zbMATH — the first resource for mathematics

Entropic uncertainty measures for large dimensional hydrogenic systems. (English) Zbl 1373.81430
Summary: The entropic moments of the probability density of a quantum system in position and momentum spaces describe not only some fundamental and/or experimentally accessible quantities of the system but also the entropic uncertainty measures of Rényi type, which allow one to find the most relevant mathematical formalizations of the position-momentum Heisenberg’s uncertainty principle, the entropic uncertainty relations. It is known that the solution of difficult three-dimensional problems can be very well approximated by a series development in \(1/D\) in similar systems with a non-standard dimensionality \(D\); moreover, several physical quantities of numerous atomic and molecular systems have been numerically shown to have values in the large-\(D\) limit comparable to the corresponding ones provided by the three-dimensional numerical self-consistent field methods. The \(D\)-dimensional hydrogenic atom is the main prototype of the physics of multidimensional many-electron systems. In this work, we rigorously determine the leading term of the Rényi entropies of the \(D\)-dimensional hydrogenic atom at the limit of large \(D\). As a byproduct, we show that our results saturate the known position-momentum Rényi-entropy-based uncertainty relations.
©2017 American Institute of Physics

81V45 Atomic physics
81V55 Molecular physics
94A17 Measures of information, entropy
62J10 Analysis of variance and covariance (ANOVA)
Full Text: DOI
[1] Witten, E., Quarks, atoms, and the 1/N expansion, Phys. Today, 33, 7, 38-43, (1980)
[2] Chatterjee, A., Large-N expansions in quantum mechanics, atomic physics and some O(N) invariant systems, Phys. Rep., 186, 249, (1990)
[3] Herschbach, D. R.; Avery, J.; Goscinski, O., Dimensional Scaling in Chemical Physics, (1993), Kluwer Academic Publishers: Kluwer Academic Publishers, London
[4] Tsipis, C. T.; Popov, V. S.; Herschbach, D. R.; Avery, J. S., New Methods in Quantum Theory, (1996), Kluwer Academic Publishers: Kluwer Academic Publishers, Dordrecht
[5] Herschbach, D. R., Dimensional interpolation for two-electron atoms, J. Chem. Phys., 84, 838, (1986)
[6] Herschbach, D. R., Fifty years in physical chemistry: Homage to mentors, methods, and molecules, Annu. Rev. Phys. Chem., 51, 1-39, (2000)
[7] Yaffe, L. G., Large N limits as classical mechanics, Rev. Mod. Phys., 54, 407, (1982)
[8] Yaffe, L. G., Large-N quantum mechanics and classical limits, Phys. Today, 36, 8, 50, (1983)
[9] Herschbach, D. R., Dimensional scaling and renormalization, Int. J. Quantum Chem., 57, 295, (1996)
[10] Yáñez, R. J.; Van Assche, W.; Dehesa, J. S., Position and momentum information entropies of the D-dimensional harmonic oscillator and hydrogen atom, Phys. Rev. A, 50, 4, 3065, (1994)
[11] Nieto, M. M., Hydrogen atom and relativistic pi-mesic atom in N-space dimensions, Am. J. Phys., 47, 12, 1067, (1979)
[12] Dehesa, J. S.; López-Rosa, S.; Martínez-Finkelshtein, A.; Yáñez, R. J., Information theory of D-dimensional hydrogenic systems: Application to circular and Rydberg states, Int. J. Quantum Chem., 110, 1529-1548, (2010)
[13] Pasternack, S., On the mean value of rs for Keplerian systems, Proc. Natl. Acad. Sci. U. S. A., 23, 91, (1937) · JFM 63.0321.04
[14] Ray, A.; Mahata, K.; Ray, P. P., Moments of probability distribution, wave functions, and their derivatives at the origin of N-dimensional central potentials, Am. J. Phys., 56, 462, (1988)
[15] Drake, G. W. F.; Swainson, R. A., Expectation values of rP for arbitrary hydrogenic states, Phys. Rev. A, 42, 1123, (1990)
[16] Andrae, D., Recursive evaluation of expectation values for arbitrary states of the relativistic one-electron atom, J. Phys. B: At., Mol. Opt. Phys., 30, 4435, (1997)
[17] Tarasov, V. F., Exact numerical values of diagonal matrix elements \(\langle r^k \rangle_{n l}\), AS \(n \leq 8\) and \(- 7 \leq k \leq 4\), and the symmetry of Appell’s function F_2(1,1), Int. J. Mod. Phys. B, 18, 3177-3184, (2004) · Zbl 1158.81331
[18] Guerrero, A.; Sánchez-Moreno, P.; Dehesa, J. S., Upper bounds on quantum uncertainty products and complexity measures, Phys. Rev. A, 84, 042105, (2011)
[19] Hey, J. D., On the momentum representation of hydrogenic wave functions: Some properties and an application, Am. J. Phys., 61, 28, (1993)
[20] van Assche, W.; Yáñez, R. J.; González-Férez, R.; Dehesa, J. S., Functionals of Gegenbauer polynomials and D-dimensional hydrogenic momentum expectation values, J. Math. Phys., 41, 6600, (2000) · Zbl 0977.33006
[21] Aptekarev, A. I.; Dehesa, J. S.; Martínez-Finkelshtein, A.; Yáñez, R. J., Quantum expectation values of D-dimensional Rydberg hydrogenic states by use of Laguerre and Gegenbauer asymptotics, J. Phys. A: Math. Theor., 43, 145204, (2010) · Zbl 1188.81183
[22] Toranzo, I. V.; Martínez-Finkelshtein, A.; Dehesa, J. S., Heisenberg-like uncertainty measures for D-dimensional hydrogenic systems at large D, J. Math. Phys., 57, 082109, (2016) · Zbl 1344.81113
[23] Buyarov, V.; Dehesa, J. S.; Martínez-Finkelshtein, A.; Sánchez-Lara, J., Computation of the entropy of polynomials orthogonal on an interval, SIAM J. Sci. Comput., 26, 2, 488-509, (2004) · Zbl 1082.33004
[24] Toranzo, I. V.; Dehesa, J. S., Rényi, Shannon and Tsallis entropies of Rydberg hydrogenic systems, Europhys. Lett., 113, 48003, (2016)
[25] Toranzo, I. V.; Puertas-Centeno, D.; Dehesa, J. S., Entropic properties of D-dimensional Rydberg systems, Physica A, 462, 1197, (2016) · Zbl 1400.82302
[26] Aptekarev, A. I.; Tulyakov, D. N.; Toranzo, I. V.; Dehesa, J. S., Rényi entropies of the highly-excited states of multidimensional harmonic oscillators by use of strong Laguerre asymptotics, Eur. Phys. J. B, 89, 85, (2016)
[27] Rényi, A.; Neyman, J., On measures of entropy and information, Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, 547-561, (1961), University of California Press: University of California Press, Berkeley
[28] Shannon, C. E., A mathematical theory of communication, Bell Syst. Tech. J., 27, 3, 379, (1948) · Zbl 1154.94303
[29] Aczel, J.; Daroczy, Z., On Measures of Information and Their Characterizations, (1975), Academic Press: Academic Press, New York · Zbl 0345.94022
[30] Romera, E.; Angulo, J. C.; Dehesa, J. S., The Hausdorff entropic moment problem, J. Math. Phys., 42, 2309, (2001), 10.1063/1.1360711; Romera, E.; Angulo, J. C.; Dehesa, J. S., The Hausdorff entropic moment problem, J. Math. Phys., 42, 2309, (2001); Erratum 44, 2354 (2003)., 10.1063/1.1360711; · Zbl 1009.44003
[31] Tsallis, C., Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys., 52, 479, (1988) · Zbl 1082.82501
[32] Leonenko, N.; Pronzato, L.; Savani, V., A class of Rényi information estimator for multi-dimensional densities, Ann. Stat., 40, 4, 2153-2182, (2008) · Zbl 1205.94053
[33] Jizba, P.; Arimitsu, T., The world according to Rényi: Thermodynamics of multifractal systems, Ann. Phys., 312, 17, (2004) · Zbl 1044.82001
[34] Dehesa, J. S.; López-Rosa, S.; Manzano, D.; Sen, K. D., Entropy and complexity analysis of d-dimension at quantum systems, Statistical Complexities: Application to Electronic Structure, (2012), Springer: Springer, Berlin
[35] Bialynicki-Birula, I.; Rudnicki, L.; Sen, K. D., Entropic uncertainty relations in quantum physics, Statistical Complexities: Application to Electronic Structure, (2011), Springer: Springer, Berlin
[36] Jizba, P.; Dunningham, J. A.; Joo, J., Role of information theoretic uncertainty relations in quantum theory, Ann. Phys., 355, 87, (2015) · Zbl 1343.81050
[37] Hall, M. J. W., Universal geometric approach to uncertainty, entropy, and information, Phys. Rev. A, 59, 4, 2602, (1999)
[38] Bialynicki-Birula, I., Formulation of the uncertainty relations in terms of the Rényi entropies, Phys. Rev. A, 74, 052101, (2006)
[39] Zozor, S.; Vignat, C., On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles, Physica A, 375, 499, (2007)
[40] Zozor, S.; Portesi, M.; Vignat, C., Some extensions of the uncertainty principle, J. Phys. A, 387, 4800, (2008)
[41] Koornwinder, T. H.; Wong, R.; Koekoek, R.; Swarttouw, R. F., Orthogonal polynomials, NIST Handbook of Mathematical Functions, (2010), Cambridge University Press: Cambridge University Press, New York
[42] Avery, J., Hyperspherical Harmonics and Generalized Sturmmians, (2002), Kluwer Academic Publishers: Kluwer Academic Publishers, New York
[43] Temme, N. M.; Toranzo, I. V.; Dehesa, J. S., Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters, J. Phys. A: Math. Theor., 50, 21, 215206, (2017) · Zbl 1374.33015
[44] Temme, N. M., Uniform asymptotic methods for integrals, Indagationes Math., 24, 739-765, (2013) · Zbl 1296.41028
[45] Temme, N. M., Asymptotic Methods for Integrals, (2015), World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ · Zbl 1312.41002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.