On moments-based Heisenberg inequalities

ZOZOR, S.; PORTESI, M.; SANCHEZ MORENO, P.; DEHESA, J. S.

Chamonix

Workshop; 30th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering; 2010

In this paper we revisit the quantitative formulation of the Heisenberg uncertainty principle. The primary version of this principle establishes the impossibility of refined simultaneous measurement of position x and momentum u for a (1-dimensional) quantum particle in terms of variances: ¡¬x¡¬2¡¬u¡¬21/4. Since this inequality applies provided each variance exists, some authors proposed entropic versions of this principle as an alternative (employing Shannon´s or R¨¦nyi´s entropies). As another alternative, we consider moments-based formulations and show that inequalities involving moments of orders other than 2 can be found. Our procedure is based on the R¨¦nyi entropic versions of the Heisenberg relation together with the search for the maximal entropy under statistical moments´ constraints (¡¬x¡¬a and ¡¬u¡¬b). Our result improves a relation proposed very recently by Dehesa et al.. [1] where the same approach was used but starting with the Shannon version of the entropic uncertainty relation. Furthermore, we show that when a =b, the best bound we can find with our approach coincides with that of Ref. [1] and, in addition, for a = b = 2 the variance-based Heisenberg relation is recovered. Finally, we illustrate our results in the cases of d-dimensional hydrogenic systems