Generalized position-momentum uncertainty products: Inclusion of moments with negative order and application to atoms

Angulo, J. C.

doi:10.1103/physreva.83.062102

Cited by 9 publications

(11 citation statements)

References 26 publications

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“…First we derive the bounds based on position and momentum expectation values with positive order, and then the corresponding ones involving momentum expectation values with a negative order. These results extend, generalize, and/or improve similar results from various authors (see, e.g., [10,13,39,[59][60][61][62][63][64] and references therein).…”

Section: Heisenberg-like Uncertainty Relationssupporting

confidence: 91%

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Heisenberg-like and Fisher-information-based uncertainty relations forN-electrond-dimensional systems

et al. 2015

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Heisenberg-like and Fisher-information-based uncertainty relations which extend and generalize previous similar expressions are obtained for N -fermion d-dimensional systems. The contributions of both spatial and spin degrees of freedom are taken into account. The accuracy of some of these generalized spinned uncertainty-like relations is numerically examined for a large number of atomic and molecular systems.

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Section: Heisenberg-like Uncertainty Relationssupporting

confidence: 91%

“…Stronger uncertaintylike relations based either on moments of order other than 2 [6,9,59] or on some information-theoretic quantities have been developed. Among the latter ones, the entropic uncertainty relations based on the Shannon entropy and on the Rényi entropy are well known [74][75][76][77].…”

Section: Discussionmentioning

confidence: 99%

Heisenberg-like and Fisher-information-based uncertainty relations forN-electrond-dimensional systems

et al. 2015

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“…In particular, an interesting Rényi-based bound on the product r −1 −2 p 2 , has been recently published [42] , where the electron density has been averaged for spin and normalized to unity.…”

Section: Mininf-based Heisenberg-like Uncertainty Relationmentioning

confidence: 99%

Extremum-entrop y-based Heisenberg-like uncertainty relations

Toranzo

López-Rosa

Esquivel

et al. 2015

J. Phys. A: Math. Theor.

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In this work we use the extremization method of various information-theoretic measures (Fisher information, Shannon entropy, Tsallis entropy) for d-dimensional quantum systems, which complementary describe the spreading of the quantum states of natural systems.Under some given constraints, usually one or two radial expectation values, this variational method allows us to determine an extremum-entropy distribution, which is is the least-biased one to characterize the state among all those compatible with the known data. Then we use it, together with the spin-dependent uncertainty-like relations of Daubechies-Thakkar type, as a tool to obtain relationships between the position and momentum radial expectation values of the type r α k α p k ≥ f (k, α, q, N ), q = 2s + 1, for d-dimensional systems of N fermions with spin s. The resulting uncertainty-like products, which take into account both spatial and spin degrees of freedom of the fermionic constituents of the system, are shown to often improve the best corresponding relationships existing in the literature.

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“…Additionally, different uncertainty-like inequalities have been derived by using information-theoretical tools. 27,44 It is worth mentioning that some of these expectation values are physically relevant and/or experimentally accessible in three-dimensional N-electron atoms. Some examples are 36 Concerning the frequency moments and their corresponding Rényi entropies (see Sec.…”

Section: Particular Cases Of Physical Interestmentioning

confidence: 99%

Uncertainty inequalities among frequency moments and radial expectation values: Applications to atomic systems

Angulo

Bouvrie

Antolín

2012

Journal of Mathematical Physics

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Rigorous and universal bounds on frequency moments of one-particle densities in terms of radial expectation values in the conjugate space are obtained. The results, valid for any d-dimensional quantum-mechanical system, are derived by using Rényi-like position-momentum inequalities in an information-theoretical framework. Especially interesting are the upper bounds on the Dirac exchange and Thomas-Fermi kinetic energies, as well as the disequilibrium or self-similarity of both position and momentum distributions. A variety of bounds for these functionals in a given space are known, but most usually in terms of quantities defined within the same space. Very few results including a density functional on one space, and expectation values on the conjugate one, are found in the literature. A pioneering bound on the disequilibrium in terms of the kinetic energy is improved in this work. A numerical study of the aforementioned relationships is carried out for atomic systems in their ground state. Some results are given in terms of relevant physical quantities, including the kinetic and electron-nucleus attraction energies, the diamagnetic susceptibility and the height of the peak of the Compton profile, among others. C 2012 American Institute of Physics. [http://dx.are also known. Although these relationships are usually applied in three-dimensional systems (i.e., with vectors of three components r and p), all of them are valid for arbitrary dimensionality. 16,17 Such uncertainty relations are physically relevant, not only because of their importance in a theoretical quantum-mechanical framework, 18-20 but also in the development of quantum information and computation. 21,22 In this sense, the studies of entropic uncertainty relations 23 and their connection with entanglement 24 are also remarkable.The aim of this study is to present uncertainty inequalities, in the form of bounds on a frequency moment (quantity defined in Sec. II) in a given space (position or momentum) in terms of two radial expectation values in the conjugate space. Such inequalities can handle radial expectation values of positive or negative orders. The bounds provided here are of universal validity (i.e., for any d-dimensional quantum mechanical system). By way of example, we carry out a numerical study for selected inequalities of physical interest in atomic systems, and the results are interpreted taking into account that some radial expectation values for atomic densities, in both position and momentum spaces, are physically relevant and/or experimentally accessible.The paper is structured as follows: Section II is devoted to the definition of the one-particle densities from the wave function, as well as the main quantities we will deal with (radial expectation values, frequency moments, Rényi entropies). In Sec. III, the uncertainty relations associated to those quantities are provided, as universal bounds on frequency moments in terms of radial expectation values in the conjugate space. Particular cases of physical interest are detailed in Se...

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Generalized position-momentum uncertainty products: Inclusion of moments with negative order and application to atoms

Cited by 9 publications

References 26 publications

Heisenberg-like and Fisher-information-based uncertainty relations forN-electrond-dimensional systems

Heisenberg-like and Fisher-information-based uncertainty relations forN-electrond-dimensional systems

Extremum-entrop y-based Heisenberg-like uncertainty relations

Uncertainty inequalities among frequency moments and radial expectation values: Applications to atomic systems

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