Relative Shannon entropies (RSE) and Relative Renyi entropies (RRE) are used as a tool
to study the Confined Hydrogen Atom (CHA) in position as well as momentum space.
The choice of the reference state is important as we will see that it helps in revealing important
features in plots of RSE and RRE while studying the CHA in various confinement radius
(rc). RRE of various orders is taken to observe any pattern in the plots which arises
due to variation in the order of RRE.
Relative Shannon entropies (RSE) and Relative Renyi entropies (RRE) are used as a tool
to study the Confined Hydrogen Atom (CHA) in position as well as momentum space.
The choice of the reference state is important as we will see that it helps in revealing important
features in plots of RSE and RRE while studying the CHA in various confinement radius
(rc). RRE of various orders is taken to observe any pattern in the plots which arises
due to variation in the order of RRE.
Abstract Shannon entropy (S), Rényi entropy (R), Tsallis entropy (T), Fisher information (I), and Onicescu energy (E) have been explored extensively in both free H atom (FHA) and confined H atom (CHA). For a given quantum state, accurate results are presented by employing respective exact analytical wave functions in r space. The p‐space wave functions are generated from respective Fourier transforms—for FHA these can be expressed analytically in terms of Gegenbauer polynomials, whereas in CHA these are computed numerically. Exact mathematical expressions of , are derived for circular states of a FHA. Pilot calculations are done taking order of entropic moments (α, β) as in r and p spaces. A detailed, systematic analysis is performed for both FHA and CHA with respect to state indices n, l, and with confinement radius (rc) for the latter. In a CHA, at small rc, kinetic energy increases, whereas decrease with growth of n, signifying greater localization in high‐lying states. At moderate rc, there exists an interplay between two mutually opposing factors: (i) radial confinement (localization) and (ii) accumulation of radial nodes with growth of n (delocalization). Most of these results are reported here for the first time, revealing many new interesting features. Comparison with literature results, wherever possible, offers excellent agreement.
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.
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 Rényi and Shannon entropies are information-theoretic measures, which have enabled to formulate the position–momentum uncertainty principle in a much more adequate and stringent way than the (variance-based) Heisenberg-like relation. Moreover, they are closely related to various energetic density functionals of quantum systems. Here we derive upper bounds on these quantities in terms of the second-order moment 〈r2〉 for general central potentials. This improves previous results of this type. The proof uses the Rényi maximization procedure with a covariance constraint due to Costa et al. [in Proceedings of the Fourth International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR), edited by A. Rangarajan, M. A. T. Figueiredo, and J. Zerubia (Springer-Verlag, Lisbon, 2003), [Lect. Notes Comput. Sci. 52, 211 (2003).]] The contributions to these bounds coming from the radial and angular parts of the physical wave functions are taken into account. Finally, the application to the d-dimensional (d ⩾ 3) hydrogenic and oscillator-like systems is provided.
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.
Abstract The position and momentum spreading of the electron distribution of the two‐dimensional confined hydrogenic atom, which is a basic prototype of the general multidimensional confined quantum systems, is numerically studied in terms of the confinement radius for the 1s, 2s, 2p, and 3d quantum states by means of the main entropy and complexity information‐theoretical measures. First, the Shannon entropy and the Fisher information, as well as the associated uncertainty relations, are computed and discussed. Then, the Fisher‐Shannon, lopezruiz‐mancini‐alvet, and LMC‐Rényi complexity measures are examined and mutually compared. We have found that these entropy and complexity quantities reflect the rich properties of the electron confinement extent in the two conjugated spaces.