  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.
