Entropic uncertainty measures for large dimensional hydrogenic systems (original) (raw)

Exact Rényi entropies of D-dimensional harmonic systems

The European Physical Journal Special Topics

The determination of the uncertainty measures of multidimensional quantum systems is a relevant issue per se and because these measures, which are functionals of the single-particle probability density of the systems, describe numerous fundamental and experimentally accessible physical quantities. However, it is a formidable task (not yet solved, except possibly for the ground and a few lowest-lying energetic states) even for the small bunch of elementary quantum potentials which are used to approximate the mean-field potential of the physical systems. Recently, the dominant term of the Heisenberg and Rényi measures of the multidimensional harmonic system (i.e., a particle moving under the action of a D-dimensional quadratic potential, D > 1) has been analytically calculated in the high-energy (i.e., Rydberg) and the high-dimensional (i.e., pseudoclassical) limits. In this work we determine the exact values of the Rényi uncertainty measures of the D-dimensional harmonic system for all ground and excited quantum states directly in terms of D, the potential strength and the hyperquantum numbers.

Information-entropic measures in free and confined hydrogen atom

International Journal of Quantum Chemistry

Shannon entropy (S), Rényi entropy (R), Tsallis entropy (T), Fisher information (I) and Onicescu energy (E) have been explored extensively in both free H atom (FHA) and confined H atom (CHA). For a given quantum state, accurate results are presented by employing respective exact analytical wave functions in r space. The p-space wave functions are generated from respective Fourier transforms−for FHA these can be expressed analytically in terms of Gegenbauer polynomials, whereas in CHA these are computed numerically. Exact mathematical expressions of R α r , R β p , T α r , T β p , E r , E p are derived for circular states of a FHA. Pilot calculations are done taking order of entropic moments (α, β) as (3 5 , 3) in r and p spaces. A detailed, systematic analysis is performed for both FHA and CHA with respect to state indices n, l, and with confinement radius (r c) for the latter. In a CHA, at small r c , kinetic energy increases, whereas S r , R α r decrease with growth of n, signifying greater localization in high-lying states. At moderate r c , there exists an interplay between two mutually opposing factors: (i) radial confinement (localization) and (ii) accumulation of radial nodes with growth of n (delocalization). Most of these results are reported here for the first time, revealing many new interesting features. Comparison with literature results, wherever possible, offers excellent agreement.

Entropic uncertainty relations in multidimensional position and momentum spaces

Physical Review A, 2011

Commutator-based entropic uncertainty relations in multidimensional position and momentum spaces are derived, twofold generalizing previous entropic uncertainty relations for one-mode states. The lower bound in the new relation is optimal, and the new entropic uncertainty relation implies the famous variance-based uncertainty principle for multimode states. The article concludes with an open conjecture.

Information-entropic measures for non-zero l states of confined hydrogen-like ions

The European Physical Journal D

Rényi entropy (R), Tsallis entropy (T), Shannon entropy (S), and Onicescu energy (E) are studied in a spherically confined H atom (CHA), in conjugate space, with special emphasis on non-zero l states. This work is a continuation of our recently published work [1]. Representative calculations are done by employing exact analytical wave functions in r space. Accurate p space-wave functions are generated numerically by performing Fourier transform on respective r-space counterparts. Further, these are extended for H-isoelectronic series by applying the scaling relations. R, T are evaluated by choosing the order of entropic moments (α, β) as (3 5 , 3) in r and p spaces. Detailed, systematic results of all these measures with respect to variations of confinement radius r c are offered here for arbitrary n, l quantum numbers. For a given n, at small r c , R α r , T α r , S r collapse with rise of l, attain a minimum, then again grow up. Growth in r c shifts the point of inflection towards higher l values. An increase in Z enhances localization of a particular state. Several other new interesting inferences are uncovered. Comparison with literature results (available only for S in 2p, 3d states), offers excellent agreement.

Relativistic effects on information measures for hydrogen-like atoms

Journal of computational and applied mathematics, 2010

Position and momentum information measures are evaluated for the ground state of the relativistic hydrogen-like atoms. Consequences of the fact that the radial momentum operator is not self-adjoint are explicitly studied, exhibiting fundamental shortcomings of the conventional uncertainty measures in terms of the radial position and momentum variances. The Shannon and Rényi entropies, the Fisher information measure, as well as several related information measures, are considered as viable alternatives. Detailed results on the onset of relativistic effects for low nuclear charges, and on the extreme relativistic limit, are presented. The relativistic position density decays exponentially at large r, but is singular at the origin. Correspondingly, the momentum density decays as an inverse power of p. Both features yield divergent Rényi entropies away from a finite vicinity of the Shannon entropy. While the position space information measures can be evaluated analytically for both the nonrelativistic and the relativistic hydrogen atom, this is not the case for the relativistic momentum space. Some of the results allow interesting insight into the significance of recently evaluated Dirac-Fock vs. Hartree-Fock complexity measures for many-electron neutral atoms. * Electronic address: sensc@uohyd.ernet.in † Electronic address: jkatriel@tx.technion.ac.il

Entropic Uncertainty Relations in Quantum Physics

Statistical Complexity, 2011

Uncertainty relations have become the trademark of quantum theory since they were formulated by Bohr and Heisenberg. This review covers various generalizations and extensions of the uncertainty relations in quantum theory that involve the Rényi and the Shannon entropies. The advantages of these entropic uncertainty relations are pointed out and their more direct connection to the observed phenomena is emphasized. Several remaining open problems are mentioned.

Momentum and uncertainty relations in the entropic approach to quantum theory

2012

In the Entropic Dynamics (ED) approach to quantum theory the particles have well-defined positions but since they follow non differentiable Brownian trajectories they cannot be assigned an instantaneous momentum. Nevertheless, four different notions of momentum can be usefully introduced. We derive relations among them and the corresponding uncertainty relations. The main conclusion is that momentum is a statistical concept: in ED the momenta are not properties of the particles; they are attributes of the probability distributions.

General entropic uncertainty relations for NNN-level systems

2013

We revisit entropic formulations of the uncertainty principle for an arbitrary pair of quantum observables in N -dimensional Hilbert space. Generalized entropies, including Shannon, Rényi and Tsallis' ones among others, are used as uncertainty measures associated with the distribution probabilities corresponding to the outcomes of the observables. We obtain a lower bound for the sum of generalized entropies for any couple of (positive) entropic indices. The bound depends on the overlap c between the observables' eigenbases, and is valid for pure states as well as for mixed states. Our approach is inspired by that of de Vicente and Sánchez-Ruiz [Phys. Rev. A 77, 04110 (2008)] and consists in a minimization of the entropy sum subject to the Landau-Pollak inequality that links the maximum probabilities for both observables. We solve the constrained optimization problem in a geometrical way and furthermore we overcome the Hölder conjugacy constraint imposed on the entropic indices by Riesz-Thorin theorem generally used for when dealing with entropic formulations of the uncertainty principle. We show that (i) for given c ≥ 1 √ 2 , the bound obtained is optimal; and that, in the case of Rényi entropies, (ii) our bound improves Deutsch's one, but (iii) Maassen-Uffink's bound prevails when c ≤ 1 2 . Finally, we compare our bound with known previous results in the cases of Rényi and Tsallis entropies. arXiv:1311.5602v1 [quant-ph]

Formulation of the uncertainty relations in terms of the Rényi entropies

Physical Review A, 2006

Quantum-mechanical uncertainty relations for position and momentum are expressed in the form of inequalities involving the Rényi entropies. The proof of these inequalities requires the use of the exact expression for the ͑p , q͒-norm of the Fourier transformation derived by Babenko and Beckner. Analogous uncertainty relations are derived for angle and angular momentum and also for a pair of complementary observables in N-level systems. All these uncertainty relations become more attractive when expressed in terms of the symmetrized Rényi entropies.

Tight entropic uncertainty relations for systems with dimension three to five

We consider two (natural) families of observables O_k for systems with dimension d = 3, 4, 5: the spin observables S_x , S_y and S_z , and the observables that have mutually unbiased bases as eigenstates. We derive tight entropic uncertainty relations for these families, in the form Sum_k H(O_k ) > α_d , where H(O_k) is the Shannon entropy of the measurement outcomes of O_k and α_d is a constant. We show that most of our bounds are stronger than previously known ones. We also give the form of the states that attain these inequalities.