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Shannon entropy: a study of confined hydrogenic-like atoms

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Wallas Nascimento Universidade Federal da Bahia

Frederico V. Prudente Universidade Federal da Bahia

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Shannon entropy: a study of confined hydrogenic-like atoms

Wallas S. Nascimento1, Frederico V. Prudente2 Instituto de Fısica, Universidade Federal da Bahia, 40170-115 Salvador, BA, Brasil

Abstract

The Shannon entropy in the atomic, molecular and chemical physics context is presented by using as test cases the hydrogenic-like atoms Hc, He+c and Li2+ c confined by an impenetrable spherical box. Novel expressions for entropic un- certainty relation and Shannon entropies Sr and Sp are proposed to ensure their physical dimensionless characteristic. The electronic ground state energy and the quantities Sr, Sp and St are calculated for the hydrogenic-like atoms to different confinement radii by using a variational method. The global behav- ior of these quantities and different conjectures are analyzed. The results are compared, when available, with those previously published.

Keywords: Information Theory, Shannon Entropies, Entropic Uncertainty Relation, Confined Quantum Systems.

1. Introduction

The basis of the mathematical theory of communication or information theory was established by Shannon in 1948 [1], where it was analyzed substantially the communication process, i.e., how is the interaction between different mechanisms. This connection between mechanisms, usually by sending and receiving a message (oral or written forms, pictures, music, etc.), happened to have a statistical and probabilistic approach, making their semantic aspects irrelevant with regard to the engineering ones. Information in this field is not the meaning, but it is a measure of freedom to select a message among many others [2]. A fundamental concept is the information entropy or Shannon entropy which has complementary interpretations - quantity of information (after measurement) or uncertainty (before measurement) - in a given probability distribution [3].

Information theory, through of the Shannon entropies in the position (Sr) and momentum (Sp) spaces, has been applied in the context of the quantum

1wallassantos@gmail.com 2prudente@ufba.br

Preprint submitted to Chemical Physics Letters November 28, 2017 arXiv:1704.04874v2 [physics.atom-ph] 27 Nov 2017 theory from the 1980s [4–6]. More specifically, we can cite the derivation of analytical relations in systems such as a particle confined in a box [7, 8], harmonic oscillator [9, 10] and hydrogen atom [1]. Moreover, the treatment of electronic excitations [12, 13] and complexity of systems [14, 15] and characterization of chemical processes [16, 17] have also used the Shannon entropy in their analysis. And in recent years the Shannon entropy has been studied from the perspective of spatially confined quantum systems [18–23].

Confined systems have their physical and chemical properties modified in relation for the free system. In a first approximation, atoms and molecules under high pressure, quantum dots and dense astrophysical objects can be treated as confined systems in spherical barrier of infinite potential (see Refs.[24, 25] and references therein). Thus, such systems have been the subject of numerous studies with different methods to impose the confinement, making this a relevant field of research [26–31]. A particularly interesting situation is when the confinement is rigorous (strong confinement regime). For example, it was shown that in such regime the energies of confined hydrogenic-like atoms tends to the values of a particle confined in a spherical cage [32]. Recent studies for the harmonic potential showed that such analysis can be made based on Shannon entropy [3, 34].

The entropy sum (St), quantity defined as the sum of the Shannon entropies in the position and momentum spaces, has occupied a privileged place in the study of quantum systems in the information theory context. For instance, an entropic uncertainty relation that it has been treated as a stronger version of the Heisenberg uncertainty relation can be derived from the entropy sum [35]. Additionally, we can cite the conjectures pointing to possible universal relationships of that quantity [36], as well as the promising study of electron correlation based on the value of the entropy sum [37]. More specifically, studies have shown that improvements in basis function set (according to variational energy criteria) leads to an increase of the entropy sum for atomic systems. This leads to the conjecture that St can be regarded as an indicator of the quality of basis functions [38–40], proposal known as Gadre-Sears-Chakravorty-Bendale

(GSCB) conjecture. Furthermore, previous works show that the value of the entropy sum not depend of the atomic number for fundamental or excited state of free hydrogenic-like atoms, while in an isoelectronic multielectronic atoms series its value depends of the atomic number [41].

Despite the success of using the Shannon entropy in atomic, molecular and chemical physics, one aspect that still requires discussion is its dimensional problem. Originally, the informational entropy proposed by Shannon is defined in terms of a logarithmic function from a discrete or continuous dimensionless probability distribution, being expressed in units of information bits, hartleys, nats and etc, according to the base of logarithm employed (a base change results only in a scale change). In this sense Shannon’s logarithmic quantities acquire information-theoretic units. For the other hand, in the quantum theory the probability density functions in the position and momentum spaces have dimensional characteristics. In this framework the use of the quantum probability density functions to build the Shannon entropy requires adaptations in the expression originally proposed because of their physical dimensionality.

Recent works have already proposed redefinitions of the Shannon entropy in the position spaces to solve this dubiousness explicitly. For example it is inserted into the logarithmic argument a function having the dimensions of reciprocal volume (expressed in the appropriate units) [42] or of electronic densities measured on the nucleus of the atom [43]. Note that these proposals were presented with specific objectives, the first one to study problems with a delimited contour of the electronic probability density, while the second one to define an expression for the Shannon entropy non-negative under all space.

In the present work, we have developed a theoretical study on one-electron atomic systems confined within impenetrable walls, more precisely the hydrogen

(Hc), helium cation (He+c ) and lithium dication (Li2+c ), by using the Shannon entropies and entropy sum. Initially, we suggest modified and more general expressions to the entropy sum (and the correspondent entropic uncertainty relation) and Shannon entropies Sr and Sp that ensure their physical dimensionless characteristic. Subsequently, we have performed the calculations of the energy, Sr, Sp and St of Hc, He+c and Li2+c in the ground state for different confinement radii by using a variational method. The global behavior of these quantities are analyzed and two conjectures are tested, the first on the dependence of them due the atomic number and the second on the minimum value of

St with respect to the confinement radius. Additionally, we have analyzed the entropic uncertainty relation validity and the use of St as a measure of basis function set quality, GSCB conjecture, in confined environments (for neutral and ionized atoms). And, finally, we have examined the strong confinement regime.

The paper is organized as follows: in section 2 are presented the concepts and definitions of the information theory in the quantum mechanics context and the novel expressions of St, Sr and Sp. In section 3 the systems of interest are presented and the trial ground state wave functions are proposed. The results for the confined hydrogenic-like atoms are shown in section 4, when pertinent analysis are done. And, finally, in the last section the main aspects of the present work are summarized and the concluding remarks are presented.

2. Concepts

In the context of the quantum theory is known that the measure of any two observables A and B can only be done within of a limit of accuracy, resulting at the standard uncertainty relation. In particular, in the case of the position and momentum observables we have the Heisenberg uncertainty principle. On the union of the quantum mechanics and information theory is obtained the entropic uncertainty relation, being treated as a stronger version of the Heisenberg uncertainty relation. Usually the entropic uncertainty relation is derived of the entropy sum, a quantity defined as St = Sr + Sp, being Sr and Sp the Shannon entropies in the position and momentum spaces, respectively. The Sr and Sp quantities are defined by replacing the dimensionless probability distribution in the original Shannon entropy formulation with the quantum probability densities in the position and momentum spaces. However, in quantum theory the probability densities ρ(~r) and γ(~p), respectively, in the n-dimensional position and momentum spaces are normalized to unity and have, respectively, dimension of inverse of n-dimensional position and momentum volume. More precisely, ρ(~r) and γ(~p)

wave functions in position and momentum spaces.

Thus, to avoid the dimensionality problem of ρ(~r) and γ(~p), we propose the following form for the entropic uncertainty relation in the context of the atoms and molecules study:

where n is the dimension on the position (and the momentum) space. The relation (1) reaches equality when it is used a Gaussian wave functions. Moreover, this relation has the property of being invariant when subjected to a scale transformation, in other words, is unaffected by a uniform elongation or compression of the atom [39].

Note that the relation (1) can be decomposed in the sum of the Sr and Sp. Defining hn = λ.κ, we can write,

Using the properties of the logarithmic function we have

Thus, we can define

and

where the quantities Sr and Sp are, respectively, the Shannon entropies in the position and momentum spaces.

The λ and κ constants ensure that the arguments of the logarithm function are dimensionless. An interpretation for Sr is to be a measure of the uncertainty in the location of the particle position, while Sp can be considered as a measure of the uncertainty in the determining of the particle momentum. Still, the Shannon entropy is a measure more satisfactory of the uncertainty (hence of the spread of the probability distribution) when compared to the standard deviation [4, 45].

Note that the informational entropy introduced by Shannon is based on discrete or continuous probability distribution and has the characteristic of being dimensionless in the physical point of view. The Shannon Sr and Sp entropies [Eqs. (4) and (5)] proposed by us differ from those previously reported in the literature by the introduction of the constants λ and κ, respectively, with dimensions of position and momentum volume. The natural choice for these constants in the scope of the quantum theory is the following: λ = a0n and κ = ( h a ing a0 the Bohr radius. We point out that the proposed forms are independent of the units system used and of the quantum system to be treated (including systems of arbitrary dimensions, see Ref. [46]). In this sense they are more gen- eral in relation to the usual expressions. Furthermore, by using atomic units (a0 and h equal to 1 atomic unit), the relation (1) and the Eqs. (4) and (5) assume the current forms of the literature, however, now with a consistent dimensional formulation.

3. Methodology

The time independent radial Schrodinger equation for confined hydrogeniclike atoms in atomic units, assuming an infinite mass for the nucleus and putting it on the center of the hard sphere with radius rc, is given by

with being ψ(r) the radial wave function solution, l the angular momentum quantum number, E the energy for the stationary state and Z the atomic number (Z = 1,

Z = 2 and Z = 3, respectively, for the Hc, He+c e Li2+c confined atoms). The variational method [47] is employed to obtain the (l = 0) ground state solution of Eq. (6). For this, we have employed a trial wave function of the type:

where Ωc(r) is a cut-off function that satisfies Ωc(rc) = 0 condition, A is a normalization constant and α is the parameter to be determined minimizing the total energy functional.

In the literature there are several proposals for the cut-off function [48, 49] and we adopted three possible ones. The first one is the polynomial function where q (= 1, 2 and 3) defines the polynomial degree. The other two types are the trigonometric functions as follows:

and pi2 r rc

Thus, the trial wave functions proposed here to describe the confined hydrogeniclike atoms are in the form of the notation and with the appropriate labels given

4. Results and discussion

The ground state wave functions and energies of confined Hc, He+c e Li2+ c were determined for each type of trial wave functions and different values of rc by the choice of the optimal variational parameter α. From the knowledge of ψ(r), we were able to determine Shannon entropies Sr and Sp [Eqs. (4) and (5)] and the entropy sum St [Eq. (1)]. The results obtained are summarized in Tables 1 and 2 for all systems of interest.

The analysis of them is organized as follows: the results for the ground state energy are discussed in subsection 4.1, while in subsection 4.2 the global behavior of Sr and Sp are presented. In subsection 4.3 the results of St are summarized with the discussion of the entropic uncertainty relation, suggestion of conjectures and use of St as a measure of basis function set quality in confined environments. Finally we have examined the limit for the strong confinement regime in subsection 4.4.

4.1. Energy

The ground state energy values of confined Hc, He+c e Li2+c for each type of trial wave functions and different values of rc are presented in Table 1, and their general behaviors are displayed in Figure 1. In general, the energy values of these confined systems tend to the ones of the free systems when rc goes to infinity and increases when rc decreases, reaching positive values for small rc. In addition, one can clearly see that the one electron atomic system with the highest atomic number has lower energies for the same confinement radius.

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