minimal momentum uncertainties may lead to loss of Hermiticity of some operators that generate this algebra. Consequently, the Hamiltonian operator constructed from these operators is also not Hermitian. In this paper, with an appropriate positive-definite Dyson map, we establish the Hermiticity of these operators by means of a similarity transformation. We then construct Hilbert space representations associated with these Hermitian operators that generate a Hermitian Heisenberg algebra. With the help of these representations, we establish the path integral formulation of any systems in this Hermitian algebra. Finally, using the path integral of a free particle as an example, we demonstrate that the Euclidean propagator, action, and kinetic energy of this system are constrained by the standard classical mechanics limits.
Abstract We develop a deformation quantization approach to quantum mechanics that exhibits a nonzero minimal uncertainty in position through the modification of commutation relations for the operators of position and momentum. Deformation quantization is a method of quantizing a classical Hamiltonian system by suitably deforming the Poisson algebra of the system. The developed theory of deformation quantization is non-formal. An appropriate integral formula for the star-product is introduced, along with a suitable space of functions on which the star-product is well defined. A C*-algebra of observables and a space of states are constructed. Moreover, an operator representation in momentum space is presented. Finally, an example of states of maximal localization is given in terms of generalized Wigner functions.