Fisher's information for the position-dependent mass Schr\"odinger system

TL;DR

This paper investigates Fisher information in the position-dependent mass Schrödinger equation with a hyperbolical potential, analyzing quantum states and uncertainty relations to understand molecular system properties.

Contribution

It provides a novel analysis of Fisher information for the position-dependent mass Schrödinger system with hyperbolical potential, including numerical results and uncertainty relations.

Findings

Fisher information analyzed for ground and excited states.

Fisher-information-based uncertainty relation and Cramer-Rao inequality hold.

Cramer-Rao and Heisenberg products measure anharmonicity.

Abstract

This study presents the Fisher information for the position-dependent mass Schr\"odinger equation with hyperbolical potential {$V(x)=-V_0{\rm csch}^2(ax)$}. The analysis of the quantum-mechanical probability for the ground and exited states $(n=0, 1, 2)$ has been obtained via the Fisher's information. This controls both chemical and physical properties of some molecular systems. The Fisher information is considered only for $x>0$ due to the singular point at $x=0$. We found that Fisher-information-based uncertainty relation and the Cramer-Rao inequality holds. Some relevant numerical results are presented. The results presented shows that the Cramer-Rao and the Heisenberg products in both spaces provide a natural measure for anharmonicity of {$-V_0{\rm csch}^2(ax)$}