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Molecular Physics

Stam's principleD-dimensional uncertainty-like relationships and some atomic properties

E. Romera

Published Oct 7, 2002• 36 citations
Publisher

Abstract

Several D-dimensional uncertainty-like relationships for N-body systems are obtained by means of the Fisher's information entropies in position and momentum spaces and the Stam's uncertainty principle. In addition, these relationships, the Fisher's entropies and the Stam's inequality are analysed numerically for all ground state neutral atoms from hydrogen (Z = 1) to lawrencium (Z = 103) using highly accurate Roothaan-Hartree-Fock wavefunctions.

DOI
10.1080/00268970210148778

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Several D-dimensional uncertainty-like relationships for N-body systems are obtained by means of the Fisher's information entropies in position and momentum spaces and the Stam's uncertainty principle. In addition, these relationships, the Fisher's entropies and the Stam's inequality are analysed numerically for all ground state neutral atoms from hydrogen (Z = 1) to lawrencium (Z = 103) using highly accurate Roothaan-Hartree-Fock wavefunctions.
General model-independent relationships among radial expectation values of the one-particle densities in position and momentum spaces for any quantum-mechanical system are obtained. They are derived from the Stam uncertainty principle and the recently reported lower bounds to the Fisher information entropy of both densities. The results are usually expressed in terms of some uncertainty products of the system. The accuracy of the bounds is numerically analyzed for neutral atoms within a Hartree-Fock framework.
Rigorous relationships among radial and logarithmic expectation values of an N-body system are obtained by means of information-theoretic methods. Especially interesting are the uncertainty expressions in terms of the product ((r alpha ) 1 alpha /(p beta ) 1 beta /, which generalize the well known relation (r 2 ))(p 2 )>or=9N 2 /4. Additionally, an inequality involving the uncertainty in the logarithmic radius and the logarithmic momentum is shown. For illustration, the accuracy of the inequalities found is numerically analysed in a Hartree-Fock framework for atomic systems. Finally, it is pointed out how these inequalities are improved in atoms by taking into account other physical quantities or by considering some semiempirical results on atomic information entropies.
The Fisher information functional occupies a very special position among the uncertainty measures of quantum systems of any dimensionality. In contrast with (global) entropic measures of Shannon, Renyi or Tsallis types, it contitutes a local measure in the sense that it is sensitive to the oscillatory character of the probability density distribution. Moreover, the Fisher information is closely related to various macroscopic properties of the system described by different density functionals. Here, we review its main analytical properties such as, inequality-based relationships with some radial expectation values and its uncertainty relation. Tighter versions of these inequalities are shown for those systems subject to an arbitrary central potential. Moreover, the Fisher information is computed for D-dimensional hydrogenic systems explicitly in terms of the radial and angular hyperquantum numbers characterizing the quantum states of the system in both position and momentum spaces. Finally, the utility of this quantity is discussed for various physico-chemical processes of recent interest: Abstraction and nucleophilic substitution reactions.
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.
The purpose of this article is to calculate the numerical values of the Shannon (S) and Fisher (I) information entropies along with their Shannon power (J) and Fisher-Shannon product (P) for a free particle in three dimensions. In both coordinate and momentum space, the numerical values have been calculated for various values of the cubical box's length (L). Further, the validity of the entropic uncertainty relations such as the Bialynicki-Birula and Myceilski (BBM) inequality relation, Stam-Cramer-Rao inequalities (better known as the Fisher based uncertainty relation) and the Fisher-Shannon product relation have been investigated. The Fisher-Shannon product has been introduced and analyzed, which can be considered as a new correlation measure. The article is basically based on the concept, measurements and on the behaviour of the Shannon entropy, Fisher information entropy, Shannon power and the Fisher-Shannon product. Here, the plane wave solution of the free particle in three dimensions is considered and the wave function is normalized in an arbitrarily large but finite cube. The Fourier transforms of the coordinate space wave function provided the momentum space wave function. The probability densities corresponding to the respective wave functions have been employed to compute the numerical values of the information theoretic quantities such as Shannon entropy, Fisher entropy, Shannon power and the Fisher-Shannon product both in the coordinate and momentum space and the computed values have been presented in the respective tables. Finally, the entropic uncertainty relations investigated have been found consistently satisfying the corresponding inequalities such as Bialynicki-Birula and Mycielski (BBM) inequality, the Stam-Cramer-Rao inequalities and the Fisher-Shannon product relation. The study establishes the validity of the entropic uncertainty relations for the motion of a free particle in three dimensions.
The universal property for the information entropy S = a + h In Ζ is verified for atoms using a systematic study with Roothaan-Hartree-Fock (RHF) wave functions with atomic number Ζ — 2 — 54. The above relation was proposed previously for atoms, nuclei, atomic clusters and correlated atoms in a trap. Kullback-Leibler relative entropy Κ and Jensen-Shannon divergence J are employed to compare RHF with Thomas-Fermi (TF) density of atoms as well as another phenomenological density obtained by Sagar et al. Two-body density distributions in position- and momentum-space are used to calculate and compare the corresponding information entropies for correlated and uncorrelated nuclei and bosonic systems (correlated atoms in a trap). It is seen that short-range correlations (SRC) increase the values of S. One-body information entropy entropy S\ is compared with two-body information entropy and a conjecture is made for TV-body information entropy SN- The entropy Κ and the divergence J are also used to evaluate the information distance between correlated and uncorrelated densities both at the one- and the two-body levels for nuclei and trapped Bose gases.