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.

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Shannon entropy for lower position and momentum eigenstates of Pöschl—Teller-like potential is evaluated. Based on the entropy densities demonstrated graphically, we note that the wave through of the position information entropy density ρ(x) moves right when the potential parameter V1 increases and its amplitude decreases. However, its wave through moves left with the increase in the potential parameter |V2|. Concerning the momentum information entropy density ρ(p), we observe that its amplitude increases with increasing potential parameter V1, but its amplitude decreases with increasing |V2|. The Bialynicki—Birula—Mycielski (BBM) inequality has also been tested for a number of states. Moreover, there exist eigenstates that exhibit squeezing in the momentum information entropy. Finally, we note that position information entropy increases with V1, but decreases with |V2|. However, the variation of momentum information entropy is contrary to that of the position information entropy.

The bound states of the Schrödinger equation for a second Pöschl–Teller like potential are obtained exactly using the Nikiforov–Uvarov method. It is found that the solutions can be explicitly expressed in terms of the Jacobi functions or hypergeometric functions. The complicated normalized wavefunctions are found.

Shannon entropy for the position and momentum eigenstates of the symmetrically trigonometric Rosen-Morse potential for the lower states n = 1-4 is evaluated. The position information entropies S x for n = 1, 2 are presented analytically. Some interesting features of the information entropy densities ρ s (x) and ρ s ( p) are demonstrated graphically. We find that the ρ s ( p) is inversely proportional to the range of potential a and the S x decreases with increasing the potential depth D. In particular, we note that the S x might become negative for some given parameters a and D. The Bialynicki-Birula-Mycielski inequality is also tested for a number of states and is found to generally hold well.

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