Quantum information measures of the Dirichlet and Neumann hyperspherical dots

TL;DR

This paper analytically investigates quantum information measures such as Shannon, Rényi, Tsallis entropies, Onicescu energies, and Fisher informations for hyperspherical quantum dots with Dirichlet and Neumann boundary conditions, revealing how boundary conditions and dimensionality influence these measures.

Contribution

It provides explicit analytic expressions for quantum information measures in hyperspherical quantum dots with different boundary conditions and analyzes their dependence on dimension and boundary type.

Findings

Lower thresholds for Rényi/Tsallis entropy existence depend on boundary conditions.

Uncertainty relations are constrained by the dimension and boundary conditions.

Analytic parallels are drawn between quantum dots and hydrogen atom properties.

Abstract

$\mathtt{d}$-dimensional hyperspherical quantum dot with either Dirichlet or Neumann boundary conditions (BCs) allows analytic solution of the Schr\"{o}dinger equation in position space and the Fourier transform of the corresponding wave function leads to the analytic form of its momentum counterpart too. This paves the way to an efficient computation in either space of Shannon, R\'{e}nyi and Tsallis entropies, Onicescu energies and Fisher informations; for example, for the latter measure, some particular orbitals exhibit simple expressions in either space at any BC type. A comparative study of the influence of the edge requirement on the quantum information measures proves that the lower threshold of the semi-infinite range of the dimensionless R\'{e}nyi/Tsallis coefficient where one-parameter momentum entropies exist is equal to $\mathtt{d}/(\mathtt{d}+3)$ for the Dirichlet hyperball and $\mathtt{d}/(\mathtt{d}+1)$ for the Neumann one what means that at the unrestricted growth of the dimensionality both measures have their Shannon fellow as the lower verge. Simultaneously, this imposes the restriction on the upper value of the interval $[1/2,\alpha_R)$ inside which the R\'{e}nyi uncertainty relation for the sum of the position $R_\rho(\alpha)$ and wave vector $R_\gamma\left(\frac{\alpha}{2\alpha-1}\right)$ components is defined: $\alpha_R$ is equal to $\mathtt{d}/(\mathtt{d}-3)$ for the Dirichlet geometry and to $\mathtt{d}/(\mathtt{d}-1)$ for the Neumann BC. Some other properties are discussed from mathematical and physical points of view. Parallels are drawn to the corresponding properties of the hydrogen atom and similarities and differences are explained based on the analysis of the associated wave functions.