Position and momentum information entropies of the<i>D</i>-dimensional harmonic oscillator and hydrogen atom

Yáñez, R. J.; Assche, Walter Van; Dehesa, J. S.

doi:10.1103/physreva.50.3065

Cited by 258 publications

(313 citation statements)

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“…The calculation of position and momentum entropies for physically interesting quantum states has been the subject of considerable effort in recent years. It has been shown 19,20 that, for the stationary states of many important systems, such as D-dimensional harmonic oscillator and hydrogen atom, the entropies can be expressed in terms of the integrals…”

Section: Introductionmentioning

confidence: 99%

Logarithmic potential of Hermite polynomials and information entropies of the harmonic oscillator eigenstates

Sánchez-Ruiz¹

1997

Journal of Mathematical Physics

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The problem of calculating the information entropy in both position and momentum spaces for the nth stationary state of the one-dimensional quantum harmonic oscillator reduces to the evaluation of the logarithmic potential V n (t) ϭϪ͐ Ϫϱdx at the zeros of the Hermite polynomial H n (x). Here, a closed analytical expression for V n (t) is obtained, which in turn yields an exact analytical expression for the entropies when the exact location of the zeros of H n (x) is known. An inequality for the values of V n (t) at the zeros of H n (x) is conjectured, which leads to a new, nonvariational, upper bound for the entropies. Finally, the exact formula for V n (t) is written in an alternative way, which allows the entropies to be expressed in terms of the even-order spectral moments of the Hermite polynomials. The asymptotic (nӷ1) limit of this alternative expression for the entropies is discussed, and the conjectured upper bound for the entropies is proved to be asymptotically valid.

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Section: Introductionmentioning

confidence: 99%

Logarithmic potential of Hermite polynomials and information entropies of the harmonic oscillator eigenstates

Sánchez-Ruiz¹

1997

Journal of Mathematical Physics

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“…As has already been pointed out in relation to (2), the existence of a non-zero asymptotic difference between classical and quantum entropies in equations (17) and (21) can be interpreted in terms of the so-called configuration form of Bohr's correspondence principle (4), with the correction term f (q) arising from the fact that Rényi entropies can be written as the expected value of a quantity that depends explicitly on the probability distribution itself (see equation (12)). It is also worth noting that all the results given in this paper for Rényi entropies may be written in terms of Tsallis entropies using (14), although then they take a more cumbersome and hence less appealing form.…”

Section: Conclusion and Open Problemsmentioning

confidence: 54%

“…Accordingly, there has been a growing interest in the calculation of S Q for physically interesting quantum states. However, the exact calculation of S Q is a very difficult mathematical problem, even for simple systems as the harmonic oscillator and hydrogen atom [2], which has attracted interest to its approximate calculation, specially for very excited or Rydberg stationary states [3,4].…”

Section: S = − P (X) Ln P (X) DX = − Ln P (X)mentioning

confidence: 99%

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Asymptotic formulae for the quantum Rényi entropies of position: application to the infinite well

Sánchez-Ruiz

1999

J. Phys. A: Math. Gen.

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Abstract. General asymptotic formulae are derived by means of the WKB approximation for the continuous and discrete Rényi entropies of position of one-dimensional quantum systems in energy eigenstates, in terms of the corresponding entropies for a microcanonical ensemble of analogous classical systems. These results are checked in the simplest particular case of the infinite potential well, where the asymptotic formula for continuous entropies holds as an exact identity. For the discrete entropies, analytical expressions are obtained from which the asymptotic formulae given for the limiting cases of large and small measurement resolution can both be verified.

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“…The Shannon entropy in the position space, and the so-called Leipnik entropy, which coincides with the sum S q + S p , were investigated numerically for excited states of various quantum systems [23][24][25]. The typical observation is that the entropy values, evaluated for the ground state probability densities, are minimal.…”

Section: Shannon Entropy In Quantum Systemsmentioning

confidence: 99%

Information Functionals and the Notion of (Un)Certainty: Random Matrix Theory - Inspired Case

Garbaczewski

2007

Acta Phys. Pol. A

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Information functionals allow one to quantify the degree of randomness of a given probability distribution, either absolutely (through min/max entropy principles) or relative to a prescribed reference one. Our primary aim is to analyze the "minimum information" assumption, which is a classic concept (R. Balian, 1968) in the random matrix theory. We put special emphasis on generic level (eigenvalue) spacing distributions and the degree of their randomness, or alternatively -information/organization deficit. MotivationThe statistical theory of random-matrix spectra (RMT) [1,2] provides an ideal playground to test workings of the Shannon and Kullback-Leibler (K-L) entropies in diverse contexts. That pertains to a direct analysis of spectral data for complex quantum systems (semiclasically chaotic case included), but as well to the statistics of Gaussian matrix ensembles and random-matrix diffusion processes. Dyson's interacting Brownian motion model can be interpreted as a non--equilibrium dynamical process, whose asymptotic distribution is related to the thermodynamical equilibrium state of a Coulomb gas (RMT as equilibrium statistical mechanics). Ultimately one may pass to probability densities inferred from the ground state(s) of singular Calogero-type quantum systems: Shannon and K-L entropies prove to be proper tools in the quantum case as well.Before embarking on these issues, let us indicate that there are ambiguities involved in the very concept of information and (un)certainty. To stay on a solid ground [3-6], we must accept a specific lore of semantic games, where baffling synonyms quite often appear and their specific meaning is under scrutiny. Examples are: information vs. entropy notions, (un)certainty and randomness vs. information deficit, entropic measures of surprise vs. information functionals,

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Position and momentum information entropies of theD-dimensional harmonic oscillator and hydrogen atom

Cited by 258 publications

References 19 publications

Logarithmic potential of Hermite polynomials and information entropies of the harmonic oscillator eigenstates

Logarithmic potential of Hermite polynomials and information entropies of the harmonic oscillator eigenstates

Asymptotic formulae for the quantum Rényi entropies of position: application to the infinite well

Information Functionals and the Notion of (Un)Certainty: Random Matrix Theory - Inspired Case

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