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Journal of Mathematical Chemistry

, Volume 54, Issue 9, pp 1777–1782

First Online: 06 June 2016Received: 30 March 2016Accepted: 25 May 2016


Classical and nonclassical contributions to Author’s resultant Shannon- and Fisher-type measures of the information content in general electronic state \\varphi {\varvec{r}} =R {\varvec{r}} \hbox { exp}\hbox {i}\phi {\varvec{r}} \, due to the state probability density \p {\varvec{r}} =R {\varvec{r}} ^{2}\ and its phase \\phi {\varvec{r}} \ or current \{\varvec{j}} {\varvec{r}} =\left \hbar -m ight p {\varvec{r}} abla \phi \left {\varvec{r}} ight \ distributions, respectively, are reexamined. The components of the overall entropy, $$\begin{aligned} S\varphi \equiv -\int {p{\varvec{r}}\ln p{\varvec{r}}+ 2\phi {\varvec{r}} \,d{\varvec{r}}\equiv Sp+S\phi }, \end{aligned}$$are shown to determine the real and imaginary parts of the state complex Shannon entropy, $$\begin{aligned} H\varphi \equiv -2\left\langle {\varphi |\ln \varphi |\varphi } ight angle =S\left p ight +\hbox {i}S\phi , \end{aligned}$$a natural quantum-amplitude generalization of the classical Shannon entropy. Its contributions are related to the associated terms in the state resultant Fisher information, $$\begin{aligned} I\varphi \equiv and {} -4\langle {\varphi | abla ^{2}|\varphi } angle \equiv \int {p{\varvec{r}}\{} abla \ln p{\varvec{r}}^{2}+2 abla \phi \left {\varvec{r}} ight ^{2}\}\,d{\varvec{r}}\equiv Ip+I\phi \\= and {} Ip+\int {p{\varvec{r}}2m-\hbar {\varvec{j}}{\varvec{r}}-p{\varvec{r}}^{2}\,d{\varvec{r}}\equiv Ip+I{\varvec{j}},} \\ \end{aligned}$$and the gradient entropy: $$\begin{aligned} \tilde{I}\varphi \equiv \langle {\varphi | abla \ln p^{2}+\hbox {i}2 abla \phi ^{2}|\varphi } angle =Ip-I\phi =\tilde{I}p+\tilde{I}\phi . \end{aligned}$$KeywordsComplex entropy Fisher information Information theory Nonclassical information Resultant information measures Shannon entropy  Download fulltext PDF

Author: Roman F. Nalewajski


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