Position and momentum information-theoretic measures of a D-dimensional particle-in-a-box

Abstract

The main information-theoretic measures of a one-dimensional particle-in-a-box (also known as the infinite potential well or the infinite square well) in both position and momentum spaces, as well as their associated uncertainty relations, are calculated and discussed. The power and entropic moments, the Shannon, Renyi and Tsallis entropies and the Fisher information together with two composite measures (Fisher–Shannon and LMC shape complexities) are considered. Moreover, the associated information-theoretic spreading lengths, which characterize the spread/delocalization of the particle beyond (but complementarily) the standard deviation, and their corresponding uncertainty relations are given and mutually compared. It is found, in particular, that the Fisher length is the proper measure of uncertainty for the infinite well, mainly because it grasps the oscillatory nature of the wavefunctions. Finally, this study is extended to a D-dimensional box.