Baixe Shannon entropy a study of confined hydrogenic like atoms e outras Trabalhos em PDF para Física, somente na Docsity! See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/316184684 Shannon entropy: a study of confined hydrogenic-like atoms Article in Chemical Physics Letters · November 2017 DOI: 10.1016/j.cplett.2017.11.048 CITATIONS 0 READS 56 2 authors: Some of the authors of this publication are also working on these related projects: Statistical Physics View project Wallas Nascimento Universidade Federal da Bahia 2 PUBLICATIONS 1 CITATION SEE PROFILE Frederico V. Prudente Universidade Federal da Bahia 69 PUBLICATIONS 630 CITATIONS SEE PROFILE All content following this page was uploaded by Wallas Nascimento on 28 November 2017. The user has requested enhancement of the downloaded file. Shannon entropy: a study of confined hydrogenic-like atoms Wallas S. Nascimento1, Frederico V. Prudente2 Instituto de F́ısica, 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 Li 2+ 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 the- ory was established by Shannon in 1948 [1], where it was analyzed substantially the communication process, i.e., how is the interaction between different mech- anisms. 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 ar X iv :1 70 4. 04 87 4v 2 [ ph ys ic s. at om -p h] 2 7 N ov 2 01 7 are defined by replacing the dimensionless probability distribution in the origi- nal 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 momen- tum 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) are given, respectively, in terms of |ψ(~r)|2 and |ψ̃(~p)| 2 , being ψ(~r) and ψ̃(~p) the 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: St = − ∫ ∫ d~r d~p ρ(~r) γ(~p) ln( h̄n ρ(~r) γ(~p)) ≥ n(1 + lnπ) , (1) where n is the dimension on the position (and the momentum) space. The rela- tion (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 trans- formation, 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 h̄n = λ.κ, we can write, St = − ∫ ∫ d~r d~p ρ(~r) γ(~p) ln( λ.κ ρ(~r)γ(~p)) . (2) Using the properties of the logarithmic function we have St = − ∫ ∫ d~r d~p ρ(~r) γ(~p) [ln( λρ(~r)) + ln( κγ(~p))] = − ∫ d~r ρ(~r) ln(λ ρ(~r))− ∫ d~p γ(~p) ln(κ γ(~p)) . (3) Thus, we can define Sr = − ∫ d~r ρ(~r) ln(λ ρ(~r)) (4) and Sp = − ∫ d~p γ(~p) ln(κ γ(~p)) , (5) 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 [44, 45]. 4 Note that the informational entropy introduced by Shannon is based on dis- crete 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 dimen- sions of position and momentum volume. The natural choice for these constants in the scope of the quantum theory is the following: λ = a0 n and κ = ( h̄a0 ) n , be- 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 h̄ 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 Schrödinger equation for confined hydrogenic- like 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 [− 1 2r d2 dr2 r + l(l + 1) 2r2 + V (r)]ψ(r) = Eψ(r) , (6) with V (r) = { −Zr r < rc ∞ r ≥ rc , (7) 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 Li 2+ 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: ψ̃(r) = φc(r) = Ae−αZrΩc(r) , (8) 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 Ωc1q(r) = [1− ( r rc )q] =⇒ φc1q(r) (9) where q (= 1, 2 and 3) defines the polynomial degree. The other two types are the trigonometric functions as follows: Ωc2(r) = [sin(1− r rc )] =⇒ φc2(r) (10) 5 and Ωc3(r) = [cos( π 2 r rc )] =⇒ φc3(r). (11) Thus, the trial wave functions proposed here to describe the confined hydrogenic- like atoms are in the form of the notation and with the appropriate labels given by φc11(r), φ c 12(r) , φ c 13(r), φ c 2(r) and φ c 3(r). 4. Results and discussion The ground state wave functions and energies of confined Hc, He + c e Li 2+ 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 ψ̃(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 Li 2+ 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. Analyzing the quality of the trial wave functions considered here by using the energy variational criterion, we can note that not necessarily the same trial wave function provides the best values for all rc studied. In Table 1 we indicated in bold and italic the best variational energy value for each value of rc and the corresponding value of St. For Li 2+ c , for example, the trial wave function φ c 13(r) provides the best energy value for rc comprised between 1.5 a.u. and 3.5 a.u., while for rc = 1.0 a.u. the trial wave function φ c 11(r) presents the best result and, finally, for rc = 0.5 a.u. the trial wave function with the best energy value is the φc12(r). 4.2. Shannon entropies Sr and Sp The Shannon entropies in the position (Sr) and momentum (Sp) spaces of confined Hc, He + c and Li 2+ c in the ground state for different values of confinement radius (rc) are presented in Table 2, and their general behaviors are displayed 6 E (a .u .) −5 0 5 10 15 rc(a.u.) 0 1 2 3 4 5 6 7 1 2 3 4 5 Lic2+ Hec+ Hc Figure 1: Ground state energy E as a function of the confinement radius rc for the confined Hc, He + c and Li 2+ c . All values are in atomic units (a.u.). Subtitles: 1=φ c 11(r), 2=φ c 12(r), 3=φc13(r), 4=φ c 2(r) and 5=φ c 3(r). in Figure 2. Based on such data, we have verified a decrease of Sr value and an increase of Sp values when rc decreases and the confinement effect is enhanced. Additionally, we can observe that Sr is smaller (and Sp is greater) for larger val- ues of Z, that is, such quantities have a dependence on atomic nucleus number, corroborating with the analytical work presented in Ref. [41]. These results evidence the interpretation that Sr indicates a measure of the uncertainty in the spatial location of the particle, and the global behavior of the quantities Sr and Sp are justifiable considering Eq. (1). An interesting result for Hc case is the intersection of Sr and Sp curves to rc ≈ 3.0 a.u.. At this point Sr and Sp assume the value near to 3.25. Moreover, the numerical values of Sr (and Sp) for these three hydrogenic- like atoms approximate when the confinement becomes more intense. In the region where the confinement effect is rigorous the quantity Sr may assume negative values, as shown in Table 2 and in Refs. [18, 22, 33]. This result has a simple explanation in the quantum context [19]: when rc is reasonably small, the probability density becomes large and an0ρ(~r) > 1. In this situation, −ρ(~r) ln an0ρ(~r) < 0 and, then, Sr may be negative. Remember that the wave- function normalization is in relation to the integral ∫ ρ(~r)d~r. Note that in the original Shannon’s work the entropy values for continuous distributions may be negative (see pg. 631 in Ref. [1]). 4.3. The entropy sum The entropy sum (St), from the Sr and Sp values presented in Table 2, of confined Hc, He + c and Li 2+ c in the ground state for different values of confine- 9 S r , S p −2 0 2 4 6 8 10 rc(a.u.) 0 1 2 3 4 5 6 7 1 2 3 4 5 Sr - Lic2+ Sp - Lic2+ Sr - Hec+ Sp - Hec+ Sr - Hc Sp - Hc Figure 2: Shannon entropies Sr e Sp as a function of the confinement radius rc for the confined Hc, He + c and Li 2+ c in the ground states. Subtitles: 1=φ c 11(r), 2=φ c 12(r), 3=φ c 13(r), 4=φ c 2(r) and 5=φc3(r). ment radius (rc) and trial wave functions are presented in Table 1, and their general behavior are displayed in Figure 3. An immediate observation is that the entropic uncertainty relation (St ≥ 6.4342) is valid for all studied confined systems. Still, based on this Table, we conjecture that the values of St for con- fined H-like atoms considered here retain a dependence on the atomic number Z and are invariant through of the relation rcZ . For the three free hydrogenic-like systems the value of entropy sum is invariant (St = 6.5666), corroborating with the analytical study presented in Ref. [41]. Moreover, we can identified in the Figure 3 that St assumes a minimum value at rc ≈ 1.0 a.u., rc ≈ 1.5 a.u. and rc ≈ 3.0 a.u. for the Li2+c , He+c and Hc, respectively. This is not observed in Ref. [18] to Hc case, where the authors found that the minimal value of St is to rc ≈ 2.5 a.u.. However, we have believed in some kind of numerical problem in those results. Note from present results that the rc value when the entropy sum is minimal decreases with increasing of the atomic number, maintaining the following relationship: rc = 3 Z . For this confinement radius, the Coulomb and the confining potentials are compensated by creating a state of lower entropic uncertainty. Additionally, we study the Gadre-Sears-Chakravorty-Bendale (GSCB) con- jecture, originally analyzed for free neutral atoms, in confined environments and also to ionized atoms, from results presented on Table 1. In particular, the GSCB conjecture establishes that better basis functions (smaller values for energy) present higher values for the entropic sum. In relation to Hc, the wave function that provides the best value of E also provides the largest value of St to rc values between 2.0 a.u. and 6.0 a.u.. There is an exception at rc = 3.5 a.u. 10 S t 6,46 6,48 6,5 6,52 6,54 6,56 6,58 6,6 0 1 2 3 4 5 6 7 1 2 3 4 5 Hc Hec+ Lic2+ rc(a.u.) Figure 3: Entropy sum St as a function of the confinement radius rc for the confined 1Hc, He+c and Li 2+ c in the ground states. Subtitles: 1=φ c 11(r), 2=φ c 12(r), 3=φ c 13(r), 4=φ c 2(r) and 5=φc3(r). where there is a inversion between φc11(r) and φ c 13(r) wave functions but the results are identical up to four figures. Similar results are observed for He+c and Li2+c to rc values between 1.0 a.u. and 5.0 a.u. and 1.0 a.u. and 3.5 a.u., respectively. However, the GSCB conjecture fails when the ground state energy of the confined H-like system is positive, and, for various rc values, the best wave function (according to energy criteria) comes from the lowest value of St. 4.4. Strong confinement regime In previous study of confined harmonic oscillator [33], we proposed that the entropy sum can be employed to determine a strong or rigorous confinement regime. This regime is defined when the influence of the confining potential becomes greater than the free-system potential to specific configurations (or rc values). 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