Heisenberg-like and Fisher-information-based uncertainty relations for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>-electron<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>-dimensional systems

Toranzo, I. V.; López-Rosa, S.; Esquivel, Rodolfo O.; Dehesa, J. S.

doi:10.1103/physreva.91.062122

Cited by 10 publications

(16 citation statements)

References 74 publications

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“…Our results in Equation's (14) and (15) are the same as those obtained by means of the Fisher's information entropies [32] [33], by Stamp's principle [51] and by Shannon's entropy [26]. Our method is more straight forward and simpler.…”

Section: Isotropic Harmonic Oscillator In N-dimensionssupporting

confidence: 72%

“…Such variances do not necessarily exist, and if they do, they describe the quantum probability distribution relative to a specific point of the probability domain. Therefore, various alternative formulations have been suggested by the use of information-theoretic uncertainty measures like the Shannon entropy [26] [27], Renyi entropies [28] [29], Tsallis entropies [30], entropic moments [31] [32] and Fisher information [32]- [36]. During the past years, the generalization of three dimensional quantum problems to higher space dimensions receives a considerable development in theoretical and mathematical physics.…”

Section: Introductionmentioning

confidence: 99%

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Uncertainty Relations for Some Central Potentials in N-Dimensional Space

Al-Jaber¹

2016

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We study the uncertainty relation for three quantum systems in the N-dimensional space by using the virial theorem (VT). It is shown that this relation depends on the energy spectrum of the system as well as on the space dimension N. It is pointed out that the form of lower bound of the inequality, which is governed by the ground state, depends on the system and on the space dimension N. A comparison between our result for the lower bound and recent results, based on information-theoretic approach, is pointed out. We examine and analyze these derived uncertainties for different angular momenta with a special attention made for the large N limit.

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Section: Isotropic Harmonic Oscillator In N-dimensionssupporting

confidence: 72%

Section: Introductionmentioning

confidence: 99%

Uncertainty Relations for Some Central Potentials in N-Dimensional Space

Al-Jaber¹

2016

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“…Let us also mention that a number of authors have published some rigorous d-dimensional bounds of the same type [3,30] with much less accuracy. Furthermore, let us point out that the inclusion of the spin s in the lower bound (3) for the expected value of p 2 was considered by Hundertmark [30] and applied to obtain uncertainty-like relations in [31]; the extension to p k has been recently used [26] in a similar sense.…”

Section: Has Been Rigorously Proved Bymentioning

confidence: 99%

“…[ρ], the Patterson function of x-ray crystallography W 3 [ρ], etc. In fact, the energetic quantities of the many-electron systems can be expressed in terms of these entropic moments as already pointed out [23,24] in the framework of the density theory functional [7] Recently, it has been argued that the the momentum expectation values and the position entropic moments for d-dimensional systems of N fermions with spin s fulfil the following semiclassical spin-dependent uncertainty-like relations of Daubechies-Thakkar type [9,11,25] (see also [26]):…”

mentioning

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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“…In quantum mechanics, one of its main applications is due to the fact that it enters into the expression for the kinetic energy of the manyparticle system [57] and in this way establishes the link between, for example, density functional methods and information theory. Due to its importance, a research on the Fisher information and its relation to the quantum entropy is a very vigorous one and it discovers a lot of the new results [40,54,[58][59][60][61][62][63][64][65][66][67]. For our subsequent analysis, we provide here easily derivable expressions for the position and momentum Fisher informations of the bound state of the negative Robin wall:…”

Section: Introductionmentioning

confidence: 99%

Theory of the Robin quantum wall in a linear potential. I. Energy spectrum, polarization and quantum‐information measures

Olendski

2016

Annalen der Physik

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Information-theoretical concepts are employed for the analysis of the interplay between a transverse electric field E applied to a one-dimensional surface and Robin boundary condition (BC), which with the help of the extrapolation length Λ zeroes at the interface a linear combination of the quantum mechanical wave function and its spatial derivative, and its influence on the properties of the structure. For doing this, exact analytical solutions of the corresponding Schrödinger equation are derived and used for calculating energies, dipole moments, position Sx and momentum S k quantum information entropies and their Fisher information Ix and I k and Onicescu information energies Ox and O k counterparts. It is shown that the weak (strong) electric field changes the Robin wall into the Dirichlet, Λ = 0 (Neumann, Λ = ∞), surface. This transformation of the energy spectrum and associated waveforms in the growing field defines an evolution of the quantum-information measures; for example, it is proved that for the Dirichlet and Neumann BCs the position (momentum) quantum information entropy varies as a positive (negative) natural logarithm of the electric intensity what results in their field-independent sum Sx + S k . Analogously, at Λ = 0 and Λ = ∞ the position and momentum Fisher informations (Onicescu energies) depend on the applied voltage as E 2/3 (E 1/3 ) and its inverse, respectively, leading to the field-independent product IxI k (OxO k ). Peculiarities of their transformations at the finite nonzero Λ are discussed and similarities and differences between the three quantum-information measures in the electric field are highlighted with the special attention being paid to the configuration with the negative extrapolation length.

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

Cited by 10 publications

References 74 publications

Uncertainty Relations for Some Central Potentials in N-Dimensional Space

Uncertainty Relations for Some Central Potentials in N-Dimensional Space

Extremum-entrop y-based Heisenberg-like uncertainty relations

Theory of the Robin quantum wall in a linear potential. I. Energy spectrum, polarization and quantum‐information measures

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