In this paper, a quantum cascade laser has been treated as a three-level system, and the calculation of the spontaneous emission and gain spectra has been given. In the calculation, the conduction band nonparabolicity and the injection and exit of electrons have been considered. Results have shown that with increasing injection current, the spontaneous emission peak blue shifts, and the peak intensity increases near linearly with current. With increasing temperatures, the broadening of the spontaneous emission spectra has been attributed to the electron-optical phonon interactions. The peak gain of the stimulated emission has been shown to be determined mainly by the subband lifespans. We have pointed out that it is essential to obtain a long lifespan for the second excited state and short lifespan for the first excited state in order to obtain efficient population inversion and high peak gain for quantum cascade lasers.