Relativistic Klein-Gordon charge effects by information-theoretic measures

The charge spreading of ground and excited states of Klein-Gordon particles moving in a Coulomb potential is quantitatively analyzed by means of the ordinary moments and the Heisenberg measure as well as by use of the most relevant information-theoretic measures of global (Shannon entropic power) and local (Fisher's information) types. The dependence of these complementary quantities on the nuclear charge Z and the quantum numbers characterizing the physical states is carefully discussed. The comparison of the relativistic Klein-Gordon and non-relativistic Schrodinger values is made. The non-relativistic limits at large principal quantum number n and for small values of Z are also reached.


Introduction
The interplay of quantum mechanics, relativity theory and information theory is a most important topic in the present-day theoretical physics [1][2][3][4][5][6][7]. Special relativity provokes both important restrictions on the transfer of information between distant systems [2] and severe changes on the integral structure of physical systems [8]. This is mainly because the relativistic effects produce a spatial redistribution of the single-particle density ρ( r) of the corresponding quantum-mechanical states, which substantially alter the spectroscopic and macroscopic properties of the systems.
The quantitative study of the relativistic modification of the spatial extent of the charge density of atomic and molecular systems by information-theoretic means is a widely open field [4]. The only works published up to now have calculated the groundstate relativistic effects on hydrogenic [4] and many-electron neutral atoms [5,7] in different 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 of the spinless single-particle charge spreading by the comparison of the Klein-Gordon and Schrödinger 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 the charge 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 exponential of the Shannon entropy S [ρ] = − log ρ( r) , measures the total extent to which the distribution 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 a measure of uncertainty the use of the Shannon entropic power allows a wider quantitative range of applicability than the Heisenberg measure [14]. Contrary to the Shannon and Heisenberg measures, which are insensitive to electronic oscillations, (translationally invariant) the Fisher information [10] has a locality property because it is a gradient functional of the density, so that it measures the pointwise concentration of the electronic cloud and quantifies its gradient content, providing a quantitative estimation of the oscillatory character of the density. Moreover, the Fisher information measures the bias to 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 quantummechanical system we have to tackle the problem of the very concept of quantum probability consistent with Lorentz covariance.
The general formulation and interpretation of this problem is still a currently discussed issue [15]. In this work we avoid this problem following the relativistic quantum mechanics [8] by restricting ourselves to study the stationary states of a spinless relativistic particle with a negative electric charge in a spherically symmetric Coulomb potential V (r) = − Z e 2 r , which are the solutions of the relativistic scalar wave equation, usually called the Klein-Gordon equation [16], appropriately normalized to the particle charge. The symbols m 0 and ǫ denote the mass and the relativistic energy eigenvalue, respectively. We will work in spherical coordinates, taking the ansatz ψ(r, θ, φ) = r −1 u(r)Y lm (θ, φ), where Y lm (θ, φ) denotes the spherical harmonics of order (l, m). Then, to highlight the resemblance with the non-relativistic Schrödinger equation, we let and substitute the radial variable r by the dimensionless variable s through the transformation r → s : So, the radial Klein-Gordon equation satisfied by u(s) can be written in the form where we have used the notation being α = e 2 c the fine structure constant. The physical solutions corresponding to the bound states (whose energy eigenvalues fulfill |ǫ| < m 0 c 2 ) require that the radial eigenfunctions u nl (r) vanish both at the origin and at infinity [9], so that they have the form where 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 wavefunctions Ψ nlm ( r, t) = ψ nlm ( r)exp(− i ǫt) are known to have the form [9] The constant N is determined not by the normalization of the wavefunction to unity as in the non-relativistic case, but by the charge conservation carried out by R 3 ρ( r)d 3 r = e to preserve the Lorentz invariance [8], where the charge density of the negatively charged particle (e.g., a π − -meson; q = −e) is given by Then, the charge normalization imposes the following restriction on the radial eigenfunctions The substitution of the expression (7) for u ǫl (s) into Eq. (10) provides the following normalization constant where we have used for the second equality the relation Let us emphasize that the resulting Lorentz-invariant charge density ρ LI ( r) given by Eq. (9) is always (i.e. for any observer's velocity v) appropriately normalized while the density ρ N LI ( r) = |ψ nlm ( r)| 2 (used in [17]) is not. This is numerically illustrated in Figure 1 for a pionic atom with nuclear charge Z = 68 in the infinite nuclear mass approximation (π − -meson mass=273.132054 a.u.). For completeness we have plotted in Figure 2 the radial density of the charge distribution for two diferent states (n = 1, l = 0) and (n = 4, l = 1) of a pionic system with nuclear charge Z = 68 in the infinite nuclear mass approximation, respectively. Moreover, we have also made in these figures a comparision with the corresponding Schrödinger density functions [18]. We observe that the relativistic effects other than spin (i) tend to compress the charge towards the origin, and (ii) they are most apparent for states S. In this paper we quantify this relativistic charge compression by three different means. First, in Section II, we compute the ordinary moments or radial expectation values r k for general (n, l, m) states, making emphasis in the Heisenberg measure for circular (l = n − 1) and S-states (l = 0). Then, in Section III, we study numerically the most relevant charge information-theoretic measures of the system; namely the Shannon entropy and the Fisher information.

Radial expectation values and Heisenberg's measure.
The charge distribution of the Klein-Gordon particles in a Coulomb potential can be completely characterized by means of the ordinary radial expectation values r k , k ∈ N, given by where we have used Eqs. (7) and (9), and the symbol J nl (k) denotes the integral [9] J nl (k) : For the lowest values of k we have Then, besides the normalization r 0 = 1, we have the following value for the centroid of the charge density, and for the second-order moment, so that the Heisenberg measure σ nl which quantifies the charge spreading around the centroid is given by To gain insight into these general expressions we are going to discuss two particular classes of quantum-mechanical states, the circular (i.e., l = n − 1) states and the nsstates (i.e., l = 0) states.
For circular states we have that and the integrals Then, the centroid of the charge distribution is, according to Eq. (15), and the second-order moment, according to equation (16), becomes so that the Heisenberg measure for circular states σ 2 n = σ 2 n,n−1 has the following value These expressions for circular states and the corresponding ones for ns-states are discussed and compared with the Schrödinger values as a function of the principal quantum 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, being the rate of this behaviour much faster for circular than for S-states. This indicates that the 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 the system in Figure 5 for the states 1S, 2S and 2P . We find that both the centroid and the variance ratios monotonically decrease as the nuclear charge Z increases. Moreover, the decreasing rate is much faster for the states 1S, than for the states 2S and 2P . These two observations illustrate that the relativistic charge compression effect is bigger in heavier systems for a given (nl)-state. Moreover, we see here again that for a given system 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 density is not a function of it.
Finally, let highlight that in all figures the Klein-Gordon values tend towards the Schrödinger values in the non-relativistic limit of large n or small Z.

Shannon and Fisher information measures
Here we study numerically the relativistic effects on the charge spreading of pionic systems of hydrogenic type by means of the following information-theoretic measures of the associated charge distribution ρ nlm ( r) given by Eq (9): The Shannon entropy power and the Fisher information. The Shannon entropic power of a negatively-charged Klein-Gordon particle characterized by the charge density ρ nlm ( r) is defined by [12] where S nlm is the Shannon entropy of ρ nlm ( r) given by the expectation value of −log (ρ nlm ( r)), i. e. which quantifies the total extent of the charge spreading of the system. Taking into account the above-mentioned ansatz for ψ( r) and Eqs. (7), (9) and (22), this expresion can be separated out into radial and angular parts as it is explained in full detaill in [19], being R nl and Y lm the radial and angular parts of the density. We should keep in mind that the angular part is the same for both Klein-Gordon and Schrödinger cases. The Fisher information is defined by [10] Remark that we are not using here the parameter dependent Fisher information originally introduced (and so much used) by statisticians [11], but its translationally invariant 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 content of the charge distribution: so, when ρ( r) has a discontinuity at a certain point, the local slope value drastically changes and the Fisher information strongly varies. This indicates that it is a local quantity in contrast to the Heisenberg measure σ 2 nl and the Shannon entropy S ρ (and its associated power), which have a global character because they are powerlike and logarithmic functionals of the density, respectively.
Unlike the moment-based quantities discussed in the previous section, these complementary measures do not depend on a special point, either the origin as the ordinary moments or the centroid as the Heisenberg measure. These quantities, first used by statisticians and electrical engineers and later by quantum physicists, have been shown to be measures of disorder or smoothness of the density ρ nlm ( r) [10,12]. Let us highlight that the Fisher information does not only measure the charge spreading of the system in a complementary and qualitatively different manner as the Heisenberg and Shannon measures but also it quantifies their oscillatory character, indicating the local charge concentration all over the space [10].
The relativistic (Klein-Gordon) and non-relativistic (Schrödinger) values of the Shannon entropic power are numerically discussed and compared in Figure 6 for the pionic system. Therein, on the left, we plot the ratio N nl (KG)/N nl (Sch) between these two values as a function of the principal quantum number n for the system with nuclear charge Z = 68. We notice that the Shannon ratio systematically increases when n is increasing, approaching to unity for large n, for both circular and S-states. Moreover, we find that this approach is much faster for circular states, what indicates once more that the relativistic effects are much more important for states S. In addition, on the right of Figure 6, we show the dependence of the Shannon ratio with the nuclear charge Z for the 1S, 2S ant 2P states. We observe, here again, that the ratio is a decreasing function of Z for any state, indicating that the relativistic effects are much more important for heavy systems. Moreover, for a given system (i.e. fixed Z) the relativistic effects increase when n (l) decreases for fixed l (n). The quantum number m affects the absolute value of the Shannon entropic power but it doesn't affect the ratio.   Figure 7 shows the dependence of the ratio of the non-relativistic and relativistic values 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 involved integral diverges. First we should remark that here, contrary to the previous quantities considered in this work, the Schrödinger values are always less than the Klein-Gordon ones; this is strongly related to the local character of the Fisher information, indicating that the localized internodal charge concentration is always larger in the relativistic case. Second, we observe that for fixed l the Fisher ratio I nl (Sch)/I nl (KG) monotonically increases when n is getting bigger, approaching to unity at a rate which grows as l is increasing. Third, we find that the Fisher ratio decreases for all states in a systematic way as the nuclear charge increases. Moreover, for a given Z value this ratio increases as either the quantum numbers n and/or l increase. For completeness, the behavior of the Shannon and Fisher ratios in terms of the orbital quantum number l for a fixed n is more explicitly shown of the left and right graphs, respectively, of Figure 8.
Finally, in Figure 9, the dependence of the Fisher ratio on the magnetic quantum number m is studied. Notice that the ratio is bigger when |m| is increasing, indicating that the lower |m| is, the more concentrated is the charge density of the state and the more important are the relativistic effects.

Conclusions
The relativistic charge compression of spinless Coulomb particles has been quantitatively investigated by means of the Heisenberg, Shannon and Fisher spreading measures. These three complementary quantities show that the relativity effects are larger (i. e. the charge compresses more towards the origin) for the lower energetic states and when the Coulomb strength (i. e. the nuclear charge Z) increases. Moreover, a detailed analysis of these quantities on the quantum numbers (n, l, m) characterising the physical states of a given system (i. e. for a fixed Z) indicate that the relativistic effects increase when n (l) decreases for fixed l (n). Furthermore, the study of the Fisher information shows that the relativistic effects also increase when the magnetic quantum number |m| is increasing for fixed (n, l).

Acknowlegments
We are very grateful to Junta de Andalucia for the grants FQM-2445 and FQM-1735, and the Ministerio de Ciencia e Innovación for the grant FIS2008-02380/FIS. We belong to the research group FQM-207. Daniel Manzano acknowledges the fellowship BES-2006-13234.