Quantum theory of large amplitude collective motion and the Born-Oppenheimer method

We study the quantum foundations of a theory of large amplitude collective motion for a Hamiltonian expressed in terms of canonical variables. In previous work the separation into slow and fast (collective and non-collective) variables was carried out without the explicit intervention of the Born Oppenheimer approach. The addition of the Born Oppenheimer assumption not only provides support for the results found previously in leading approximation, but also facilitates an extension of the theory to include an approximate description of the fast variables and their interaction with the slow ones. Among other corrections, one encounters the Berry vector and scalar potential. The formalism is illustrated with the aid of some simple examples, where the potentials in question are actually evaluated and where the accuracy of the Born Oppenheimer approximation is tested. Variational formulations of both Hamiltonian and Lagrangian type are described for the equations of motion for the slow variables.