R\’enyi complexity ratio of two density functions is introduced for three and multidimensional quantum systems. Localization property of several density functions are defined and five theorems about near continuous property of R\’enyi complexity ratio are proved by Lebesgue measure. Some properties of R\’enyi complexity ratio are demonstrated and investigated for different quantum systems. Exact analytical forms of R\’enyi entropy, R\’enyi complexity ratio, statistical complexities based on R\’enyi entropy for integral order have been presented for solutions of pseudoharmonic and a family of isospectral potentials. Some properties of R\’enyi complexity ratio are verified for six diatomic molecules (CO, NO, N$_2$, CH, H$_2$, and ScH) and for other quantum systems.
R\’enyi complexity ratio of two density functions is introduced for three and multidimensional quantum systems. Localization property of several density functions are defined and five theorems about near continuous property of R\’enyi complexity ratio are proved by Lebesgue measure. Some properties of R\’enyi complexity ratio are demonstrated and investigated for different quantum systems. Exact analytical forms of R\’enyi entropy, R\’enyi complexity ratio, statistical complexities based on R\’enyi entropy for integral order have been presented for solutions of pseudoharmonic and a family of isospectral potentials. Some properties of R\’enyi complexity ratio are verified for six diatomic molecules (CO, NO, N2, CH, H2, and ScH) and for other quantum systems.
Abstract Rényi complexity ratio of two density functions is introduced for three and multidimensional quantum systems. Localization property of several density functions are defined and five theorems about near continuous property of Rényi complexity ratio are proved by Lebesgue measure. Some properties of Rényi complexity ratio are demonstrated and investigated for different quantum systems. Exact analytical forms of Rényi entropy, Rényi complexity ratio, statistical complexities based on Rényi entropy for integral order have been presented for solutions of pseudoharmonic and a family of isospectral potentials. Some properties of Rényi complexity ratio are verified for six diatomic molecules (CO, NO, N2, CH, H2, and ScH) and for other quantum systems.
Rényi complexity ratio of two density functions is introduced for three and multidimensional quantum systems. Localization property of several density functions are defined and five theorems about near continuous property of Rényi complexity ratio are proved by Lebesgue measure. Some properties of Rényi complexity ratio are demonstrated and investigated for different quantum systems. Exact analytical forms of Rényi entropy, Rényi complexity ratio, statistical complexities based on Rényi entropy for integral order have been presented for solutions of pseudoharmonic and a family of isospectral potentials. Some properties of Rényi complexity ratio are verified for some diatomic molecules (CO, NO, N2, CH, H2, and ScH) and for some other quantum systems.
In this Thesis we use the notions of information entropy, statistical complexity and\nquantum entanglement, together with the methods of Classical and Quantum Information\nand the algebraic and asymptotic techniques of the theory of orthogonal polynomials and special functions of Applied Mathematics and Mathematical Physics, to study and quantify the multiple facets of the spatial delocalization of\nthe charge and matter distributions of the multidimensional quantum systems of bosonic\nand fermionic character. These facets, which determine both the uncertainty measures of\nthe systems and their physical and chemical properties, are manifested in the enormous\ndiversity of multidimensional geometries of the single-particle densities which characterize\nthe non-relativistic mechano-quantically allowed states of such systems, according to\nthe Density Functional Theories based on the Hohenberg-Kohn theorem and generalizations.\nThe aim of this Thesis is to contribute to the emergent informational representation\nof the quantum systems, which complements the standard representation\nbased on energetic concepts. The analytical determination of the entropy-,\ncomplexity- and entanglement-like measures and their associated uncertainty relations is a fundamental task (not yet solved except for the ground state and\na few low-lying excited states) even for the very small number of elementary quantummechanical\npotentials which are utilized to approach and model the self-consistent mean-\n eld of the physical systems (atoms, molecules,...), as this work shows. Basically, this\nis because the quantum manifestations of the multidimensional physical systems are\ngenerally inaccessible as the associated Schr odinger equation cannot be exactly solved\nexcept for a few cases where the quantum-mechanical potential has a known symmetry.
A two-parameter family of complexity measures C̃(α,β) based on the Rényi entropies is introduced and characterized by a detailed study of its mathematical properties. This family is the generalization of a continuous version of the Lopez-Ruiz–Mancini–Calbet complexity, which is recovered for α=1 and β=2. These complexity measures are obtained by multiplying two quantities bringing global information on the probability distribution defining the system. When one of the parameters, α or β, goes to infinity, one of the global factors becomes a local factor. For this special case, the complexity is calculated on different quantum systems: H-atom, harmonic oscillator, and square well.
Abstract Exact solutions of a pseudoharmonic oscillator and a family of isospectral potentials are investigated in spherical coordinates. The entropic moment, generalized quantum similarity index, and some quantum information measures are investigated analytically and numerically for two density functions of two quantum systems with same energy and one quantum system with different energies. Analytical results are compared for 19 selected molecules and verified by some physical and artificial values of the spectroscopic parameters.
Information-theoretic inequalities play a fundamental role in numerous scientific and technological areas (e.g., estimation and communication theories, signal and information processing, quantum physics, …) as they generally express the impossibility to have a complete description of a system via a finite number of information measures. In particular, they gave rise to the design of various quantifiers (statistical complexity measures) of the internal complexity of a (quantum) system. In this paper, we introduce a three-parametric Fisher–Rényi complexity, named ( p , β , λ ) -Fisher–Rényi complexity, based on both a two-parametic extension of the Fisher information and the Rényi entropies of a probability density function ρ characteristic of the system. This complexity measure quantifies the combined balance of the spreading and the gradient contents of ρ , and has the three main properties of a statistical complexity: the invariance under translation and scaling transformations, and a universal bounding from below. The latter is proved by generalizing the Stam inequality, which lowerbounds the product of the Shannon entropy power and the Fisher information of a probability density function. An extension of this inequality was already proposed by Bercher and Lutwak, a particular case of the general one, where the three parameters are linked, allowing to determine the sharp lower bound and the associated probability density with minimal complexity. Using the notion of differential-escort deformation, we are able to determine the sharp bound of the complexity measure even when the three parameters are decoupled (in a certain range). We determine as well the distribution that saturates the inequality: the ( p , β , λ ) -Gaussian distribution, which involves an inverse incomplete beta function. Finally, the complexity measure is calculated for various quantum-mechanical states of the harmonic and hydrogenic systems, which are the two main prototypes of physical systems subject to a central potential.