skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Entropy and complexity analysis of hydrogenic Rydberg atoms

Abstract

The internal disorder of hydrogenic Rydberg atoms as contained in their position and momentum probability densities is examined by means of the following information-theoretic spreading quantities: the radial and logarithmic expectation values, the Shannon entropy, and the Fisher information. As well, the complexity measures of Cramer-Rao, Fisher-Shannon, and Lopez Ruiz-Mancini-Calvet types are investigated in both reciprocal spaces. The leading term of these quantities is rigorously calculated by use of the asymptotic properties of the concomitant entropic functionals of the Laguerre and Gegenbauer orthogonal polynomials which control the wavefunctions of the Rydberg states in both position and momentum spaces. The associated generalized Heisenberg-like, logarithmic and entropic uncertainty relations are also given. Finally, application to linear (l= 0), circular (l=n- 1), and quasicircular (l=n- 2) states is explicitly done.

Authors:
 [1];  [2]; ;  [1];  [2];  [1];  [2]
  1. Instituto Carlos I de Fisica Teorica y Computacional, Universidad de Granada, 18071-Granada (Spain)
  2. (Spain)
Publication Date:
OSTI Identifier:
22163004
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 54; Journal Issue: 5; Other Information: (c) 2013 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
74 ATOMIC AND MOLECULAR PHYSICS; ASYMPTOTIC SOLUTIONS; ATOMS; DENSITY; ENTROPY; EXPECTATION VALUE; HYDROGEN; POLYNOMIALS; PROBABILITY; RYDBERG STATES; WAVE FUNCTIONS

Citation Formats

Lopez-Rosa, S., Departamento de Fisica Aplicada II, Universidad de Sevilla, 41012-Sevilla, Toranzo, I. V., Dehesa, J. S., Departamento de Fisica Atomica, Molecular y Nuclear, Universidad de Granada, 18071-Granada, Sanchez-Moreno, P., and Departamento de Matematica Aplicada, Universidad de Granada, 18071-Granada. Entropy and complexity analysis of hydrogenic Rydberg atoms. United States: N. p., 2013. Web. doi:10.1063/1.4807095.
Lopez-Rosa, S., Departamento de Fisica Aplicada II, Universidad de Sevilla, 41012-Sevilla, Toranzo, I. V., Dehesa, J. S., Departamento de Fisica Atomica, Molecular y Nuclear, Universidad de Granada, 18071-Granada, Sanchez-Moreno, P., & Departamento de Matematica Aplicada, Universidad de Granada, 18071-Granada. Entropy and complexity analysis of hydrogenic Rydberg atoms. United States. doi:10.1063/1.4807095.
Lopez-Rosa, S., Departamento de Fisica Aplicada II, Universidad de Sevilla, 41012-Sevilla, Toranzo, I. V., Dehesa, J. S., Departamento de Fisica Atomica, Molecular y Nuclear, Universidad de Granada, 18071-Granada, Sanchez-Moreno, P., and Departamento de Matematica Aplicada, Universidad de Granada, 18071-Granada. Wed . "Entropy and complexity analysis of hydrogenic Rydberg atoms". United States. doi:10.1063/1.4807095.
@article{osti_22163004,
title = {Entropy and complexity analysis of hydrogenic Rydberg atoms},
author = {Lopez-Rosa, S. and Departamento de Fisica Aplicada II, Universidad de Sevilla, 41012-Sevilla and Toranzo, I. V. and Dehesa, J. S. and Departamento de Fisica Atomica, Molecular y Nuclear, Universidad de Granada, 18071-Granada and Sanchez-Moreno, P. and Departamento de Matematica Aplicada, Universidad de Granada, 18071-Granada},
abstractNote = {The internal disorder of hydrogenic Rydberg atoms as contained in their position and momentum probability densities is examined by means of the following information-theoretic spreading quantities: the radial and logarithmic expectation values, the Shannon entropy, and the Fisher information. As well, the complexity measures of Cramer-Rao, Fisher-Shannon, and Lopez Ruiz-Mancini-Calvet types are investigated in both reciprocal spaces. The leading term of these quantities is rigorously calculated by use of the asymptotic properties of the concomitant entropic functionals of the Laguerre and Gegenbauer orthogonal polynomials which control the wavefunctions of the Rydberg states in both position and momentum spaces. The associated generalized Heisenberg-like, logarithmic and entropic uncertainty relations are also given. Finally, application to linear (l= 0), circular (l=n- 1), and quasicircular (l=n- 2) states is explicitly done.},
doi = {10.1063/1.4807095},
journal = {Journal of Mathematical Physics},
number = 5,
volume = 54,
place = {United States},
year = {Wed May 15 00:00:00 EDT 2013},
month = {Wed May 15 00:00:00 EDT 2013}
}
  • Oscillations are shown to exist in the inversion symmetry of the electronic wave function of a hydrogenic atom coherently excited to a Rydberg state by a short pulse of laser radiation in a uniform electric field. The dependence of these oscillations on field strength is shown to scale as {ital n}{sup 2} where {ital n} is the principal quantum number. The possibility of using these oscillations to measure electric signals on picosecond timescales (terahertz frequencies) is suggested. {copyright} {ital 1996 American Institute of Physics.}
  • The theory of collisions of Rydberg atoms with rare-gas atoms is generalized to include the effects of an applied static electric field. Two effects occur. First, the field causes the states to split, and the increased energy separations lead to a decrease in the inelastic cross sections. Second, there is a change in the form of the wave functions, because the various Stark states in the field have electron charge distributions that may be localized in very different regions of space. The present work focuses on this second effect. With the description of the initial and final Stark states ofmore » the Rydberg atom by the proper electronic wave functions obtained in parabolic coordinates, cross sections are obtained for a variety of specific transitions of Rydberg atoms (n = 10, 15, and 20) caused by collisions. The calculations are strictly valid for hydrogen, but exhibit general features that are expected to apply to other Rydberg atoms as well. The results show a dramatic effect on the cross section for particular transitions depending on the degree of overlap of the initial and final charge distributions. In addition, it is observed that transitions are favored that involve small changes in the azimuthal quantum number m. The origin of this tendency is discussed.« less
  • The ionization probabilities of hydrogenic Rydberg states in intense fields are calculated using a trajectory method, which was previously shown to be accurate for ionization of the ground-state hydrogen atom [J. S. Cohen, Phys. Rev. A 64, 043412 (2001)]. It is found that the ionization probability approaches the classical over-the-barrier probability for sufficiently large n quantum numbers, but that tunneling still significantly decreases the onset field strengths at surprisingly high n. Calculations are done for ns, np{sub 0}, and np{sub {+-}} targets, subjected to sudden and adiabatically ramped pulses in the long-wavelength limit. The dependence on the angular-momentum projection mmore » along the field axis is also examined for circular orbi0008.« less
  • Cited by 17
  • We describe the calculation of hydrogenic (one-loop) Bethe logarithms for all states with principal quantum numbers n{<=}200. While, in principle, the calculation of the Bethe logarithm is a rather easy computational problem involving only the nonrelativistic (Schroedinger) theory of the hydrogen atom, certain calculational difficulties affect highly excited states, and in particular states for which the principal quantum number is much larger than the orbital angular momentum quantum number. Two evaluation methods are contrasted. One of these is based on the calculation of the principal value of a specific integral over a virtual photon energy. The other method relies directlymore » on the spectral representation of the Schroedinger-Coulomb propagator. Selected numerical results are presented. The full set of values is available at arXiv.org/quant-ph/0504002.« less