A new approximation scheme to the centrifugal term is proposed to obtain the $l eq 0$ solutions of the Schr{o}dinger equation with the Manning-Rosen potential. We also find the corresponding normalized wave functions in terms of the Jacobi polynomials. To show the accuracy of the new approximation scheme, we calculate the energy eigenvalues numerically for arbitrary quantum numbers $n$ and $l$ with two different values of the potential parameter $alpha .$ The bound state energies of various states for a few $% HCl,$ $CH,$ $LiH$ and $CO$ diatomic molecules are also calculated. The numerical results are in good agreement with those obtained by using program based on a numerical integration procedure. Our solution can be also reduced to the s-wave ($l=0$) case and to the Hulth{e}n potential case.
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