Bound state solutions of the SchrӦdinger equation with second PӦschl-Teller like potential model and the vibrational partition function, mean energy and mean free energy

•A Schrӧdinger equation is proposed to be solved with Second Pӧschl-Teller potential.•The energy equation and the corresponding wave function are obtained explicitly.•A particle in this system exhibits the same behaviour for s-wave and ℓ-wave equations.•The energy is purely negative when the potential parameter V2 increases.•The partition function increases as the potential range α increases.

By employing two different methods and suitable approximation schemes to deal with the centrifugal term, we solved approximately the Schrȍdinger equation with the Second Pȍschl-Teller like potential model. The bound state energy equation and the unnormalized radial wave functions have been obtained. The vibrational partition function, mean energy and vibrational mean free energy are also obtained.