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

Physical Review A

Quantum-information entropies for highly excited states of single-particle systems with power-type potentials

J. S. Dehesa, Andrei Martı́nez-Finkelshtein, Vladimir Nikolaevich Sorokin

Published Dec 20, 2002• 32 citations
Publisher

Abstract

The asymptotics of the Boltzmann-Shannon information entropy as well as the Renyi entropy for the quantum probability density of a single-particle system with a confining (i.e., bounded below) power-type potential ${V(x)=x}^{2k}$ with $k\ensuremath{\in}N$ and $x\ensuremath{\in}R,$ is investigated in the position and momentum spaces within the semiclassical (WKB) approximation. It is found that for highly excited states both physical entropies, as well as their sum, have a logarithmic dependence on its quantum number not only when $k=1$ (harmonic oscillator), but also for any fixed k. As a by-product, the extremal case $\stackrel{\ensuremath{\rightarrow}}{k}\ensuremath{\infty}$ (the infinite well potential) is also rigorously analyzed. It is shown that not only the position-space entropy has the same constant value for all quantum states, which is a known result, but also that the momentum-space entropy is constant for highly excited states.

DOI
10.1103/physreva.66.062109

Related papers

Rényi Entropies of Multidimensional Oscillator and Hydrogenic Systems with Applications to Highly Excited Rydberg States

J. S. Dehesa

The various facets of the internal disorder of quantum systems can be described by means of the Rényi entropies of their single-particle probability density according to modern density functional theory and quantum information techniques. In this work, we first show the lower and upper bounds for the Rényi entropies of general and central-potential quantum systems, as well as the associated entropic uncertainty relations. Then, the Rényi entropies of multidimensional oscillator and hydrogenic-like systems are reviewed and explicitly determined for all bound stationary position and momentum states from first principles (i.e., in terms of the potential strength, the space dimensionality and the states's hyperquantum numbers). This is possible because the associated wavefunctions can be expressed by means of hypergeometric orthogonal polynomials. Emphasis is placed on the most extreme, non-trivial cases corresponding to the highly excited Rydberg states, where the Rényi entropies can be amazingly obtained in a simple, compact, and transparent form. Powerful asymptotic approaches of approximation theory have been used when the polynomial's degree or the weight-function parameter(s) of the Hermite, Laguerre, and Gegenbauer polynomials have large values. At present, these special states are being shown of increasing potential interest in quantum information and the associated quantum technologies, such as e.g., quantum key distribution, quantum computation, and quantum metrology.

Entropy•Nov 2, 2022•4 citations

Quantum information-entropic measures for exponential-type potential

A. N. Ikot, Gaotsiwe Joel Rampho, P. O. Amadi +3 more

The approximate solutions of the Schrödinger equation with the central generalized Hulthén and Yukawa potential are investigated within the framework of the functional method. The obtained wave function and the energy levels are used to study the Shannon entropy, Renyi entropy, Fisher information, Shannon Fisher complexity, Shannon power and Fisher-Shannon product in both position and momentum spaces for the ground and first excited states. The theoretic information theories such as Shannon entropies Sr,Sp, Fisher information Ip,Ip and Renyi entropies Rβ as well as the information entropy probability densities ρ(r),ϕ(p) are illustrated graphically. The Bialynicki-Birula – Mycielski (BBM) inequality for the Shannon entropies and the Cramer-Rao inequality for the Fisher information are satisfied for the potential model.

Results in Physics•Jun 29, 2020•21 citations

SHANNON ENTROPY OF THE WIGNER FUNCTION AND POSITION-MOMENTUM CORRELATION IN MODEL SYSTEMS

Humberto G. Laguna, Robin P. Sagar

Shannon entropies of the Wigner function are calculated for ground and excited stationary states of the Particle-In-A-Box and Harmonic Oscillator model systems and examined as a measure of the localization of the phase-space distribution. We show that their behavior is consistent with that of the sum of the position and momentum space entropies as a function of quantum number. Position-momentum correlation is then analyzed in these systems by defining mutual information between position and momentum variables. This mutual information yields non-zero values, in contrast to the quantum covariance, and increases with quantum number.

International Journal of Quantum Information•Oct 1, 2010•29 citations

Shannon information entropies for rectangular multiple quantum well systems with constant total lengths*

M. Solaimani, Guo-Hua Sun, Shi‐Hai Dong

We first study the Shannon information entropies of constant total length multiple quantum well systems and then explore the effects of the number of wells and confining potential depth on position and momentum information entropy density as well as the corresponding Shannon entropy. We find that for small full width at half maximum (FWHM) of the position entropy density, the FWHM of the momentum entropy density is large and vice versa. By increasing the confined potential depth, the FWHM of the position entropy density decreases while the FWHM of the momentum entropy density increases. By increasing the potential depth, the frequency of the position entropy density oscillation within the quantum barrier decreases while that of the position entropy density oscillation within the quantum well increases. By increasing the number of wells, the frequency of the position entropy density oscillation decreases inside the barriers while it increases inside the quantum well. As an example, we might localize the ground state as well as the position entropy densities of the 1st, 2nd, and 6th excited states for a four-well quantum system. Also, we verify the Bialynicki–Birula–Mycieslki (BBM) inequality.

Chinese Physics B•Apr 1, 2018•11 citations

Quantum information measures of infinite spherical well

Ye-Jiao Shi, Guo-Hua Sun, Farida Tahir +3 more

In this work, we study the Shannon information entropies [Formula: see text] and [Formula: see text] of an infinite spherical well. The Shannon entropy [Formula: see text] is calculated numerically in terms of the analytical result of the wave function in momentum space. Some typical features of the position and momentum probability densities [Formula: see text] and [Formula: see text] as well as the information entropy densities [Formula: see text] and [Formula: see text] are demonstrated. We find that the position entropy [Formula: see text] increases with the radius a of the spherical well for given quantum numbers l, m and n. It is interesting to note that the position entropy [Formula: see text] decreases with the quantum numbers l and n for a fixed radius a and quantum number m. The position entropy [Formula: see text] is almost independent of the quantum numbers l, m and n. The momentum entropy [Formula: see text] first increases and then decreases with respect to the radius a. We also note that the [Formula: see text] increases with the radius a and finally arrives at a constant. In addition, the Bialynicki–Birula–Mycielski (BBM) inequality is verified and also hold for this confined system.

Modern Physics Letters A•May 25, 2018•24 citations

Position- and Momentum-Space Quantum Information Measures of the Double-Morse Oscillator

Firoz Chogle, Ernesto Damiani, Berihu Teklu

We investigate the quantum-information properties of a particle confined by the double Morse potential in position and momentum spaces. The quasi-exact solvability of the model gives analytical expressions for the first two bound states, allowing the corresponding probability densities to be analyzed directly. Shannon entropy, Onicescu energy, Fisher information, statistical complexity, and Fisher-Shannon products are evaluated as functions of the parameter $A$, which controls the transition from a well-separated double well to a merged single-well profile. The position distribution is more delocalized and structurally complex when the double-well character is pronounced, whereas the momentum distribution exhibits the complementary trend. As the wells merge, the ground-state Fisher--Shannon product approaches its Gaussian reference value, whereas the excited state retains stronger non-Gaussian structure.

quant-pharXiv:2608.25953•Physics Letters A•Aug 27, 2026•0 citations
The asymptotics of the Boltzmann-Shannon information entropy as well as the Renyi entropy for the quantum probability density of a single-particle system with a confining (i.e., bounded below) power-type potential V(x)=x2k{V(x)=x}^{2k}V(x)=x2k with k\ensuremath∈Nk\ensuremath{\in}Nk\ensuremath∈N and x\ensuremath∈R,x\ensuremath{\in}R,x\ensuremath∈R, is investigated in the position and momentum spaces within the semiclassical (WKB) approximation. It is found that for highly excited states both physical entropies, as well as their sum, have a logarithmic dependence on its quantum number not only when k=1k=1k=1 (harmonic oscillator), but also for any fixed k. As a by-product, the extremal case k\ensuremath→\ensuremath∞\stackrel{\ensuremath{\rightarrow}}{k}\ensuremath{\infty}k\ensuremath→​\ensuremath∞ (the infinite well potential) is also rigorously analyzed. It is shown that not only the position-space entropy has the same constant value for all quantum states, which is a known result, but also that the momentum-space entropy is constant for highly excited states.
The various facets of the internal disorder of quantum systems can be described by means of the Rényi entropies of their single-particle probability density according to modern density functional theory and quantum information techniques. In this work, we first show the lower and upper bounds for the Rényi entropies of general and central-potential quantum systems, as well as the associated entropic uncertainty relations. Then, the Rényi entropies of multidimensional oscillator and hydrogenic-like systems are reviewed and explicitly determined for all bound stationary position and momentum states from first principles (i.e., in terms of the potential strength, the space dimensionality and the states's hyperquantum numbers). This is possible because the associated wavefunctions can be expressed by means of hypergeometric orthogonal polynomials. Emphasis is placed on the most extreme, non-trivial cases corresponding to the highly excited Rydberg states, where the Rényi entropies can be amazingly obtained in a simple, compact, and transparent form. Powerful asymptotic approaches of approximation theory have been used when the polynomial's degree or the weight-function parameter(s) of the Hermite, Laguerre, and Gegenbauer polynomials have large values. At present, these special states are being shown of increasing potential interest in quantum information and the associated quantum technologies, such as e.g., quantum key distribution, quantum computation, and quantum metrology.
The approximate solutions of the Schrödinger equation with the central generalized Hulthén and Yukawa potential are investigated within the framework of the functional method. The obtained wave function and the energy levels are used to study the Shannon entropy, Renyi entropy, Fisher information, Shannon Fisher complexity, Shannon power and Fisher-Shannon product in both position and momentum spaces for the ground and first excited states. The theoretic information theories such as Shannon entropies Sr,Sp, Fisher information Ip,Ip and Renyi entropies Rβ as well as the information entropy probability densities ρ(r),ϕ(p) are illustrated graphically. The Bialynicki-Birula – Mycielski (BBM) inequality for the Shannon entropies and the Cramer-Rao inequality for the Fisher information are satisfied for the potential model.
Shannon entropies of the Wigner function are calculated for ground and excited stationary states of the Particle-In-A-Box and Harmonic Oscillator model systems and examined as a measure of the localization of the phase-space distribution. We show that their behavior is consistent with that of the sum of the position and momentum space entropies as a function of quantum number. Position-momentum correlation is then analyzed in these systems by defining mutual information between position and momentum variables. This mutual information yields non-zero values, in contrast to the quantum covariance, and increases with quantum number.
We first study the Shannon information entropies of constant total length multiple quantum well systems and then explore the effects of the number of wells and confining potential depth on position and momentum information entropy density as well as the corresponding Shannon entropy. We find that for small full width at half maximum (FWHM) of the position entropy density, the FWHM of the momentum entropy density is large and vice versa. By increasing the confined potential depth, the FWHM of the position entropy density decreases while the FWHM of the momentum entropy density increases. By increasing the potential depth, the frequency of the position entropy density oscillation within the quantum barrier decreases while that of the position entropy density oscillation within the quantum well increases. By increasing the number of wells, the frequency of the position entropy density oscillation decreases inside the barriers while it increases inside the quantum well. As an example, we might localize the ground state as well as the position entropy densities of the 1st, 2nd, and 6th excited states for a four-well quantum system. Also, we verify the Bialynicki–Birula–Mycieslki (BBM) inequality.
In this work, we study the Shannon information entropies [Formula: see text] and [Formula: see text] of an infinite spherical well. The Shannon entropy [Formula: see text] is calculated numerically in terms of the analytical result of the wave function in momentum space. Some typical features of the position and momentum probability densities [Formula: see text] and [Formula: see text] as well as the information entropy densities [Formula: see text] and [Formula: see text] are demonstrated. We find that the position entropy [Formula: see text] increases with the radius a of the spherical well for given quantum numbers l, m and n. It is interesting to note that the position entropy [Formula: see text] decreases with the quantum numbers l and n for a fixed radius a and quantum number m. The position entropy [Formula: see text] is almost independent of the quantum numbers l, m and n. The momentum entropy [Formula: see text] first increases and then decreases with respect to the radius a. We also note that the [Formula: see text] increases with the radius a and finally arrives at a constant. In addition, the Bialynicki–Birula–Mycielski (BBM) inequality is verified and also hold for this confined system.
We investigate the quantum-information properties of a particle confined by the double Morse potential in position and momentum spaces. The quasi-exact solvability of the model gives analytical expressions for the first two bound states, allowing the corresponding probability densities to be analyzed directly. Shannon entropy, Onicescu energy, Fisher information, statistical complexity, and Fisher-Shannon products are evaluated as functions of the parameter AAA, which controls the transition from a well-separated double well to a merged single-well profile. The position distribution is more delocalized and structurally complex when the double-well character is pronounced, whereas the momentum distribution exhibits the complementary trend. As the wells merge, the ground-state Fisher--Shannon product approaches its Gaussian reference value, whereas the excited state retains stronger non-Gaussian structure.