A model of genome evolution is proposed. Based on several general assumptions the evolutionary theory of a genome is formulated. Both the deterministic classical equation and the stochastic quantum equation are proposed. The classical equation is written in a form of of second-order differential equations on nucleotide frequencies varying in time. It is proved that the evolutionary equation can be put in a form of the least action principle and the latter can be used for obtaining the quantum generalization of the evolutionary law. The wave equation and uncertainty relation for the quantum evolution are deduced logically. Two fundamental constants of time dimension, the quantization constant and the evolutionary inertia, are introduced for characterizing the genome evolution. During speciation the large-scale rapid change of nucleotide frequency makes the evolutionary inertia of the dynamical variables of the genome largely decreasing or losing. That leads to the occurrence of quantum phase of the evolution. The observed smooth/sudden evolution is interpreted by the alternating occurrence of the classical and quantum phases. In this theory the probability of new-species formation is calculable from the first-principle. To deep the discussions we consider avian genome evolution as an example. More concrete forms on the assumed potential in fundamental equations, namely the diversity and the environmental potential, are introduced. Through the numerical calculations we found that the existing experimental data on avian macroevolution are consistent with our theory. Particularly, the law of the rapid post-Cretaceous radiation of neoavian birds can be understood in the quantum theory. Finally, the present work shows the quantum law may be more general than thought, since it plays key roles not only in atomic physics, but also in genome evolution.