Approximate non-relativistic s-wave energy spectra with non-polynomial potentials within the framework of the WKB approximation

The harmonic oscillator and the Coulomb potential energy perturbed by rational functions are studied using the Wentzel–Kramers–Brillouin (WKB) approximation method. Using the proper quantization conditions, the bound state eigenvalue solutions to the Schrodinger equation were obtained in terms of the complete elliptic integrals of the first, second and third kinds. The results obtained numerically, are in excellent agreement compared to the results obtained via the shifted-1/N expansion and the generalized pseudo-spectral (GPS) methods. Also, the s-wave bound state energy eigenvalue solutions of the harmonic and Coulomb’s potentials are obtained as special cases. For the perturbed harmonic oscillator, we showed that the convergence of the WKB approximation by the asymptotic approximations of the energy-dependent elliptic functions led to improved results for constrained values of potential parameters.