The two-dimensional layered materials such as graphene and transition metal dichalcogenides reveal the unique structure of the electron-hole spectrum that generates discrete degrees of freedom related to the electron spin and the valley pseudospin. Optical phonons with the angular momentum may induce the transitions between the quantum levels in this system. Therefore, the investigation of these phonons is of particular importance now. One of the most popular methods for the determination of the angular momentum of optical phonons is circularly polarized Raman spectroscopy, common features of which were not entirely discovered so far due to a restricted set of studied materials. Here, we use this method to examine optical phonons in Si as an example of a high-symmetry model crystal. The decomposition of scattered light into the circularly polarized components with the subsequent angle-resolved analysis is employed. This allows us to find that Raman scattering by longitudinal optical phonons occurs with conservation of the photon helicity in accordance to the angular momentum conservation law, since the longitudinal optical phonon angular momentum is zero. In contrast, transverse optical phonon reverses photon helicity, meaning that its angular momentum is 2ħ. It is shown that Raman scattering of circularly polarized light is described well taking into account the angular momentum conservation law in addition to the main conclusions of the classical theory developed for the case of linearly polarized light.