Rényi, Shannon and Tsallis entropies of Rydberg hydrogenic systems

Abstract

The Rényi entropies $R_p[\rho ],0<p<\infty$ of the probability density ${\rho }_{n,l,m}(\vec{r})$ 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 ($n\to \infty$) of these Laguerre norms. Finally, these quantities are numerically examined in terms of the quantum numbers and the nuclear charge.

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• Rényi and Tsallis entropies of the Aharonov-Bohm ring in uniform magnetic fields
• Analytical Shannon information entropies for all discrete multidimensional hydrogenic states
• Exact Shannon entropies for the multidimensional harmonic states
• Rényi entropies for multidimensional hydrogenic systems in position and momentum spaces
• Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters
• Heisenberg-like uncertainty measures for $D$-dimensional hydrogenic systems at large D
• Rényi and Tsallis entropies of the Dirichlet and Neumann one-dimensional quantum wells
• Quantum information measures of the Dirichlet and Neumann hyperspherical dots
• The Moments of the Hydrogen Atom by the Method of Brackets