Fisher information for the position-dependent mass Schrödinger system

This study presents the Fisher information for the position-dependent mass Schrodinger equation with hyperbolic potential V ( x ) = − V 0 csch 2 ( a x ) . 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 − V 0 csch 2 ( a x ) .