The bound state solutions of the s-wave radial Schrödinger equation with Manning-Rosen potential are presented exactly by the standard method. It is found that the solutions can be expressed by the generalized hypergeometric functions 2 F 1 (a, b; c; z). The intractable normalized wavefunctions are derived. We also study the special case for α = 0 and find that this potential will reduce to the Hulthén potential.

The bound-state solutions of the Schrödinger equation for an exponential-type potential with the centrifugal term are presented approximately. It is shown that the complicated normalization wavefunctions can be expressed by the generalized hypergeometric functions 2 F 1 (a, b; c; z). To show the accuracy of our results, we calculate the energy eigenvalues numerically for arbitrary quantum numbers n and l with two different values of the parameter σ . It is found that the results are in good agreement with those obtained by another method for short-range potential. Two special cases for s-wave case (l = 0) and σ = 1 are also studied briefly.

By using the exact quantization rule, we present analytical solutions of the Schrödinger equation for the deformed harmonic oscillator in one dimension, the Kratzer potential and pseudoharmonic oscillator in three dimensions. The energy levels of all the bound states are easily calculated from this quantization rule. The normalized wavefunctions are also obtained. It is found that the present approach can simplify the calculations.

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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