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Journal of Mathematical Physics

Uncertainty inequalities among frequency moments and radial expectation values: Applications to atomic systems

Juan Carlos Angulo, Peter Alexander Bouvrie, J. Antolín

Published Apr 1, 2012• 7 citations
Publisher

Abstract

Rigorous and universal bounds on frequency moments of one-particle densities in terms of radial expectation values in the conjugate space are obtained. The results, valid for any d-dimensional quantum-mechanical system, are derived by using Rényi-like position-momentum inequalities in an information-theoretical framework. Especially interesting are the upper bounds on the Dirac exchange and Thomas-Fermi kinetic energies, as well as the disequilibrium or self-similarity of both position and momentum distributions. A variety of bounds for these functionals in a given space are known, but most usually in terms of quantities defined within the same space. Very few results including a density functional on one space, and expectation values on the conjugate one, are found in the literature. A pioneering bound on the disequilibrium in terms of the kinetic energy is improved in this work. A numerical study of the aforementioned relationships is carried out for atomic systems in their ground state. Some results are given in terms of relevant physical quantities, including the kinetic and electron-nucleus attraction energies, the diamagnetic susceptibility and the height of the peak of the Compton profile, among others.

DOI
10.1063/1.4705275

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Rigorous and universal bounds on frequency moments of one-particle densities in terms of radial expectation values in the conjugate space are obtained. The results, valid for any d-dimensional quantum-mechanical system, are derived by using Rényi-like position-momentum inequalities in an information-theoretical framework. Especially interesting are the upper bounds on the Dirac exchange and Thomas-Fermi kinetic energies, as well as the disequilibrium or self-similarity of both position and momentum distributions. A variety of bounds for these functionals in a given space are known, but most usually in terms of quantities defined within the same space. Very few results including a density functional on one space, and expectation values on the conjugate one, are found in the literature. A pioneering bound on the disequilibrium in terms of the kinetic energy is improved in this work. A numerical study of the aforementioned relationships is carried out for atomic systems in their ground state. Some results are given in terms of relevant physical quantities, including the kinetic and electron-nucleus attraction energies, the diamagnetic susceptibility and the height of the peak of the Compton profile, among others.
Rigorous and universal relationships among radial expectation values of any DDD-dimensional quantum-mechanical system are obtained, using R\'enyi-like position-momentum inequalities in an information-theoretical framework. Although the results are expressed in terms of four moments (two in position space and two in the momentum one), especially interesting are the cases that provide expressions of uncertainty in terms of products $\ensuremath{\langle}{r}^{a}\ensuremath{\rangle}{}^{1/a}\ensuremath{\langle}{p}^{b}\ensuremath{\rangle}{}^{1/b}$, widely considered in the literature, including the famous Heisenberg relationship $\ensuremath{\langle}{r}^{2}\ensuremath{\rangle}\ensuremath{\langle}{p}^{2}\ensuremath{\rangle}\ensuremath{\geqslant}{D}^{2}/4$. Improved bounds for these products have recently been provided, but are always restricted to positive orders a,b>0a,b>0a,b>0. The interesting part of this work are the inequalities for negative orders. A study of these relationships is carried out for atomic systems in their ground state. Some results are given in terms of relevant physical quantities, including the kinetic and electron-nucleus attraction energies, the diamagnetic susceptibility, and the height of the peak of the Compton profile, among others.
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 bounds to atomic information entropies and their sum in position and momentum spaces have been derived in terms of radial and momentum expectation values ⟨ r n ⟩ and ⟨ p n ⟩. Numerical studies on atomic systems show that the bounds presented in terms of ⟨ r ⟩ and ⟨ p ⟩, the first moments of the position and momentum - space densities are sharper than those given by Gadre and Bendale in terms of ⟨ r 2 ⟩ and ⟨ p 2 ⟩, the second moments of the position and momentum-space densities. Within the BKCM procedure, several relationships between the information entropies and the average electron densities ⟨ρ⟩ and ⟨γ⟩ in position and momentum spaces have been established.
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 Rényi and Shannon entropies are information-theoretic measures, which have enabled to formulate the position–momentum uncertainty principle in a much more adequate and stringent way than the (variance-based) Heisenberg-like relation. Moreover, they are closely related to various energetic density functionals of quantum systems. Here we derive upper bounds on these quantities in terms of the second-order moment 〈r2〉 for general central potentials. This improves previous results of this type. The proof uses the Rényi maximization procedure with a covariance constraint due to Costa et al. [in Proceedings of the Fourth International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR), edited by A. Rangarajan, M. A. T. Figueiredo, and J. Zerubia (Springer-Verlag, Lisbon, 2003), [Lect. Notes Comput. Sci. 52, 211 (2003).]] The contributions to these bounds coming from the radial and angular parts of the physical wave functions are taken into account. Finally, the application to the d-dimensional (d ⩾ 3) hydrogenic and oscillator-like systems is provided.