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Quantum information theoretic analysis on the hydrogen atom trapped within a penetrable repulsive barrier potential

Koustav D. Chakladar, Santanu Mondal, K. D. Sen, Jayanta K. Saha

Published Feb 17, 2026• 1 citation
Publisher

Abstract

In this study, we aim to investigate the oscillatory nature of the probability density in position and momentum space of a hydrogen atom under a penetrable repulsive single-barrier (RSB) potential and the related information-theoretic measures. We have employed the Ritz variational method, utilizing a Slater-type trial wavefunction to determine the eigenvalues and eigenvectors of the hydrogen atom. The momentum space wavefunction is then estimated by employing the Fourier transformation of the position space counterpart. Both of these conjugate space wavefunctions are used to estimate the Shannon entropy, which is then utilized to verify the well-known Bialynicki-Birula and Mycielski (BBM) inequality. Our investigation reveals confinement-induced oscillatory features in the radial probability density of position space and corresponding oscillations in momentum space. These oscillations are strongly correlated with atomic swelling, orbital fusion, fission, and collapse, which manifest as discontinuities in the variations of the Shannon entropy. For non-zero angular momentum states, additional oscillatory features arise from the combined effect of the centrifugal term and the barrier potential. Altogether, the findings highlight the intricate interplay between external confinement and orbital deformation, their impact on information-theoretic measures, and the consequent emergence of quantum phase transitions.

DOI
10.1007/s44464-025-00003-7

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In this study, we aim to investigate the oscillatory nature of the probability density in position and momentum space of a hydrogen atom under a penetrable repulsive single-barrier (RSB) potential and the related information-theoretic measures. We have employed the Ritz variational method, utilizing a Slater-type trial wavefunction to determine the eigenvalues and eigenvectors of the hydrogen atom. The momentum space wavefunction is then estimated by employing the Fourier transformation of the position space counterpart. Both of these conjugate space wavefunctions are used to estimate the Shannon entropy, which is then utilized to verify the well-known Bialynicki-Birula and Mycielski (BBM) inequality. Our investigation reveals confinement-induced oscillatory features in the radial probability density of position space and corresponding oscillations in momentum space. These oscillations are strongly correlated with atomic swelling, orbital fusion, fission, and collapse, which manifest as discontinuities in the variations of the Shannon entropy. For non-zero angular momentum states, additional oscillatory features arise from the combined effect of the centrifugal term and the barrier potential. Altogether, the findings highlight the intricate interplay between external confinement and orbital deformation, their impact on information-theoretic measures, and the consequent emergence of quantum phase transitions.
Abstract The effect of impenetrable spherical confinement on alkali systems (H, Li) has been thoroughly examined, emphasizing the distinctive high-momentum oscillatory behavior of the momentum space radial density. We have utilized the Ritz variational framework with a Slater-type basis set to derive the position-space wavefunction and Fourier–Dirac transformation of the former to find the momentum-space wavefunction, analytically. The derived momentum-space density has been analyzed in four asymptotic limits ( R → ∞ , R → 0 , p → ∞ , and p → 0 ; R being the confinement radius and p being the radial component of linear momentum) and its oscillatory behavior in strong spatial confinement region is critically investigated. Oscillatory behavior is also noticed in the Compton profile of the compressed atomic system. Furthermore, the effects of confinement on Shannon information entropy in both position and momentum spaces are investigated, offering insights into the Bialynicki–Birula–Mycielski inequality and the local variations of information measures in terms of Shannon entropy density.
Abstract This study investigates the impact of spatial confinement and the Hellmann potential on the Shannon entropy of a hydrogenic atom. A hydrogenic atom screened by the Hellmann potential and confined within an impenetrable spherical potential is analyzed. The Schrödinger equation is solved numerically using the finite difference method to determine energy eigenvalues and wavefunctions. These wavefunctions are examined in both position and momentum spaces to calculate the Shannon entropy in position space , the Shannon entropy in momentum space , and the total Shannon entropy . How the confinement radius and the screening parameter affect these entropies is investigated, and the results are compared to those of unconstrained systems. The findings are contrasted with previously reported results and discussed in relation to the Beckner–Bialynicki–Birula–Mycielski (BBM) inequality, offering insights into the behavior of quantum states under confinement. Significant variations in Shannon entropy and energy levels are observed with changes in confinement and screening parameters, providing a deeper understanding of quantum mechanics in confined systems. The study reveals a critical screening parameter at which the behavior of entropy transitions, shedding light on the interplay between spatial confinement, screening effects, and quantum uncertainties.
The effect of a penetrable repulsive single-barrier potential on the structural properties of hydrogen atom in ground and different excited (n,l) states [n=1-3, l=0-2] is studied. The Lagrange mesh method is adopted to solve the corresponding Schrodinger equation numerically for energy eigenvalues and eigenfunctions. Different novel features and phenomena e.g. shrinking the size of the atom, atomic swelling, orbital fusion and fission etc. are noted when the strength of the barrier is changed by tuning its position and height. It is remarkable that all such alterations of the atomic orbital are well articulated from the Shannon entropy profile.
A straightforward way to study the electronic structure and certain physical properties of atoms and molecules under high pressures is through the confinement (caging) model, in which the atom is placed at the center of a spherical box with impenetrable walls. The hydrogen atom confined inside penetrable and impenetrable spherical cavities is perhaps the most studied compressed system, and exact solutions exist for both types of confinement. The physical properties of the hydrogen atom confined within impenetrable-wall cavities (AHCI) have been widely analyzed; however, the same system embedded in penetrable spherical cavities (AHCP) has received less attention. A more physical model suggests using a spherical box with penetrable walls. Therefore, in this work, we compute via the variational method, the Shannon and Fisher information entropy in position space,SrS_{r}Sr​andIrI_{r}Ir​, for the2p2 \mathbf{p}2pand 3d states of a hydrogen atom subjected to spatial confinement by penetrable and impenetrable spherical cavities. Additionally, we calculate the tunneling probability for the penetrable confinement and seek to relate it to the overlaps of the radial probability densities as a function of the cavity radius and the height of the confining barrier. To model the penetrable confinement, we employ a logistic potential which, unlike earlier reports that used a discontinuous step potential, is a continuous function. The wavefunction is expanded in terms of Slater functions that satisfy appropriate boundary conditions; in some cases, we find notable differences with respect to the impenetrable potential.
Abstract Information-theoretic measures for n l ( 2 L ) states of a H atom (with n = 1 − 10 and l = 0 − 2 , where n and l denote principal and angular momentum quantum numbers) have been investigated within a quantum dot by utilizing the Ritz variational principle, with the help of a Slater-type basis set. A well-established two-parameter (depth and width) model of finite oscillator potential is used to simulate the dot environment. The variationally optimized position (r)-space wave function is utilized to determine the momentum (p)-space wave function, leading to the generation of p-space radial density distribution. We explore the impact of cavity parameters on quantum information theoretic measures, such as Shannon (S) and Fisher information (I) entropy, in the ground as well as the excited state. The results of S were also used to test the Bialynicki–Birula–Mycielski inequality, related to the entropic uncertainty principle for the confined H atom. Some simple new fitting laws pertaining to S and I have been proposed. Furthermore, the p-space radial density is employed to derive the Compton profile of the confined H atom. Possible tunability of S , I and Compton profiles with respect to the parameters is noted.
Various information theoretic measures, complexities and their inequalities are studied for hydrogen atom under modified Hulthen potential which is a combination of Hulthen potential and inverse square potential. In addition, spherical confinement is also considered. Few of these results are compared with the results are available in the literature. In particular the Shannon entropy for free and confined atom is detailed for many states. The effects of inverse square potential parameter (‘a’ which is either positive or negative) on Shannon entropy and Fisher information is studied in confined and free region of atom. The Bialynicki-Birula-Mycielski (BBM) inequality is verified for the confined system.