Fisher information and uncertainty relations for potential family

Abstract

An approximate bound state solution of the three-dimensional Schrödinger equation for potential family was obtained together with the corresponding wave function, after which the Fisher information for a potential family was explicitly obtained via the methodology of expectation value and radial expectation value. Some uncertainty relations that are closely related to Heisenberg-like uncertainty were obtained and numerical results were generated to justify the relations and inequalities. Finally, we have numerically studied the Fisher information for two special cases from the result of the potential family that has applications to prototype systems. It was deduced that the Fisher information of the potential family and that of the two special cases exhibit similar features.bound state solutions, expectation value, Fisher information, potential term, uncertainty relations 1 | INTRODUCTION A particle in a D-dimensional spherical box is one of the basic fundamental ideas of quantum physics such as the effect of boundary condition and quantization energy. A particle confined in a quantum system exhibits a distinguishable change in both the physical and chemical properties. [1,2] According to Laguna and Sagar, [3] the confined quantum-mechanical models have proved to be suitable first approximations for establishing the effect of pressure on the spectral lines of atoms and molecules. Thus, the behavior of confined systems with different potential models required vigorous attention because of its applications such as in quantum dots, astrophysics, condensed matter and semiconductor physics. [4][5][6] A onedimensional confined Harmonic Oscillator model has been studied in connection with the effect of distant boundaries on the energy level, [7,8] effect on the separation of variables, [9] information entropies in position space [10] and energies and Einstein coefficients. [11] Similarly, in the introductory courses of quantum mechanics, several models proposed were useful for the discussion of semiclassical approaches, variational methods, and perturbation theory. Motivated by this, the authors aimed to study the Fisher information of a particle subjected to a system confined in a potential family.Fisher information introduced in the theory of statistical estimation is the main theoretic tool of the extreme physical information principle that allows numerous derivations of fundamental equations of physics. [12][13][14][15] Fisher information is the gradient functional of probability density, which is a local measure as it is sensitive to local rearrangements of the density. The higher the Fisher information, the smaller the uncertainty in a system. This gives a higher accuracy in predicting the localization of a particle. Fisher information has been shown to be closely connected with other density functional that characterizes a number of microscopic properties of fundamental character as well as physical observables. [16,17] This, however, results to the formulation of uncertainty relations that is stronger than the C...

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