<p>The Nikiforov–Uvarov functional analysis method was employed in combination with the Deng–Fan and Eckart potential models to derive energy equations from the solutions of the Schrödinger equation. This study investigates the relationship between the variance of a quantum system and the information-theoretic measures specifically, Fisher information and Shannon entropy in both position and momentum spaces. Variance was calculated from expectation values in conjugate spaces, allowing for the determination of uncertainty products, Fisher information products, and Shannon entropic sums. The findings demonstrate that these information measures satisfy their fundamental inequality bounds: an increase in Fisher information corresponds to reduced uncertainty and higher information content, while a decrease in Fisher information results in increased uncertainty. The Deng–Fan potential model was also applied to compute the energy spectra of CO (X¹Σ⁺) and HCl (X¹Σ⁺) diatomic molecules, with results closely aligning with those reported in the literature through alternative analytical methods. The observed trend of increasing energy with higher principal quantum numbers confirms the quantized nature of energy levels. Furthermore, the interaction potential energy for the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(a^{3} \sum\nolimits_{u}^{ + } {}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>a</mi> <mn>3</mn> </msup> <msubsup> <mo>∑</mo> <mrow> <mi>u</mi> </mrow> <mo>+</mo> </msubsup> <mrow /> </mrow> </math></EquationSource> </InlineEquation> state of ⁷Li₂ molecule was modeled. The predicted vibrational energy levels show good agreement with Rydberg–Klein–Rees (RKR) data and previous studies. These results are crucial for understanding molecular spectra, energy transitions, and the dynamics of molecular systems in spectroscopy, chemical reactions, and molecular physics.</p>