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Entropy

Shannon Entropy of a Hydrogenic Impurity on a Conical Surface: Confinement and Aharonov–Bohm Effects

Luis Manuel Arvizu, Eleuterio Castaño, N. Aquino

Published Mar 22, 2026• 0 citations
Publisher

Abstract

In this work, we solve the Schrödinger equation for a hydrogenic impurity located at the apex of a right circular cone, with the electron constrained to move on the conical surface of semi-aperture angle θ0 and subjected to an Aharonov–Bohm magnetic flux along the symmetry axis. Analytical expressions for the energy eigenvalues and normalized radial wave functions are obtained in terms of the principal quantum number n and the angular quantum number m, the magnetic flux ν, and the cone angle. The Shannon entropy is evaluated in both configuration and momentum spaces for several low-lying states, and its variation with ν and θ0 is analyzed in detail. When the magnetic flux vanishes, pairs of states n, m and n, −m share the same entropic behavior; for finite flux, this degeneracy is lifted and the entropies depend explicitly on the state, the cone geometry, and the flux strength. Finally, we verify that the entropic sum Sr+Sp fulfills the Bialynicki-Birula–Mycielski bound, providing an information-theoretic consistency check for the model.

DOI
10.3390/e28030356

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Results in Physics•Jun 3, 2025•4 citations
In this work, we solve the Schrödinger equation for a hydrogenic impurity located at the apex of a right circular cone, with the electron constrained to move on the conical surface of semi-aperture angle θ0 and subjected to an Aharonov–Bohm magnetic flux along the symmetry axis. Analytical expressions for the energy eigenvalues and normalized radial wave functions are obtained in terms of the principal quantum number n and the angular quantum number m, the magnetic flux ν, and the cone angle. The Shannon entropy is evaluated in both configuration and momentum spaces for several low-lying states, and its variation with ν and θ0 is analyzed in detail. When the magnetic flux vanishes, pairs of states n, m and n, −m share the same entropic behavior; for finite flux, this degeneracy is lifted and the entropies depend explicitly on the state, the cone geometry, and the flux strength. Finally, we verify that the entropic sum Sr+Sp fulfills the Bialynicki-Birula–Mycielski bound, providing an information-theoretic consistency check for the model.
Abstract In this work, we solve the Schrödinger equation of a particle restricted to move on a cone surface of finite height under the influence of an Aharonov–Bohm magnetic field. The energy eigenvalues and their respective wave functions are obtained analytically as a function of the radial distance from the apex and the apex angle. We compute the Shannon entropy in configuration and momentum space for the ground and the first excited states as a function of the slant height and the apex angle. In the absence of magnetic flux the states (n,m) and (n,−m) have identical Shannon entropies, but for non-zero fluxes the Shannon entropies of these states are different, their values depending on the state, the apex angle and the value of the magnetic flux.
Abstract In this work we solve the time-independent Schrödinger equation of a particle restricted to move on the surface of a circular cone of finite height. The energy eigenvalues, as well as the corresponding wave functions, are obtained analytically as a function of, r and ϕ , the radial distance to the apex, 0 ≤ r ≤ r 0 , and the angular variable around the axis of the cone. We compute the Shannon entropy of this system in both configuration and momentum space as a function of r 0 and θ 0 , the angular semi-aperture of the cone. In configuration space, the Shannon entropy decreases, signalling a more pronounced localization, as either r 0 or θ 0 diminish; in momentum space, an opposite behaviour happens, i.e., the Shannon entropy increases when either, r 0 or θ 0 , decrease. We also compute the radial standard deviation; we find that the Shannon entropy better describes the localization-delocalization phenomena. The present results agree with those previously published for a particle confined to a circle of radius r 0 , which corresponds to θ 0 = π / 2 in the present case.
In this research article, the influence of magnetic and Aharonov-Bohm flux fields on Shannon entropy with improved Kratzer potential for Scandium fluoride (ScF) and hydrogen (H2) molecules are studied analytically and numerically for three low-lying states. The wave function and probability density which controls the chemical and physical properties of atomic and molecular systems have been obtained by solving the Schrodinger wave equation coupled with charged particle Hamiltonian via the Nikiforov-Uvarov functional analysis method. The total normalized wave function was expressed in terms of the hypergeometric function of Jacobi polynomials and then utilized to evaluate the Shannon entropy for two diatomic molecules in position and momentum spaces. The wave function and electronic densities were graphically analyzed. Our numerical results of Shannon entropies sum as applied to ScF and H2 were verified to obey the Shannon-entropy-based uncertainty relation. Observations from our results also reveal negative values of Shannon entropies in the position coordinates depicting high localization of particles and stability. In Addition, the effects of the external fields on the quantum systems were observed to influence the quantum information-theoretic measure by small decrements in the Shannon entropy values when compared to the case of the absence of the external fields, indicating a further improvement in the localization of particles and system's stability. This work has applications in atomic and molecular dynamic, molecular drug design, nanostructure processes, information technology, amongst others.
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 Shannon information entropy for scandium fluoride (ScF) and hydrogen (H2) molecules in the presence of magnetic and Aharanov–Bohm (AB) fields with modified Kratzer plus screened Coulomb potential in the position and momentum coordinate is investigated. The wavefunction and probability density are obtained by solving the two-dimensional (2D) Schrödinger wave equation using the Nikiforov–Uvarov functional analysis method and then applied to calculate the Shannon entropy. The graphical behaviour of the 2D wavefunction and probability density is showcased. Our results depict the effects of the external fields on Shannon entropy for ScF and (H2), the negative values of the position Shannon entropies obtained imply higher localisation of the particles and molecules indicating more stability of the system. Another interesting part of this work is that Bialynicki–Birula–Mycielski inequality is tested and proven bounded for both diatomic molecules. A few to mention, this research will enhance the apprehension of the dynamics of quantum molecules and systems subjected to external fields, quantum information signal processing and communication.
In this paper, Fisher and Shannon information entropies in two-dimensional spaces were analysed for hydrogen diatomic molecules using a non-central exponential-type potential, both in the presence and absence of magnetic and Aharanov-Bohm (AB) flux fields. By solving the two-dimensional Schrödinger equation in the presence of charged Hamiltonian using the parametric Nikiforov-Uvarov (NU) method, we obtained numerical solutions for different potential depths for Fisher and Shannon information entropies. These results adhere to Bialynicki-Birula and Mycielski (BBM) and Fisher-based inequalities. The findings have significant applications in chemical physics, aiding in the analysis of molecular electronic properties, stability and reactivity under varying external influences. Additionally, the study contributes to quantum information by offering insights into the localisation and delocalisation phenomena, with direct applications to quantum computation, entanglement, and communication technologies.