Abstract The spreading of the quantum-mechanical probability distribution density of the three-dimensional system is quantitatively determined by means of the local information-theoretic quantity of the Shannon information and information energy in both position and momentum spaces. The complexity measure which is equivalent to Cramer–Rao uncertainty product is determined. We have obtained the information content stored, the concentration of quantum system and complexity measure numerically for n = 0, 1, 2 and 3 respectively.
Abstract The spreading of the quantum-mechanical probability distribution density of the three-dimensional system is quantitatively determined by means of the local information-theoretic quantity of the Shannon information and information energy in both position and momentum spaces. The complexity measure which is equivalent to Cramer–Rao uncertainty product is determined. We have obtained the information content stored, the concentration of quantum system and complexity measure numerically for n = 0, 1, 2 and 3 respectively.
The spreading of the quantum-mechanical probability cloud for the ground state of the Morse and modified Pöschl–Teller potentials, which controls the chemical and physical properties of some molecular systems, is studied in position and momentum space by means of global (Shannon's information entropy, variance) and local (Fisher's information) information-theoretic measures. We establish a general relation between variance and Fisher's information, proving that, in the case of a real-valued and symmetric wavefunction, the well-known Cramer–Rao and Heisenberg uncertainty inequalities are equivalent. Finally, we discuss the asymptotics of all three information measures, showing that the ground state of these potentials saturates all the uncertainty relations in an appropriate limit of the parameter.
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 purpose of this article is to calculate the numerical values of the Shannon (S) and Fisher (I) information entropies along with their Shannon power (J) and Fisher-Shannon product (P) for a free particle in three dimensions. In both coordinate and momentum space, the numerical values have been calculated for various values of the cubical box's length (L). Further, the validity of the entropic uncertainty relations such as the Bialynicki-Birula and Myceilski (BBM) inequality relation, Stam-Cramer-Rao inequalities (better known as the Fisher based uncertainty relation) and the Fisher-Shannon product relation have been investigated. The Fisher-Shannon product has been introduced and analyzed, which can be considered as a new correlation measure. The article is basically based on the concept, measurements and on the behaviour of the Shannon entropy, Fisher information entropy, Shannon power and the Fisher-Shannon product. Here, the plane wave solution of the free particle in three dimensions is considered and the wave function is normalized in an arbitrarily large but finite cube. The Fourier transforms of the coordinate space wave function provided the momentum space wave function. The probability densities corresponding to the respective wave functions have been employed to compute the numerical values of the information theoretic quantities such as Shannon entropy, Fisher entropy, Shannon power and the Fisher-Shannon product both in the coordinate and momentum space and the computed values have been presented in the respective tables. Finally, the entropic uncertainty relations investigated have been found consistently satisfying the corresponding inequalities such as Bialynicki-Birula and Mycielski (BBM) inequality, the Stam-Cramer-Rao inequalities and the Fisher-Shannon product relation. The study establishes the validity of the entropic uncertainty relations for the motion of a free particle in three dimensions.
A Schrödinger equation is solved for a cotangent hyperbolic potential using the elegant parametric Nikiforov–Uvarov method. The results obtained are used to calculate the Fisher information for position space and momentum space as well as the Shannon entropy for position space and momentum space at the ground state. The energies of the cotangent potential are observed to be fully bounded for different quantum states. The Fisher information satisfied the Fisher relation known as Cramer–Rao inequality for the Fisher product. Its result also obeyed the Heisenberg principle as its alternative for uncertainty measurement. However, the Shannon entropy only satisfied the Bialynick-Birula, Mycielski inequality for higher values of the screening parameter. Its relation to the Heisenberg principle failed as both the position space and the momentum space are parallel with high entropy squeezing. This potential has not been reported for any system to the best of our understanding.
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.