E. Romera

The Fisher-Shannon information plane, an electron correlation tool

A new correlation measure, the product of the Shannon entropy power and the Fisher information of the electron density, is introduced by analyzing the Fisher-Shannon information plane of some two-electron systems (He-like ions, Hookes atoms). The uncertainty and scaling properties of this information product are pointed out. In addition, the Fisher and Shannon measures of a finite many-electron system are shown to be bounded by the corresponding single-electron measures and the number of electrons of the system.

Uncertainty relation for Fisher information of D-dimensional single-particle systems with central potentials

An uncertainty Fisher information relation in quantum mechanics is derived for multidimensional single-particle systems with central potentials. It is based on the concept of Fisher information in the two complementary position and momentum spaces, which is a gradient functional of the corresponding probability distributions. The lower bound of the product of position and momentum Fisher informations is shown to depend on the orbital and magnetic quantum numbers of the physical state and the space dimensionality. Applications to various elementary systems is discussed.

Identifying Wave-Packet Fractional Revivals by Means of Information Entropy

Wave-packet fractional revivals is a relevant feature in the long time-scale evolution of a wide range of physical systems, including atoms, molecules, and nonlinear systems. We show that the sum of information entropies in both position and momentum conjugate spaces is an indicator of fractional revivals by analyzing three different model systems: (i) the infinite square well, (ii) a particle bouncing vertically against a wall in a gravitational field, and (iii) the vibrational dynamics of hydrogen iodide molecules. This description in terms of information entropies complements the usual one in terms of the autocorrelation function.

A generalized statistical complexity measure: Applications to quantum systems

A two-parameter family of complexity measures C(α,β) based on the Renyi entropies is introduced and characterized by a detailed study of its mathematical properties. This family is the generalization of a continuous version of the Lopez-Ruiz–Mancini–Calbet complexity, which is recovered for α=1 and β=2. These complexity measures are obtained by multiplying two quantities bringing global information on the probability distribution defining the system. When one of the parameters, α or β, goes to infinity, one of the global factors becomes a local factor. For this special case, the complexity is calculated on different quantum systems: H-atom, harmonic oscillator, and square well.

The Hausdorff entropic moment problem

Our aim in this paper is twofold. First, to find the necessary and sufficient conditions to be satisfied by a given sequence of real numbers {ωn}n=0∞ to represent the “entropic moments” ∫[0,a][ρ(x)]ndx of an unknown non-negative, decreasing and differentiable (a.e.) density function ρ(x) with a finite interval support. These moments are called entropic moments because they are closely connected with various information entropies (Renyi, Tsallis, …). Second, we outline an efficient method for the reconstruction of the density function from the knowledge of its first N entropic moments.

Electron momentum densities of atoms

Spherically averaged electron momentum densities Π(p) are constructed by the numerical Hartree–Fock method for all 103 atoms from hydrogen (atomic number Z=1) to lawrencium (Z=103) in their experimental ground states. We find three different types of momentum densities spread across the periodic table in a very simple manner for the 98 atoms other than He, N, Mn, Ge, and Pd. Atoms in groups 1–6, 13, and 14, and all lanthanides and actinides have a unimodal momentum density with a maximum at p=0, atoms in groups 15–18 have a unimodal momentum density with a local minimum at p=0 and a maximum at p>0, and atoms in groups 7–12 have a bimodal momentum density with a primary maximum at p=0 and a small secondary maximum at p>0. Our results confirm the existence of nonmonotonic momentum densities reported in the literature, but also reveal some errors in the previous classification of atomic momentum densities. The physical origin for the appearance of the three different modalities in Π(p) is clarified by analys...

Entropic uncertainty and the quantum phase transition in the Dicke model

We show that the description of the quantum phase transition in terms of the entropic uncertainty relation turns out to be more suitable than in terms of the standard variance-based uncertainty relation. The entropic uncertainty relation detects the quantum phase transition in the Dicke model and it provides a correct description of the quantum fluctuations or quantum uncertainty of the system.

Inverse atomic densities and inequalities among density functionals

Rigorous relationships among physically relevant quantities of atomic systems (e.g., kinetic, exchange, and electron–nucleus attraction energies, information entropy) are obtained and numerically analyzed. They are based on the properties of inverse functions associated to the one-particle density of the system. Some of the new inequalities are of great accuracy and/or improve similar ones previously known, and their validity extends to other many-fermion systems and to arbitrary dimensionality.

A generalized relative complexity measure

A class of two-parameter relative complexity measures is presented. Several important properties are proved. As an example we consider the Dicke model that describes a single-mode bosonic field interacting with an ensemble of N two-level atoms. There is a quantum phase transition in the limit. It is found that the relative complexity is an excellent marker of the quantum phase transition.

Reconstruction of a density from its entropic moments

We describe a general reconstruction method of the density of a physical system from a finite number of entropic moments. These statistical quantities, which are integrals of the density to a power α∈R, may also represent some fundamental and/or experimentally accessible quantities of quantum-mechanical systems for specific values of α. We take advantage of the strategy recently used by us to solve the Hausdorff and Stieltjes entropic moment problems, where the main role is played by the inverse function of the density. In our method we first calculate such inverse function by use of an algorithm of minimization of the Fisher information measure of the density, and then we invert it. Two particular cases are discussed to illustrate the applicability of the method.