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.
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.
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 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.
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.
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 In this research work, the quantum information‐theoretic analysis of the static screened Coulomb potential has been carried out by studying both analytically and numerically the entropic measures, Fisher information as well as the Onicescu information energy of its wave function. Explicit expressions of these information‐theoretic measures were obtained. Using the Srivastava–Daoust linearization formula in terms of the multivariate Lauricella hypergeometric function, the Rényi entropy, Tsallis entropy, Onicescu information energy were analytically obtained. From the results obtained, it is observed that some of the Shannon entropies are negative, indicating that, negative entropies exists for the probability densities that are highly localized. The trends in the variation of the information‐theoretic measures with the potential screening parameter a for this atomic model are discussed. The Bialynicki‐Birula, Mycielski inequality (BBM), and the Fisher information based uncertainty relation are also verified.
Abstract Confinement of atoms inside various cavities has been studied for nearly eight decades. However, the Compton profile (CP) for such systems has not yet been investigated. Here we construct the CP for a H atom radially confined inside a hard spherical enclosure, as well as in a free condition . Some exact analytical relations for the CP’s of circular or nodeless states of free atoms is presented. By means of a scaling idea, this has been further extended to the study of a H-like atom trapped inside an impenetrable cavity. The accuracy of these constructed CPs has been confirmed by computing various momentum moments. Apart from that, several information theoretical measures, like Shannon entropy ( S ) and Onicescu energy ( E ) have been exploited to characterize these profiles. Exact closed-form expressions are derived for S and E using the ground state CP in free H-like atoms. A detailed study reveals that, increase in confinement inhibits the rate of dissipation of kinetic energy. At a fixed ℓ , this rate diminishes with a rise in n . However, at a certain n , this rate accelerates with progress in ℓ . Similar analysis on the respective free counterpart displays an exactly opposite trend as that in a confined system. However, in both free and confined environments, the CP generally gets broadened with rise in Z . Representative calculations are done numerically for low-lying states of the confined systems, taking two forms of position-space wave functions: (a) exact (b) highly accurate eigenfunctions through a generalized pseudospectral method. In essence, CPs are reported for confined H atoms (and isoelectronic series) and investigated adopting an information-theoretic framework.