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

Journal of Mathematical Physics

Fisher information of D-dimensional hydrogenic systems in position and momentum spaces

J. S. Dehesa, Sheila López‐Rosa, Beatriz Olmos, Rafael J. Yáñez

Published May 1, 2006• 73 citations
Publisher

Abstract

The spreading of the quantum-mechanical probability distribution density of D-dimensional hydrogenic orbitals is quantitatively determined by means of the local information-theoretic quantity of Fisher in both position and momentum spaces. The Fisher information is found in closed form in terms of the quantum numbers of the orbital.
DOI
10.1063/1.2190335

Related papers

Information theory of D-dimensional hydrogenic systems: Application to circular and Rydberg states

J. S. Dehesa, Sheila López‐Rosa, Andrei Martı́nez-Finkelshtein +1 more

The analytic information theory of quantum systems includes the exact determination of their spatial extension or multidimensional spreading in both position and momentum spaces by means of the familiar variance and its generalization, the power and logarithmic moments, and, more appropriately, the Shannon entropy and the Fisher information. These complementary uncertainty measures have a global or local character, respectively, because they are power-like (variance, moments), logarithmic (Shannon) and gradient (Fisher) functionals of the corresponding probability distribution. Here we explicitly discuss all these spreading measures (and their associated uncertainty relations) in both position and momentum for the main prototype in D-dimensional physics, the hydrogenic system, directly in terms of the dimensionality and the hyperquantum numbers which characterize the involved states. Then, we analyze in detail such measures for s-states, circular states (i.e., single-electron states of highest angular momenta allowed within an electronic manifold characterized by a given principal hyperquantum number), and Rydberg states (i.e., states with large radial hyperquantum numbers n). © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010

International Journal of Quantum Chemistry•Oct 6, 2009•87 citations

The Fisher Information: Properties and Physico-Chemical Applications

The Fisher information functional occupies a very special position among the uncertainty measures of quantum systems of any dimensionality. In contrast with (global) entropic measures of Shannon, Renyi or Tsallis types, it contitutes a local measure in the sense that it is sensitive to the oscillatory character of the probability density distribution. Moreover, the Fisher information is closely related to various macroscopic properties of the system described by different density functionals. Here, we review its main analytical properties such as, inequality-based relationships with some radial expectation values and its uncertainty relation. Tighter versions of these inequalities are shown for those systems subject to an arbitrary central potential. Moreover, the Fisher information is computed for D-dimensional hydrogenic systems explicitly in terms of the radial and angular hyperquantum numbers characterizing the quantum states of the system in both position and momentum spaces. Finally, the utility of this quantity is discussed for various physico-chemical processes of recent interest: Abstraction and nucleophilic substitution reactions.

BENTHAM SCIENCE PUBLISHERS eBooks•Dec 11, 2013•1 citation

Fisher Information and Complexity Measure of Generalized Morse Potential Model

C. A. Onate, J. O. A. Idiodi

Abstract The spreading of the quantum-mechanical probability distribution density of the three-dimensional system is quantitatively determined by means of the local information-theoretic quantity of the Shannon information and information energy in both position and momentum spaces. The complexity measure which is equivalent to Cramer–Rao uncertainty product is determined. We have obtained the information content stored, the concentration of quantum system and complexity measure numerically for n = 0, 1, 2 and 3 respectively.

Communications in Theoretical Physics•Sep 1, 2016•26 citations

Uncertainty relation for Fisher information of D-dimensional single-particle systems with central potentials

E. Romera, Pablo Sánchez-Moreno, J. S. Dehesa

An uncertainty Fisher information relation in quantum mechanics is derived for multidimensional single-particle systems with central potentials. It is based on the concept of Fisher information in the two complementary position and momentum spaces, which is a gradient functional of the corresponding probability distributions. The lower bound of the product of position and momentum Fisher informations is shown to depend on the orbital and magnetic quantum numbers of the physical state and the space dimensionality. Applications to various elementary systems is discussed.

Journal of Mathematical Physics•Oct 1, 2006•93 citations

Entropy and complexity analysis of hydrogenic Rydberg atoms

Sheila López‐Rosa, I. V. Toranzo, Pablo Sánchez-Moreno +1 more

The internal disorder of hydrogenic Rydberg atoms as contained in their position and momentum probability densities is examined by means of the following information-theoretic spreading quantities: the radial and logarithmic expectation values, the Shannon entropy, and the Fisher information. As well, the complexity measures of Crámer-Rao, Fisher-Shannon, and López Ruiz-Mancini-Calvet types are investigated in both reciprocal spaces. The leading term of these quantities is rigorously calculated by use of the asymptotic properties of the concomitant entropic functionals of the Laguerre and Gegenbauer orthogonal polynomials which control the wavefunctions of the Rydberg states in both position and momentum spaces. The associated generalized Heisenberg-like, logarithmic and entropic uncertainty relations are also given. Finally, application to linear (l = 0), circular (l = n − 1), and quasicircular (l = n − 2) states is explicitly done.

arXiv:1305.1149•Journal of Mathematical Physics•May 1, 2013•21 citations

Fisher information for quasi-one-dimensional hydrogen atom

Saha, Aparna, B. Talukdar, Supriya Chatterjee

The coordinate-space wave function $ψ(x)$ of quasi-one-dimensional atoms is defined in the $x\geq 0$ region only. This poses a typical problem to write a physically acceptable momentum-space wave function $ϕ(p)$ from the Fourier transform of $ψ(x)$. We resolve the problem with special attention to the behavior of real and imaginary parts of the complex-valued function $ϕ(p)$ as a function of $p$ and confirm that $ϕ_i(p)$ (the imaginary part of $ϕ(p)$) represents the correct momentum-space wave function. We make use of the results for $ψ(x)$ and $ϕ_i(p)$ to express the position- and momentum-space Fisher information in terms of the principal quantum number and energy eigen value of the system and provide some useful checks on the result presented with particular attention on the information theoretic uncertainty relation.

arXiv:1703.03578•Mar 10, 2017•3 citations
The analytic information theory of quantum systems includes the exact determination of their spatial extension or multidimensional spreading in both position and momentum spaces by means of the familiar variance and its generalization, the power and logarithmic moments, and, more appropriately, the Shannon entropy and the Fisher information. These complementary uncertainty measures have a global or local character, respectively, because they are power-like (variance, moments), logarithmic (Shannon) and gradient (Fisher) functionals of the corresponding probability distribution. Here we explicitly discuss all these spreading measures (and their associated uncertainty relations) in both position and momentum for the main prototype in D-dimensional physics, the hydrogenic system, directly in terms of the dimensionality and the hyperquantum numbers which characterize the involved states. Then, we analyze in detail such measures for s-states, circular states (i.e., single-electron states of highest angular momenta allowed within an electronic manifold characterized by a given principal hyperquantum number), and Rydberg states (i.e., states with large radial hyperquantum numbers n). © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010
The Fisher information functional occupies a very special position among the uncertainty measures of quantum systems of any dimensionality. In contrast with (global) entropic measures of Shannon, Renyi or Tsallis types, it contitutes a local measure in the sense that it is sensitive to the oscillatory character of the probability density distribution. Moreover, the Fisher information is closely related to various macroscopic properties of the system described by different density functionals. Here, we review its main analytical properties such as, inequality-based relationships with some radial expectation values and its uncertainty relation. Tighter versions of these inequalities are shown for those systems subject to an arbitrary central potential. Moreover, the Fisher information is computed for D-dimensional hydrogenic systems explicitly in terms of the radial and angular hyperquantum numbers characterizing the quantum states of the system in both position and momentum spaces. Finally, the utility of this quantity is discussed for various physico-chemical processes of recent interest: Abstraction and nucleophilic substitution reactions.
Abstract The spreading of the quantum-mechanical probability distribution density of the three-dimensional system is quantitatively determined by means of the local information-theoretic quantity of the Shannon information and information energy in both position and momentum spaces. The complexity measure which is equivalent to Cramer–Rao uncertainty product is determined. We have obtained the information content stored, the concentration of quantum system and complexity measure numerically for n = 0, 1, 2 and 3 respectively.
An uncertainty Fisher information relation in quantum mechanics is derived for multidimensional single-particle systems with central potentials. It is based on the concept of Fisher information in the two complementary position and momentum spaces, which is a gradient functional of the corresponding probability distributions. The lower bound of the product of position and momentum Fisher informations is shown to depend on the orbital and magnetic quantum numbers of the physical state and the space dimensionality. Applications to various elementary systems is discussed.
The internal disorder of hydrogenic Rydberg atoms as contained in their position and momentum probability densities is examined by means of the following information-theoretic spreading quantities: the radial and logarithmic expectation values, the Shannon entropy, and the Fisher information. As well, the complexity measures of Crámer-Rao, Fisher-Shannon, and López Ruiz-Mancini-Calvet types are investigated in both reciprocal spaces. The leading term of these quantities is rigorously calculated by use of the asymptotic properties of the concomitant entropic functionals of the Laguerre and Gegenbauer orthogonal polynomials which control the wavefunctions of the Rydberg states in both position and momentum spaces. The associated generalized Heisenberg-like, logarithmic and entropic uncertainty relations are also given. Finally, application to linear (l = 0), circular (l = n − 1), and quasicircular (l = n − 2) states is explicitly done.
The coordinate-space wave function ψ(x)ψ(x)ψ(x) of quasi-one-dimensional atoms is defined in the x≥0x\geq 0x≥0 region only. This poses a typical problem to write a physically acceptable momentum-space wave function ϕ(p)ϕ(p)ϕ(p) from the Fourier transform of ψ(x)ψ(x)ψ(x). We resolve the problem with special attention to the behavior of real and imaginary parts of the complex-valued function ϕ(p)ϕ(p)ϕ(p) as a function of ppp and confirm that ϕi(p)ϕ_i(p)ϕi​(p) (the imaginary part of ϕ(p)ϕ(p)ϕ(p)) represents the correct momentum-space wave function. We make use of the results for ψ(x)ψ(x)ψ(x) and ϕi(p)ϕ_i(p)ϕi​(p) to express the position- and momentum-space Fisher information in terms of the principal quantum number and energy eigen value of the system and provide some useful checks on the result presented with particular attention on the information theoretic uncertainty relation.