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:1904.06076 · Annalen der Physik

Quantum confinement in an asymmetric double‐well potential through energy analysis and information entropic measure

Neetik Mukherjee, Amlan K. Roy

Published Dec 9, 2015• 37 citations
arXivHTMLPDF

Abstract

Localization of a particle in the wells of an asymmetric double‐well (DW) potential is investigated here. Information entropy‐based uncertainty measures, such as Shannon entropy, Fisher information, Onicescu energy, etc., and phase‐space area, are utilized to explain the contrasting effect of localization‐delocalization and role of asymmetric term in such two‐well potentials. In asymmetric situation, two wells behaves like two different potentials. A general rule has been proposed for arrangement of quasi‐degenerate pairs, in terms of asymmetry parameter. Further, it enables to describe the distribution of particle in either of the deeper or shallow wells in various energy states. One finds that, all states eventually get localized to the deeper well, provided the asymmetry parameter attains certain threshold value. This generalization produces symmetric DW as a natural consequence of asymmetric DW. Eigenfunctions, eigenvalues are obtained by means of a simple, accurate variation‐induced exact diagonalization method. In brief, information measures and phase‐space analysis can provide valuable insight toward the understanding of such potentials.

DOI
10.1002/andp.201500301

Related papers

Information entropy as a measure of tunneling and quantum confinement in a symmetric double‐well potential

Neetik Mukherjee, Arunesh Roy, Amlan K. Roy

Information entropic measures such as Fisher information, Shannon entropy, Onicescu energy and Onicescu Shannon entropy of a symmetric double‐well potential are calculated in both position and momentum space. Eigenvalues and eigenvectors of this system are obtained through a variation‐induced exact diagonalization procedure. The information entropy‐based uncertainty relation is shown to be a better measure than conventional uncertainty product in interpreting purely quantum mechanical phenomena, such as, tunneling and quantum confinement in this case. Additionally, the phase‐space description provides a semi‐classical explanation for this feature. Total information entropy and phase‐space area show similar behavior with increasing barrier height. image

arXiv:1904.11172•Annalen der Physik•Aug 24, 2015•37 citations

Information entropic measures of a quantum harmonic oscillator in symmetric and asymmetric confinement within an impenetrable box

Abhisek Ghosal, Neetik Mukherjee, Amlan K. Roy

Information‐based uncertainty measures like Shannon entropy, Onicescu energy and Fisher information (in position and momentum space) are employed to understand the effect of symmetric and asymmetric confinement in a quantum harmonic oscillator. Also, the transformation of the Hamiltonian into a dimensionless form gives an idea of the composite effect of force constant and confinement length ( x c ). In the symmetric case, a wide range of x c has been taken up, whereas asymmetric confinement is dealt with by shifting the minimum of the potential from the origin keeping box length and boundary fixed. Eigenvalues and eigenvectors for these systems are obtained quite accurately via an imaginary‐time propagation scheme. For asymmetric confinement, a variation‐induced exact diagonalization procedure is also introduced, which produces very high quality results. One finds that, in symmetric confinement, after a certain characteristic x c , all these properties converge to respective values of a free harmonic oscillator. In the asymmetric situation, excited‐state energies always pass through a maximum. For this potential, the classical turning point decreases, whereas well depth increases with the strength of asymmetry. A study of these uncertainty measures reveals that localization increases with an increase of the asymmetry parameter. image

arXiv:1904.06071•Annalen der Physik•Sep 28, 2016•40 citations

Correlation and Information‐Theoretic Approaches in Quantum Coupled Double‐Well Systems

Kirtee Kumar, Vinod Prasad

Abstract This study investigates the quantum correlation and information‐theoretic measures of coupled double‐well systems, modeled using Razavy's hyperbolic potential. By employing numerical techniques, including the finite difference method and Kronecker tensor product, the time‐independent equation is solved to determine energy eigenvalues and eigenfunctions for quantum states, such as , , , and of the bipartite quantum system. The analysis focuses on quantum information measures, such as Shannon entropy, mutual information, and statistical correlation coefficients, to understand how potential parameters and and the coupling strength influence correlations. The results reveal intricate dependencies between quantum states, highlighting symmetry and asymmetry in correlation measures for mixed states under opposite coupling potentials. Further, the different behaviors of the statistical correlation coefficients are found in position and momentum space and their information‐theoretic measures as functions of potential parameters, which is very helpful for understanding the dynamics of quantum particles in the coupled system. Additionally, the study investigates the entanglement properties of bipartite quantum states, employing von Neumann, and linear entropy to measure quantum correlations. The results provide critical insights into the dynamics of quantum particles in confined systems, connecting quantum mechanics and information theory. This work opens new avenues to exploring quantum information dynamics in Razavy double‐well systems, which gives novel insights into the role of spatial and momentum correlations in particle interactions.

Advanced Theory and Simulations•Oct 28, 2025•1 citation

Quantum Information Measures of the One‐Dimensional Robin Quantum Well

Oleg Olendski

Abstract Shannon quantum information entropies , Fisher informations , Onicescu energies , and statistical complexities are calculated both in the position (subscript x) and momentum (k) representations for the Robin quantum well characterized by the extrapolation lengths and at the two confining surfaces. The analysis concentrates on finding and explaining the most characteristic features of these quantum information measures in the whole range of variation of the Robin distance Λ for the symmetric, , and antisymmetric, , geometries. Analytic results obtained in the limiting cases of the extremely large and very small magnitudes of the extrapolation parameter are corroborated by the exact numerical computations that are extended to the arbitrary length Λ. It is confirmed, in particular, that the entropic uncertainty relation and general inequality , which is valid both in the position and momentum spaces, hold true at any Robin distance and for every quantum state n. For either configuration, there is a range of the extrapolation lengths where the rule that is correct for the Neumann ( ) or Dirichlet ( ) boundary conditions, is violated. Other analytic and numerical results for all measures are discussed too and their physical meaning is highlighted.

arXiv:1805.09807•Annalen der Physik•May 31, 2018•12 citations

Information‐Entropic Measures in Confined Isotropic Harmonic Oscillator

Neetik Mukherjee, Amlan K. Roy

Abstract Information‐based uncertainty measures like Rényi entropy (R), Shannon entropy (S) and Onicescu energy (E) are employed to understand the influence of radial confinement in an isotropic harmonic oscillator (IHO). The transformation of the Hamiltonian into a dimensionless form gives an idea of the composite effect of oscillation frequency (ω) and confinement radius (). For a given quantum state, accurate results are provided by applying the respective exact analytical wave function in the r‐space. The p‐space wave functions are produced from Fourier transforms of radial functions. Pilot calculations are done taking order of entropic moments () as in r‐ and p‐spaces. A detailed, systematic analysis is performed for a 3D confined harmonic oscillator (CHO) with respect to state indices , and . It has been found that the CHO acts as a bridge between particles in a spherical box and the 3D free IHO. In the low‐ region, increases while decrease with the rise in . At moderate , there exists an interaction between two competing factors: i) radial confinement (localization) and ii) accumulation of radial nodes with growth of (delocalization). Most of these results are reported here for the first time, revealing many new interesting features.

arXiv:1904.07063•Advanced Theory and Simulations•Sep 13, 2018•9 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
Localization of a particle in the wells of an asymmetric double‐well (DW) potential is investigated here. Information entropy‐based uncertainty measures, such as Shannon entropy, Fisher information, Onicescu energy, etc., and phase‐space area, are utilized to explain the contrasting effect of localization‐delocalization and role of asymmetric term in such two‐well potentials. In asymmetric situation, two wells behaves like two different potentials. A general rule has been proposed for arrangement of quasi‐degenerate pairs, in terms of asymmetry parameter. Further, it enables to describe the distribution of particle in either of the deeper or shallow wells in various energy states. One finds that, all states eventually get localized to the deeper well, provided the asymmetry parameter attains certain threshold value. This generalization produces symmetric DW as a natural consequence of asymmetric DW. Eigenfunctions, eigenvalues are obtained by means of a simple, accurate variation‐induced exact diagonalization method. In brief, information measures and phase‐space analysis can provide valuable insight toward the understanding of such potentials.
Information entropic measures such as Fisher information, Shannon entropy, Onicescu energy and Onicescu Shannon entropy of a symmetric double‐well potential are calculated in both position and momentum space. Eigenvalues and eigenvectors of this system are obtained through a variation‐induced exact diagonalization procedure. The information entropy‐based uncertainty relation is shown to be a better measure than conventional uncertainty product in interpreting purely quantum mechanical phenomena, such as, tunneling and quantum confinement in this case. Additionally, the phase‐space description provides a semi‐classical explanation for this feature. Total information entropy and phase‐space area show similar behavior with increasing barrier height. image
Information‐based uncertainty measures like Shannon entropy, Onicescu energy and Fisher information (in position and momentum space) are employed to understand the effect of symmetric and asymmetric confinement in a quantum harmonic oscillator. Also, the transformation of the Hamiltonian into a dimensionless form gives an idea of the composite effect of force constant and confinement length ( x c ). In the symmetric case, a wide range of x c has been taken up, whereas asymmetric confinement is dealt with by shifting the minimum of the potential from the origin keeping box length and boundary fixed. Eigenvalues and eigenvectors for these systems are obtained quite accurately via an imaginary‐time propagation scheme. For asymmetric confinement, a variation‐induced exact diagonalization procedure is also introduced, which produces very high quality results. One finds that, in symmetric confinement, after a certain characteristic x c , all these properties converge to respective values of a free harmonic oscillator. In the asymmetric situation, excited‐state energies always pass through a maximum. For this potential, the classical turning point decreases, whereas well depth increases with the strength of asymmetry. A study of these uncertainty measures reveals that localization increases with an increase of the asymmetry parameter. image
Abstract This study investigates the quantum correlation and information‐theoretic measures of coupled double‐well systems, modeled using Razavy's hyperbolic potential. By employing numerical techniques, including the finite difference method and Kronecker tensor product, the time‐independent equation is solved to determine energy eigenvalues and eigenfunctions for quantum states, such as , , , and of the bipartite quantum system. The analysis focuses on quantum information measures, such as Shannon entropy, mutual information, and statistical correlation coefficients, to understand how potential parameters and and the coupling strength influence correlations. The results reveal intricate dependencies between quantum states, highlighting symmetry and asymmetry in correlation measures for mixed states under opposite coupling potentials. Further, the different behaviors of the statistical correlation coefficients are found in position and momentum space and their information‐theoretic measures as functions of potential parameters, which is very helpful for understanding the dynamics of quantum particles in the coupled system. Additionally, the study investigates the entanglement properties of bipartite quantum states, employing von Neumann, and linear entropy to measure quantum correlations. The results provide critical insights into the dynamics of quantum particles in confined systems, connecting quantum mechanics and information theory. This work opens new avenues to exploring quantum information dynamics in Razavy double‐well systems, which gives novel insights into the role of spatial and momentum correlations in particle interactions.
Abstract Shannon quantum information entropies , Fisher informations , Onicescu energies , and statistical complexities are calculated both in the position (subscript x) and momentum (k) representations for the Robin quantum well characterized by the extrapolation lengths and at the two confining surfaces. The analysis concentrates on finding and explaining the most characteristic features of these quantum information measures in the whole range of variation of the Robin distance Λ for the symmetric, , and antisymmetric, , geometries. Analytic results obtained in the limiting cases of the extremely large and very small magnitudes of the extrapolation parameter are corroborated by the exact numerical computations that are extended to the arbitrary length Λ. It is confirmed, in particular, that the entropic uncertainty relation and general inequality , which is valid both in the position and momentum spaces, hold true at any Robin distance and for every quantum state n. For either configuration, there is a range of the extrapolation lengths where the rule that is correct for the Neumann ( ) or Dirichlet ( ) boundary conditions, is violated. Other analytic and numerical results for all measures are discussed too and their physical meaning is highlighted.
Abstract Information‐based uncertainty measures like Rényi entropy (R), Shannon entropy (S) and Onicescu energy (E) are employed to understand the influence of radial confinement in an isotropic harmonic oscillator (IHO). The transformation of the Hamiltonian into a dimensionless form gives an idea of the composite effect of oscillation frequency (ω) and confinement radius (). For a given quantum state, accurate results are provided by applying the respective exact analytical wave function in the r‐space. The p‐space wave functions are produced from Fourier transforms of radial functions. Pilot calculations are done taking order of entropic moments () as in r‐ and p‐spaces. A detailed, systematic analysis is performed for a 3D confined harmonic oscillator (CHO) with respect to state indices , and . It has been found that the CHO acts as a bridge between particles in a spherical box and the 3D free IHO. In the low‐ region, increases while decrease with the rise in . At moderate , there exists an interaction between two competing factors: i) radial confinement (localization) and ii) accumulation of radial nodes with growth of (delocalization). Most of these results are reported here for the first time, revealing many new interesting features.
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.