Configuration Complexities of Hydrogenic Atoms

TL;DR

This paper analyzes various information-theoretic measures of hydrogenic atoms, revealing their dependence on quantum numbers and providing bounds, thus deepening understanding of atomic complexity in quantum physics.

Contribution

It explicitly expresses key complexity measures in terms of quantum numbers and provides bounds, advancing the quantitative analysis of hydrogenic atomic states.

Findings

Complexity measures are independent of nuclear charge Z.

Fisher-Shannon measure depends quadratically on quantum number n.

Upper bounds for complexity measures are derived in terms of quantum numbers.

Abstract

The Fisher-Shannon and Cramer-Rao information measures, and the LMC-like or shape complexity (i.e., the disequilibrium times the Shannon entropic power) of hydrogenic stationary states are investigated in both position and momentum spaces. First, it is shown that not only the Fisher information and the variance (then, the Cramer-Rao measure) but also the disequilibrium associated to the quantum-mechanical probability density can be explicitly expressed in terms of the three quantum numbers (n, l, m) of the corresponding state. Second, the three composite measures mentioned above are analytically, numerically and physically discussed for both ground and excited states. It is observed, in particular, that these configuration complexities do not depend on the nuclear charge Z. Moreover, the Fisher-Shannon measure is shown to quadratically depend on the principal quantum number n. Finally, sharp upper bounds to the Fisher-Shannon measure and the shape complexity of a general hydrogenic orbital are given in terms of the quantum numbers.