Quantum information measures of the Dirichlet and Neumann hyperspherical dots

Abstract

$\mathtt{d}$-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 $\mathtt{d}/(\mathtt{d}+3)$ 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 Rényi 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.

arXiv comment5 figures, 2 tables

References in corpus (35)

• Measuring entanglement entropy through the interference of quantum many-body twins
• Quantum thermalization through entanglement in an isolated many-body system
• Entropic Uncertainty Relations and their Applications
• Probing entanglement entropy via randomized measurements
• Entropic uncertainty relations - A survey
• Formulation of the uncertainty relations in terms of the Renyi entropies
• Features of the Extension of a Statistical Measure of Complexity to Continuous Systems
• Quantum-memory-assisted entropic uncertainty relations
• Continuous-variable entropic uncertainty relations
• The Large D Limit of Einstein's Equations
• On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles
• On the Role of Information Theoretic Uncertainty Relations in Quantum Theory
• Entropic measures of Rydberg-like harmonic states
• Rényi entropies of the highly-excited states of multidimensional harmonic oscillators by use of strong Laguerre asymptotics
• Rényi, Shannon and Tsallis entropies of Rydberg hydrogenic systems
• The Shannon-entropy-based uncertainty relation for $D$-dimensional central potentials
• Entropic properties of $D$-dimensional Rydberg systems
• Entropic uncertainty measures for large dimensional hydrogenic systems
• Complexity of D-dimensional hydrogenic systems in position and momentum spaces
• Rényi and Tsallis entropies: three analytic examples
• Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems
• Rényi and Tsallis entropies of the Aharonov-Bohm ring in uniform magnetic fields
• Uncertainty Relations for coarse-grained measurements: an overview
• Exact Shannon entropies for the multidimensional harmonic states
• Analytical Shannon information entropies for all discrete multidimensional hydrogenic states
• Shannon entropy as a measure of uncertainty in positions and momenta
• Heisenberg and entropic uncertainty measures for large-dimensional harmonic systems
• Comparative analysis of information measures of the Dirichlet and Neumann two-dimensional quantum dots
• Rényi entropies for multidimensional hydrogenic systems in position and momentum spaces
• Exact Rényi entropies of $D$-dimensional harmonic systems
• One-parameter Fisher-Rényi complexity: Notion and hydrogenic applications
• Entropy and complexity analysis of the D-dimensional rigid rotator and hyperspherical harmonics
• Quantum information measures of the one-dimensional Robin quantum well
• Heisenberg-like uncertainty measures for $D$-dimensional hydrogenic systems at large D
• Rényi and Tsallis entropies of the Dirichlet and Neumann one-dimensional quantum wells

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