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arXiv:1708.07299 · Journal of Statistical Mechanics Theory and Experiment

Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems

Nahual Sobrino, David Puertas‐Centeno, I. V. Toranzo, J. S. Dehesa

Published Aug 1, 2017• 24 citations
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Abstract

Abstract In this work we find that not only the Heisenberg-like uncertainty products and the Rényi-entropy-based uncertainty sum have the same first-order values for all the quantum states of the D -dimensional hydrogenic and oscillator-like systems, respectively, in the pseudoclassical ( D → ∞ ) limit but a similar phenomenon also happens for both the Fisher-information-based uncertainty product and the Shannon-entropy-based uncertainty sum, as well as for the Crámer–Rao and Fisher–Shannon complexities. Moreover, we show that the López–Ruiz–Mancini–Calvet (LMC) and LMC-Rényi complexity measures capture the hydrogenic-harmonic difference in the high dimensional limit already at first order.

DOI
10.1088/1742-5468/aa7df4

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Abstract In this work we find that not only the Heisenberg-like uncertainty products and the Rényi-entropy-based uncertainty sum have the same first-order values for all the quantum states of the D -dimensional hydrogenic and oscillator-like systems, respectively, in the pseudoclassical ( D → ∞ ) limit but a similar phenomenon also happens for both the Fisher-information-based uncertainty product and the Shannon-entropy-based uncertainty sum, as well as for the Crámer–Rao and Fisher–Shannon complexities. Moreover, we show that the López–Ruiz–Mancini–Calvet (LMC) and LMC-Rényi complexity measures capture the hydrogenic-harmonic difference in the high dimensional limit already at first order.
The position-momentum Shannon and R\'enyi uncertainty products of general quantum systems are shown to be bounded not only from below (through the known uncertainty relations), but also from above in terms of the Heisenberg-Kennard product $\ensuremath{\langle}{r}^{2}\ensuremath{\rangle}\ensuremath{\langle}{p}^{2}\ensuremath{\rangle}$. Moreover, the Cram\'er-Rao, Fisher-Shannon, and L\'opez-Ruiz, Mancini, and Calbet shape measures of complexity (whose lower bounds have been recently found) are also bounded from above. The improvement of these bounds for systems subject to spherically symmetric potentials is also explicitly given. Finally, applications to hydrogenic and oscillator-like systems are done.
The D-dimensional harmonic system (i.e., a particle moving under the action of a quadratic potential) is, together with the hydrogenic system, the main prototype of the physics of multidimensional quantum systems. In this work, we rigorously determine the leading term of the Heisenberg-like and entropy-like uncertainty measures of this system as given by the radial expectation values and the Rényi entropies, respectively, at the limit of large D. The associated multidimensional position-momentum uncertainty relations are discussed, showing that they saturate the corresponding general ones. A conjecture about the Shannon-like uncertainty relation is given, and an interesting phenomenon is observed: the Heisenberg-like and Rényi-entropy-based equality-type uncertainty relations for all of the D-dimensional harmonic oscillator states in the pseudoclassical ( D → ∞ ) limit are the same as the corresponding ones for the hydrogenic systems, despite the so different character of the oscillator and Coulomb potentials.
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.
In a recent paper [S. López-Rosa et al., J. Math. Phys. 54, 052109 (2013)], the authors made detailed investigations on the information-theoretic quantities for hydrogenic atoms in Rydberg states. Asymptotic forms of different types of complexity measures (Crámer-Rao, Fisher-Shannon, and LMC types) in both position and momentum spaces were obtained for hydrogenic states (n,l,m) with fixed l and increasingly large n. However, the expressions for the Crámer-Rao complexity are incorrect due to the wrong definition of variance. The correct form of this complexity in position space is presented exactly in this work. In momentum space, two different definitions of variance are provided and corresponding Crámer-Rao complexities are discussed separately. In either case, the momentum complexity increases much faster than that in position space when increasing principal quantum number n.
The Cramér-Rao, Fisher-Shannon and LMC shape complexity measures have been recently shown to play a relevant role to study the internal disorder of finite many-body systems (e.g., atoms, molecules, nuclei). They highlight amongst the bunch of complexities appeared in the literature by their mathematical simplicity and transparent physical interpretation. They are composed by two spreading measures (variance, Fisher information and Shannon entropies) of the single-particle probability density of the systems. Here we discuss the physico-mathematical knowledge of these two-component complexities for hydrogenic systems with standard (D=3D = 3D=3) and non-standard dimensionalities. These systems include not only the hydrogen atom and its isoelectronic series but also a large diversity of quantum systems such as, e.g. exotic atoms, antimatter atoms, some qubits systems and Rydberg atoms, ions and molecules.
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 Coulomb-like 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.