© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Journal of Physics A Mathematical and Theoretical

Information theoretic measures for the Dunkl–Coulomb problem

Sami Ghazouani, Aymen Mannai

Published Jan 29, 2025• 2 citations

Abstract

Abstract Motivated by the conclusion section of Ghazouani and Sboui (2020 J. Phys. A: Math. Theor. 53 035202), we merge together the invariance algebra of the Dunkl–Coulomb problem and the Lie algebra so ( 2 , 1 ) into a bigger new algebras. Next, we introduce several information theoretic measures of the Dunkl–Coulomb problem: Using the Dunkl transform and the canonical Dunkl transform, we obtain an analytical expression for the momentum wave functions of the Dunkl–Coulomb problem. We give an analytical expression for the expectation values of the position ⟨ r a ⟩ n m and momentum ⟨ λ a ⟩ n m in various bound states of the Dunkl–Coulomb problem. With these, a Heisenberg-type inequality for the three-dimensional linear canonical Dunkl transform is derived and is applied to the bound states of the Dunkl–Coulomb problem.

DOI: 10.1088/1751-8121/adaff0

Related papers

A Clifford Algebra of Signature (n,3n) and the Density Operators of Quantum Information Theory

Nolmar Melo, Carlile Lavor

Advances in Applied Clifford Algebras•Jun 5, 2012•3 citations

Dunkl–Klein–Gordon equation in higher dimensions

B. Hamil, B. C. Lütfüoğlu, M. Merad

In this study, we replace the standard partial derivatives in the Klein–Gordon equation with Dunkl derivatives and obtain exact analytical solutions for the eigenvalues and eigenfunctions of the Dunkl–Klein–Gordon equation in higher dimensions. We apply this formalism to two key quantum mechanical systems: the d-dimensional harmonic oscillator and the Coulomb potential. First, we introduce Dunkl quantum mechanics in d-dimensional hyperspherical coordinates, followed by an analysis of the d-dimensional Dunkl–Klein–Gordon oscillator. Subsequently, we derive the energy spectrum and eigenfunctions, which are expressed using confluent hypergeometric functions. Furthermore, we examine the impact of the Dunkl formalism on both the eigenvalues and eigenfunctions. In the second case, we explore both the bound-state solutions and scattering scenarios of the Dunkl–Klein–Gordon equation with the Coulomb potential. The bound-state solutions are represented in terms of confluent hypergeometric functions, while the scattering states enable us to compute the particle creation density and probability using the Bogoliubov transformation method.

arXiv:2409.12655•Modern Physics Letters A•May 28, 2025•10 citations

Information theoretic measures in one-dimensional Dunkl oscillator

Debraj Nath, Niladri Ghosh, Amlan K. Roy

We consider the solution of one dimensional Schrödinger Dunkl equation for energies and eigenfunctions. Then we provide analytical expressions for various information theoretic measures. For a given density function, quantities such as position expectation value, entropic moment, disequilibrium, Rényi entropy, Shannon entropy, Tsallis entropy, Fisher information are presented. Next, a few relative information measures corresponding to two density functions, like relative entropy, relative Fisher, relative Rényi, relative Tsallis, along with their associated Jensen divergences such as Jensen–Shannon divergence, Jensen–Fisher divergence, Jensen–Rényi divergence, Jensen–Tsallis divergence are treated. Sample results are provided in graphical form. Dependence of these quantities on the Dunkl parameter μ shows distinct features for μ < 0 and μ > 0.

Journal of Mathematical Physics•Aug 1, 2024•11 citations

Dynamical symmetries of the Calogero-Coulomb model

Tigran Hakobyan

We construct the dynamical symmetry of the quantum Calogero model with particle exchange in a confining Coulomb field. This symmetry is governed by the algebra $so(N+1,2)$, deformed by exchange (Dunkl) operators, with its invariant sector generated by the Dunkl angular momentum tensor and the modified Laplace-Runge-Lenz vector. The equidistant analogue of the Hamiltonian, with a linear spectrum, is expressed in terms of the conformal subalgebra $so(1,2)$. In addition, the wave functions of the Calogero-Coulomb Hamiltonian are classified into infinite-dimensional lowest-weight $so(1,2)$ multiplets.

hep-thCross-listedarXiv:2603.21325•Mar 24, 2026

Quantum Information Theory

Yorick Hardy, Willi-Hans Steeb

The concepts of classical information theory can be extended to quantum information theory. Since in general measurement yields a result with probability, we may suggest using these probabilities in classical information theory. However the probabilities do not contain phase information, which cannot be neglected. Thus the definitions are given in terms of the density operator. These probabilities depend on the basis used for measurement. A density operator ρ over a n-dimensional Hilbert space H is a positive operator with unit trace. The trace tr(A) is defined as $$tr(A): = \sum\limits_{j = 1}^n {\left\langle {{\beta _j}|A|{\beta _j}} \right\rangle } $$ where β j for j = 1,..., n is any orthonormal basis in HThus tr(P)=1. The eigenvalues of a density operator are greater than zero. By the spectral theorem every density operator can be represented as a mixture of pure states $$\rho = \sum\limits_{j = 1}^n {Pj|} \left\langle {aj} \right.\langle aj|$$ where α j for j = 1,..., n are the orthonormal eigenvectors of ρ (which form a basis in H and $$pj \in R,{p_j}0,\sum\limits_{j = 1}^n {{p_j}} = 1$$

Birkhäuser Basel eBooks•Jan 1, 2001•8 citations

Correlations in quantum many-body systems and quantum information theoretic measures

晃太郎加藤

•Mar 23, 2017•0 citations

Abstract Motivated by the conclusion section of Ghazouani and Sboui (2020 J. Phys. A: Math. Theor. 53 035202), we merge together the invariance algebra of the Dunkl–Coulomb problem and the Lie algebra so ( 2 , 1 ) into a bigger new algebras. Next, we introduce several information theoretic measures of the Dunkl–Coulomb problem: Using the Dunkl transform and the canonical Dunkl transform, we obtain an analytical expression for the momentum wave functions of the Dunkl–Coulomb problem. We give an analytical expression for the expectation values of the position ⟨ r a ⟩ n m and momentum ⟨ λ a ⟩ n m in various bound states of the Dunkl–Coulomb problem. With these, a Heisenberg-type inequality for the three-dimensional linear canonical Dunkl transform is derived and is applied to the bound states of the Dunkl–Coulomb problem.

In this study, we replace the standard partial derivatives in the Klein–Gordon equation with Dunkl derivatives and obtain exact analytical solutions for the eigenvalues and eigenfunctions of the Dunkl–Klein–Gordon equation in higher dimensions. We apply this formalism to two key quantum mechanical systems: the d-dimensional harmonic oscillator and the Coulomb potential. First, we introduce Dunkl quantum mechanics in d-dimensional hyperspherical coordinates, followed by an analysis of the d-dimensional Dunkl–Klein–Gordon oscillator. Subsequently, we derive the energy spectrum and eigenfunctions, which are expressed using confluent hypergeometric functions. Furthermore, we examine the impact of the Dunkl formalism on both the eigenvalues and eigenfunctions. In the second case, we explore both the bound-state solutions and scattering scenarios of the Dunkl–Klein–Gordon equation with the Coulomb potential. The bound-state solutions are represented in terms of confluent hypergeometric functions, while the scattering states enable us to compute the particle creation density and probability using the Bogoliubov transformation method.

We consider the solution of one dimensional Schrödinger Dunkl equation for energies and eigenfunctions. Then we provide analytical expressions for various information theoretic measures. For a given density function, quantities such as position expectation value, entropic moment, disequilibrium, Rényi entropy, Shannon entropy, Tsallis entropy, Fisher information are presented. Next, a few relative information measures corresponding to two density functions, like relative entropy, relative Fisher, relative Rényi, relative Tsallis, along with their associated Jensen divergences such as Jensen–Shannon divergence, Jensen–Fisher divergence, Jensen–Rényi divergence, Jensen–Tsallis divergence are treated. Sample results are provided in graphical form. Dependence of these quantities on the Dunkl parameter μ shows distinct features for μ < 0 and μ > 0.

We construct the dynamical symmetry of the quantum Calogero model with particle exchange in a confining Coulomb field. This symmetry is governed by the algebra

so(N+1,2)

, deformed by exchange (Dunkl) operators, with its invariant sector generated by the Dunkl angular momentum tensor and the modified Laplace-Runge-Lenz vector. The equidistant analogue of the Hamiltonian, with a linear spectrum, is expressed in terms of the conformal subalgebra

so(1,2)

. In addition, the wave functions of the Calogero-Coulomb Hamiltonian are classified into infinite-dimensional lowest-weight

so(1,2)

multiplets.

The concepts of classical information theory can be extended to quantum information theory. Since in general measurement yields a result with probability, we may suggest using these probabilities in classical information theory. However the probabilities do not contain phase information, which cannot be neglected. Thus the definitions are given in terms of the density operator. These probabilities depend on the basis used for measurement. A density operator ρ over a n-dimensional Hilbert space H is a positive operator with unit trace. The trace tr(A) is defined as

tr(A): = \sum\limits_{j = 1}^n {\left\langle {{\beta _j}|A|{\beta _j}} \right\rangle }

where β j for j = 1,..., n is any orthonormal basis in HThus tr(P)=1. The eigenvalues of a density operator are greater than zero. By the spectral theorem every density operator can be represented as a mixture of pure states

\rho = \sum\limits_{j = 1}^n {Pj|} \left\langle {aj} \right.\langle aj|

where α j for j = 1,..., n are the orthonormal eigenvectors of ρ (which form a basis in H and

pj \in R,{p_j}0,\sum\limits_{j = 1}^n {{p_j}} = 1