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.
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.
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.
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.
Abstract The information‐theoretic measure of confined hydrogen atom has been investigated extensively in the literature. However, most of them were focused on the ground state and accurate values of information entropies, such as Shannon entropy, for confined hydrogen are still not determined. In this work, we establish the benchmark results of the Shannon entropy for confined hydrogen atom in a spherical impenetrable sphere, in both position and momentum spaces. This is done by examining the bound state energies, the normalization of wave functions, and the scaling property with respect to isoelectronic hydrogenic ions. The angular and radial parts of Shannon entropy in two conjugate spaces are provided in detail for both free and confined hydrogen atom in ground and several excited states. The entropies in position space decrease logarithmically with decreasing the size of confinement, while those in momentum space increase logarithmically. The Shannon entropy sum, however, approaches to finite values when the confinement radius closes to zero. It is also found that the Shannon entropy sum shares same trend for states with similar density distributions. Variations of entropy for nodeless bound states are significantly distinct form those owning nodes when changing the confinement radius.
Fisher information (FI) and Shannon entropy (SP), as fundamental theoretical constructs, play crucial roles in quantum information, offering significant insights into communication systems. This work addresses the radial Schrödinger equation for a cotangent hyperbolic potential, solved using the parametric Nikiforov–Uvarov method. The derived wave functions are utilized to explore FI and SP within a one-dimensional system. The analysis focuses on the influence of the screening parameter and potential depth on these quantities, examining their behavior for the ground state and the first excited state. The study complies with key theoretical principles, including the Cramer–Rao inequality, the Beckner-Bialynicki-Birula-Mycielski (BBM) inequality, and the Heisenberg uncertainty principle. The results shed light on the dynamics of quantum states, emphasizing the relationship between quantum uncertainties and information measures. Significant variations in FI and SP emerge with changes in the screening parameter and potential depth, enhancing the understanding of quantum mechanics in confined systems. The findings also reveal the phenomenon of entropy squeezing, underscoring the complex nature of quantum uncertainties. The model demonstrates realistic behavior, with distinct transition points in SP as a function of potential depth, highlighting unique and intriguing characteristics. These contributions enrich the theoretical framework for investigating quantum systems, providing valuable perspectives on their intricate behaviors.
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.
In this research, the radial Schrödinger equation for a newly proposed screened Kratzer–Hellmann potential model was studied via the conventional Nikiforov–Uvarov method. The approximate bound state solution of the Schrödinger equation was obtained using the Greene–Aldrich approximation in addition to the normalized eigenfunction for the new potential model, both analytically and numerically. These results were employed to evaluate the rotational–vibrational partition function and other thermodynamic properties for the screened Kratzer–Hellmann potential. The results obtained have been graphically discussed. Also, the normalized eigenfunction has been used to calculate some information-theoretic measures including Shannon entropy and Fisher information for low-lying states in both position and momentum spaces numerically. The Shannon entropy results obtained agreed with the Bialynicki-Birula and Mycielski inequality, while the Fisher information results obtained agreed with the Stam, Cramér–Rao inequality. Also, alternating increasing and decreasing localization across the screening parameter for both eigenstates was observed.