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

International Journal of Quantum Chemistry

Complexity analysis of two families of orthogonal functions

Piu Ghosh, Debraj Nath

Published May 6, 2019• 10 citations
Publisher

Abstract

Abstract We study the shape LMC (López‐Ruiz, Mancini and Calvet), Fisher‐Shannon (FS) and Cramér‐Rao (CR) complexities of two families of orthogonal functions associated with the solutions of isospectral deformations of the Pöschl‐Teller and Harmonic oscillator potentials. We have compared the behavior of these complexities for the orthogonal functions with the complexities associated with Pöschl‐Teller and Harmonic oscillator potentials whose solutions are given in terms of the classical orthogonal polynomials. All these complexities are discussed in terms of the quantum number n and isospectrality parameter λ.

DOI
10.1002/qua.25964

Related papers

Parameter and q asymptotics of Lq‐norms of hypergeometric orthogonal polynomials

Nahual Sobrino, J. S. Dehesa

Abstract The three canonical families of the hypergeometric orthogonal polynomials in a continuous real variable (Hermite, Laguerre, and Jacobi) control the physical wavefunctions of the bound stationary states of a great number of quantum systems [Correction added after first online publication on 21 December, 2022. The sentence has been modified.]. The algebraic ‐norms of these polynomials describe many chemical, physical, and information theoretical properties of these systems, such as, for example, the kinetic and Weizsäcker energies, the position and momentum expectation values, the Rényi and Shannon entropies and the Cramér‐Rao, the Fisher‐Shannon and LMC measures of complexity. In this work, we examine review and solve theq‐asymptotics and the parameter asymptotics (i.e., when the weight function's parameter tends towards infinity) of the unweighted and weighted ‐norms for these orthogonal polynomials. This study has been motivated by the application of these algebraic norms to the energetic, entropic, and complexity‐like properties of the highly excited Rydberg and high‐dimensional pseudo‐classical states of harmonic (oscillator‐like) and Coulomb (hydrogenic) systems, and other quantum systems subject to central potentials of anharmonic type (such as, e.g., some molecular systems) [Correction added after first online publication on 21 December, 2022. Oscillatorlike has been changed to oscillator‐like.].

International Journal of Quantum Chemistry•Sep 14, 2022•2 citations

Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems

Nahual Sobrino, David Puertas‐Centeno, I. V. Toranzo +1 more

Abstract In this work we find that not only the Heisenberg-like uncertainty products and the Rényi-entropy-based uncertainty sum have the same first-order values for all the quantum states of the D -dimensional hydrogenic and oscillator-like systems, respectively, in the pseudoclassical ( D → ∞ ) limit but a similar phenomenon also happens for both the Fisher-information-based uncertainty product and the Shannon-entropy-based uncertainty sum, as well as for the Crámer–Rao and Fisher–Shannon complexities. Moreover, we show that the López–Ruiz–Mancini–Calvet (LMC) and LMC-Rényi complexity measures capture the hydrogenic-harmonic difference in the high dimensional limit already at first order.

arXiv:1708.07299•Journal of Statistical Mechanics Theory and Experiment•Aug 1, 2017•24 citations

Research Article Orthogonal polynomials attached to coherent states for the symmetric Pöschl–Teller oscillator

Khalid Ahbli, Patrick Kayupe Kikodio, Zouhaı̈r Mouayn

We consider a one-parameter family of nonlinear coherent states by replacing the factorial in coefficients zn/n! of the canonical coherent states by a specific generalized factorial xnγ!, γ≥0. These states are superposition of eigenstates of the Hamiltonian with a symmetric Pöschl–Teller potential depending on a parameter ν>1. The associated Bargmann-type transform is defined for γ=ν. Some results on the infinite square well potential are also derived. For some different values of γ, we discuss two sets of orthogonal polynomials that are naturally attached to these coherent states.

Integral Transforms and Special Functions•Jul 20, 2016•3 citations

Features of time-independent Wigner functions

Thomas Curtright, D. B. Fairlie, COSMAS K. ZACHOS

The Wigner phase-space distribution function provides the basis for Moyal's deformation quantization alternative to the more conventional Hilbert space and path integral quantizations. The general features of time-independent Wigner functions are explored here, including the functional (``star'') eigenvalue equations they satisfy; their projective orthogonality spectral properties; their Darboux (``supersymmetric'') isospectral potential recursions; and their canonical transformations. These features are illustrated explicitly through simple solvable potentials: the harmonic oscillator, the linear potential, the P\"oschl-Teller potential, and the Liouville potential.

arXiv:hep-th/9711183•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jun 8, 1998•240 citations

Complexity Study of a Single Particle Under q-Deformed Potentials

Ferhat Nutku, K. D. Sen, Ekrem Aydıner

We have studied the variation of the position space statistical complexity measure defined by López-Ruiz, Mancini, and Calbet such as the product of exponential of the Shannon information entropy and the disequilibrium by using the 1D-normalized probability densities derived from solutions of the Schrödinger equation corresponding to the q-deformed harmonic oscillator and q-deformed Morse potentials. An analysis of the numerical results in terms of Shannon information entropy, disequilibrium and complexity measure are presented. In q-deformed harmonic oscillator, q-dependence of the complexity shows a minimum point for all excited energy levels. In the case of q-deformed Morse Potential, complexity decreases with increasing $q$ for the investigated diatomic molecules.

arXiv:1806.09371•Jun 25, 2018•0 citations

Calculation of moments and uncertainty in position space

Anil Kumar

The moments are calculated for some levels of Poschl-Teller potentials. Using isospectral Hamiltonians, the calculations are exte nded to deformed wave functions and their properties are graphically demonstrated. Two Hamiltonians are said to be strictly isospectral, if they have exactly same eigenvalue s pectrum and S-matrix. The wave functions and their dependent quantities are differ ent but related. This property is utilized to obtain the uncertainty in position space. Due to the presence of a deformation parameter, the uncertainty in position space is rea rranged.

•Jan 1, 2014•0 citations
Abstract We study the shape LMC (López‐Ruiz, Mancini and Calvet), Fisher‐Shannon (FS) and Cramér‐Rao (CR) complexities of two families of orthogonal functions associated with the solutions of isospectral deformations of the Pöschl‐Teller and Harmonic oscillator potentials. We have compared the behavior of these complexities for the orthogonal functions with the complexities associated with Pöschl‐Teller and Harmonic oscillator potentials whose solutions are given in terms of the classical orthogonal polynomials. All these complexities are discussed in terms of the quantum number n and isospectrality parameter λ.
Abstract The three canonical families of the hypergeometric orthogonal polynomials in a continuous real variable (Hermite, Laguerre, and Jacobi) control the physical wavefunctions of the bound stationary states of a great number of quantum systems [Correction added after first online publication on 21 December, 2022. The sentence has been modified.]. The algebraic ‐norms of these polynomials describe many chemical, physical, and information theoretical properties of these systems, such as, for example, the kinetic and Weizsäcker energies, the position and momentum expectation values, the Rényi and Shannon entropies and the Cramér‐Rao, the Fisher‐Shannon and LMC measures of complexity. In this work, we examine review and solve theq‐asymptotics and the parameter asymptotics (i.e., when the weight function's parameter tends towards infinity) of the unweighted and weighted ‐norms for these orthogonal polynomials. This study has been motivated by the application of these algebraic norms to the energetic, entropic, and complexity‐like properties of the highly excited Rydberg and high‐dimensional pseudo‐classical states of harmonic (oscillator‐like) and Coulomb (hydrogenic) systems, and other quantum systems subject to central potentials of anharmonic type (such as, e.g., some molecular systems) [Correction added after first online publication on 21 December, 2022. Oscillatorlike has been changed to oscillator‐like.].
Abstract In this work we find that not only the Heisenberg-like uncertainty products and the Rényi-entropy-based uncertainty sum have the same first-order values for all the quantum states of the D -dimensional hydrogenic and oscillator-like systems, respectively, in the pseudoclassical ( D → ∞ ) limit but a similar phenomenon also happens for both the Fisher-information-based uncertainty product and the Shannon-entropy-based uncertainty sum, as well as for the Crámer–Rao and Fisher–Shannon complexities. Moreover, we show that the López–Ruiz–Mancini–Calvet (LMC) and LMC-Rényi complexity measures capture the hydrogenic-harmonic difference in the high dimensional limit already at first order.
We consider a one-parameter family of nonlinear coherent states by replacing the factorial in coefficients zn/n! of the canonical coherent states by a specific generalized factorial xnγ!, γ≥0. These states are superposition of eigenstates of the Hamiltonian with a symmetric Pöschl–Teller potential depending on a parameter ν>1. The associated Bargmann-type transform is defined for γ=ν. Some results on the infinite square well potential are also derived. For some different values of γ, we discuss two sets of orthogonal polynomials that are naturally attached to these coherent states.
The Wigner phase-space distribution function provides the basis for Moyal's deformation quantization alternative to the more conventional Hilbert space and path integral quantizations. The general features of time-independent Wigner functions are explored here, including the functional (``star'') eigenvalue equations they satisfy; their projective orthogonality spectral properties; their Darboux (``supersymmetric'') isospectral potential recursions; and their canonical transformations. These features are illustrated explicitly through simple solvable potentials: the harmonic oscillator, the linear potential, the P\"oschl-Teller potential, and the Liouville potential.
We have studied the variation of the position space statistical complexity measure defined by López-Ruiz, Mancini, and Calbet such as the product of exponential of the Shannon information entropy and the disequilibrium by using the 1D-normalized probability densities derived from solutions of the Schrödinger equation corresponding to the q-deformed harmonic oscillator and q-deformed Morse potentials. An analysis of the numerical results in terms of Shannon information entropy, disequilibrium and complexity measure are presented. In q-deformed harmonic oscillator, q-dependence of the complexity shows a minimum point for all excited energy levels. In the case of q-deformed Morse Potential, complexity decreases with increasing qqq for the investigated diatomic molecules.
The moments are calculated for some levels of Poschl-Teller potentials. Using isospectral Hamiltonians, the calculations are exte nded to deformed wave functions and their properties are graphically demonstrated. Two Hamiltonians are said to be strictly isospectral, if they have exactly same eigenvalue s pectrum and S-matrix. The wave functions and their dependent quantities are differ ent but related. This property is utilized to obtain the uncertainty in position space. Due to the presence of a deformation parameter, the uncertainty in position space is rea rranged.