The process of nonresonant photon scattering on a hydrogen atom is considered as a microscopic model for a quantum-mechanical description of the reflection of light on a mirror. Multiple reflections of this kind then can serve as a model for light storage within a system of mirrors. In combination with modern high-precision measurements of atomic transition frequencies, such a storage of light could allow for new information on the time variation of fundamental constants, such as the fine-structure constant. The intensity distribution shift arising due to the inherent asymmetry of the natural line profile sets an ultimate limit for the preservation of the photon light frequency during storage. In the case of a 'hydrogen mirror' this shift appears to be much larger than the relative accuracy of about 10-15 required in measurements of a time dependence of the fine-structure constant. Possibilities of achieving the required accuracy are discussed.