An idealised model to treat the effect of spherically confining the electron in a hydrogen-like atom is studied, where the potential is infinite in all space except for a spherical shell. The exact solution of the Schrödinger equation is obtained in terms of two independent solutions of the Kummer equations. It is found that, in some cases, it is necessary to use the standard Kummer M function and a non-standard second solution. In other cases we may use the Kummer U function and in a limiting case the two standard solutions of Bessel's equation. The effect of an imposed dipole field on the shell is treated using the first-order perturbation equation from which the polarisability can be calculated. In addition, the exact wavefunction is used to calculate the Shannon entropies of both position and momentum and it is shown that these measures give insight into the form of the wavefunction.
An idealised model to treat the effect of spherically confining the electron in a hydrogen-like atom is studied, where the potential is infinite in all space except for a spherical shell. The exact solution of the Schrödinger equation is obtained in terms of two independent solutions of the Kummer equations. It is found that, in some cases, it is necessary to use the standard Kummer M function and a non-standard second solution. In other cases we may use the Kummer U function and in a limiting case the two standard solutions of Bessel's equation. The effect of an imposed dipole field on the shell is treated using the first-order perturbation equation from which the polarisability can be calculated. In addition, the exact wavefunction is used to calculate the Shannon entropies of both position and momentum and it is shown that these measures give insight into the form of the wavefunction.
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.
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 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 chapter reiterates the quantum numbers for atomic orbitals, known from general chemistry, and places them into the context developed so far. It is sketched how the Schrodinger equation (SE) for the hydrogen atom hydrogen-like systems (one electron plus a nucleus of charge Z) is set up. When the nucleus is treated as a fixed point charge, the SE is only for the electron. The solutions of the SE can be obtained by switching to spherical polar coordinates, such that the variables are separable in terms of the electron distance from the nucleus, r, and two angles. The kinetic energy of the electron then has a radial component, and an angular component. The latter is associated with the angular momentum quantum number, which is codified by the letters s, p, d, f, and so forth. A step by step solution of the SE is provided later, in chapter 19.
We review how external electric fields reshape the information content of spatially confined atoms, and we contrast the response to a uniform static field with that to a monochromatic, time-dependent laser field. The central quantities are the Shannon entropies in position and momentum space, Sr and Sp, and their sum ST = Sr +Sp, which obeys the BBM bound. Using the confined hydrogen atom (CHA) and the particle in a spherical box (PISB), its tight confinement limit, as reference systems, we first show that the field-free total entropy of the CHA depends non-monotonically on the confining radius whereas that of the PISB is strictly scale invariant. We then present Stark maps of the entropies for the ten lowest m = 0 levels at the confining radius of 6 a.u. The ground-state polarizability is reduced from the free-atom value 4.5 a.u. to 4.06 a.u. in the CHA but is 46.8 a.u. in the PISB, and the entropic response of the two systems differs correspondingly.
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.