R\'{e}nyi and Tsallis entropies: three analytic examples

Abstract

A comparative study of one-dimensional quantum structures which allow analytic expressions for the position and momentum R\'{e}nyi $R(\alpha)$ and Tsallis $T(\alpha)$ entropies, focuses on extracting the most characteristic physical features of these one-parameter functionals. Consideration of the harmonic oscillator reconfirms a special status of the Gaussian distribution: at any parameter $\alpha$ it converts into the equality both R\'{e}nyi and Tsallis uncertainty relations removing for the latter an additional requirement $1/2\leq\alpha\leq1$ that is a necessary condition for all other geometries. It is shown that the lowest limit of the semi infinite range of the dimensionless parameter $\alpha$ where \emph{momentum} components exist strongly depends on the \emph{position} potential and/or boundary condition for the \emph{position} wave function. Asymptotic limits reveal that in either space the entropies $R(\alpha)$ and $T(\alpha)$ approach their Shannon counterpart, $\alpha=1$, along different paths. Similarities and differences between the two entropies and their uncertainty relations are exemplified. Some unsolved problems are pointed at too.