The bound-state solutions of the Schrödinger equation with the Eckart potential with the centrifugal term are obtained approximately. It is shown that the solutions can be expressed in terms of the generalized hypergeometric functions 2 F 1 (a, b; c; z). The intractable normalized wavefunctions are also derived. To show the accuracy of our results, we calculate the eigenvalues numerically for arbitrary quantum numbers n and l. It is found that the results are in good agreement with those obtained by other methods for short-range potential (large a). Two special cases for l = 0 and β = 0 are also studied briefly.

We present an analytical solution of the radial Dirac equation for the rotational Morse potential through the Pekeris approximation. The bound state energy eigenvalues are obtained by using an exact quantization rule for non-zero κ values of the Dirac equation. As an application of the rule, we give the numerical solutions of the results for special values of the potential parameters.

The bound state solutions of the Schrödinger equation for a second Pöschl-Teller-like potential with the centrifugal term are obtained approximately. It is found that the solutions can be expressed in terms of the hypergeometric functions 2 F 1 (a, b; c; z). To show the accuracy of our results, we calculate the eigenvalues numerically for arbitrary quantum numbers n and l. It is found that the results are in good agreement with those obtained by other method for short-range potential. Two special cases for l = 0 and V 1 = V 2 are also studied briefly.

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We proposed a new approximate scheme for a centrifugal term. Using new approximate formula for 1/r2, we obtained the bound state and scattering state solutions of the Manning–Rosen potential with centrifugal terms. All approximate analytical formulae of energy eigenvalues, normalized wavefunctions and scattering phase shifts are presented. In addition, we also suggested another much better approximate formula to 1/r2 for bound states. All data calculated by the above approximate analytical formulae are compared with those obtained by using the numerical integration method in the bound state and scattering state cases. Furthermore, the complete s-wave scattering state solutions for the Manning–Rosen potential are also naturally derived.

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