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arXiv:quant-ph/0411016

Information Entropy and Correlation of the Hooke's Atom

Rajneesh Atre, Chandra Mohapatra, Prasanta Kumar Panigrahi

Published Nov 2, 2004• 1 citation
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Abstract

We provide an algebraic procedure to find the eigenstates of two-charged particles in an oscillator potential, known as {\it{Hooke's}} atom. For the planar Hooke's atom, the exact eigenstates and single particle densities for arbitrary azimuthal quantum number, are obtained analytically. Information entropies associated with the wave functions for the relative motion are then studied systematically, since the same incorporates the effect of the Coulomb interaction. The {\it{quantum pottery}} of the information entropy density reveals a number of intricate structures, which differ significantly for the attractive and repulsive cases. We indicate the procedure to obtain the approximate eigen states. Making use of the relationship of this dynamical system with the quasi-exactly solvable systems, one can also develop a suitable perturbation theory, involving the Coulomb coupling $Z$, for the approximate wave functions.

DOI
10.48550/arxiv.quant-ph/0411016

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We provide an algebraic procedure to find the eigenstates of two-charged particles in an oscillator potential, known as {\it{Hooke's}} atom. For the planar Hooke's atom, the exact eigenstates and single particle densities for arbitrary azimuthal quantum number, are obtained analytically. Information entropies associated with the wave functions for the relative motion are then studied systematically, since the same incorporates the effect of the Coulomb interaction. The {\it{quantum pottery}} of the information entropy density reveals a number of intricate structures, which differ significantly for the attractive and repulsive cases. We indicate the procedure to obtain the approximate eigen states. Making use of the relationship of this dynamical system with the quasi-exactly solvable systems, one can also develop a suitable perturbation theory, involving the Coulomb coupling ZZZ, for the approximate wave functions.
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The sums of position- and momentum-space mutual information measures are used to examine the pairwise and higher-order statistical correlation in the ground states of N-particle coupled oscillators. Analytical expressions for these measures are shown to be related to the logarithmic interaction energies of these states, plus those of mirror states where the intensities of the one- and two-body potentials are interchanged, and the nature of the attractive or repulsive interaction is opposite to that in the original state. The measures separate the contributions from the interactions and those from the effective interactions due to marginalization into different terms. The pairwise mutual information sum is linearly related to the Shannon entropy sum in two particle systems, while the total correlation sum exhibits a similar relationship in three particle ones. In the latter instance, the interaction information sum can be related to entropy differences. This illustrates how entropy sums are connected to correlation measure sums in these systems. All measures approach zero with large N, when the magnitudes of the one- and two-body potentials are fixed. The pair mutual information and total correlation sums decay monotonically with N in the presence of an attractive potential and monotonically increase with a repulsive potential. On the other hand, the interaction information sum exhibits a minimum at small N, with an attractive potential. This is a consequence of the higher-order correlations governing behavior at smaller N while the pairwise ones dominate at larger N. Results are presented when the magnitude of the one-body potential is set to N.
We study the properties of the Hooke's law correlation energy (Ec{E}_{\mathrm{c}}Ec​), defined as the correlation energy when two electrons interact via a harmonic potential in a DDD-dimensional space. More precisely, we investigate the 1S{}^{1}S1S ground-state properties of two model systems: the Moshinsky atom (in which the electrons move in a quadratic potential) and the spherium model (in which they move on the surface of a sphere). A comparison with their Coulombic counterparts is made that highlights the main differences of the Ec{E}_{\mathrm{c}}Ec​ in both the weakly and strongly correlated limits. Moreover, we show that the Schr\"odinger equation of the spherium model is exactly solvable for two values of the dimension (D=1and3D=1 \text{and} 3D=1and3) and that the exact wave function is based on Mathieu functions.
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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.