The energy levels of the Schrödinger equation under the Eckart-Hellmann potential (EHP) energy function are studied by the Nikiforov-Uvarov-Functional Analysis (NUFA) method. We obtained the analytic solution of the energy spectra and the wave function in closed form with the help of Greene-Aldrich approximation. The numerical bound states energy for various screening parameters at different quantum states and vibrational energies of EHP for CuLi, TiH, VH, and TiC diatomic molecules were computed. Four exceptional cases of this potential were achieved. To test the accuracy of our results, we computed the bound states energy eigenvalues of Hellmann potential which are in excellent agreement with the report of other researchers.

In this study, the Schrödinger equation with the Hulthén plus screened Kratzer potentials (HSKP) are solved via the Nikiforov-Uvarov (NU) and the series expansion methods. We obtained the energy equation and the wave function in closed form with Greene-Aldrich approximation via the NU method. The series expansion method was also used to obtain the energy equation of HSKP. Three distinct cases were obtained from the combined potentials. The energy eigenvalues of HSKP for HCl, LiH, H2, and NO diatomic molecules were computed for various quantum states. To test the accuracy of our results, we computed the bound states energy of HCl and LiH, for a special case of Kratzer and screened Kratzer potentials, which are in excellent agreement with the report of other researchers.

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The solutions of the Schrödinger equation are obtained with an inversely quadratic Yukawa potential using the Nikiforov-Uvarov-Functional-analysis method. The energy spectrum and wave function were obtained in closed form. The energy equation was used to predict the masses of the heavy mesons such as charmonium (сĉ) and bottomonium (bḃ) for different quantum numbers. The results obtained agreed with other theoretical predictions and experimental data with a percentage error of 1.68 % and 0.50 % for charmonium (сĉ) and bottomonium (bḃ) respectively.

In this study, the energy equation and normalized wave function were obtained by solving the Schrödinger equation analytically utilizing the Eckart-Hellmann potential and the Nikiforov-Uvarov method. Fisher information and Shannon entropy were investigated. Our results showed higher-order characteristic behavior for position and momentum space. Our numerical results showed an increase in the accuracy of the location of the predicted particles occurring in the position space. Also, our results show that the sum of the position and momentum entropies satisfies the lower-bound Berkner, Bialynicki-Birula, and Mycieslki inequality and Fisher information was also satisfied for the different eigenstates. This study's findings have applications in quantum chemistry, atomic and molecular physics, and quantum physics.