By using the Hartree approximation and the simplified method of quasidiscreteness multiple scales, we have studied quantum solitary wave solutions for a one-dimensional ferromagnetic chain with exchange interaction and classical magnetic moment interaction. In this chain there are both traveling and stationary quantum solilary waves. With the help of the obtained quantum solitary wave solution, the energy levels and magnetic moments of the quantum solitary wave have been investigated further. It is shown that the energy and magnetic moments of the quantum solitary wave are quantized. These novel results provide a possible way for understanding macroscopic quantum effects such as quantum steps of the hysteresis loop in magnetic materials.