High Dimensional Atomic States of Hydrogenic Type: Heisenberg-like and Entropic Uncertainty Measures

This work has been partially supported by the Grant PID2020-113390GB-I00 of the Agencia Estatal de Investigacion (Spain)) and the European Regional Development Fund (FEDER), and the Grant FQM-207 of the Agencia de Innovacion y Desarrollo de Andalucia.High dimensional atomic states play a relevant role in a broad range of quantum fields, ranging from atomic and molecular physics to quantum technologies. The D-dimensional hydrogenic system (i.e., a negatively-charged particle moving around a positively charged core under a Coulomblike potential) is the main prototype of the physics of multidimensional quantum systems. In this work, we review the leading terms of the Heisenberg-like (radial expectation values) and entropy-like (Rényi, Shannon) uncertainty measures of this system at the limit of high D. They are given in a simple compact way in terms of the space dimensionality, the Coulomb strength and the state’s hyperquantum numbers. The associated multidimensional position–momentum uncertainty relations are also revised and compared with those of other relevant systems.Agencia Estatal de Investigacion (Spain)) PID2020-113390GB-I00European Commission FQM-207Agencia de Innovacion y Desarrollo de Andalucia FQM-20

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

Funding: Research partially supported by the grants P20-00082 (Junta de Andalucía), PID2020- 113390GB-I00 (Agencia Estatal de Investigación (Spain), the European Regional Development Fund (FEDER)), and the Grant FQM-207 of the Agencia de Innovación y Desarrollo de Andalucía.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.Grant P20-00082 (Junta de Andalucía)PID2020- 113390GB-I00 (Agencia Estatal de Investigación (Spain)European Regional Development Fund (FEDER)Grant FQM-207 of the Agencia de Innovación y Desarrollo de Andalucí

Fisher information of special functions and second-order differential equations

We investigate a basic question of information theory, namely the evaluation of the Fisher information and the relative Fisher information with respect to a nonnegative function, for the probability distributions obtained by squaring the special functions of mathematical physics which are solutions of second-order differential equations. Emphasis is made in the Nikiforov-Uvarov hypergeometric-type functions. We obtain explicit expressions for these information-theoretic properties via the expectation values of the coefficients of the differential equation. We illustrate our approach for various special functions of physico-mathematical interest

Entropy-Like Properties and L-q-Norms of Hypergeometric Orthogonal Polynomials: Degree Asymptotics

This work was partially supported by the Agencia Estatal de Investigacion (Spain) and the European Regional Development Fund (FEDER) under the grant PID2020-113390GB-I00.In this work, the spread of hypergeometric orthogonal polynomials (HOPs) along their orthogonality interval is examined by means of the main entropy-like measures of their associated Rakhmanov’s probability density—so, far beyond the standard deviation and its generalizations, the ordinary moments. The Fisher information, the Rényi and Shannon entropies, and their corresponding spreading lengths are analytically expressed in terms of the degree and the parameter(s) of the orthogonality weight function. These entropic quantities are closely related to the gradient functional (Fisher) and the Lq-norms (Rényi, Shannon) of the polynomials. In addition, the degree asymptotics for these entropy-like functionals of the three canonical families of HPOs (i.e., Hermite, Laguerre, and Jacobi polynomials) are given and briefly discussed. Finally, a number of open related issues are identified whose solutions are both physico-mathematically and computationally relevant.Agencia Estatal de Investigacion (Spain)European Commission PID2020-113390GB-I0

Algebraic Lq-norms and complexity-like properties of Jacobi polynomials: Degree and parameter asymptotics

Agencia Andaluza del Conocimiento of the Junta de Andalucia (Spain), Grant/Award Number: PY20-00082; European Regional Development Fund (FEDER), Grant/Award Number: PID2020-113390GB-I00; Agencia Estatal de Investigacion (Spain), Grant/Award Number: FIS2017-89349P; Basque Government and UPV/EHU, Grant/Award Number: IT1249-19The Jacobi polynomials Poa,ss THORN n oxTHORN conform the canonical family of hypergeometric orthogonal polynomials (HOPs) with the two-parameter weight function o1 xTHORNao1thornxTHORN ss, a,ss > 1, on the interval 1/21, thorn1. The spreading of its associated probability density (i.e., the Rakhmanov density) over the support interval has been quantified, beyond the dispersion measures (moments around the origin, variance), by the algebraic Lq-norms (Shannon and Renyi entropies) and the monotonic complexity-like measures of Cramer-Rao, Fisher-Shannon, and LMC (LopezRuiz, Mancini, and Calbet) types. These quantities, however, have been often determined in an analytically highbrow, non-handy way; specially when the degree or the parameters oa, ss THORN are large. In this work, we determine in a simple, compact form the leading term of the entropic and complexity-like properties of the Jacobi polynomials in the two extreme situations: (n!8; fixed a, ss) and (a!8; fixed n, ss). These two asymptotics are relevant per se and because they control the physical entropy and complexity measures of the high energy (Rydberg) and high dimensional (pseudoclassical) states of many exactly, conditional exactly, and quasi-exactly solvable quantum- mechanical potentials which model numerous atomic and molecular systems.Junta de Andalucia PY20-00082European Commission PID2020-113390GB-I00Agencia Estatal de Investigacion (Spain) FIS2017-89349PBasque Government IT1249-19UPV/EHU IT1249-1

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أصدر المجلس الأعلى للتربية العدد 6 من دفاتر المجلس و هي، دفاتر تجمع وقائع الملتقيات و الأيام الدراسية و الندوات التي يشرف على تنظيمها المجلس الأعلى للتربية. و قد خصص هذا العدد لجمع أعمال الملتقى الذي عقده المجلس حول التعليم و التكوين في مرحلة ما بعد الأساسي أيام 8 إلى 10 ديسمبر 1998، بقصر الأمم بنادي الصنوبر، بالجزائر. و قد توخى الملتقى أهدافا تمثلت في : حصر إشكاليات التعليم و التكوين في مرحلة ما بعد الأساسي، من خلال تشخيص الوضعية الحالية للتعليم. الإطلاع على بعض التجارب الإقليمية و العالمية ..