North - Holland BORN - OPPENHEIMER REVISITED

The Born-Oppenheimer approximation [1] can be used whenever the hamilto..ian of a fast system depends on the coordinates of a slow system . In some textbooks, the problem is solved by freezing the coordinates of the slow system and solving the hamiltonian of the fast system; then the energy of the fast system enters the effective hamiltonian for the slow system as a potential energy term. This solution was found by Mead and Truhlar [2] to be insufficient : a vector potential must be inserted to adjust for the separation of the system into two parts. Berry [3] derived the general form of this vector potential . Even when the vector potential is included, however, errors and unjustified approximations are sometimes made. We have found a conceptually simple and direct approach to solving the BornOppenheimer approximation. We treat the problem via degenerate perturbation theory and solve it by modifying a subset of the operators, without introducing trial wave functions. In this approach, no geometrical tools are needed . The application of degenerate perturbation theory leads to an inhomogeneous set of linear equations, the solution of which contains the desired vector potential . We present the algebraic method in sect . 2 of this paper . Sect . 3 extends the method to classical mechanics. A non-abelian example is given in sect . 4, and an application to field theory is worked out in sect . 5 . Sect . 6 shows that the slow variables need not commute . Our last example, in sect . 7, suggests how to treat the non-adiabatic case along these lines.