Abstract This study presents the Shannon and Renyi information entropy for both position and momentum space and the Fisher information for the position-dependent mass Schrödinger equation with the Frost-Musulin potential. The analysis of the quantum mechanical probability has been obtained via the Fisher information. The variance information of this potential is equally computed. This controls both the chemical properties and physical properties of some of the molecular systems. We have observed the behaviour of the Shannon entropy. Renyi entropy, Fisher information and variance with the quantum number n respectively.
Abstract This study presents the Shannon and Renyi information entropy for both position and momentum space and the Fisher information for the position-dependent mass Schrödinger equation with the Frost-Musulin potential. The analysis of the quantum mechanical probability has been obtained via the Fisher information. The variance information of this potential is equally computed. This controls both the chemical properties and physical properties of some of the molecular systems. We have observed the behaviour of the Shannon entropy. Renyi entropy, Fisher information and variance with the quantum number n respectively.
The bound state solutions of the Schrödinger equation (SE) under a deformed hyperbolic potential are used in this paper to investigate the correlation between the information content, such as Shannon entropies and Fisher information, and the variance of a quantum system in both momentum and position spaces. The variance was obtained from the expectation moments in conjugated spaces and used to get the uncertainty products, Fisher information products and Shannon entropic sums for different potential parameters. These information measures were observed to vary with the potential parameters and obey their lowest bound inequalities. An increase in the Fisher information leads to a lower uncertainty and less information while decreases in the Fisher information result in higher uncertainties. We proposed a relationship between the Shannon entropies, Fisher information and the variance where an increase in the Fisher information leads to a lower Shannon entropy or a lower spread of the probability distribution and vice versa. The numerical results obtained with the proposed relations agree with the most utilized method in the literature. This current work simplifies the calculation of the Shannon entropies notably for complex potential functions.
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 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.
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.
There has been an interest in studies on informationtheoretic and complexity measures for quantum model systems in the recent years. The information entropy for the trigonometric Rosen-Morse potential, harmonic oscillator, hydrogenic atoms, Morse potential, Yukawa and Hulthen potentials have been investigated. The Kratzer potential plays an important role in atomic and molecular physics and quantum chemistry. Cheng and Dai proposed a new potential consisting of the modified Kratzer potential plus a new proposed ring-shaped potential which has possible applications to ring-shaped organic molecules like cyclic polyenes and benzene. In this work, we shall obtain the Fisher information, Fisher length, Shannon entropy, Shannon length, Renyi entropy, Renyi length, Tsallis entropy and the Fisher-Shannon complexity for the ring-shaped modified Kratzer potential. In this section, we present the definition of the information theoretical measures used in this work which are based on the 1-normalized electron density in the position and momentum space. The Shannon information entropy in the position space is given by Eq. (1). ( ) ( ) log ( )d S ρ ρ ρ = −∫ r r r ... (1)
The approximate solutions of the Schrödinger equation with the central generalized Hulthén and Yukawa potential are investigated within the framework of the functional method. The obtained wave function and the energy levels are used to study the Shannon entropy, Renyi entropy, Fisher information, Shannon Fisher complexity, Shannon power and Fisher-Shannon product in both position and momentum spaces for the ground and first excited states. The theoretic information theories such as Shannon entropies Sr,Sp, Fisher information Ip,Ip and Renyi entropies Rβ as well as the information entropy probability densities ρ(r),ϕ(p) are illustrated graphically. The Bialynicki-Birula – Mycielski (BBM) inequality for the Shannon entropies and the Cramer-Rao inequality for the Fisher information are satisfied for the potential model.