Analytical Shannon information entropies for all discrete multidimensional hydrogenic states

TL;DR

This paper derives analytical formulas for the Shannon entropies of all discrete states in multidimensional hydrogenic systems, providing insights into electronic delocalization and uncertainty measures in quantum systems.

Contribution

It provides the first analytical expressions for radial and angular Shannon entropies for all discrete hydrogenic states in any dimension, based on quantum numbers and system parameters.

Findings

Explicit formulas for total Shannon entropies of quasi-spherical states.

Analytical expressions depend on quantum numbers, dimensionality, and nuclear charge.

Includes formulas for the ground state and specific classes of states.

Abstract

The entropic uncertainty measures of the multidimensional hydrogenic states quantify the multiple facets of the spatial delocalization of the electronic probability density of the system. The Shannon entropy is the most adequate uncertainty measure to quantify the electronic spreading and to mathematically formalize the Heisenberg uncertainty principle, partially because it does not depend on any specific point of their multidimensional domain of definition. In this work the radial and angular parts of the Shannon entropies for all the discrete stationary states of the multidimensional hydrogenic systems are obtained from first principles; that is, they are given in terms of the states' principal and magnetic hyperquantum numbers $(n,\mu_1,\mu_2,\ldots,\mu_{D-1})$, the system's dimensionality $D$ and the nuclear charge $Z$ in an analytical, compact form. Explicit expressions for the total Shannon entropies are given for the quasi-spherical states, which conform a relevant class of specific states of the $D$-dimensional hydrogenic system characterized by the hyperquantum numbers $\mu_1=\mu_2\ldots=\mu_{D-1}=n-1$, including the ground state.