Comparative analysis of information measures of the Dirichlet and Neumann two-dimensional quantum dots

TL;DR

This paper compares information measures like Shannon, Rényi, Tsallis entropies, Onicescu energies, and Fisher information for 2D Dirichlet and Neumann quantum dots, revealing how boundary conditions and dimensionality affect these measures and their uncertainty relations.

Contribution

It provides an analytic framework for computing and analyzing various information measures in 2D quantum dots with different boundary conditions, highlighting the impact of geometry and boundary types.

Findings

Lower limits of Rényi/Tsallis coefficients depend on boundary conditions and dimensionality.

Rényi uncertainty relation validity range differs between Dirichlet and Neumann cases.

Lowest-energy levels saturate entropic inequalities at specific coefficients.

Abstract

Analytic representation of both position as well as momentum waveforms of the two-dimensional (2D) circular quantum dots with the Dirichlet and Neumann boundary conditions (BCs) allowed an efficient computation in either space of Shannon $S$, R\'{e}nyi $R(\alpha)$ and Tsallis $T(\alpha)$ entropies, Onicescu energies $O$ and Fisher informations $I$. It is shown that a transition to the 2D geometry lifts the 1D degeneracy of the position components $S_\rho$, $O_\rho$, $R_\rho(\alpha)$. Among many other findings, it is established that the lower limit $\alpha_{TH}$ of the semi-infinite range of the dimensionless R\'{e}nyi/Tsallis coefficient where one-parameter momentum entropies exist is equal to 2/5 for the Dirichlet requirement and 2/3 for the Neumann one. Since their 1D counterparts are $1/4$ and $1/2$, respectively, this simultaneously reveals that this critical value crucially depends not only on the position BC but the dimensionality of the structure too. As the 2D Neumann threshold $\alpha_{TH}^N$ is greater than one half, its R\'{e}nyi uncertainty relation for the sum of the position and wave vector components $R_\rho(\alpha)+R_\gamma\left(\frac{\alpha}{2\alpha-1}\right)$ is valid in the range $[1/2,2)$ only with its logarithmic divergence at the right edge whereas for all other systems it is defined at any coefficient $\alpha$ not smaller than one half. For both configurations, the lowest-energy level at $\alpha=1/2$ does saturate R\'{e}nyi and Tsallis entropic inequalities. Other properties are discussed and analyzed from the mathematical and physical points of view.