The Schroedinger equation with a momentum-dependent potential has been solved to yield the single-particle energies for neutrons and protons in bound nuclear systems. A nonlocal nucleon-nucleus potential is reduced to a momentum-dependent potential which is further transformed to an energy-dependent effective potential. The effective potential is approximated by an analytically solvable potential, the well-known Morse function. The eigenvalues and eigenfunctions are obtained in closed forms in terms of the Morse parameters. From this method the following well-known results have been obtained: (i) The radii of neutron and proton distribution are different. For light nuclei the radius for proton distribution is larger than that of the neutron distribution, and for medium and heavy nuclei, the order reverses. (ii) The magnitudes of the single-particle energies increase with A but for deep bound states, they level off at large A, thus exhibiting the saturation property of nuclear forces. (iii) Within the major shells the single-particle levels cross each other, which is consistent with the observed data. (iv) The single-particle energies are found in good agreement with the experimental values. (v) The magic numbers resulting from these calculations are in complete agreement with the observed ones. (vi) The so-called semimagic numbers Z or N = 6, 16, 40, and 58 are also noted in the appropriate regions. (vii) The spin-orbit splittng increases as l increases and slightly decreases as A increases.