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

Abstract

d-dimensional hyperspherical quantum dot with either Dirichlet or Neumann boundary conditions (BCs) allows analytic solution of the Schrödinger 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, Rényi 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 Rényi/Tsallis coefficient where one-parameter momentum entropies exist is equal to d/(d + 3) for the Dirichlet hyperball and d/(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, α R ) inside which the Rényi uncertainty relation for the sum of the position R ρ (α) and wave vector R γ α 2α−1 components is defined: α R is equal to d/(d − 3) for the Dirichlet geometry and to d/(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.

Spherical-Symmetry and Spin Effects on the Uncertainty Measures of Multidimensional Quantum Systems with Central Potentials

“…Multidimensional central potentials with a specific analytical form (e.g., power-law, oscillator, Coulomb, van der Waals, Morse, Pöschl–Teller, Hulthen, Woods-Saxon, convex, Yukawa, ...) have been used to interpret a great deal of physical phenomena and chemical processes [ 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 ]. They have been applied to study the behaviour of nanotechnological systems (e.g., quantum dots and wires) and to explain the experiments of dilute systems in magnetic traps at extremely low temperatures [ 16 , 17 , 18 ], which has allowed for a fast development of a density-functional theory of independent particles in multidimensional central potentials [ 11 , 19 ]. The D -dimensional scaling method of Dudley R. Herschbach et al [ 6 ], whose starting point is the high dimensionality limit, is able to describe the physical and chemical properties of finite many-electron systems with an accuracy similar to the self-consistent Hartree–Fock approaches; keep in mind that in both Herschbach and Hartree–Fock methods the average potential which is ultimately diagonalized is spherically symmetric.…”

Quantum-information theory of a Dirichlet ring with Aharonov-Bohm field

“…reveals that the increase of |m| brings from below its normalized value closer and closer to r 1 what means, similar to the QD 34 , a stronger localization of the electron at the outer surface. Contrary, for the Volcano-type potential 36…”

“…Next, a transformation from the QD, r 0 = 0, to the thick QR with very small but nonzero inner radius does not have a conspicuous effect on the m = 0 levels, as windows (a) and (b) exemplify, since for the former geometry their position function vanishes at the center of the disc. However, since the QD axially symmetric orbitals, m = 0, are the only ones that have a finite probability of finding a particle at the origin 34 , this change of the topology of the system from simply to doubly connected domain drastically influences the evolution of the corresponding waveforms. Panel (c) exhibits ground-state function R 10 (r) at several small and extremely small r 0 .…”

“…Angular integration is performed identically to the QD geometry 34 what means that in the momentum space the variables are separated too:…”

“…All mentioned above quantities are under the vigorous theoretical analysis for different nanostructures that is additionally spurred by the experimental advances in evaluating them; e.g., Shannon 28,29 and Rényi [29][30][31][32] entropies were detected and estimated for several manybody systems. In a very recent development, Shannon, Rényi and Tsallis 33 entropies together with the Fisher informations and Onicescu energies were calculated for the circular quantum dot (QD) with the Dirichlet or Neumann boundary conditions (BCs) and their similarities and differences were pointed out 34 . Present endeavor extends this research to the case of the quantum ring (QR), i.e., a 2D planar domain where the charged particle moves inside the annular region of the width a with the inner radius r 0 that is additionally pierced at its center by the Aharonov-Bohm (AB) field 35 .…”