Heisenberg and Entropic Uncertainty Measures for Large-Dimensional Harmonic Systems

Puertas‐Centeno, David; Toranzo, I. V.; Dehesa, J. S.

doi:10.3390/e19040164

Cited by 20 publications

(36 citation statements)

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“…Then, uncertainty measures of entropic character [26][27][28][29][30][31][32][33][34] have been considered; they are much more appropriate because, contrary to the Heisenberg-like ones, they do not depend on any specific point of the system. Recently these studies have been extended by calculating the dominant term of the Heisenberglike and Rényi-entropy-based uncertainty measures for both the D-dimensional hydrogenic and harmonic systems at the quassiclassical border in the two conjugated position and momentum spaces [35,36]. It was found that the Heisenberg-like and Rényi-entropy-based equality-type uncertainty relations for all of the D-dimensional harmonic oscillator states in the pseudoclassical (D → ∞) limit are the same as the corresponding ones for the hydrogenic systems, despite the so different character of the oscillator and Coulomb potentials.…”

Section: Introductionmentioning

confidence: 99%

Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems

Sobrino¹,

Puertas‐Centeno²,

Toranzo³

et al. 2017

J. Stat. Mech.

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In this work we find that not only the Heisenberg-like uncertainty products and the Rényi-entropy-based uncertainty sum have the same first-order values for all the quantum states of the D-dimensional hydrogenic and oscillator-like systems, respectively, in the pseudoclassical (D → ∞) limit but a similar phenomenon also happens for both the Fisherinformation-based uncertainty product and the Shannon-entropy-based uncertainty sum, as well as for the Crámer-Rao and Fisher-Shannon complexities. Moreover, we show that the LMC (López-Ruiz-Mancini-Calvet) and LMC-Rényi complexity measures capture the hydrogenic-harmonic difference in the high dimensional limit already at first order.

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Section: Introductionmentioning

confidence: 99%

Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems

Sobrino¹,

Puertas‐Centeno²,

Toranzo³

et al. 2017

J. Stat. Mech.

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“…These results extend and complement various efforts about the information entropies of harmonic systems. [11,28,29,32,[36][37][38]63,69,70,73,76,77,[84][85][86][87][88][89]…”

Section: Shannon Entropy Of High-dimensional Harmonic Systemsmentioning

confidence: 99%

“…∞ limit. Here again, to determine the dependence on the hyperquantum numbers, we have to go further in the asymptotical behavior (76) and (75) for the Laguerre and Gegenbauer functionals E 1L∞ À Á and EC ∞ , respectively, which requires a deeper research in approximation theory. These results extend and complement a number of results about the multidimensional hydrogenic systems.…”

Section: Shannon Entropy Of High-dimensional Hydrogenic Systemsmentioning

confidence: 99%

“…Presently, much attention is being centered around the D-dimensional hydrogenic and harmonic systems because of their enormous interest in general quantum mechanics [55][56][57][58][59][60] and many specific areas from quantum chromodynamics and elementary particle physics [61,62] to heat transport, [63,64] atomic and molecular physics, [65][66][67][68] and classical and quantum information. [26,28,[69][70][71][72][73][74][75][76][77][78] Even for these systems, the analytical evaluation of the Shannon entropy for their stationary states is a formidable task because it is a logarithmic integral functional of some special functions of applied mathematics and mathematical physics [30,[79][80][81] (eg, orthogonal polynomials, hyperspherical harmonics, and so on), which control the associated wave functions. This issue is much more difficult than the determination of other uncertainty measures like the Heisenberg-like and Fisher information ones, not only analytically but also numerically.…”

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confidence: 99%

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The Shannon entropy of high‐dimensional hydrogenic and harmonic systems

Dehesa

Belega

Toranzo

et al. 2019

Int J of Quantum Chemistry

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There exists an increasing interest on the dimensionality dependence of the entropic properties for the stationary states of the multidimensional quantum systems in order to contribute to its emergent informational representation, which extends and complements the standard energetic representation. Nowadays, this is specially so for high-dimensional systems as they have been recently shown to be very useful in both scientific and technological fields. In this work, the Shannon entropy of the discrete stationary states of the high-dimensional harmonic (ie, oscillator-like) and hydrogenic systems is analytically determined in terms of the dimensionality, the potential strength, and the state's hyperquantum numbers. We have used an informationtheoretic methodology based on the asymptotics of some entropy-like integral functionals of the orthogonal polynomials and hyperspherical harmonics which control the wave functions of the quantum states, when the polynomial parameter is very large; this is basically because such a parameter is a linear function of the system's dimensionality. Finally, it is shown that the Shannon entropy of the D-dimensional harmonic and hydrogenic systems has a logarithmic growth rate of the type D log D when D ! ∞. K E Y W O R D S high-dimensional harmonic systems, high-dimensional hydrogenic systems, high-dimensional quantum systems, logarithmic integrals of orthogonal polynomials-asymptotics, Shannon entropy of high-dimensional quantum systems 1 | INTRODUCTION Multidimensional quantum systems consisting of many particles are a major challenge in Quantum Chemistry and Physics, as their behavior can be determined only with immense computational power. The physical solutions of their associated wave equations, and consequently all the chemical and physical properties of these systems, crucially depend on the space dimensionality D. Scientists are trying to discover elegant notions and techniques to simplify the problem (see, eg, ref. [1]). The 1986 Nobel Prize in Chemistry winner Dudley Robert Herschbach et al have developed [2][3][4][5] a very useful strategy to study the atomic and molecular systems, the D-dimensional scaling method, where the D is the basic variable. The pseudoclassical or highdimensional (D ! ∞) limit is the starting point of this strategy. Indeed, this method requires to solve a finite many-electron problem in the (D ! ∞)-limit, and then perturbation theory in 1D is used to have an approximate result for the standard dimension (D = 3), obtaining at times a quantitative accuracy comparable to the self-consistent Hartree-Fock calculations. Most important here is that the electronic structure for the (D ! ∞)-limit is beguilingly simple and exactly computable for any atom and molecule. For finite D but very large, the electrons are confined to harmonic oscillations about the fixed positions attained in the (D ! ∞)-limit. Indeed, at this high-dimensional limit, the electrons of a many-electron system assume fixed positions relative to

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“…This linearization problem, which is equivalent to the evaluation of integrals of the product of three or more HOPs of the same type, is also called Clebsch-Gordan-type problem because its structure is similar to the Clebsch-Gordan series for spherical functions [8,9,40,41]. It arises in a wide range of mathematical and physical problems, from the generalized moment problems as stated by M. G. Krein [42][43][44], stochastic processes [45] and combinatorics [46][47][48][49], information entropies [20,[50][51][52][53][54] up to quantum physics. Indeed, numerous physical and chemical properties of a given complex system (atoms, molecules,...) are determined by these overlap integrals or Krein-like functionals of various HOPs, basically because the main route for the search of the solutions of the corresponding Schrödinger equation is their expansion as linear combinations of a known basis set of functions which are controlled by the HOPs (see e.g.…”

Section: Introductionmentioning

confidence: 99%

Linearization and Krein-like functionals of hypergeometric orthogonal polynomials

Dehesa

Moreno-Balcázar

Toranzo

2018

Journal of Mathematical Physics

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The Krein-like r-functionals of the hypergeometric orthogonal polynomials {pn(x)} with kernel of the form x s [ω(x)] β pm 1 (x) . . . pm r (x), being ω(x) the weight function on the interval ∆ ∈ R, are determined by means of the Srivastava linearization method. The particular 2-functionals, which are particularly relevant in quantum physics, are explicitly given in terms of the degrees and the characteristic parameters of the polynomials. They include the well-known power moments and the novel Krein-like moments. Moreover, various related types of exponential and logarithmic functionals are also investigated.

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Heisenberg and Entropic Uncertainty Measures for Large-Dimensional Harmonic Systems

Cited by 20 publications

References 87 publications

Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems

Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems

The Shannon entropy of high‐dimensional hydrogenic and harmonic systems

Linearization and Krein-like functionals of hypergeometric orthogonal polynomials

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