Two‐dimensional confined hydrogen: An entropy and complexity approach

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. K E Y W O R D SFisher information of the two-dimensional confined hydrogen atom, Fisher-Shannon complexity measure of the two-dimensional confined hydrogen atom, LMC-Rényi complexity measure of the two-dimensional confined hydrogen atom, Shannon entropy of the two-dimensional confined hydrogen atom, two-dimensional confined hydrogen atom | INTRODUCTIONThe fundamental and practical relevance of the spherically confined quantum systems has been manifested from the early days of quantum physics [1,2] up until now. [3][4][5][6][7] They have been used as prototypes to explain numerous phenomena and systems not only in the three-dimensional world but also for nonrelativistic and relativistic D-dimensional (D ≥ 2) chemistry and physics. [8][9][10][11][12][13][14] For example, pioneered by Gerhard Ertl, the 2007 Nobel Laureate in Chemistry, and his collaborators, surface chemistry has been extensively studied and has become a key branch in chemistry. [11][12][13] In physics of materials, two-dimensional systems have been used to gain insight into the properties of semiconductors (see, eg, refs. [15, 16]), and they start to play a fundamental role in bringing strong and new forms of control to the atomic-scale limit over the dynamics of matter in the extreme confinement of electromagnetic energy by phonon polaritonics. [10] Moreover, the properties of the real fluids can be studied by means of crystalline fluids (ie, with a symmetry) with nonstandard dimensions (see, eg, ref. [17]).The idea of two-and multidimensional spherical confinement of atoms has been used not only to simulate the effect of high pressure on the static dipole polarizability in hydrogen [18] but also to model a great deal of nanotechnological objects such as quantum dots; quantum wells and quantum wires; [19,20] atoms and molecules embedded in nanocavities, for example, in fullerenes, zeolites cages, and helium droplets; [4,6,7,[21][22][23] dilute bosonic and fermionic systems in magnetic traps of extremely low temperatures; [24][25][26] and a variety of quantum-information elements. [27,28] This has provoked a fast development of a density functional theory of independent particles moving in multidimensional central potentials with various analytical forms (see, eg, refs. [5-7, 29-...

Shannon information entropy sum of the confined hydrogenic atom under the influence of an electric field

“…Furthermore, information-theoretic measures have been shown to display interesting behaviour in confined systems [55,56]. Recently, Estañón et al [48] and Salazar et al [57] have studied entropic measures of 2-D and 3-D confined hydrogenic systems. The latter study focused on the behaviour of the Shannon entropy of confined 3-D hydrogenic systems, where it was shown that the competition between Coulombic and hard wall potential leads to interesting features or structures in the entropy sum as a function of the confinement radius.…”

Charge currents and induced magnetic fields in a bounded two-dimensional hydrogen atom

“…The problem of dimensionally confined hydrogen atom is of interest in various branches of physics like semiconductor physics, material sciences, astrophysics, condensed matter physics, plasma physics and atomic and molecular physics [2]. Hydrogenic atoms confined in one and two dimensions have been well studied [1][2][3][4][5][6][7][8][9][10][11]. Historically, the 2D hydrogen problem was introduced as a leading approximation for the electron motion in a highly anisotropic crystal [12].…”

“…The present work deals with a non-relativistic 2D hydrogen atom spatially bounded in a circular region with impenetrable boundary [6][7][8][9]. Since an exciton is a hydrogen-like bound state of an electron and a hole, the motivation for this study is based on the practical applications to the systems of excitons in the heterostructures.…”

Localization–delocalization of a particle in a quantum corral in presence of a constant magnetic field

“…There exists an increasing interest in the study of diverse quantum confined systems by means of the theoretical information methods [26][27][28][29][30][31][32]39]. In information theory, entropy is a measure of the uncertainty associated with a random variable.…”