There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The task of quantizing general relativity raises serious questions about the meaning of the present formulation and interpretation of quantum mechanics when applied to so fundamental a structure as the space-time geometry itself. This paper seeks to clarify the foundations of quantum mechanics. It presents a reformulation of quantum theory in a form believed suitable for application to general relativity. The aim is not to deny or contradict the conventional formulation of quantum theory, which has demonstrated its usefulness in an overwhelming variety of problems, but rather to supply a new, more general and complete formulation, from which the conventional interpretation can be deduced. The relationship of this new formulation to the older formulation is therefore that of a metatheory to a theory, that is, it is an underlying theory in which the nature and consistency, as well as the realm of applicability, of the older theory can be investigated and clarified.

"relative State" Formulation of Quantum Mechanics

Annals of physics

Mathematica--A System for Doing Mathematics by Computer.

Preface to the second edition Preface to the first edition 1. Background 2. Orthogonal polynomials in two variables 3. General properties of orthogonal polynomials in several variables 4. Orthogonal polynomials on the unit sphere 5. Examples of orthogonal polynomials in several variables 6. Root systems and Coxeter groups 7. Spherical harmonics associated with reflection groups 8. Generalized classical orthogonal polynomials 9. Summability of orthogonal expansions 10. Orthogonal polynomials associated with symmetric groups 11. Orthogonal polynomials associated with octahedral groups and applications References Author index Symbol index Subject index.

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Orthogonal Polynomials of Several Variables

The course of the Diels-Alder reactions of cyclopentadiene and maleic anhydride were studied. Two reaction paths were modelled: endo- and exo-selective paths. All structures within the transient region were characterized and analyzed by means of geometrical descriptors, physicochemical parameters and information-theoretical measures in order to observe the linkage between chemical behavior and the carriage of information. We have shown that the information-theoretical characterization of the chemical course of the reaction is in complete agreement with its phenomenological behavior in passing from reactants to products. In addition, we were able to detect the main differences between the two reaction mechanisms. This type of informational analysis serves to provide tools to help understand the chemical reactivity of the two simplest Diels-Alder reactions, which permits the establishment of a connection between the quantum changes that molecular systems exert along reaction coordinates and standard physicochemical phenomenology. In the present study, we have shown that every reaction stage has a family of subsequent structures that are characterized not solely by their phenomenological behavior but also by informational properties of their electronic density distribution (localizability, order, uniformity). Moreover, we were able to describe the main differences between endo-adduct and exo-adduct pathways. With the advent of new experimental techniques, it is in principle possible to observe the structural changes in the transient regions of chemical reactions. Indeed, through this work we have provided the theoretical concepts needed to unveil the concurrent processes associated with chemical reactions.

Insight into the informational-structure behavior of the Diels-Alder reaction of cyclopentadiene and maleic anhydride.

The position- and momentum-space entropies of the isotropic harmonic oscillator and the hydrogen atom in D dimensions are shown to be related to some entropy integrals which involve classical orthogonal polynomials. These integrals are exactly calculated for Chebyshev polynomials and only in an approximate way for Gegenbauer polynomials. The physical entropies are explicitly obtained in the ground state and in a few low-lying excited states. Finally, the dimensionality dependence of the ground-state entropies of the two above-mentioned quantum-mechanical systems is analyzed (numerically) and the values of the entropies in a large class of excited states of the D-dimensional (D=1,2,3) harmonic oscillator and hydrogen atom are tabulated and discussed.

Position and momentum information entropies of the D-dimensional harmonic oscillator and hydrogen atom

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, Hooke’s 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.

Jesús S. Dehesa

Papers

Insight into the informational-structure behavior of the Diels-Alder reaction of cyclopentadiene and maleic anhydride.

Position and momentum information entropies of the D-dimensional harmonic oscillator and hydrogen atom

The Fisher-Shannon information plane, an electron correlation tool.

The Fisher information of single-particle systems with a central potential

Quantum information entropies and orthogonal polynomials