We derive a rigorous quantum formula called topological trajectory to describe orbital angular momentum (OAM) index based on linear momentum density of Laguerre–Gauss (L–G) light beam. By considering the correspondence between optics and quantum theory, we construct a coherent state from two L–G modes light beam. The light beams with fractional OAM are described by the coherent state. By making use of topological trajectory, we present the conditions that OAM index is fractional.