Quantum information measurements of the exact solution of the Schrödinger equation for a q-deformed Morse potential

We present a comprehensive numerical and analytical study of information-theoretic measures–specifically, the Shannon entropy in position (\(S_x\)) and momentum (\(S_{p_x}\)) spaces, for a non-relativistic fermion subject to a q-deformed Pöschl–Teller-like hyperbolic potential, including comparisons with the q-deformed Morse potential. By systematically varying the deformation parameter q, the inverse length scale \(\alpha\), and the potential depth \(V_0\), we investigate their combined influence on spatial localization, uncertainty, and the global and local information content of the quantum states. Our results show that q induces a controllable trade-off between \(S_x\) and \(S_{p_x}\), while preserving their sum; \(\alpha\) predominantly enhances total uncertainty, signaling increased delocalization; and \(V_0\) favors spatial localization at the cost of momentum spread. All configurations obey the Bialynicki-Birula–Mycielski (BBM) inequality, confirming the robustness of the approach. These findings underscore the deep connection between potential geometry and quantum information measures, with prospective implications for deformed quantum systems, relativistic extensions, and Lorentz symmetry-violating frameworks.