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.
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.
ABSTRACT The dimensional analysis of the position and momentum variances‐based quantum mechanical Heisenberg uncertainty measures was carried out for generalized Yukawa‐type potential. The entropic information measures given by Shannon entropy sums and Fisher information products were obtained out for generalized Yukawa‐type potential. It is interesting to observe that the Shannon entropy in position space, , increases and then decreases with the quantum number , suggesting a growth in delocalization due to a rapid increase in fluctuations. Likewise, the Shannon entropy in momentum space, , decreases and then increases with the quantum number , and vice versa. This unusual phenomenon may be explained by the parity restrictions inherent in the system. The Bialynicki‐Birula and Mycielski (BBM) entropic uncertainty relation was verified to be saturated; however, the sum of the entropies increases with the quantum number up to a certain point and then decreases as continues to increase. We also found that the Fisher information accurately increases with the quantum number . The numerical results that support the validity of the scaling properties for Shannon entropy sum and Fisher information product were presented. The existence of a squeezed phenomenon in the ground and first excited state for some values of the potential screening parameter in momentum space was found. The Heisenberg uncertainty relation is met when the squeeze in one coordinate is compensated by an increase in the other coordinate. The results obtained exceed the theoretical lower bounds for all the information‐theoretic measures considered, confirming their physical validity and accuracy.
The Shannon information entropy of 1-normalized electron density in position and momentum space Sr and Sp, and the sum ST, respectively, are reported for the ground-state H, He+, Li2+, H-, He, Li+, Li, and B atoms confined inside an impenetrable spherical boundary defined by radius R. We find new characteristic features in ST denoted by well-defined minimum and maximum as a function of confinement. The results are analyzed in the background of the irreducible lower bound stipulated by the entropy uncertainty principle [I. Bialynicki-Birula and J. Mycielski, Commun. Math. Phys. 44, 129 (1975)]. The spherical confinement model leads to the ST values which satisfy the lower bound up to the limits of extreme confinements with the interesting new result displaying regions over which a set of upper and lower bounds to the information entropy sum can be locally prescribed. Similar calculations on the H atom in 2s excited states are presented and their novel characteristics are discussed.
Abstract The information‐theoretic measure of confined hydrogen atom has been investigated extensively in the literature. However, most of them were focused on the ground state and accurate values of information entropies, such as Shannon entropy, for confined hydrogen are still not determined. In this work, we establish the benchmark results of the Shannon entropy for confined hydrogen atom in a spherical impenetrable sphere, in both position and momentum spaces. This is done by examining the bound state energies, the normalization of wave functions, and the scaling property with respect to isoelectronic hydrogenic ions. The angular and radial parts of Shannon entropy in two conjugate spaces are provided in detail for both free and confined hydrogen atom in ground and several excited states. The entropies in position space decrease logarithmically with decreasing the size of confinement, while those in momentum space increase logarithmically. The Shannon entropy sum, however, approaches to finite values when the confinement radius closes to zero. It is also found that the Shannon entropy sum shares same trend for states with similar density distributions. Variations of entropy for nodeless bound states are significantly distinct form those owning nodes when changing the confinement radius.
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.