R\'enyi and Tsallis entropies of the Dirichlet and Neumann one-dimensional quantum wells
Abstract
A comparative analysis of the Dirichlet and Neumann boundary conditions (BCs) of the one-dimensional (1D) quantum well extracts similarities and differences of the R\'enyi R(α) as well as Tsallis T(α) entropies between these two geometries. It is shown, in particular, that for either BC the dependencies of the R\'enyi position components on the parameter α are the same for all orbitals but the lowest Neumann one for which the corresponding functional R is not influenced by the variation of α. Lower limit αTH of the semi infinite range of the dimensionless R\'enyi/Tsallis coefficient where momentum entropies exist crucially depends on the position BC and is equal to one quarter for the Dirichlet requirement and one half for the Neumann one. At α approaching this critical value, the corresponding momentum functionals do diverge. The gap between the thresholds αTH of the two BCs causes different behavior of the R\'enyi uncertainty relations as functions of α. For both configurations, the lowest-energy level at α=1/2 does saturate either type of the entropic inequality thus confirming an earlier surmise about it. It is also conjectured that the threshold αTH of one half is characteristic of any 1D non-Dirichlet system. Other properties are discussed and analyzed from the mathematical and physical points of view.