Abstract: The energy states of the exciton in a single square GaAs-based quantum well are obtained from a numerical solution of the three-dimensional Schrödinger equation for the envelope of the exciton wave function. This equation is based on the exciton effective energy operator with a spherical approximation of the Luttinger Hamiltonian. The calculated states are classified based on the types of one-dimensional functions for the factorized form of the wave function. The upper limit for the energies of the exciton states in a quantum well is confirmed by the complex scaling method.