We fabricated ZnO-ZnMgO single quantum well (SQW) samples having different well-widths by using the atomic layer deposition technique. The QW samples exhibit different optical transition behaviors with different QW widths. We confirm that when the well-width of 1.5 nm does not have a confined quantum energy level due to the Mg diffusion into the well caused by after-thermal treatment whereas the QWs wider than 1.5 nm show optical transitions between the confined energy levels.