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Entropic uncertainty measures for large dimensional hydrogenic systems. (English) Zbl 1373.81430
Summary: The entropic moments of the probability density of a quantum system in position and momentum spaces describe not only some fundamental and/or experimentally accessible quantities of the system but also the entropic uncertainty measures of Rényi type, which allow one to find the most relevant mathematical formalizations of the position-momentum Heisenberg’s uncertainty principle, the entropic uncertainty relations. It is known that the solution of difficult three-dimensional problems can be very well approximated by a series development in \(1/D\) in similar systems with a non-standard dimensionality \(D\); moreover, several physical quantities of numerous atomic and molecular systems have been numerically shown to have values in the large-\(D\) limit comparable to the corresponding ones provided by the three-dimensional numerical self-consistent field methods. The \(D\)-dimensional hydrogenic atom is the main prototype of the physics of multidimensional many-electron systems. In this work, we rigorously determine the leading term of the Rényi entropies of the \(D\)-dimensional hydrogenic atom at the limit of large \(D\). As a byproduct, we show that our results saturate the known position-momentum Rényi-entropy-based uncertainty relations.
©2017 American Institute of Physics

MSC:
81V45 Atomic physics
81V55 Molecular physics
94A17 Measures of information, entropy
62J10 Analysis of variance and covariance (ANOVA)
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