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Physical Review A

Rigorous relationships among quantum-mechanical kinetic energy and atomic information entropies: Upper and lower bounds

Shridhar Ramachandra Gadre, Rajeev D. Bendale

Published Aug 1, 1987• 97 citations
Publisher

Abstract

An uncertainty-type lower bound [I. Bialynicki-Birula and J. Mycielski, Commun. Math. Phys. 44, 129 (1975)] to the information-entropy sum in complementary spaces has recently been reformulated by Gadre et al. [Phys. Rev. A 32, 2602 (1985)] in terms of the respective one-particle probability densities. This bound has been exploited to derive rigorous upper as well as lower bounds to the information entropies and their sum in terms of the corresponding second moments of their distributions. Thus the present work establishes a direct connection, as suggested by Sears, Parr, and Dinur [Israel J. Chem. 19, 165 (1980)], between the quantum-mechanical kinetic energy and information entropy in position space. It has also been demonstrated that given at least one arbitrary moment-type constraint in each space, it is possible to derive an upper bound to the information entropy sum in complementary spaces.

DOI
10.1103/physreva.36.1932

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An uncertainty-type lower bound [I. Bialynicki-Birula and J. Mycielski, Commun. Math. Phys. 44, 129 (1975)] to the information-entropy sum in complementary spaces has recently been reformulated by Gadre et al. [Phys. Rev. A 32, 2602 (1985)] in terms of the respective one-particle probability densities. This bound has been exploited to derive rigorous upper as well as lower bounds to the information entropies and their sum in terms of the corresponding second moments of their distributions. Thus the present work establishes a direct connection, as suggested by Sears, Parr, and Dinur [Israel J. Chem. 19, 165 (1980)], between the quantum-mechanical kinetic energy and information entropy in position space. It has also been demonstrated that given at least one arbitrary moment-type constraint in each space, it is possible to derive an upper bound to the information entropy sum in complementary spaces.
Rigorous bounds to atomic information entropies and their sum in position and momentum spaces have been derived in terms of radial and momentum expectation values ⟨ r n ⟩ and ⟨ p n ⟩. Numerical studies on atomic systems show that the bounds presented in terms of ⟨ r ⟩ and ⟨ p ⟩, the first moments of the position and momentum - space densities are sharper than those given by Gadre and Bendale in terms of ⟨ r 2 ⟩ and ⟨ p 2 ⟩, the second moments of the position and momentum-space densities. Within the BKCM procedure, several relationships between the information entropies and the average electron densities ⟨ρ⟩ and ⟨γ⟩ in position and momentum spaces have been established.
The position-space entropy Sρ and the momentum-space entropy Sγ are two increasingly important quantities in the study of the structure and scattering phenomena of atomic and molecular systems. Here, an information-theoretic method which makes use of the Bialynicki–Birula and Mycielski’s inequality is described to find rigorous upper and lower bounds to these two entropies in a compact, simple and transparent form. The upper bounds to Sρ are given in terms of radial expectation values <rα≳ and/or the mean logarithmic radii <ln r≳ and <(ln r)2≳, whereas the lower bounds depend on the momentum expectation values <pα≳ and/or the mean logarithmic momenta <ln p≳ and <(ln p)2≳. Similar bounds to Sγ are also shown in a parallel way. A near Hartree–Fock numerical analysis for all atoms with Z≤54 shows that some of these bounds are so tight that they may be used as computational values for the corresponding quantities. The role of the mean logarithmic radius <ln r≳ and the mean logarithmic momentum <ln p≳ in the improvement of accuracy of the aforementioned bounds is certainly striking.
The one-electron Shannon information entropy sum is reformulated in terms of a single entropic quantity dependent on a one-electron phase space quasiprobability density. This entropy is shown to form an upper bound for the entropy of the one-electron Wigner distribution. Two-electron entropies in position and momentum space, and their sum, are introduced, discussed, calculated, and compared to their one-electron counterparts for neutral atoms. The effect of electron correlation on the two-electron entropies is examined for the helium isoelectronic series. A lower bound for the two-electron entropy sum is developed for systems with an even number of electrons. Calculations illustrate that this bound may also be used for systems with an odd number of electrons. This two-electron entropy sum is then recast in terms of a two-electron phase space quasiprobability density. We show that the original Bialynicki-Birula and Mycielski information inequality for the N-electron wave function may also be formulated in terms of an N-electron phase space density. Upper bounds for the two-electron entropies in terms of the one-electron entropies are reported and verified with numerical calculations.
General model-independent relationships among radial expectation values of the one-particle densities in position and momentum spaces for any quantum-mechanical system are obtained. They are derived from the Stam uncertainty principle and the recently reported lower bounds to the Fisher information entropy of both densities. The results are usually expressed in terms of some uncertainty products of the system. The accuracy of the bounds is numerically analyzed for neutral atoms within a Hartree-Fock framework.
In classical statistical mechanics, C exp[ -( mv*v/2 + V(x))/T) represents density i.e. the probability for a particle to have a velocity v and be at x. If applies this probability to Shannon’s entropy formula, one obtains C/T ( mv*v/2 + V(x)) exp[ -( mv*v/2 + V(x))/T) + exp[ -( mv*v/2 + V(x))/T) ln(C). Thus, it seems energy times density + density times ln(C) is the entropy density, a well known result in classical statistical mechanics, although sometimes not explicitly stated. The first term integrated over x is essentially the kinetic energy, while the second integrated over velocity yields the potential energy term. Thus, the first term partial density seems to be a purely momentum dependent kinetic energy piece times probability and the second a purely x dependent potential energy piece. (There is still an integral over velocity for the first term and one over x for the second to obtain the full entropy.) In quantum mechanics, we argue that a(p1)exp(ip1 x) a(p2) exp(ip2 x) serves as the probability where W(x)=Sum over p a(p) exp(ipx) = wavefunction. This leads to an expression for entropy density with two terms, one of which may be immediately integrated over momentum and the other over space. If one wishes to make a link to the classical statistical mechanical case, one may ask what requirement is necessary for the momentum related piece to represent average kinetic energy and the spatial part, average potential energy. The requirement, we find, yields the wavefunction of the ground state harmonic oscillator. For other cases, the partial density is not equivalent to average energy. Thus, we try to establish a link between entropy density and energy density, but require the kinetic energy related term to be strictly a function of momentum and the potential related term strictly one of x. In quantum mechanics, one may form averages at a point x. For example, KEave(x)= [Sum over p p*p/2m a(p) exp(ipx)]/W(x). If one examines a p/-p pair with a(p)=a(-p) (for example), cos(px) terms arise which are positive at some x and negative at others. We argued in previous notes that this represents stochastic flow in the opposite direction from that expected by -dV/dx. In other words, -dV/dx is an average which consists of stochastic force addends which may give momentum hits to the right or left, but which add to yield -dV/dx. As a result, one may have negative contributions to the kinetic energy (and density as well). In this note, we argue one may also have negative contributions to the entropy density due to this phenomenon.
A direct connection of information entropy SSS and kinetic energy TTT is obtained for nuclei and atomic clusters, which establishes TTT as a measure of the information in a distribution. It is conjectured that this is a universal property for fermionic many-body systems. We also check rigorous inequalities previously found to hold between S and T for atoms and verify that they hold for nuclei and atomic clusters as well. These inequalities give a relationship of Shannon's information entropy in position-space with an experimental quantity i.e. the rms radius of nuclei and clusters.