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Physical Review A

Information-theoretical analysis of Dirac and nonrelativistic quantum oscillators

I. López-García, A. J. Macías, Sheila López‐Rosa, Juan Carlos Angulo

Published Aug 23, 2023• 6 citations
Publisher

Abstract

An information-theoretical study of the Dirac oscillator and the usual quantum oscillator is provided, in terms of the respective spatial densities. The comparative analysis is grounded on the values of Shannon entropy, Fisher information, and disequilibrium of both systems in arbitrary states. It is emphasized the dependence of these entropic functionals on the angular frequency and the quantum numbers that characterize the state of the systems, and remarkably also on the different spin-angular momentum coupling schemes. The results are interpreted accordingly with the structural patterns of the corresponding densities, providing information on their spread, uniformity, and disorder.

DOI
10.1103/physreva.108.022812

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An information-theoretical study of the Dirac oscillator and the usual quantum oscillator is provided, in terms of the respective spatial densities. The comparative analysis is grounded on the values of Shannon entropy, Fisher information, and disequilibrium of both systems in arbitrary states. It is emphasized the dependence of these entropic functionals on the angular frequency and the quantum numbers that characterize the state of the systems, and remarkably also on the different spin-angular momentum coupling schemes. The results are interpreted accordingly with the structural patterns of the corresponding densities, providing information on their spread, uniformity, and disorder.
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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 Crámer-Rao, Fisher-Shannon, and López 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.
The Fisher information functional occupies a very special position among the uncertainty measures of quantum systems of any dimensionality. In contrast with (global) entropic measures of Shannon, Renyi or Tsallis types, it contitutes a local measure in the sense that it is sensitive to the oscillatory character of the probability density distribution. Moreover, the Fisher information is closely related to various macroscopic properties of the system described by different density functionals. Here, we review its main analytical properties such as, inequality-based relationships with some radial expectation values and its uncertainty relation. Tighter versions of these inequalities are shown for those systems subject to an arbitrary central potential. Moreover, the Fisher information is computed for D-dimensional hydrogenic systems explicitly in terms of the radial and angular hyperquantum numbers characterizing the quantum states of the system in both position and momentum spaces. Finally, the utility of this quantity is discussed for various physico-chemical processes of recent interest: Abstraction and nucleophilic substitution reactions.
We study the localization-delocalization of a particle moving within a circular region of radius r0 from a theoretical information point of view. We computed the Shannon entropy, Fisher information and disequilibrium in configuration and momentum spaces for a collection of stationary states. Comparing our results of Shannon entropies with those previously published we found good agreement with those. Shannon entropy, Fisher information and the disequilibrium offer complementary results for the description of the particle localization-delocalization.
The local structure present in Wigner and Husimi phase-space distributions and their marginals are studied and quantified via information-theoretic quantities. Shannon, R\'enyi, and cumulative residual entropies of the Wigner and Husimi distributions are examined in the ground and excited states of a harmonic oscillator. The entropies of the Wigner function marginals are lower than the corresponding entropies of the Husimi function marginals, which illustrates how the nodal structure present in the Wigner function is lost upon consideration of the Husimi function. Shannon and cumulative residual entropies of the Wigner function yield entropies which are complex valued. Absolute values and real components of these quantities increase with quantum number, displaying a behavior which is consistent with their real-valued Husimi function counterparts. This comportment is also similar to the real-valued R\'enyi entropies of the Wigner function. The entropies of the Wigner function are observed to be lower than the corresponding Husimi function ones, in agreement with the results for the marginals. The real components of the Wigner function entropies are seen to be closer to the uncertainty relation bound compared to the corresponding Husimi function entropies. These real components are also closer to the bound when contrasted to the entropic sum of the marginal densities. The R\'enyi or collision entropy of the Wigner function sits exactly on the bound. Related statistical correlation measures show that the position-momentum correlation is larger in the Wigner function compared to the Husimi function and increases with quantum number.
We investigate the quantum-information properties of a particle confined by the double Morse potential in position and momentum spaces. The quasi-exact solvability of the model gives analytical expressions for the first two bound states, allowing the corresponding probability densities to be analyzed directly. Shannon entropy, Onicescu energy, Fisher information, statistical complexity, and Fisher-Shannon products are evaluated as functions of the parameter AAA, which controls the transition from a well-separated double well to a merged single-well profile. The position distribution is more delocalized and structurally complex when the double-well character is pronounced, whereas the momentum distribution exhibits the complementary trend. As the wells merge, the ground-state Fisher--Shannon product approaches its Gaussian reference value, whereas the excited state retains stronger non-Gaussian structure.