The bound state solutions to the radial Schrödinger equation are obtained in three-dimensional space using the series expansion method within the framework of a general interaction potential. The energy eigenvaluesof the pseudoharmonic and Kratzer potentials are given as special cases. The obtained analytical results are applied to several diatomic molecules, i.e. $\rm{N}_{2}$,CO,NO and CH. In order to check the accuracy of the present method, a comparison is made with similar results obtained in the literature by using other techniques.

An energy eigenvalue expression for a sextic anharmonic interaction potential is derived by solving the Schrödinger equation analytically within the framework of the Nikiforov–Uvarov (NU) method. The resultant expression is then used to calculate the mass spectra of some heavy and light mesons, viz., c$\bar{c}$, b$\bar{b}$ , b$\bar{c}$, c$\bar{s}$, b$\bar{s}$ and b$\bar{q}$, and the results are found to be in good agreement with the other theoretical and experimental studies. The energy eigenvalue expression is further used to compute some important thermodynamic quantities like partition function, specific heat capacity, free energy, mean energy, entropy and magnetisation.