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.
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 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 approximate solutions of the Schrödinger equation with the central generalized Hulthén and Yukawa potential are investigated within the framework of the functional method. The obtained wave function and the energy levels are used to study the Shannon entropy, Renyi entropy, Fisher information, Shannon Fisher complexity, Shannon power and Fisher-Shannon product in both position and momentum spaces for the ground and first excited states. The theoretic information theories such as Shannon entropies Sr,Sp, Fisher information Ip,Ip and Renyi entropies Rβ as well as the information entropy probability densities ρ(r),ϕ(p) are illustrated graphically. The Bialynicki-Birula – Mycielski (BBM) inequality for the Shannon entropies and the Cramer-Rao inequality for the Fisher information are satisfied for the potential model.
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 The Shannon entropies in both coordinate and momentum spaces for some low-lying eigenstates of the Hulthén potential (HP) are investigated over a wide range of screening parameters where bound states exist. The system wave functions are expanded in terms of multiple groups of Slater-type orbitals and the Hamiltonian is solved by employing the Rayleigh–Ritz variational method. The accuracy of the obtained bound-state eigenenergies, Shannon entropies, and radial mean values are validated by comparing with the analytical solutions of the HP in s-wave states. The Bialynicki–Birula and Mycielski inequalities for Shannon entropies in both the coordinate and momentum spaces as well as their sum are examined in both weak and strong screening situations. It is shown that the variation of Shannon entropy sum is governed by its upper bound for the system in a well-defined bound state. In the critical bound region where the system undergoes a bound-continuum transition, the Shannon entropies and entropy sum differ in states by their orbital angular momenta and their asymptotic behavior can be properly shaped by the rigorous upper and lower bounds in terms of the radial and momentum expectation values ⟨ r ⟩ and ⟨ p ⟩ . We finally extended the calculation to a modified HP (MHP) that has been widely discussed in the literature and compared our results with previous predictions. Benchmark Shannon entropies are provided for further investigation of the information measures of MHPs.
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.
In this work, we study the Shannon information entropies [Formula: see text] and [Formula: see text] of an infinite spherical well. The Shannon entropy [Formula: see text] is calculated numerically in terms of the analytical result of the wave function in momentum space. Some typical features of the position and momentum probability densities [Formula: see text] and [Formula: see text] as well as the information entropy densities [Formula: see text] and [Formula: see text] are demonstrated. We find that the position entropy [Formula: see text] increases with the radius a of the spherical well for given quantum numbers l, m and n. It is interesting to note that the position entropy [Formula: see text] decreases with the quantum numbers l and n for a fixed radius a and quantum number m. The position entropy [Formula: see text] is almost independent of the quantum numbers l, m and n. The momentum entropy [Formula: see text] first increases and then decreases with respect to the radius a. We also note that the [Formula: see text] increases with the radius a and finally arrives at a constant. In addition, the Bialynicki–Birula–Mycielski (BBM) inequality is verified and also hold for this confined system.