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.
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.
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 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.
Spatial confinements induce localization or delocalization on the electron density in atoms and molecules, and the hydrogen atom is not the exception to these results. In previous works, this system has been confined by an infinite and a finite potential where the wave-function exhibits an exact solution, and, consequently, their Shannon entropies deliver exact results. In this article, the Shannon entropy in configuration space is examined for the hydrogen atom submitted to four different potentials: (a) infinite potential; (b) Coulomb plus harmonic oscillator; (c) constant potential; and (d) dielectric continuum. For all these potentials, the Schrödinger equation admitted an exact analytic solution, and therefore the corresponding electron density has a closed-form. From the study of these confinements, we observed that the Shannon entropy in configuration space is a good indicator of localization and delocalization of the electron density for ground and excited states of the hydrogen atom confined under these circumstances. In particular, the confinement imposed by a parabolic potential induced characteristics that were not presented for other confinements; for example, the kinetic energy exhibited oscillations when the confinement radius is varied and such oscillations coincided with oscillations showed by the Shannon entropy in configuration space. This result indicates that, when the kinetic energy is increased, the Shannon entropy is decreased and vice versa.
We calculate the energy and Shannon entropy for a hydrogen atom confined in a dielectric spherical microcavity for the fist time. In contrast to the hydrogen atom in the vacuum microcavity, some unexpected and interesting phenomena appear: First, the turning radius for the bound energy changes from positive to negative depends on the dielectric constant of the spherical microcavity sensitively. With the increase of the relative dielectric constant, the turning radius gets larger. Second, the dielectric spherical microcavity impacts the rearrangement of the excited state energy, and breaks the energy degeneracy of the excited states. At some given radius, there is energy crossover between different orbital. Third, the dielectric in the spherical microcavity affects the Shannon entropy for the confined hydrogen atom greatly. The Shannon entropy in the vacuum microcavity is the smallest and the Shannon entropy increases with the relative dielectric constant. For smaller size spherical microcavity, the Shannon entropy change is always negative, which suggests that the electron density is localized. With the increase of the radius of the microcavity, the Shannon entropy change becomes positive, and the confinement of the electron density gets delocalized. Our results show that we can control the confining effect of the spherical microcavity on the atom by changing the dielectric. This work can guide the future experimental studies for trapping and manipulating of atoms and molecules in the external environment and has some practical applications in metrology and quantum information processing.
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,SrandIr, for the2pand 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 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.