In this talk, we will discuss the observability problem for Schrödinger equations. In the existing literature, the observability of Schrödinger equation on compact manifolds and bounded domains has been extensively studied. However, the observability problem on an unbounded set (by measurable control regions) is much less studied in the literature. After discussing briefly the previous results, we will present recent results of the quantitative observability result of the 1D Schrödinger equation on real line. Our proof relies on different techniques for low-frequency and high-frequency estimates. This talk is based on the joint work with Pei Su and Chenmin Sun.