A quantum-mechanical theory of the \ifmmode \check{C}\else \v{C}\fi{}erenkov radiation is developed wherein the refractive medium is treated not as a continuum but as an aggregation of atoms. The atoms and the electromagnetic field are considered as forming a single system which interacts with an incident charged particle. The \ifmmode \check{C}\else \v{C}\fi{}erenkov radiation arises in first-order transitions induced by this interaction. The nature of this process suggests a quantum-mechanical definition of the dielectric constant with the aid of which the theory yields the well-known properties of the \ifmmode \check{C}\else \v{C}\fi{}erenkov radiation. The definition of the dielectric constant is shown to lead to the same expression for this quantity as is given by the Kramers-Heisenberg dispersion formula.