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Physica Scripta

Shannon entropy of a particle on a conical surface

Luis Manuel Arvizu, Eleuterio Castaño, N. Aquino

Published Aug 15, 2024• 9 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/ad6fe0

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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.
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.
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.
We study the localization-delocalization of a particle moving within a circular region of radius r0 from a theoretical information point of view. We computed the Shannon entropy, Fisher information and disequilibrium in configuration and momentum spaces for a collection of stationary states. Comparing our results of Shannon entropies with those previously published we found good agreement with those. Shannon entropy, Fisher information and the disequilibrium offer complementary results for the description of the particle localization-delocalization.
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 We consider the quantum problem of a particle in either a spherical box or a finite spherical well confined by a circular cone with an apex angle 2θ 0 emanating from the center of the sphere, with 0 < θ 0 < π. This non-central potential can be solved by an extension of techniques used in spherically-symmetric problems. The angular parts of the eigenstates depend on azimuthal angle φ and polar angle θ as P λ m ( cos θ ) e im φ where P λ m is the associated Legendre function of integer order m and (usually noninteger) degree λ. There is an infinite discrete set of values λ = λ i m (i = 0, 1, 3, ...) that depend on m and θ 0. Each λ i m has an infinite sequence of eigenenergies E n ( λ i m ) , with corresponding radial parts of eigenfunctions. In a spherical box the discrete energy spectrum is determined by the zeros of the spherical Bessel functions. For several θ 0 we demonstrate the validity of Weyl’s continuous estimate  W for the exact number of states  up to energy E, and evaluate the fluctuations of  around  W . We examine the behavior of bound states in a well of finite depth U 0, and find the critical value U c (θ 0) when all bound states disappear. The radial part of the zero energy eigenstate outside the well is 1/r λ+1, which is not square-integrable for λ ≤ 1/2. (0 < λ ≤ 1/2) can appear for θ
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.