In this talk, we investigate the initial value problem of the quasilinear Schrödinger equation and introduce the generalized energy method, which is different from those of the series work of Kenig et al. (Invent. Math., 2004; Adv. Math., 2005; Adv. Math., 2006) and the work of Marzuola et al. (Adv. Math., 2012; Kyoto J. Math., 2014; Arch. Ration. Mech. Anal., 2021). We find that the momentum type estimates can be equally important as the energy estimates. By combining these two type bounds, we eventually close the estimates, which will lead to the desired results by artificial viscosity method. The key of the analysis is to find suitable weight to multiply momentum type conservation law equality and produce some good terms that can help the momentum type estimates and the energy estimates to control the bad terms in each other. For quadratic interaction problem, we derive the lower regularity results with small initial data in the same function spaces as in the works of Kenig et al. For cubic interaction problem, we obtain the same low regularity results as Marzuola et al. (Kyoto J. Math., 2014). This talk is based on a joint work with Yi Zhou.