Relativistic Klein-Gordon charge effects by information-theoretic measures
aa r X i v : . [ qu a n t - ph ] J a n Relativistic Klein-Gordon charge effects byinformation-theoretic measures.
D Manzano , , R J Y´a˜nez , and J S Dehesa , Instituto Carlos I de F´ısica Te´orica y Computacional, Universidad de Granada,18071-Granada, Spain Departamento de F´ısica At´omica, Molecular y Nuclear, Universidad de Granada,18071-Granada, Spain Departamento de Matem´atica Aplicada, Universidad de Granada, 18071-Granada,SpainE-mail: [email protected] , [email protected] , [email protected] Abstract.
The charge spreading of ground and excited states of Klein-Gordonparticles moving in a Coulomb potential is quantitatively analyzed by means ofthe ordinary moments and the Heisenberg measure as well as by use of the mostrelevant information-theoretic measures of global (Shannon entropic power) and local(Fisher’s information) types. The dependence of these complementary quantities onthe nuclear charge Z and the quantum numbers characterizing the physical statesis carefully discussed. The comparison of the relativistic Klein-Gordon and non-relativistic Schr¨odinger values is made. The non-relativistic limits at large principalquantum number n and for small values of Z are also reached. elativistic Klein-Gordon charge effects by information-theoretic measures.
1. Introduction
The interplay of quantum mechanics, relativity theory and information theory is a mostimportant topic in the present-day theoretical physics [1–7]. Special relativity provokesboth important restrictions on the transfer of information between distant systems [2]and severe changes on the integral structure of physical systems [8]. This is mainlybecause the relativistic effects produce a spatial redistribution of the single-particledensity ρ ( ~r ) of the corresponding quantum-mechanical states, which substantially alterthe spectroscopic and macroscopic properties of the systems.The quantitative study of the relativistic modification of the spatial extent of thecharge density of atomic and molecular systems by information-theoretic means is awidely open field [4]. The only works published up to now have calculated the ground-state relativistic effects on hydrogenic [4] and many-electron neutral atoms [5, 7] indifferent settings by use of the renowned standard desviation (or Heisenberg measure)as well as various information-theoretic measures.In this paper we quantify the relativistic effects of the ground and excited states ofthe spinless single-particle charge spreading by the comparison of the Klein-Gordon andSchr¨odinger values for three qualitatively different measures: the Heisenberg measure σ [ ρ ], the Shannon entropic power N [ ρ ] [9] and the Fisher information I [ ρ ] [10, 12].While the Heisenberg quantity gives the spreading with respect to the centroid of thecharge distribution, the Shannon and Fisher measures do not refer to any specific point.The Shannon entropic power N [ ρ ], which is essentially given by the exponentialof the Shannon entropy S [ ρ ] = − h log ρ ( ~r ) i , measures the total extent to which thedistribution is in fact concentrated [12, 13]. This quantity has various relevant features.First, it avoids the dimensionality troubles of S [ ρ ], highligting its physical meaning.Second, it exists when σ does not. Third, it is finite whenever σ is. Thus, as ameasure of uncertainty the use of the Shannon entropic power allows a wider quantitativerange of applicability than the Heisenberg measure [14]. Contrary to the Shannon andHeisenberg measures, which are insensitive to electronic oscillations, (translationallyinvariant) the Fisher information [10] has a locality property because it is a gradientfunctional of the density, so that it measures the pointwise concentration of the electroniccloud and quantifies its gradient content, providing a quantitative estimation of theoscillatory character of the density. Moreover, the Fisher information measures the biasto particular points of the space, i.e. it gives a measure to the local disorder.To calculate the measures of the charge spreading in a relativistic quantum-mechanical system we have to tackle the problem of the very concept of quantumprobability consistent with Lorentz covariance. The general formulation andinterpretation of this problem is still a currently discussed issue [15]. In this workwe avoid this problem following the relativistic quantum mechanics [8] by restrictingourselves to study the stationary states of a spinless relativistic particle with a negativeelectric charge in a spherically symmetric Coulomb potential V ( r ) = − Z e r , which arethe solutions of the relativistic scalar wave equation, usually called the Klein-Gordon elativistic Klein-Gordon charge effects by information-theoretic measures. ǫ − V ( r )] ψ ( ~r ) = ( − ~ c ∇ + m c ) ψ ( ~r ) , (1)appropriately normalized to the particle charge. The symbols m and ǫ denote themass and the relativistic energy eigenvalue, respectively. We will work in sphericalcoordinates, taking the ansatz ψ ( r, θ, φ ) = r − u ( r ) Y lm ( θ, φ ), where Y lm ( θ, φ ) denotesthe spherical harmonics of order ( l, m ). Then, to highlight the resemblance with thenon-relativistic Schr¨odinger equation, we let β ≡ ~ c ( m c − ǫ ) = 2 m c ~ c s − (cid:18) ǫm c (cid:19) , (2) λ ≡ ǫZe ~ c β , (3)and substitute the radial variable r by the dimensionless variable s through thetransformation r → s : s = βr. (4)So, the radial Klein-Gordon equation satisfied by u ( s ) can be written in the form d u ( s ) ds − (cid:20) l ′ ( l ′ + 1) s − λs + 14 (cid:21) u ( s ) = 0 , (5)where we have used the notation l ′ = s(cid:18) l + 12 (cid:19) − γ − , with γ ≡ Zα, (6)being α = e ~ c the fine structure constant. The physical solutions corresponding tothe bound states (whose energy eigenvalues fulfill | ǫ | < m c ) require that the radialeigenfunctions u nl ( r ) vanish both at the origin and at infinity [9], so that they have theform u nl ( s ) = N s ( l ′ +1) e − s e L l ′ +1 n − l − ( s ) , (7)where e L ( α ) k ( s ) denotes the orthonormal Laguerre polynomials of degree k and parameter α . The energy eigenvalues ǫ ≡ ǫ ln ( Z ) of the stationary bound states with wavefunctionsΨ nlm ( ~r, t ) = ψ nlm ( ~r ) exp ( − i ~ ǫt ) are known to have the form [9] ǫ = m c q (cid:0) γn − l + l ′ (cid:1) (8)The constant N is determined not by the normalization of the wavefunction to unity asin the non-relativistic case, but by the charge conservation carried out by R R ρ ( ~r ) d r = e to preserve the Lorentz invariance [8], where the charge density of the negatively chargedparticle (e.g., a π − -meson; q = − e ) is given by ρ nlm ( ~r ) = em c [ ǫ − V ( r )] | ψ nlm ( ~r ) | . (9) elativistic Klein-Gordon charge effects by information-theoretic measures. Z ∞ ǫ − V ( r ) m c u ǫl ( r ) dr = 1 m c Z ∞ (cid:18) ǫβ + γ ~ cρ (cid:19) u ǫl ( s ) ds. (10)The substitution of the expression (7) for u ǫl ( s ) into Eq. (10) provides the followingnormalization constant N = m c (cid:20) ǫβ ( n + l ′ − l ) + γ ~ c (cid:21) − = m c γ ~ c n + l ′ − l ) + γ , (11)where we have used for the second equality the relation ǫβ = ~ c n + l ′ − lγ (12)Let us emphasize that the resulting Lorentz-invariant charge density ρ LI ( ~r ) givenby Eq. (9) is always (i.e. for any observer’s velocity v ) appropriately normalized whilethe density ρ NLI ( ~r ) = | ψ nlm ( ~r ) | (used in [17]) is not. This is numerically illustratedin Figure 1 for a pionic atom with nuclear charge Z = 68 in the infinite nuclear massapproximation ( π − -meson mass=273.132054 a.u.). T o t a l P r obab ili t y v/c ρ LI ρ NLI
Figure 1.
Normalization of the charge density for the Lorentz invariant (LI) and thenon-Lorentz invariant (NLI) charge densities for different velocities of the observer.
For completeness we have plotted in Figure 2 the radial density of the chargedistribution for two diferent states ( n = 1 , l = 0) and ( n = 4 , l = 1) of a pionic systemwith nuclear charge Z = 68 in the infinite nuclear mass approximation, respectively.Moreover, we have also made in these figures a comparision with the correspondingSchr¨odinger density functions [18]. We observe that the relativistic effects other thanspin (i) tend to compress the charge towards the origin, and (ii) they are most apparentfor states S . elativistic Klein-Gordon charge effects by information-theoretic measures. r ρ (r) r (a.u.)KG SchZ=68n=1, l=05000100001500020000 0.0001 0.0002 0.0003 r ρ (r) r (a.u.)KG SchZ=68n=1, l=0 4008001200 0 0.001 0.002 0.003 r ρ (r) r (a.u.)KG SchZ=68n=4, l=1 Figure 2.
Comparison of the charge Klein-Gordon and Schr¨odinger radialdensity for the states 1 S (left) and 4 P (right) of the pionic system with Z = 68.Atomic units ( ~ = m e = e = 1) are used. In this paper we quantify this relativistic charge compression by three differentmeans. First, in Section II, we compute the ordinary moments or radial expectationvalues (cid:10) r k (cid:11) for general ( n, l, m ) states, making emphasis in the Heisenberg measure forcircular ( l = n −
1) and S -states ( l = 0). Then, in Section III, we study numerically themost relevant charge information-theoretic measures of the system; namely the Shannonentropy and the Fisher information.
2. Radial expectation values and Heisenberg’s measure.
The charge distribution of the Klein-Gordon particles in a Coulomb potential can becompletely characterized by means of the ordinary radial expectation values (cid:10) r k (cid:11) , k ∈ N ,given by (cid:10) r k (cid:11) := Z R r k ρ nlm ( ~r ) d r = 1 m c Z ∞ (cid:18) ǫ + Ze r (cid:19) r k u nl ( r ) dr = 1 m c β k Z ∞ (cid:18) ǫβ + γ ~ cs (cid:19) s k u nl ( s ) ds = N m c β k (cid:20) ǫβ J nl ( k ) + γ ~ c J nl ( k − (cid:21) , (13)where we have used Eqs. (7) and (9), and the symbol J nl ( k ) denotes the integral [9] elativistic Klein-Gordon charge effects by information-theoretic measures. J nl ( k ) := Z ∞ x l ′ + k +2 e − x he L (2 l ′ +1) n − l − ( x ) i dx = ( n − l − n − l + 2 l ′ + 1) × n − l − X j = n − l − k − k + 1 n − l − j − ! Γ(2 l ′ + k + j + 3) j ! (14)For the lowest values of k we have J nl (0) = 2( n + l ′ − l ) J nl (1) = 2 (cid:2) n − l ) + l ′ (6 n + 2 l ′ − l − (cid:3) J nl (2) = 4( n + l ′ − l ) × (cid:2) n − l ) + l ′ (10 n + 2 l ′ − l − (cid:3) . Then, besides the normalization h r i = 1, we have the following value h r i = N m c β (cid:20) ǫβ J nl (1) + γ ~ c J nl (0) (cid:21) (15)for the centroid of the charge density, and (cid:10) r (cid:11) = N m c β (cid:20) ǫβ J nl (2) + γ ~ c J nl (1) (cid:21) (16)for the second-order moment, so that the Heisenberg measure σ nl which quantifies thecharge spreading around the centroid is given by σ nl ≡ σ [ ρ nlm ] = (cid:10) r (cid:11) − h r i = N m c β (cid:26) ǫβ J nl (2) + γ ~ c J nl (1) −− N m c (cid:20) ǫβ J nl (1) + γ ~ c J nl (0) (cid:21) ) . (17)To gain insight into these general expressions we are going to discuss two particularclasses of quantum-mechanical states, the circular (i.e., l = n −
1) states and the ns-states (i.e., l = 0) states.For circular states we have that l ′ = q(cid:0) n − (cid:1) − γ − ǫ = m c r ( γl ′ +1 ) ; ǫβ = ~ c γ ( l ′ + 1) , elativistic Klein-Gordon charge effects by information-theoretic measures. N m c = γ ~ c l ′ + 1) + γ , and the integrals J nl (0) = 2 l ′ + 2 J nl (1) = (2 l ′ + 2)(2 l ′ + 3) J nl (2) = (2 l ′ + 2)(2 l ′ + 3)(2 l ′ + 4)Then, the centroid of the charge distribution is, according to Eq. (15), h r i = ~ c m c γ q (cid:0) γl ′ +1 (cid:1) (cid:2) (2 l ′ + 2)(2 l ′ + 3) + 4 γ (cid:3) (18)and the second-order moment, according to equation (16), becomes (cid:10) r (cid:11) = (cid:18) ~ cm c (cid:19) γ ( l ′ + 1)(2 l ′ + 3) (cid:2) ( l ′ + 1)( l ′ + 2) + γ (cid:3) , (19)so that the Heisenberg measure for circular states σ n = σ n,n − has the following value σ n = (cid:18) ~ cm c (cid:19) (cid:18) l ′ + 14 γ (cid:19) × ( l ′ + 1)(2 l ′ + 3) [( l ′ + 1) + 2 γ ] + 2 γ ( l ′ + 1) + γ . (20)These expressions for circular states and the corresponding ones for ns-states arediscussed and compared with the Schr¨odinger values as a function of the principalquantum number n for the pionic system with nuclear charge Z = 68 in Figure 3.We observe that both centroid and variance ratios increase very rapidly with n, beingthe rate of this behaviour much faster for circular than for S -states. This indicates thatthe charge compression provoked by relativity in a given system (i.e. for fixed Z ) (i)decreases when n ( l ) is increasing for fixed l ( n ). This can also be noticed in Figure 4,where the two previous ratios have been plotted as a function of l for different values of n . Then, we have plotted these two ratios in terms of the nuclear charge Z of thesystem in Figure 5 for the states 1 S , 2 S and 2 P . We find that both the centroid andthe variance ratios monotonically decrease as the nuclear charge Z increases. Moreover,the decreasing rate is much faster for the states 1 S , than for the states 2 S and 2 P .These two observations illustrate that the relativistic charge compression effect is biggerin heavier systems for a given ( nl )-state. Moreover, we see here again that for a givensystem it increases both when n decreases for fixed l and when l decreases for fixed n .The quantum number m doesn’t affect both ratios because the radial part of the densityis not a function of it.Finally, let highlight that in all figures the Klein-Gordon values tend towards theSchr¨odinger values in the non-relativistic limit of large n or small Z . elativistic Klein-Gordon charge effects by information-theoretic measures. < r > n l ( K G ) / < r > n l ( S c h ) nl=0l=n-1 Z=68 0.4 0.5 0.6 0.7 0.8 0.9 1 10 20 30 40 50 < r > n l ( K G ) / < r > n l ( S c h ) nl=0l=n-1 Z=68 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 20 30 40 50 σ l ( K G ) / σ l ( S c h ) nl=0l=n-1 Z=68 Figure 3.
Comparison of the Klein-Gordon and Schr¨odinger values for thecentroid (Left) and the variance (Right) of the pionic 1 S and the circular statesas a function of the quantum number n with Z = 68. < r > n l ( K G ) / < r > n l ( S c h ) n = < r > n l ( K G ) / < r > n l ( S c h ) n = = = = = = = = = = = σ l ( K G ) / σ l ( S c h ) n = σ l ( K G ) / σ l ( S c h ) n = = = = = = = = = = Figure 4.
Comparison of the Klein-Gordon and Schr¨odinger values for thecentroid (Left) and the variance (Right), as a function of the quantum number l varying from 0 to n −
1, for different values of n .
3. Shannon and Fisher information measures
Here we study numerically the relativistic effects on the charge spreading of pionicsystems of hydrogenic type by means of the following information-theoretic measures ofthe associated charge distribution ρ nlm ( ~r ) given by Eq (9): The Shannon entropy powerand the Fisher information.The Shannon entropic power of a negatively-charged Klein-Gordon particlecharacterized by the charge density ρ nlm ( ~r ) is defined by [12] N nlm ≡ N [ ρ nlm ] = 12 πe exp (cid:18) S nlm (cid:19) , (21)where S nlm is the Shannon entropy of ρ nlm ( ~r ) given by the expectation value of − log ( ρ nlm ( ~r )), i. e. S nlm ≡ S [ ρ nlm ] = − Z R ρ nlm ( ~r ) log ρ nlm ( ~r ) d r, (22) elativistic Klein-Gordon charge effects by information-theoretic measures. < r > n l ( K G ) / < r > n l ( S c h ) Z n=2 l=1n=2 l=0n=1 l=0 0.4 0.5 0.6 0.7 0.8 0.9 1 0 10 20 30 40 50 60 < r > n l ( K G ) / < r > n l ( S c h ) Z n=2 l=1n=2 l=0n=1 l=0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 10 20 30 40 50 60 σ l ( K G ) / σ l ( S c h ) Z n=2 l=1n=2 l=0n=1 l=0
Figure 5.
Comparison of the Klein-Gordon and Schr¨odinger values for thecentroid (Left) and the variance (Right), as a function of the nuclear charge Z for the pionic states 1 S , 2 S and 2 P . which quantifies the total extent of the charge spreading of the system. Taking intoaccount the above-mentioned ansatz for ψ ( ~r ) and Eqs. (7), (9) and (22), this expresioncan be separated out into radial and angular parts S nlm = S [ R nl ] + S [ Y lm ] , as it is explained in full detaill in [19], being R nl and Y lm the radial and angular partsof the density. We should keep in mind that the angular part is the same for bothKlein-Gordon and Schr¨odinger cases.The Fisher information is defined by [10] I nlm ≡ I [ ρ nlm ] = Z R |∇ ρ ( ~r ) | ρ ( ~r ) d r. (23)Remark that we are not using here the parameter dependent Fisher informationoriginally introduced (and so much used) by statisticians [11], but its translationallyinvariant form that does not depend on any parameter; see ref. [10,20] for further details.It is worthy to point out that the Fisher information is a measure of the gradient contentof the charge distribution: so, when ρ ( ~r ) has a discontinuity at a certain point, the localslope value drastically changes and the Fisher information strongly varies. This indicatesthat it is a local quantity in contrast to the Heisenberg measure σ nl and the Shannonentropy S ρ (and its associated power), which have a global character because they arepowerlike and logarithmic functionals of the density, respectively.Unlike the moment-based quantities discussed in the previous section, thesecomplementary measures do not depend on a special point, either the origin as theordinary moments or the centroid as the Heisenberg measure. These quantities, firstused by statisticians and electrical engineers and later by quantum physicists, have beenshown to be measures of disorder or smoothness of the density ρ nlm ( ~r ) [10, 12]. Let ushighlight that the Fisher information does not only measure the charge spreading of thesystem in a complementary and qualitatively different manner as the Heisenberg and elativistic Klein-Gordon charge effects by information-theoretic measures. N nl (KG) /N nl (Sch) between thesetwo values as a function of the principal quantum number n for the system with nuclearcharge Z = 68. We notice that the Shannon ratio systematically increases when n isincreasing, approaching to unity for large n, for both circular and S -states. Moreover, wefind that this approach is much faster for circular states, what indicates once more thatthe relativistic effects are much more important for states S. In addition, on the right ofFigure 6, we show the dependence of the Shannon ratio with the nuclear charge Z forthe 1 S , 2 S ant 2 P states. We observe, here again, that the ratio is a decreasing functionof Z for any state, indicating that the relativistic effects are much more important forheavy systems. Moreover, for a given system (i.e. fixed Z ) the relativistic effects increasewhen n ( l ) decreases for fixed l ( n ). The quantum number m affects the absolute valueof the Shannon entropic power but it doesn’t affect the ratio. N n l ( K G ) / N n l ( S c h ) nl=n-1l=0 Z=6800.20.40.60.81 0 5 10 15 20 25 N n l ( K G ) / N n l ( S c h ) nl=n-1l=0 Z=68 00.20.40.60.81 0 10 20 30 40 50 60 N n l ( K G ) / N n l ( S c h ) Z n=2 l=1n=2 l=0n=1 l=0
Figure 6.
Comparison of the Klein-Gordon and Schr¨odinger values for theShannon entropic power as a function of the principal quantum number n (Left) and the nuclear charge Z (Right). Figure 7 shows the dependence of the ratio of the non-relativistic and relativisticvalues of the Fisher information for various states with l = 0 on their quantum numbers( n, l, m ) for the pionic system with Z = 68 (left graph) and on the nuclear charge Z (right graph). The Fisher information for S -states is not defined because the involvedintegral diverges. First we should remark that here, contrary to the previous quantitiesconsidered in this work, the Schr¨odinger values are always less than the Klein-Gordonones; this is strongly related to the local character of the Fisher information, indicatingthat the localized internodal charge concentration is always larger in the relativisticcase. Second, we observe that for fixed l the Fisher ratio I nl (Sch) /I nl (KG) monotonicallyincreases when n is getting bigger, approaching to unity at a rate which grows as l isincreasing. Third, we find that the Fisher ratio decreases for all states in a systematic elativistic Klein-Gordon charge effects by information-theoretic measures. I n l ( S c h ) /I n l ( K G ) nl=n-1 m=n-1l=1 m=1l=1 m=0Z=680.880.920.961 0 5 10 15 20 25 I n l ( S c h ) /I n l ( K G ) nl=n-1 m=n-1l=1 m=1l=1 m=0Z=68 0.840.880.920.961 0 10 20 30 40 50 60 I n l ( S c h ) /I n l ( K G ) Zn=2 l=1 m=0n=3 l=1 m=0n=2 l=1 m=1
Figure 7.
Comparison of the Klein-Gordon and Schr¨odinger values for theFisher information as a function of the principal quantum number n (Left) andthe nuclear charge Z (Right). N n l ( K G ) / N n l ( S c h ) n = N n l ( K G ) / N n l ( S c h ) n = = = = = = = = = = = I n l ( S c h ) /I n l ( K G ) n = I n l ( S c h ) /I n l ( K G ) n = = = = = = = = Figure 8.
Comparison of the Klein-Gordon and Schr¨odinger values for theShannon entropic power as a function of l variying from 0 to n − l variying from 1 to n − n . way as the nuclear charge increases. Moreover, for a given Z value this ratio increasesas either the quantum numbers n and/or l increase.For completeness, the behavior of the Shannon and Fisher ratios in terms of theorbital quantum number l for a fixed n is more explicitly shown of the left and rightgraphs, respectively, of Figure 8.Finally, in Figure 9, the dependence of the Fisher ratio on the magnetic quantumnumber m is studied. Notice that the ratio is bigger when | m | is increasing, indicatingthat the lower | m | is, the more concentrated is the charge density of the state and themore important are the relativistic effects. elativistic Klein-Gordon charge effects by information-theoretic measures. I n l m ( K G ) /I n l m ( S c h ) l = I n l m ( K G ) /I n l m ( S c h ) l = l = l = l = l = l = Figure 9.
Comparison of the Klein-Gordon and Schr¨odinger values for theFisher information as a function of m varying from 0 to l , for different valuesof l .
4. Conclusions
The relativistic charge compression of spinless Coulomb particles has been quantitativelyinvestigated by means of the Heisenberg, Shannon and Fisher spreading measures. Thesethree complementary quantities show that the relativity effects are larger (i. e. thecharge compresses more towards the origin) for the lower energetic states and when theCoulomb strength (i. e. the nuclear charge Z ) increases. Moreover, a detailed analysisof these quantities on the quantum numbers ( n, l, m ) characterising the physical statesof a given system (i. e. for a fixed Z ) indicate that the relativistic effects increasewhen n ( l ) decreases for fixed l ( n ). Furthermore, the study of the Fisher informationshows that the relativistic effects also increase when the magnetic quantum number | m | is increasing for fixed ( n, l ).
5. Acknowlegments
We are very grateful to Junta de Andalucia for the grants FQM-2445 and FQM-1735, andthe Ministerio de Ciencia e Innovaci´on for the grant FIS2008-02380/FIS. We belong tothe research group FQM-207. Daniel Manzano acknowledges the fellowship BES-2006-13234. elativistic Klein-Gordon charge effects by information-theoretic measures. [1] J. Avery, Information Theory and Evolution (World Sci. Publ., N.Y.,2003).[2] A. Peres and D.R. Terno, Rev. Mod. Phys.
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