Rényi, Shannon and Tsallis entropies of Rydberg hydrogenic systems

Abstract

The Rényi entropies R p [ρ], 0 < p < ∞ of the probability density ρ n,l,m ( 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 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 → ∞) of these Laguerre norms. Finally, these quantities are numerically examined in terms of the quantum numbers and the nuclear charge. PACS numbers: 89.70.Cf, 89.70.-a, 32.80.Ee, 31.15.-p Keywords: Information theory of quantum systems, Rényi entropy of quantum systems, Rydberg states, hydrogenic atoms. I. INTRODUCTIONRecent years have witnessed a growing interest in the analytical information theory of finite quantum systems. A major goal of this theory is the explicit determination of the entropic measures (Fisher information and Shannon, Rényi and Tsallis entropies,...) in terms of the quantum numbers which characterize the state's wavefunction of the system. These quantities, which quantify the spatial delocalization of the single-particle density of the systems in various complementary ways, are most appropriate uncertainty measures because they do not make any reference to some specific point of the corresponding Hilbert space, in contrast to the variance * ivtoranzo@ugr.es † dehesa@ugr.es arXiv:1603.09494v1 [quant-ph]

Benchmark calculations of Rényi, Tsallis entropies, and Onicescu information energy for ground state helium using correlated Hylleraas wave functions

“…Moreover, both Rényi and Tsallis entropies decrease as order α increases. This behavior is also reported in the literature . Furthermore, for α > 1, Rényi entropy is larger than Tsallis entropy and have similar slopes.…”

“…This behavior is also reported in the literature. [28,70,75,76] Furthermore, for α > 1, Rényi entropy is larger than Tsallis entropy and have similar slopes. On the other hand, for α < 1, Tsallis entropy is larger and increases much more drastically than Rényi entropy.…”

Information‐entropic measures in free and confined hydrogen atom

“…Here, we restrict ourselves to a few references pertaining to H atom. Some of these for an unconfined free H atom (FHA) are: in 3D, in D‐dimension, upper bounds of S , R , S in 3D, in D‐dimension, R , T in 3D, in D ‐dimension . Relativistic effects on the information measures of FHA are also examined .…”

“…Former measures total extent of density whereas E quantifies separation of density with respect to equilibrium. They are related to in following way, …”

Quantum information‐theoretic measures for the static screened Coulomb potential

“…However, these information measures and their related uncertainty relations are basic ingredients used to identify several atomic and molecular phenomena . They are widely used in quantum physics in the analysis of quantum entanglement, quantum revivals, atomic ionization properties .…”