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arXiv:2601.07683

Quantum information and statistical complexity of hydrogen-like ions in Dunkl-Schrödinger system

Akash Halder, Amlan K. Roy, Debraj Nath

quant-phmath-phSubmitted Jan 12, 2026• Announced Jan 13, 2026• 0 citations
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Abstract

In this work, we present analytical solutions of Schrödinger equation for Coulomb potential in presence of a Dunkl reflection operator. Expressions are offered for eigenvalues, eigenfunctions and radial densities for H-isoelectronic series (Z=1-3). The degeneracy in energy in absence and presence of the reflection has been discussed. The standard deviation, Shannon entropy, Rényi entropy in position space have been derived for arbitrary quantum states. Then several important complexity measures like López-Ruiz-Mancini-Calbet (LMC), Shape-Rényi complexity (SRC), Generalized Rényi complexity (GRC), Rényi complexity ratio (RCR) are considered in the analytical framework. Representative results are given for three one-electron atomic ions in tabular and graphical format. Changes in these measures with respect to parity and Dunkl parameter have been given in detail. Most of these results are offered here for the first time.

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In this work, we present analytical solutions of Schrödinger equation for Coulomb potential in presence of a Dunkl reflection operator. Expressions are offered for eigenvalues, eigenfunctions and radial densities for H-isoelectronic series (Z=1-3). The degeneracy in energy in absence and presence of the reflection has been discussed. The standard deviation, Shannon entropy, Rényi entropy in position space have been derived for arbitrary quantum states. Then several important complexity measures like López-Ruiz-Mancini-Calbet (LMC), Shape-Rényi complexity (SRC), Generalized Rényi complexity (GRC), Rényi complexity ratio (RCR) are considered in the analytical framework. Representative results are given for three one-electron atomic ions in tabular and graphical format. Changes in these measures with respect to parity and Dunkl parameter have been given in detail. Most of these results are offered here for the first time.
We consider the solution of one dimensional Schrödinger Dunkl equation for energies and eigenfunctions. Then we provide analytical expressions for various information theoretic measures. For a given density function, quantities such as position expectation value, entropic moment, disequilibrium, Rényi entropy, Shannon entropy, Tsallis entropy, Fisher information are presented. Next, a few relative information measures corresponding to two density functions, like relative entropy, relative Fisher, relative Rényi, relative Tsallis, along with their associated Jensen divergences such as Jensen–Shannon divergence, Jensen–Fisher divergence, Jensen–Rényi divergence, Jensen–Tsallis divergence are treated. Sample results are provided in graphical form. Dependence of these quantities on the Dunkl parameter μ shows distinct features for μ < 0 and μ > 0.
An information theoretic analysis is done for the isotropic harmonic oscillator potential within the Dunkl-Schrödinger framework in spherical coordinates. Starting from the exact analytical eigensolution, various quantum information measures such as Shannon entropy, Rényi information, Tsallis entropy are derived. Besides, their relative measures like relative Shannon, relative Rényi, relative Tsallis as well as corresponding divergences (Jensen-Shannon, Jensen-Rényi, Jensen-Tsallis) are also obtained. In order to get Shannon entropy, a novel factorization method is introduced. This is facilitated through the use of well-known weighted Lebesgue measure. The results from the Dunkl case agree exactly with the non-Dunkl scenario, when Dunkl parameters vanish. The reflection operators and Dunkl parameters considerably influence the above measures. These are portrayed in graphical forms.
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 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.
A two-parameter family of complexity measures C̃(α,β) based on the Rényi entropies is introduced and characterized by a detailed study of its mathematical properties. This family is the generalization of a continuous version of the Lopez-Ruiz–Mancini–Calbet complexity, which is recovered for α=1 and β=2. These complexity measures are obtained by multiplying two quantities bringing global information on the probability distribution defining the system. When one of the parameters, α or β, goes to infinity, one of the global factors becomes a local factor. For this special case, the complexity is calculated on different quantum systems: H-atom, harmonic oscillator, and square well.