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Rényi, Shannon and Tsallis entropies of Rydberg hydrogenic systems

The Rényi entropies $R_{p}[ρ], 0<p<\infty$ of the probability density $ρ_{n,l,m}(\vec{r})$ of a physical system completely characterize the chemical and physical properties of the quantum state described by the three integer quantum numbers $(n,l,m)$. The analytical determination of these quantities is practically impossible up until now, even for the very few systems where their Schrödinger equation is exactly solved. In this work, the Rényi entropies of Rydberg (highly-excited) hydrogenic states are explicitly calculated in terms of the quantum numbers and the parameter $p$. To do that we use a methodology which first connects these quantities to the $\mathcal{L}_{p}$-norms $N_{n,l}(p)$ of the Laguerre polynomials which characterize the state's wavefunction. Then, the Rényi, Shannon and Tsallis entropies of the Rydberg states are determined by calculating the asymptotics ($n\rightarrow\infty$) of these Laguerre norms. Finally, these quantities are numerically examined in terms of the quantum numbers and the nuclear charge.