53 research outputs found
Spreading lengths of Hermite polynomials
The Renyi, Shannon and Fisher spreading lengths of the classical or
hypergeometric orthogonal polynomials, which are quantifiers of their
distribution all over the orthogonality interval, are defined and investigated.
These information-theoretic measures of the associated Rakhmanov probability
density, which are direct measures of the polynomial spreading in the sense of
having the same units as the variable, share interesting properties: invariance
under translations and reflections, linear scaling and vanishing in the limit
that the variable tends towards a given definite value. The expressions of the
Renyi and Fisher lengths for the Hermite polynomials are computed in terms of
the polynomial degree. The combinatorial multivariable Bell polynomials, which
are shown to characterize the finite power of an arbitrary polynomial, play a
relevant role for the computation of these information-theoretic lengths.
Indeed these polynomials allow us to design an error-free computing approach
for the entropic moments (weighted L^q-norms) of Hermite polynomials and
subsequently for the Renyi and Tsallis entropies, as well as for the Renyi
spreading lengths. Sharp bounds for the Shannon length of these polynomials are
also given by means of an information-theoretic-based optimization procedure.
Moreover, it is computationally proved the existence of a linear correlation
between the Shannon length (as well as the second-order Renyi length) and the
standard deviation. Finally, the application to the most popular
quantum-mechanical prototype system, the harmonic oscillator, is discussed and
some relevant asymptotical open issues related to the entropic moments
mentioned previously are posed.Comment: 16 pages, 4 figures. Journal of Computational and Applied Mathematics
(2009), doi:10.1016/j.cam.2009.09.04
Information measures of hydrogenic systems, Laguerre polynomials and spherical harmonics
AbstractFisher's information and Shannon's entropy are two complementary information measures of a probability distribution. Here, the probability distributions which characterize the quantum-mechanical states of a hydrogenic system are analyzed by means of these two quantities. These distributions are described in terms of Laguerre polynomials and spherical harmonics, whose characteristics are controlled by the three integer quantum numbers of the corresponding states. We have found the explicit expression for the Fisher information, and a lower bound for the Shannon entropy with the help of an isoperimetric inequality
Configuration Complexities of Hydrogenic Atoms
The Fisher-Shannon and Cramer-Rao information measures, and the LMC-like or
shape complexity (i.e., the disequilibrium times the Shannon entropic power) of
hydrogenic stationary states are investigated in both position and momentum
spaces. First, it is shown that not only the Fisher information and the
variance (then, the Cramer-Rao measure) but also the disequilibrium associated
to the quantum-mechanical probability density can be explicitly expressed in
terms of the three quantum numbers (n, l, m) of the corresponding state.
Second, the three composite measures mentioned above are analytically,
numerically and physically discussed for both ground and excited states. It is
observed, in particular, that these configuration complexities do not depend on
the nuclear charge Z. Moreover, the Fisher-Shannon measure is shown to
quadratically depend on the principal quantum number n. Finally, sharp upper
bounds to the Fisher-Shannon measure and the shape complexity of a general
hydrogenic orbital are given in terms of the quantum numbers.Comment: 22 pages, 7 figures, accepted i
Three strongly correlated charged bosons in a one-dimensional harmonic trap: natural orbital occupancies
We study a one-dimensional system composed of three charged bosons confined
in an external harmonic potential. More precisely, we investigate the
ground-state correlation properties of the system, paying particular attention
to the strong-interaction limit. We explain for the first time the nature of
the degeneracies appearing in this limit in the spectrum of the reduced density
matrix. An explicit representation of the asymptotic natural orbitals and their
occupancies is given in terms of some integral equations.Comment: 6 pages, 4 figures, To appear in European Physical Journal
Scaling properties of composite information measures and shape complexity for hydrogenic atoms in parallel magnetic and electric fields
The scaling properties of various composite information-theoretic measures
(Shannon and R\'enyi entropy sums, Fisher and Onicescu information products,
Tsallis entropy ratio, Fisher-Shannon product and shape complexity) are studied
in position and momentum spaces for the non-relativistic hydrogenic atoms in
the presence of parallel magnetic and electric fields. Such measures are found
to be invariant at the fixed values of the scaling parameters given by and . Numerical results which support the validity of the scaling
properties are shown by choosing the representative example of the position
space shape complexity. Physical significance of the resulting scaling
behaviour is discussed.Comment: 10 pages, 2 figure
Ground-state correlation properties of charged bosons trapped in strongly anisotropic harmonic potentials
We study systems of a few charged bosons contained within a strongly
anisotropic harmonic trap. A detailed examination of the ground-state
correlation properties of two-, three-, and four-particle systems is carried
out within the framework of the single-mode approximation of the transverse
components. The linear correlation entropy of the quasi-1D systems is discussed
in dependence on the confinement anisotropy and compared with a strictly 1D
limit. Only at weak interaction the correlation properties depend strongly on
the anisotropy parameter.Comment: 5 pages, 6 figure
Entanglement in helium
Using a configuration-interaction variational method, we accurately compute
the reduced, single-electron von Neumann entropy for several low-energy,
singlet and triplet eigenstates of helium atom. We estimate the amount of
electron-electron orbital entanglement for such eigenstates and show that it
decays with energy.Comment: 5 pages, 2 figures, added references and discussio
Parameter-based Fisher's information of orthogonal polynomials
AbstractThe Fisher information of the classical orthogonal polynomials with respect to a parameter is introduced, its interest justified and its explicit expression for the Jacobi, Laguerre, Gegenbauer and Grosjean polynomials found
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