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International Journal of Quantum Chemistry

Net information measures for modified Yukawa and Hulthén potentials

S. H. Patil, K. D. Sen

Published Jan 1, 2007• 51 citations
Publisher

Abstract

Abstract The dimensional analyses of the position and momentum variances‐based quantum mechanical Heisenberg uncertainty measure, as well as the entropic information measures given by the Shannon information entropy sum and the product of Fisher information measures are carried out for two widely used nonrelativistic isotropic exponential‐cosine screened Coulomb potentials generated by multiplying the superpositions of (i) Yukawa‐like, −Z(e−μr/r), and (ii) Hulthén‐like, −Zμ(1/(eμr − 1)), potentials by cos(bμr) followed by addition of the term a/r2, where a and b ≥ 0, μ are the screening parameters and Z, in case of atoms, denotes the nuclear charge. Under the spherical symmetry, all the information measures considered are shown to be independent of the scaling of the set [μ, Z] at a fixed value of μ/Z, a, and b and the other parameters defining the superpositions of the potentials. Numerical results are presented, which support the validity of the scaling properties. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

DOI
10.1002/qua.21333

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Abstract The dimensional analyses of the position and momentum variances‐based quantum mechanical Heisenberg uncertainty measure, as well as the entropic information measures given by the Shannon information entropy sum and the product of Fisher information measures are carried out for two widely used nonrelativistic isotropic exponential‐cosine screened Coulomb potentials generated by multiplying the superpositions of (i) Yukawa‐like, −Z(e−μr/r), and (ii) Hulthén‐like, −Zμ(1/(eμr − 1)), potentials by cos(bμr) followed by addition of the term a/r2, where a and b ≥ 0, μ are the screening parameters and Z, in case of atoms, denotes the nuclear charge. Under the spherical symmetry, all the information measures considered are shown to be independent of the scaling of the set [μ, Z] at a fixed value of μ/Z, a, and b and the other parameters defining the superpositions of the potentials. Numerical results are presented, which support the validity of the scaling properties. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007
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Abstract This work presents analytical and numerical results for the position‐ and momentum‐space information entropies of the 1s2 state of helium‐like ions using different interaction potentials. The potentials that we used are the Yukawa potential and the exponential‐cosine‐screened Coulomb potential. The investigated studies allow us to relate the position‐space information with the momentum‐space information of Shannon and Fisher, as well as Shannon entropy power and the Fisher‐Shannon information product, through different famous relations. The calculation is performed using the one‐electron charge density of entangled two‐parameter wavefunction. On one hand, the results that are presented for 10 members in the helium isoelectronic sequence demonstrate with precision the effect of correlation on bare charge distributions. On the other hand, it leads to some very important results for both the correlated and uncorrelated values of the informatic entropies. Analytical formulas for the momentum‐space information entropies are given. The effect of the nuclear charge and the screening parameter on the information expressions has been studied for both potentials. Detailed computational and numerical values and characteristics of these information quantities, as a function of the screening parameter, are reported here for the first time. New inequality has been proposed with Fisher's total value to measure the correlation of two electrons.
The dimensional analyses of the position and momentum variance based quantum mechanical Heisenberg uncertainty measure and the other useful net entropic information measures for the bound states of two constrained Coulomb potentials are reported for the first time. The potentials describe an electron moving in the central field due to a nucleus of charge Z with radius R defining the constraints as (a) the truncated potential given by , and (b) the radius of the impenetrable spherical wall. The net information measures for the two potentials are explicitly shown to be independent of the scaling of the set [ Z , R ] at a fixed value of ZR . Analytic proof is presented, for the first time, showing the presence of a characteristic extremum in the variation of the net information entropy as a function of the radius R with its location scaling as Z −1 . Numerical results are presented which support the validity of the scaling properties.
This work presents analytical and numerical results for the position- and momentum-space information entropies, of the 1s 2 -state of helium-like ions, using different interaction potentials. The potentials that we used are the Yukawa potential (YP), and the exponential-cosine-screened Coulomb potential (ECSCP). The investigated studies allow us to relate the position-space information with the momentum-space information of Shannon and Fisher, as well as Shannon entropy power, and the Fisher-Shannon information product, through different famous relations. The calculation is done using the one-electron charge density of entangled two-parameter wave function. On one hand, the results that are presented for ten members in the helium isoelectronic sequence demonstrate with precision the effect of correlation on bare charge distributions. On the other hand, it leads to some very important results for both the correlated and uncorrelated values of the informatic entropies. Analytical formula for the momentum-space information entropies are given. The effect of the nuclear charge and the screening parameter on the information expressions has been studied for both potentials. Detailed computational and numerical values and characteristics of these information quantities, as a function of the screening parameter, are reported here for the first time. New inequality has been proposed with Fisher’s total value to measure the correlation of two electrons.
This work present analytical results for the position- and momentum-space information entropies, of the 1s2-state of helium-like ions, using different interaction potentials. The potentials that we used are the Yukawa potential (YP), and the exponential-cosine-screened Coulomb potential (ECSCP). The investigated studies allow us to relate the position-space information with the momentum-space information of Shannon and Fisher, as well as Shannon entropy power, and the Fisher-Shannon information product, through different famous relations. The calculation is done using one-electron charge density of entangled two-parameter wave function. On one hand, the results that are presented for fifteen members in the helium iso-electronic sequence demonstrate with precision the effect of correlation on bare charge distributions. On the other hand, it leads to some very important results for both the correlated and uncorrelated values of the informatic entropies. Analytical formula for the momentum-space information entropies are given. The effect of the nuclear charge and the screening parameter on the informatic expressions has been studied for both potentials. Detailed computational and numerical values and characteristics of these quantities are reported here for the first time. New inequality has been proposed with Fisher’s total value to measure the correlation of two electrons.
Yukawa potential and Hulthẻn potential are very useful potential models with applications in different areas of physics. The present study examined theoretic measure and thermodynamic properties of energy levels and wave functions for a combination of these two potentials. The effect of screening parameter on Fisher information and Shannon entropy as well as the effect of temperature on the various thermodynamic properties of the combined potentials and its subsets potentials are well studied. The effects of screening parameter on the thermodynamic properties of the combined potentials are also studied. This study shows equivalence of temperature effect on the combined potential and Hulthẻn potential but a different effect on Yukawa potential. Fisher information and the Shannon entropy for both position and momentum spaces as a function of the screening parameter obey Heisenberg principle, satisfy Cramer-Rao inequality and Bialynick-Birula, Mycielski inequality. The Shannon entropy shows the stability of the system for low screening parameter. It is also shown that the strength of Hulthẻn potential outweighed the strength of Yukawa potential in their combination for thermodynamic properties.