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Entropy

Phase-Space Position-Momentum Correlation and Potentials

Humberto G. Laguna, Robin P. Sagar

Published Apr 25, 2013• 10 citations
Publisher

Abstract

Solutions to the radial Schr¨odinger equation of a particle in a quantum corral are used to probe how the statistical correlation between the position, and The momentum of the particle depends on the effective potential. The analysis is done via the Wigner function and its Shannon entropy. We show by comparison to the particle-in-a-box model that the attractive potential increases the magnitude of the correlation, while a repulsive potential decreases the magnitude of this correlation. Varying the magnitude of the repulsive potential yields that the correlation decreases with a stronger repulsive potential.

DOI
10.3390/e15051516

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Solutions to the radial Schr¨odinger equation of a particle in a quantum corral are used to probe how the statistical correlation between the position, and The momentum of the particle depends on the effective potential. The analysis is done via the Wigner function and its Shannon entropy. We show by comparison to the particle-in-a-box model that the attractive potential increases the magnitude of the correlation, while a repulsive potential decreases the magnitude of this correlation. Varying the magnitude of the repulsive potential yields that the correlation decreases with a stronger repulsive potential.
Position–momentum correlation in the Moshinsky atom is studied by examining the mutual information between Wigner functions and position/momentum space densities, at the one- and two-particle levels. We show that this correlation is independent of the strength of the interparticle potential at the two-particle level. At the one-particle level, the correlation decreases as the potential is increased. Shannon entropies of one- and two-particle Wigner functions are also discussed as measures of the localization in phase-space distributions.
Shannon entropies of the Wigner function are calculated for ground and excited stationary states of the Particle-In-A-Box and Harmonic Oscillator model systems and examined as a measure of the localization of the phase-space distribution. We show that their behavior is consistent with that of the sum of the position and momentum space entropies as a function of quantum number. Position-momentum correlation is then analyzed in these systems by defining mutual information between position and momentum variables. This mutual information yields non-zero values, in contrast to the quantum covariance, and increases with quantum number.
We study the influence of the interparticle and confining potentials on statistical correlation via the correlation coefficient and mutual information in ground and some excited states of the Moshinsky atom in position and momentum space. The magnitude of the correlation between positions and between momenta is equal in the ground state. In excited states, the correlation between the momenta of the particles is greater than between their positions when they interact through an attractive potential whereas for repulsive interparticle potentials the opposite is true. Shannon entropies, and their sums (entropic formulations of the uncertainty principle), are also analyzed, showing that the one-particle entropy sum is dependent on the interparticle potential and thus able to detect the correlation between particles.
Interaction information, a measure of higher-order statistical correlations, is examined in quantum systems consisting of three coupled and uncoupled oscillators, in position and in momentum space. The magnitudes of the interaction information are greater than zero, and equal in both spaces, in the uncoupled case. This is interpreted as a dominance of synergic effects due to the indistinguishability of particles, and is larger for symmetric wave functions compared to antisymmetric ones. In symmetric wave functions, the magnitude of interaction information increases with the coupling strength, and the inclusion of coupling reveals differences in behavior between the spaces. With attractive potentials, interaction information is positive in momentum space (synergic) and negative in position space (redundant). This is reversed with repulsive potentials. It is now position space which yields positive values, while momentum space gives negative ones. The inclusion of spin provides antisymmetric wave functions which yield that the overall tendencies are maintained. However, there are regions for both attractive and repulsive potentials where the sign of interaction information changes, with the interpretation that the balance changes from a dominance of synergic to that of redundant interactions, and vice versa. This suggests that the nature of the interactions in these systems can be tuned with the coupling strength.
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.
In this study, the energy equation and normalized wave function were obtained by solving the Schrödinger equation analytically utilizing the Eckart-Hellmann potential and the Nikiforov-Uvarov method. Fisher information and Shannon entropy were investigated. Our results showed higher-order characteristic behavior for position and momentum space. Our numerical results showed an increase in the accuracy of the location of the predicted particles occurring in the position space. Also, our results show that the sum of the position and momentum entropies satisfies the lower-bound Berkner, Bialynicki-Birula, and Mycieslki inequality and Fisher information was also satisfied for the different eigenstates. This study's findings have applications in quantum chemistry, atomic and molecular physics, and quantum physics.