The optically active energy states of the exciton in GaAs/Al_0.3Ga_0.7As quantum wells of various widths are calculated by the direct solution of the three-dimensional Schrodinger equation. A dependence of energies on the quantum well width as a parameter is obtained for widths up to 200 nm. The results are compared with the model of quantization of the exciton as a whole in wide quantum wells. The radiative decay rates for several lowest exciton states are also found.