Spherical-Symmetry and Spin Effects on the Uncertainty Measures of Multidimensional Quantum Systems with Central Potentials

Dehesa, J. S.

doi:10.3390/e23050607

Cited by 5 publications

(3 citation statements)

References 163 publications

(235 reference statements)

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“…The most relevant property of the Rényi entropies ( 1 ) and ( 2 ) of general D -dimensional quantum systems is the entropic uncertainty relation

which is saturated by the Gaussian distributions. This relation was proved by Zozor, Portesi and Vignat [ 21 ] for arbitrary indices, extending the one-dimensional relation previously found by Bialynicki-Birula [ 20 ] and Zozor and Vignat [ 90 ] for conjugated indices (i.e., when

); see [ 17 , 91 ] for further details. See [ 94 , 95 ] for other related inequality relations.…”

Section: Rényi Entropies Of General and Central-potential Quantum Sys...supporting

confidence: 72%

“…In this section, we first describe the lower and upper bounds [ 48 , 89 ] on the Rényi entropies ( 1 ) and ( 2 ) of general multidimensional quantum systems in position and momentum spaces, and the corresponding entropic uncertainty relations [ 17 , 20 , 21 , 90 , 91 ]. Then, we show the improvement of these properties for central potentials, and we point out some open problems [ 89 , 92 , 93 ].…”

Section: Rényi Entropies Of General and Central-potential Quantum Sys...mentioning

confidence: 99%

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Rényi Entropies of Multidimensional Oscillator and Hydrogenic Systems with Applications to Highly Excited Rydberg States

Dehesa

2022

Entropy

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The various facets of the internal disorder of quantum systems can be described by means of the Rényi entropies of their single-particle probability density according to modern density functional theory and quantum information techniques. In this work, we first show the lower and upper bounds for the Rényi entropies of general and central-potential quantum systems, as well as the associated entropic uncertainty relations. Then, the Rényi entropies of multidimensional oscillator and hydrogenic-like systems are reviewed and explicitly determined for all bound stationary position and momentum states from first principles (i.e., in terms of the potential strength, the space dimensionality and the states’s hyperquantum numbers). This is possible because the associated wavefunctions can be expressed by means of hypergeometric orthogonal polynomials. Emphasis is placed on the most extreme, non-trivial cases corresponding to the highly excited Rydberg states, where the Rényi entropies can be amazingly obtained in a simple, compact, and transparent form. Powerful asymptotic approaches of approximation theory have been used when the polynomial’s degree or the weight-function parameter(s) of the Hermite, Laguerre, and Gegenbauer polynomials have large values. At present, these special states are being shown of increasing potential interest in quantum information and the associated quantum technologies, such as e.g., quantum key distribution, quantum computation, and quantum metrology.

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“…The most relevant property of the Rényi entropies ( 1 ) and ( 2 ) of general D -dimensional quantum systems is the entropic uncertainty relation

); see [ 17 , 91 ] for further details. See [ 94 , 95 ] for other related inequality relations.…”

Section: Rényi Entropies Of General and Central-potential Quantum Sys...supporting

confidence: 72%

Section: Rényi Entropies Of General and Central-potential Quantum Sys...mentioning

confidence: 99%

Rényi Entropies of Multidimensional Oscillator and Hydrogenic Systems with Applications to Highly Excited Rydberg States

Dehesa

2022

Entropy

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“…The interest in the asymptotics (λ ! ∞) of the Gegenbauer polynomial themselves and their algebraic norms has been a long-standing problem [35,35,61,62,80,100,[102][103][104][105] because of fundamental and quantum applications; this is basically because the Gegenbauer polynomials control the angular part of the quantum wavefunctions of central potentials in position space and the momentum wavefunctions of Coulomb systems (see e.g., the reviews [65,106,107].…”

Section: Parameter Asymptotics Formentioning

confidence: 99%

Parameter andqasymptotics of Lq‐norms of hypergeometric orthogonal polynomials

Sobrino

Dehesa

2022

Int J of Quantum Chemistry

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The three canonical families of the hypergeometric orthogonal polynomials in a continuous real variable (Hermite, Laguerre, and Jacobi) control the physical wavefunctions of the bound stationary states of a great number of quantum systems [Correction added after first online publication on 21 December, 2022. The sentence has been modified.]. The algebraic frakturLq‐norms of these polynomials describe many chemical, physical, and information theoretical properties of these systems, such as, for example, the kinetic and Weizsäcker energies, the position and momentum expectation values, the Rényi and Shannon entropies and the Cramér‐Rao, the Fisher‐Shannon and LMC measures of complexity. In this work, we examine review and solve the q‐asymptotics and the parameter asymptotics (i.e., when the weight function's parameter tends towards infinity) of the unweighted and weighted frakturLq‐norms for these orthogonal polynomials. This study has been motivated by the application of these algebraic norms to the energetic, entropic, and complexity‐like properties of the highly excited Rydberg and high‐dimensional pseudo‐classical states of harmonic (oscillator‐like) and Coulomb (hydrogenic) systems, and other quantum systems subject to central potentials of anharmonic type (such as, e.g., some molecular systems) [Correction added after first online publication on 21 December, 2022. Oscillatorlike has been changed to oscillator‐like.].

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Monotone complexity measures of multidimensional quantum systems with central potentials

Dehesa

2023

Journal of Mathematical Physics

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In this work, we explore the (inequality-type) properties of the monotone complexity-like measures of the internal complexity (disorder) of multidimensional non-relativistic electron systems subject to a central potential. Each measure quantifies the combined balance of two spreading facets of the electron density of the system. We show that the hyperspherical symmetry (i.e., the multidimensional spherical symmetry) of the potential allows Cramér–Rao, Fisher–Shannon, and Lopez-Ruiz, Mancini, Calbet–Rényi complexity measures to be expressed in terms of the space dimensionality and the hyperangular quantum numbers of the electron state. Upper bounds, mutual complexity relationships, and complexity-based uncertainty relations of position–momentum type are also found by means of the electronic hyperangular quantum numbers and, at times, the Heisenberg–Kennard relation. We use a methodology that includes a variational approach with a covariance matrix constraint and some algebraic linearization techniques of hyperspherical harmonics and Gegenbauer orthogonal polynomials.

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Spherical-Symmetry and Spin Effects on the Uncertainty Measures of Multidimensional Quantum Systems with Central Potentials

Cited by 5 publications

References 163 publications

Rényi Entropies of Multidimensional Oscillator and Hydrogenic Systems with Applications to Highly Excited Rydberg States

Rényi Entropies of Multidimensional Oscillator and Hydrogenic Systems with Applications to Highly Excited Rydberg States

Parameter andqasymptotics of Lq‐norms of hypergeometric orthogonal polynomials

Monotone complexity measures of multidimensional quantum systems with central potentials

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