Expectation values in one- and two-electron atomic systems

Tsapline, B.

doi:10.1016/0009-2614(70)85235-6

Cited by 41 publications

(20 citation statements)

References 7 publications

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“…This particular result has been recently found by the authors (Angulo and Dehesa [9]) and shown to generalize all the known inequalities involving three successive radial expectation values (Tsapline [16], Blau et al [17], Gadre [18], Gadre and Matcha [19], G~ilvez [20]). It is important to remark that for completely monotonic electron densities, i.e.…”

Section: Charge Monotonicity Effects: Inequalities Among Radial Exsupporting

confidence: 75%

Charge monotonicity of atomic systems and radial expectation values

Angulo

Dehesa

1993

Z Phys D - Atoms, Molecules and Clusters

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A function f(r) is monotone of order p if its pth-derivative f(P)(r) fulfils that ( -1) p f(v) (r)>0. So, e.g. the monotonicity properties of order p = 0, 1, 2 describe the non-negativity (p = 0), the monotonic decreasing from the origin (p = 1) and the convexity (p = 2) of the function, respectively. Here, the monotonicity properties of the electron function g,,(r; ~)=(--t)" p(n)(r)r-~, ~>0, of the ground state of atomic systems are analysed both analytically and numerically. The symbol p(r) denotes the spherically averaged electron density. First of all, the condition which specifies, if exists, a value en; such that g,(r; c%) be monotone of order p is obtained. In particular, it is found that %1=max {rp'(r)/p(r)}, %2=max {q0(r)}, cql = max {rp"(r)/p'(r)} and cq2 = max {ql(r)}, where qo(r) and q~(r) are simple combinations of the first few derivatives of p(r). Secondly, numerical calculations of the first few values c% in a Hartree-Fock framework for all ground-state atoms with nuclear charge Z < 54 are performed. In doing so, the pioneering work of Weinstein, Politzer and Srebrenik about the monotonically decreasing behavior of p(r) is considerably extended. Also, it is found that Hydrogen and Helium are the only two atoms having the functions p(r), -p'(r) and p"(r) with the property of convexity. Thirdly, it is analytically shown that the charge function r ~ p(r) with e > [(1 + 4Z2/I) 1/2-1]/2,

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Section: Charge Monotonicity Effects: Inequalities Among Radial Exsupporting

confidence: 75%

Charge monotonicity of atomic systems and radial expectation values

Angulo

Dehesa

1993

Z Phys D - Atoms, Molecules and Clusters

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“…For p → ∞ the variable x, defined in Eq. (18), satisfies x → (γ + 1) π 2 . As long as γ < 1 one finds that F (p), defined by Eq.…”

Section: Rényi Momentum Entropiesmentioning

confidence: 99%

“…Since localization (i.e., low uncertainty) means high values of the Fisher information measures, the counterpart of the Heisenberg or Shannon bound should be an upper bound on the product of position and momentum Fisher information measures. For a single particle under the influence of a central potential, Dehesa et al [17] have very recently reported a lower bound on the Heisenberg product [18][19][20][21] which can be directly related to the Fisher information.…”

Section: Introductionmentioning

confidence: 98%

Relativistic effects on information measures for hydrogen-like atoms

Katriel

Sen

2010

Journal of Computational and Applied Mathematics

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Dedicated to Professor Jesús S. Dehesa on the occasion of his 60th birthday. Keywords:Relativistic H atom Heisenberg uncertainty relation Shannon and Rényi entropies Fisher information measure Statistical complexity a b s t r a c t Position and momentum information measures are evaluated for the ground state of the relativistic hydrogen-like atoms. Consequences of the fact that the radial momentum operator is not self-adjoint are explicitly studied, exhibiting fundamental shortcomings of the conventional uncertainty measures in terms of the radial position and momentum variances. The Shannon and Rényi entropies, the Fisher information measure, as well as several related information measures, are considered as viable alternatives. Detailed results on the onset of relativistic effects for low nuclear charges, and on the extreme relativistic limit, are presented. The relativistic position density decays exponentially at large r, but is singular at the origin. Correspondingly, the momentum density decays as an inverse power of p. Both features yield divergent Rényi entropies away from a finite vicinity of the Shannon entropy. While the position space information measures can be evaluated analytically for both the nonrelativistic and the relativistic hydrogen atom, this is not the case for the relativistic momentum space. Some of the results allow interesting insight into the significance of recently evaluated Dirac-Fock vs. Hartree-Fock complexity measures for many-electron neutral atoms.

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“…Equation (1) has been successfully used to study uncertainty-like relationship in case of spherically symmetric potentials including radial position-momentum uncertainties in Klein-Gordon Hlike atoms [5][6][7][8][9][10]. Uncertainty in position space is known to provide a good measure of spatial delocalization [11] of a quantum particle.…”

Section: Introductionmentioning

confidence: 99%

Information entropy as a measure of tunneling and quantum confinement in a symmetric double‐well potential

Mukherjee

Roy

2015

Annalen der Physik

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Information entropic measures such as Fisher information, Shannon entropy, Onicescu energy and Onicescu Shannon entropy of a symmetric double-well potential are calculated in both position and momentum space. Eigenvalues and eigenvectors of this system are obtained through a variation-induced exact diagonalization procedure. The information entropy-based uncertainty relation is shown to be a better measure than conventional uncertainty product in interpreting purely quantum mechanical phenomena, such as, tunneling and quantum confinement in this case. Additionally, the phase-space description provides a semiclassical explanation for this feature. Total information entropy and phase-space area show similar behavior with increasing barrier height.

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Expectation values in one- and two-electron atomic systems

Cited by 41 publications

References 7 publications

Charge monotonicity of atomic systems and radial expectation values

Charge monotonicity of atomic systems and radial expectation values

Relativistic effects on information measures for hydrogen-like atoms

Information entropy as a measure of tunneling and quantum confinement in a symmetric double‐well potential

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