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:1005.1903 · Europhysics Letters (EPL)

Complexity analysis of Klein-Gordon single-particle systems

Daniel Manzano, Sheila López‐Rosa, J. S. Dehesa

Published May 1, 2010• 6 citations
arXivHTMLPDF

Abstract

The Fisher-Shannon complexity is used to quantitatively estimate the contribution of relativistic effects to the internal disorder of Klein-Gordon single-particle Coulomb systems which is manifest in the rich variety of three-dimensional geometries of its corresponding quantum-mechanical probability density. It is observed that, contrary to the non-relativistic case, the Fisher-Shannon complexity of these relativistic systems does depend on the potential strength (nuclear charge). This is numerically illustrated for pionic atoms. Moreover, its variation with the quantum numbers ( n , l , m ) is analysed in various ground and excited states. It is found that the relativistic effects enhance when n and/or l are decreasing.

DOI
10.1209/0295-5075/90/48001

Related papers

Relativistic Klein–Gordon charge effects by information-theoretic measures

Daniel Manzano, Rafael J. Yáñez, J. S. Dehesa

The charge spreading of the ground and excited states of Klein-Gordon particles moving in a Coulomb potential is quantitatively analysed by means of ordinary moments and the Heisenberg measure as well as by using the most relevant information-theoretic measures of global (Shannon entropic power) and local (Fisher information) types.The dependence of these complementary quantities on the nuclear charge Z and the quantum numbers characterizing the physical states is carefully discussed.The comparison of relativistic Klein-Gordon and non-relativistic Schrödinger values is made.Nonrelativistic limits at large principal quantum number n and for small values of Z are also reached.Contents 1. Introduction 2 2. Radial expectation values and Heisenberg's measure 5 3. Shannon and Fisher information measures 8 4. Conclusions 12

arXiv:1001.1228•New Journal of Physics•Feb 11, 2010•13 citations

Relativistic investigation of fisher information and Shannon entropy for Schioberg plus Yukawa potential

I. B. Okon, Akaninyene Daniel Antia, Cecilia N. Isonguyo +6 more

Quantum information entropies have been properly studied within the confines of nonrelativistic quantum mechanics ( QM ) by solving Schrödinger wave equation for different considerable potential models to obtained the energy and total normalised wave function which are essential tools for determining various information entropies. However, not much research has been done on evaluation of information entropies within the relativistic quantum mechanics because of its complexities and existence of both multiple and quadratic eigenvalues for fixed quantum state ( n ) and orbital angular quantum number \((\ell )\) . In this work, we investigate Fisher information and Shannon entropy in three dimensional space within the relativistic regime by solving Klein-Gordon ( KG ) equation for equal scalar and vector potential with Schioberg plus Yukawa potential ( SPYP ) using the Parametric Nikiforov-Uvarov ( PNU ) method and compare the results to its nonrelativistic counterpart. It is important to note that the numerical solutions for the three dimensional Fisher information and Shannon entropy obtained for both the relativistic and nonrelativistic wave equations using SPYP adequately satisfies Bialynicki-Birula and Mycieslski (BBM) uncertainty inequality. The radial wave function, radial probability density, momentum space wave function and momentum probability density graphical plots were all obtained for both cases respectively. The numerical computations for Fisher information and Shannon entropy in three dimensional space were carried out for \(\ell = 0\) with different quantum states \(n= 0\) and 1. The investigation of Fisher information and Shannon entropy are relevance in describing localisation and delocalisation mechanism of particles in atomic orbitals within a potential well, relativistic confinement effect and entropy-energy correlation.

Discover Applied Sciences•Sep 24, 2026•0 citations

Entropic system in the relativistic Klein-Gordon Particle

C. A. Onate, M. C. Onyeaju

The solutions of Kratzer potential plus Hellmann potential was obtained under the Klein-Gordon equation via the parametric Nikiforov-Uvarov method. The relativistic energy and its corresponding normalized wave functions were fully calculated. The theoretic quantities in terms of the entropic system under the relativistic Klein-Gordon equation (a spinless particle) for a Kratzer-Hellmann’s potential model were studied. The effects of a and b respectively (the parameters in the potential that determine the strength of the potential) on each of the entropy were fully examined. The maximum point of stability of a system under the three entropies was determined at the point of intersection between two formulated expressions plotted against a as one of the parameters in the potential. Finally, the popular Shannon entropy uncertainty relation known as Bialynick-Birula, Mycielski inequality was deduced by generating numerical results.

Journal of the Nigerian Society of Physical Sciences•Aug 29, 2021•2 citations

Shannon entropy and Fisher information of the one-dimensional Klein–Gordon oscillator with energy-dependent potential

Abdelmalek Boumali, Malika Labidi

In this paper, we studied, at first, the influence of the energy-dependent potentials on the one-dimensionless Klein–Gordon oscillator. Then, the Shannon entropy and Fisher information of this system are investigated. The position and momentum information entropies for the low-lying states n = 0, 1, 2 are calculated. Some interesting features of both Fisher and Shannon densities, as well as the probability densities, are demonstrated. Finally, the Stam, Cramer–Rao and Bialynicki–Birula–Mycielski (BBM) inequalities have been checked, and their comparison with the regarding results have been reported. We showed that the BBM inequality is still valid in the form [Formula: see text], as well as in ordinary quantum mechanics.

arXiv:1706.07701•Modern Physics Letters A•Feb 28, 2018•34 citations

The Study of One-Dimensional Relativistic Shannon Entropy with Hulthen–Coulomb Interaction Using Klein-Gordon Equation

I. B. Okon, Akaninyene Daniel Antia, Cecilia N. Isonguyo +6 more

The study of quantum information entropies is significant in quantum computation and signal analysis. Much research work has been carried out in quantum information entropies under nonrelativistic wave equation. However, not much research has been reported in evaluating quantum information entropies using relativistic wave equation. In this work, we evaluate Shannon information entropy in one-dimensional space with generalised Hulthen plus Coulomb potential (GHPCP) in Klein-Gordon equation using parametric Nikiforov-Uvarov method. The numerical computations obtained both for position and momentum spaces as well as the Shannon sum obeys Bialynicki-Birula and Mycieslski (BBM) uncertainty inequality in order to ascertain the degree of accuracy of the analytical and numerical calculation. This research has relevance in atomic and molecular physics especially in localisation and delocalisation of electrons or particle in atomic orbitals.

International Journal of Research and Innovation in Applied Science•Jan 1, 2026•0 citations

Quantifying Dirac hydrogenic effects via complexity measures

Peter Alexander Bouvrie, Sheila López‐Rosa, J. S. Dehesa

The primary dynamical Dirac relativistic effects can only be seen in hydrogenic systems without the complications introduced by electron-electron interactions in many-electron systems. They are known to be the contraction towards the origin of the electronic charge in hydrogenic systems and the nodal disappearance (because of the raising of all the nonrelativistic minima) in the electron density of the excited states of these systems. In addition we point out the (largely ignored) gradient reduction of the charge density near and far from the nucleus. In this work we quantify these effects by means of single (Fisher information) and composite [Fisher-Shannon complexity and plane, L\'opez-Ruiz, Mancini, and Calbet (LMC) complexity] information-theoretic measures. While the Fisher information measures the gradient content of the density, the (dimensionless) composite information-theoretic quantities grasp twofold facets of the electronic distribution: The Fisher-Shannon complexity measures the combined balance of the gradient content and the total extent of the electronic charge, and the LMC complexity quantifies the disequilibrium jointly with the spreading of the density in the configuration space. Opposite to other complexity notions (e.g., computational and algorithmic complexities), these two quantities describe intrinsic properties of the system because they do not depend on the context but are functionals of the electron density. Moreover, they are closely related to the intuitive notion of complexity because they are minimum for the two extreme (or least complex) distributions of perfect order and maximum disorder.

arXiv:1408.7056•Physical Review A•Jul 17, 2012•7 citations
The Fisher-Shannon complexity is used to quantitatively estimate the contribution of relativistic effects to the internal disorder of Klein-Gordon single-particle Coulomb systems which is manifest in the rich variety of three-dimensional geometries of its corresponding quantum-mechanical probability density. It is observed that, contrary to the non-relativistic case, the Fisher-Shannon complexity of these relativistic systems does depend on the potential strength (nuclear charge). This is numerically illustrated for pionic atoms. Moreover, its variation with the quantum numbers ( n , l , m ) is analysed in various ground and excited states. It is found that the relativistic effects enhance when n and/or l are decreasing.
The charge spreading of the ground and excited states of Klein-Gordon particles moving in a Coulomb potential is quantitatively analysed by means of ordinary moments and the Heisenberg measure as well as by using the most relevant information-theoretic measures of global (Shannon entropic power) and local (Fisher information) types.The dependence of these complementary quantities on the nuclear charge Z and the quantum numbers characterizing the physical states is carefully discussed.The comparison of relativistic Klein-Gordon and non-relativistic Schrödinger values is made.Nonrelativistic limits at large principal quantum number n and for small values of Z are also reached.Contents 1. Introduction 2 2. Radial expectation values and Heisenberg's measure 5 3. Shannon and Fisher information measures 8 4. Conclusions 12
Quantum information entropies have been properly studied within the confines of nonrelativistic quantum mechanics ( QM ) by solving Schrödinger wave equation for different considerable potential models to obtained the energy and total normalised wave function which are essential tools for determining various information entropies. However, not much research has been done on evaluation of information entropies within the relativistic quantum mechanics because of its complexities and existence of both multiple and quadratic eigenvalues for fixed quantum state ( n ) and orbital angular quantum number (ℓ)(\ell )(ℓ) . In this work, we investigate Fisher information and Shannon entropy in three dimensional space within the relativistic regime by solving Klein-Gordon ( KG ) equation for equal scalar and vector potential with Schioberg plus Yukawa potential ( SPYP ) using the Parametric Nikiforov-Uvarov ( PNU ) method and compare the results to its nonrelativistic counterpart. It is important to note that the numerical solutions for the three dimensional Fisher information and Shannon entropy obtained for both the relativistic and nonrelativistic wave equations using SPYP adequately satisfies Bialynicki-Birula and Mycieslski (BBM) uncertainty inequality. The radial wave function, radial probability density, momentum space wave function and momentum probability density graphical plots were all obtained for both cases respectively. The numerical computations for Fisher information and Shannon entropy in three dimensional space were carried out for ℓ=0\ell = 0ℓ=0 with different quantum states n=0n= 0n=0 and 1. The investigation of Fisher information and Shannon entropy are relevance in describing localisation and delocalisation mechanism of particles in atomic orbitals within a potential well, relativistic confinement effect and entropy-energy correlation.
The solutions of Kratzer potential plus Hellmann potential was obtained under the Klein-Gordon equation via the parametric Nikiforov-Uvarov method. The relativistic energy and its corresponding normalized wave functions were fully calculated. The theoretic quantities in terms of the entropic system under the relativistic Klein-Gordon equation (a spinless particle) for a Kratzer-Hellmann’s potential model were studied. The effects of a and b respectively (the parameters in the potential that determine the strength of the potential) on each of the entropy were fully examined. The maximum point of stability of a system under the three entropies was determined at the point of intersection between two formulated expressions plotted against a as one of the parameters in the potential. Finally, the popular Shannon entropy uncertainty relation known as Bialynick-Birula, Mycielski inequality was deduced by generating numerical results.
In this paper, we studied, at first, the influence of the energy-dependent potentials on the one-dimensionless Klein–Gordon oscillator. Then, the Shannon entropy and Fisher information of this system are investigated. The position and momentum information entropies for the low-lying states n = 0, 1, 2 are calculated. Some interesting features of both Fisher and Shannon densities, as well as the probability densities, are demonstrated. Finally, the Stam, Cramer–Rao and Bialynicki–Birula–Mycielski (BBM) inequalities have been checked, and their comparison with the regarding results have been reported. We showed that the BBM inequality is still valid in the form [Formula: see text], as well as in ordinary quantum mechanics.
The study of quantum information entropies is significant in quantum computation and signal analysis. Much research work has been carried out in quantum information entropies under nonrelativistic wave equation. However, not much research has been reported in evaluating quantum information entropies using relativistic wave equation. In this work, we evaluate Shannon information entropy in one-dimensional space with generalised Hulthen plus Coulomb potential (GHPCP) in Klein-Gordon equation using parametric Nikiforov-Uvarov method. The numerical computations obtained both for position and momentum spaces as well as the Shannon sum obeys Bialynicki-Birula and Mycieslski (BBM) uncertainty inequality in order to ascertain the degree of accuracy of the analytical and numerical calculation. This research has relevance in atomic and molecular physics especially in localisation and delocalisation of electrons or particle in atomic orbitals.
The primary dynamical Dirac relativistic effects can only be seen in hydrogenic systems without the complications introduced by electron-electron interactions in many-electron systems. They are known to be the contraction towards the origin of the electronic charge in hydrogenic systems and the nodal disappearance (because of the raising of all the nonrelativistic minima) in the electron density of the excited states of these systems. In addition we point out the (largely ignored) gradient reduction of the charge density near and far from the nucleus. In this work we quantify these effects by means of single (Fisher information) and composite [Fisher-Shannon complexity and plane, L\'opez-Ruiz, Mancini, and Calbet (LMC) complexity] information-theoretic measures. While the Fisher information measures the gradient content of the density, the (dimensionless) composite information-theoretic quantities grasp twofold facets of the electronic distribution: The Fisher-Shannon complexity measures the combined balance of the gradient content and the total extent of the electronic charge, and the LMC complexity quantifies the disequilibrium jointly with the spreading of the density in the configuration space. Opposite to other complexity notions (e.g., computational and algorithmic complexities), these two quantities describe intrinsic properties of the system because they do not depend on the context but are functionals of the electron density. Moreover, they are closely related to the intuitive notion of complexity because they are minimum for the two extreme (or least complex) distributions of perfect order and maximum disorder.