Semiclassical kinetic equation and the WKB approximation

Abstract

We study the connection between the classical response function of a nucleus described by the Vlasov equation and the corresponding quantum response function. In the limit of large quantum numbers, the Fourier coefficients which appear in the Vlasov theory correspond to the quantum matrix elements evaluated in the WKB approximation. The classical frequencies of motion give the quantum excitation energies according to the correspondence principle. For a central potential, we identify the normal modes of the classical systems with the corresponding shell-model transitions. Thus we can improve the Vlasov theory by excluding excitation modes which correspond to forbidden quantum transitions. Also, we study the effect of a spin-orbit force in the context of the Vlasov theory. Finally, always by exploiting the close analogy between the classical and the quantum response functions, we introduce exchange (Fock) term in the semiclassical RPA equation given by the Vlasov theory.