Crámer‐Raocomplexity of the confined two‐dimensional hydrogen

Abstract

The internal disorder of the confined two-dimensional hydrogenic atom is numerically studied in terms of the confinement radius for the 1s, 2s, 2p, and 3d quantum states by means of the statistical Crámer-Rao complexity measure. First, the confinement dependence of the variance and the Fisher information of the position and momentum spreading of its electron distribution are computed and discussed. Then, the Crámer-Rao complexity measure (which quantifies the combined balance of the charge concentration around the mean value and the gradient content of the electron distribution) is investigated in position and momentum spaces. We found that confinement does distinguish complexity of the system for all quantum states by means of these two component measures.

The confined helium atom: An information–theoretic approach

“…Fisher–Shannon complexity measure for a probability density $$$ \rho $$$ is defined jointly by the Fisher information $$$ {F}_r\left[\rho \right] $$$ and the Shannon entropic power. The Fisher information [49, 51–53, 55, 83, 88, 93, 100, 101] is a point‐to‐point measure of the electron cloud distribution since it is a gradient functional of $$$ \rho \left(\overrightarrow{r}\right) $$$ and in configuration space is tightly connected to the kinetic energy due to its dependence on the gradient of the distribution. It is interpreted as a measure of the tendency toward disorder, meaning that the larger this quantity is, the more ordered the distribution will be.…”

Crámer-Rao complexity of the two-dimensional confined hydrogen

“…The higher this quantity is, the more localized is the density, the smaller is the uncertainty and the higher is the accuracy in estimating the localization of the particle. Recently (see Figure 3 of [54]) the Fisher information of the 2D-HA has been computed for the 1s, 2s, 2p and 3d states. Therein, we found that the Fisher information decreases (position) and increases (momentum) when r 0 is increasing, so that they tend broadly and fastly to the free values (analytically calculated in section II and numerically given in Table I) in such a way that the Fisher-information-based uncertainty relation for the 1s and 2s states is always fulfilled because they are described by real wavefunctions.…”

“…Up until now, contrary to the stationary states of the confined 3D-HA where both the (energy-dependent) spectroscopic and the (eigenfunction-dependent) information-theoretic properties have received much attention [27,[38][39][40][41][42][43][44][45][46][47], the knowledge of these properties for the confined 2D-HA is quite scarce [48-50, 52, 53]. Just recently, the authors have determined [54] the entropy-like (Shannon, Fisher) and complexity-like (Fisher-Shannon, LMC) measures for a few low-lying stationary states of the confined 2D-HA. The aim of this work is to extend this informational approach by means of the calculation of the confinement dependence of the variance and the Crámer-Rao complexity measure for the 1s, 2s, 2p and 3d quantum states of the 2D-HA in the two conjugated spaces.…”

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.…”