Fisher information for the position-dependent mass Schrödinger system

This study presents the Fisher information for the position-dependent mass Schrödinger equation with hyperbolic potential V(x)=-_V0^csch2(ax). The analysis of the quantum-mechanical probability for the ground and exited states (n=0,1,2) has been obtained via the Fisher 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 show that the Cramer-Rao and the Heisenberg products in both spaces provide a natural measure for anharmonicity of -_V0^csch2(ax).