Physical information entropy and probability Shannon entropy

Craig S. Lent

J. O. A. Idiodi, C. A. Onate

Abstract This study presents the Shannon and Renyi information entropy for both position and momentum space and the Fisher information for the position-dependent mass Schrödinger equation with the Frost-Musulin potential. The analysis of the quantum mechanical probability has been obtained via the Fisher information. The variance information of this potential is equally computed. This controls both the chemical properties and physical properties of some of the molecular systems. We have observed the behaviour of the Shannon entropy. Renyi entropy, Fisher information and variance with the quantum number n respectively.

David Ellerman

We live in the information age. Claude Shannon, as the father of the information age, gave us a theory of communications that quantified an "amount of information," but, as he pointed out, "no concept of information itself was defined." Logical entropy provides that definition. Logical entropy is the natural measure of the notion of information based on distinctions, differences, distinguishability, and diversity. It is the (normalized) quantitative measure of the distinctions of a partition on a set--just as the Boole-Laplace logical probability is the normalized quantitative measure of the elements of a subset of a set. And partitions and subsets are mathematically dual concepts--so the logic of partitions is dual in that sense to the usual Boolean logic of subsets, and hence the name "logical entropy." The logical entropy of a partition has a simple interpretation as the probability that a distinction or dit (elements in different blocks) is obtained in two independent draws from the underlying set. The Shannon entropy is shown to also be based on this notion of information-as-distinctions; it is the average minimum number of binary partitions (bits) that need to be joined to make all the same distinctions of the given partition. Hence all the concepts of simple, joint, conditional, and mutual logical entropy can be transformed into the corresponding concepts of Shannon entropy by a uniform non-linear dit-bit transform. And finally logical entropy linearizes naturally to the corresponding quantum concept. The quantum logical entropy of an observable applied to a state is the probability that two different eigenvalues are obtained in two independent projective measurements of that observable on that state. Keywords: logical entropy, Shannon entropy, partitions, MaxEntropy, quantum logical entropy, von Neumann entropy

Thomas Wallek, Andreas Pfennig

Roger Balian

A. N. Ikot, Gaotsiwe Joel Rampho, P. O. Amadi +3 more

The approximate solutions of the Schrödinger equation with the central generalized Hulthén and Yukawa potential are investigated within the framework of the functional method. The obtained wave function and the energy levels are used to study the Shannon entropy, Renyi entropy, Fisher information, Shannon Fisher complexity, Shannon power and Fisher-Shannon product in both position and momentum spaces for the ground and first excited states. The theoretic information theories such as Shannon entropies Sr,Sp, Fisher information Ip,Ip and Renyi entropies Rβ as well as the information entropy probability densities ρ(r),ϕ(p) are illustrated graphically. The Bialynicki-Birula – Mycielski (BBM) inequality for the Shannon entropies and the Cramer-Rao inequality for the Fisher information are satisfied for the potential model.