By using the Renyi entropy, and following the same scheme that in the Fisher-Renyi entropy product case, a generalized statistical complexity is defined. Several properties of it, including inequalities and lower and upper bounds are derived. The hydrogen atom is used as a test system where to quantify these two different statistical magnitudes, the Fisher-Renyi entropy product and the generalized statistical complexity. For each level of energy, both indicators take their minimum values on the orbitals that correspond to the highest orbital angular momentum. Hence, in the same way as happens with the Fisher-Shannon and the statistical complexity, these generalized Renyi-like statistical magnitudes break the energy degeneration in the H-atom.
Deep Dive into Generalized statistical complexity and Fisher-Renyi entropy product in the $H$-atom.
By using the Renyi entropy, and following the same scheme that in the Fisher-Renyi entropy product case, a generalized statistical complexity is defined. Several properties of it, including inequalities and lower and upper bounds are derived. The hydrogen atom is used as a test system where to quantify these two different statistical magnitudes, the Fisher-Renyi entropy product and the generalized statistical complexity. For each level of energy, both indicators take their minimum values on the orbitals that correspond to the highest orbital angular momentum. Hence, in the same way as happens with the Fisher-Shannon and the statistical complexity, these generalized Renyi-like statistical magnitudes break the energy degeneration in the H-atom.
Nowadays the study of statistical magnitudes in quantum systems has a role of growing importance. So, information entropies and statistical complexities have been calculated on different atomic systems [1,2]. In particular, the H-atom is a natural test system where to quantify all this kind of magnitudes [3,4,5,6,7]. This system displays a remarkable property when Fisher-Shannon information [8,9,10] and the so called LMC complexity [11,12] are computed on it. Namely, for each energy level, the minimum values of those statistical measures are taken on the wave functions with the highest orbital angular momentum, those orbitals that correspond to the Bohr-like orbits in the pre-quantum image [7].
Shannon information [13] plays an important role in both entropic products, the Fisher-Shannon information and the LMC statistical complexity. Different generalizations of the Shannon information dependent on a parameter α can be found in the literature [14,15]. Implementing these α-entropies in both entropic products, different families of statistical indicators can be generated [16]. In particular, if Rényi entropies are taken in exponential form for this purpose, the Fisher-Rényi entropy product [17] and a new generalized LMC statistical complexity can be defined.
In this work, an information theoretical analysis of the same quantum system, the Hatom, is presented in terms of the Fisher-Rényi entropy product and the generalized LMC statistical measure. These α-dependent magnitudes show a similar behavior to that found in the limit α → 1, that correspond to the already studied case of the Fisher-Shannon information and LMC complexity [7]. That is, the degeneration of the energy is also broken by these statistical magnitudes.
The paper is organized as follows. In section 2, the Fisher-Rényi entropy product is recalled and the generalized LMC complexity measure is introduced. Some of their properties are also presented. The calculation of these magnitudes in the H-atom is presented in section 3. Our conclusions are contained in the last section.
Consider a D-dimensional distribution function f (r), with f (r) nonnegative and f (r)dr = 1 and define Rényi entropy power of index α blue as in [17] by
being
a decreasing function from 1 to 0 when α runs in (0.5, ∞) and η α=1 = e -1 , and the Rényi entropy of order α given by
where r stands for r 1 , …, r D .
Rényi entropy power is an extension of Shannon entropy power [18] and verifies that when
A scaling property is verified by Rényi entropy power [17], which transforms as
under scaling of the function Ψ λ (r 1 , …, r D ) = λ D/2 Ψ(λr 1 , …, λr D ). Rényi entropy power also has the property [17]
Fisher information [19] of the probability density function f is given by
A. Fisher-Rényi Entropy Product P (α) f
The Fisher-Rényi entropy product is defined by
It displays the following important properties [17]:
is invariant under scaling transformation f λ = λ D f (λr), i. e. P (α)
f , (ii) it verifies the inequality P
The measure of complexity C f introduced in [11,12], the so-called LMC complexity, is defined by
It can be generalized to the α-dependent measure of complexity,
f , which is defined by
that tends to the measure of complexity C f in the limit α → 1. It satisfies the next properties:
f , (iii) taking into account that Rényi entropy is a nonincreasing function of α [18], it is straightforward to see that
Let us finally point out that when α goes to 1, the lower bound (that takes the value 1) for the original LMC complexity is recovered [12].
The probability density, ρ(r), for a bound state, Ψ nlm (r), with quantum numbers (n, l, m) of the non-relativistic H-atom is given by ρ(r) = |Ψ nlm (r)| 2 in position space (r = (r, Ω) with r the radial distance and Ω the solid angle), with
The radial part, R nl (r), is expressed as [20] R
with L β α (r) the associated Laguerre polynomials, and Y lm (Ω) the spherical harmonic of the atomic state. Let us recall at this point the range of the quantum numbers: n ≥ 1, 0 ≤ l ≤ n -1, and -l ≤ m ≤ l. Atomic units are used through the text.
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