Topological quantum mechanics and the first Chern class

Topological properties of quantum system is directly associated with the wave function. Based on the decomposition theory of gauge potential, a new comprehension of topological quantum mechanics is discussed. One shows that a topological invariant, the first Chern class, is inherent in the Schr\"odinger system, which is only associated with the Hopf index and Brouwer degree of the wave function. This relationship between the first Chern class and the wave function is the topological source of many topological effects in quantum system.