Position and momentum information‐theoretic measures of the pseudoharmonic potential

In this study, the information‐theoretic measures in both the position and momentum spaces for the pseudoharmonic potential using Fisher information, Shannon entropy, Renyi entropy, Tsallis entropy, and Onicescu information energy are investigated analytically and numerically. The results obtained are applied to some diatomic molecules. The Renyi and Tsallis entropies are analytically obtained in position space using Srivastava–Niukkanen linearization formula in terms of the Lauricella hypergeometric function. Also, they are obtained in the momentum space in terms of the multivariate Bell polynomials of Combinatorics. We observed that the Fisher information increases with n in both the position and momentum spaces, but decreases with for all the diatomic molecules considered. The Shannon entropy also increases with increasing n in the position space and decreases with increasing . The variations of the Renyi and Tsallis entropies with are also discussed. The exact and numerical values of the Onicescu information energy are also obtained, after which the ratio of information‐theoretic impetuses to lengths for Fisher, Shannon, and Renyi are obtained. © 2015 Wiley Periodicals, Inc.     

Theoretic quantum information entropies for the generalized hyperbolic potential

The Shannon entropy (S) and the Fisher Information (I) entropies are investigated for a generalized hyperbolic potential in position and momentum spaces. First, the Schrodinger equation is solved exactly using the Nikiforov-Uvarov-Functional Analysis method to obtain the energy spectra and the corresponding wave function. By Fourier transforming the position space wave function, the corresponding momentum wave function was obtained for the low-lying states corresponding to the ground and first excited states. The positions and momentum of Shannon entropy and Fisher Information entropies were calculated numerically. Finally, the Bialynicki-Birula and Mycielski and the Stam-Cramer-Rao inequalities for the Shannon entropy and Fisher Information entropies, respectively, were tested and were found to be satisfied for all cases considered.     

Net information measures for modified Yukawa and Hulthén potentials

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/(e^μr ? 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     

Quantum information‐theoretic measures for the static screened Coulomb potential

In this research work, the quantum information‐theoretic analysis of the static screened Coulomb potential has been carried out by studying both analytically and numerically the entropic measures, Fisher information as well as the Onicescu information energy of its wave function. Explicit expressions of these information‐theoretic measures were obtained. Using the Srivastava–Daoust linearization formula in terms of the multivariate Lauricella hypergeometric function, the Rényi entropy, Tsallis entropy, Onicescu information energy were analytically obtained. From the results obtained, it is observed that some of the Shannon entropies are negative, indicating that, negative entropies exists for the probability densities that are highly localized. The trends in the variation of the information‐theoretic measures with the potential screening parameter a for this atomic model are discussed. The Bialynicki‐Birula, Mycielski inequality (BBM), and the Fisher information based uncertainty relation are also verified.     

Quantum information entropy of Eckart potential

In this work, the position and momentum space information densities of the Eckart potential are graphically demonstrated and their properties are studied. The position space information densities have quite an asymmetric shape depending on the values of quantum numbers. The information entropy is obtained and Bialynicki‐Birula and Mycielski inequality is numerically saturated for some parameters of the potential. It is shown that the inequality is saturated with increasing potential depth.     

Studies on some exponential‐screened coulomb potentials

The generalized pseudospectral method is used to study the bound‐state spectra of some of the exponentially screened Coulomb potentials, namely, the exponential cosine screened Coulomb (ECSC) and general exponential screened Coulomb (GESC) potential, with special emphasis on higher states and stronger interaction. Eigenvalues accurate up to 11 significant figures are obtained through a nonuniform optimal spatial discretization of the radial Schrödinger equation. All the 55 eigenstates of ECSC potential with n ≤ 10 and 36 eigenstates of GESC potential with n ≤ 8 are considered for arbitrary values of the screening parameter, covering a wide range of interaction. Excited states as high as up to n = 18 have been computed with high accuracy for the first time. Excellent agreement with the literature data has been observed in all cases. All the GESC eigenstates are calculated with much greater accuracy than the existing methods available in literature. Many l ≠ 0 states of this potential are reported here. In both cases, a detailed variation of energies with respect to the parameters in potential is monitored. © 2012 Wiley Periodicals, Inc.     

Atomic quantum similarity indices in position and momentum spaces

Quantum similarity for atoms is investigated using electron densities in position and momentum spaces. Contrary to the results in position space, the analysis in the momentum space shows how the momentum density carries fundamental information about periodicity and structure of the system and reveals the pattern of Mendeleev's table. A global analysis in the joint r-p space keeps this result.     

Electron pair density information measures in atomic systems

Shannon entropies of the pair density, conditional entropies, and mutual information are studied in position and in momentum space for ground state neutral atoms and selected excited states at the Hartree‐Fock level. We show that the mutual information, a measure of correlation, is larger in position space than in momentum space. This result also holds for a mutual information defined in terms of the exchange density; however, these quantities display much more structure than the corresponding ones based on the pair densities. The interpretation of this behavior is that exchange effects are smaller in momentum space than in position space in these systems. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Complex‐scaling calculations for resonance states of he with screened coulomb potentials

The doubly excited ¹S^e, ¹P^o, and ¹D^e resonance states of helium atom with screened Coulomb potentials are investigated. The complex scaling method with configuration interaction type basis functions are employed to extract the resonances associated with the He⁺(N = 2, 3, and 4) thresholds. 18 resonances (six below each of the He⁺ threshold) for each angular‐momentum state are calculated. The results lying below the He⁺(N = 2) threshold are in good agreement with previous calculations by the stabilization method with correlated basis wave functions. The ¹P^o and ¹D^e resonance states lying below the He⁺(N = 3 and 4) thresholds in the screening environment are reported for the first time. © 2013 Wiley Periodicals, Inc.     

Quantum information entropy of modified Hylleraas plus exponential Rosen Morse potential and squeezed states

In this article, some information theoretic concepts are analyzed for modified Hylleraas plus exponential Rosen Morse potential in position and momentum space. The angular and radial contributions of the information density are graphically demonstrated for different states. The entropy densities have asymmetric shape which depends on the values of quantum numbers. The information entropy is analytically obtained for ground state of the potential whereas the numerical calculations have been performed for the higher states and Bialynicki‐Birula and Mycielski inequality is tested for various states using different parameters of the potential. It is shown that the information entropy is reduced, both in position and momentum space, for careful selection of some parameters. Further, it is found that there exist eigenstates exhibiting squeezing in information entropy of modified Hylleraas plus exponential Rosen Morse and Eckart potential. Interestingly, in case of Eckart potential, the squeezed states are obtained in position as well as momentum space and are attempted to saturate for some values of the parameters.     

Mutual information and electron correlation in momentum space

Mutual information and information entropies in momentum space are proposed as measures of the nonlocal aspects of information. Singlet and triplet state members of the helium isoelectronic series are employed to examine Coulomb and Fermi correlations, and their manifestations, in both the position and momentum space mutual information measures. The triplet state measures exemplify that the magnitude of the spatial correlations relative to the momentum correlations depends on and may be controlled by the strength of the electronic correlation. The examination of one- and two-electron Shannon entropies in the triplet state series yields a crossover point, which is characterized by a localized momentum density. The mutual information density in momentum space illustrates that this localization is accompanied by strong correlation at small values of p.     

Information theory of D‐dimensional hydrogenic systems: Application to circular and Rydberg states

The analytic information theory of quantum systems includes the exact determination of their spatial extension or multidimensional spreading in both position and momentum spaces by means of the familiar variance and its generalization, the power and logarithmic moments, and, more appropriately, the Shannon entropy and the Fisher information. These complementary uncertainty measures have a global or local character, respectively, because they are power‐like (variance, moments), logarithmic (Shannon) and gradient (Fisher) functionals of the corresponding probability distribution. Here we explicitly discuss all these spreading measures (and their associated uncertainty relations) in both position and momentum for the main prototype in D‐dimensional physics, the hydrogenic system, directly in terms of the dimensionality and the hyperquantum numbers which characterize the involved states. Then, we analyze in detail such measures for s‐states, circular states (i.e., single‐electron states of highest angular momenta allowed within an electronic manifold characterized by a given principal hyperquantum number), and Rydberg states (i.e., states with large radial hyperquantum numbers n). © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Critical parameters and spherical confinement of H atom in screened Coulomb potential

Critical parameters in three screened potentials, namely, Hulthén, Yukawa, and exponential cosine screened Coulomb potential are reported. Accurate estimates of these parameters are given for each of these potentials, for all states having . Comparison with literature results is made, wherever possible. Present values compare excellently with reference values; for higher n, ?, our results are slightly better. Some of these are presented for first time. Further, we investigate the spherical confinement of H atom embedded in a dense plasma modeled by an exponential cosine screened potential. Accurate energies along with their variation with respect to box size and screening parameter are calculated and compared with reference results in literature. Sample dipole polarizabilities are also provided in this case. The generalized pseudospectral method is used for accurate determination of eigenvalues and eigenfunctions for all calculations. © 2016 Wiley Periodicals, Inc.     

Bound states of a more general exponential screened Coulomb potential

An alternative approximation scheme has been used in solving the Schr?dinger equation to the more general case of exponential screened Coulomb potential, V(r) = −(a/r)[1 + (1 + br)e ^−2br ]. The bound state energies of the 1s, 2s and 3s-states, together with the ground state wave function are obtained analytically upto the second perturbation term.      

Fisher information and uncertainty relations for potential family

An approximate bound state solution of the three-dimensional Schrödinger equation for potential family was obtained together with the corresponding wave function, after which the Fisher information for a potential family was explicitly obtained via the methodology of expectation value and radial expectation value. Some uncertainty relations that are closely related to Heisenberg-like uncertainty were obtained and numerical results were generated to justify the relations and inequalities. Finally, we have numerically studied the Fisher information for two special cases from the result of the potential family that has applications to prototype systems. It was deduced that the Fisher information of the potential family and that of the two special cases exhibit similar features.     

Contracted Schrödinger equation in quantum phase‐space

The phase space formulation of quantum mechanics is equivalent to standard quantum mechanics where averages are calculated by way of phase space integration as in the case of classical statistical mechanics. We derive the quantum hierarchy equations, often called the contracted Schrödinger equation, in the phase space representation of quantum mechanics which involves quasi‐distributions of position and momentum. We use the Wigner distribution for the phase space function and the Moyal phase space eigenvalue formulation to derive the hierarchy. We show that the hierarchy equations in the position, momentum, and position‐momentum representations are very similar in structure. © 2017 Wiley Periodicals, Inc.     

Atomic and molecular intracules for excited states

Intracules in position space, momentum space and phase space have been calculated for low-lying excited states of the He atom, Be atom, formaldehyde and butadiene. The phase-space intracules (Wigner intracules) provide significantly more information than the position- and momentum-space intracules, particularly for the Be atom. Exchange effects are investigated through the differences between corresponding singlet and triplet states.     

Atomic complexity measures in position and momentum spaces

Fisher-Shannon (FS) and Lopez-Ruiz, Mancini, and Calbet (LMC) complexity measures, detecting not only randomness but also structure, are computed by using near Hartree-Fock wave functions for neutral atoms with nuclear charge Z=1-103 in position, momentum, and product spaces. It is shown that FS and LMC complexities are qualitatively and numerically equivalent for these systems. New complexity candidates are defined, computed, and compared by using the following information-theoretic magnitudes: Shannon entropy, Fisher information, disequilibrium, and variance. Localization-delocalization planes are constructed for each complexity measure, where the subshell pattern of the periodic table is clearly shown. The complementary use of r and p spaces provides a compact and more complete understanding of the information content of these planes.     

Benchmark values of Shannon entropy for spherically confined hydrogen atom

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.