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Perturbation theory for projected states

For quantum systems with a symmetry it is often convenient to start from a simple soluble model lacking the symmetry, and restore the symmetry by projection. If this is done for all eigenstates of the model system, one obtains an overcomplete set of basis functions, which is not suitable for standard perturbation theory. The paper develops a method by which one can do perturbation theory in this situation. The method is illustrated by application to the problem of the centre-of-mass motion of a nucleus, the symmetry being the translational and Galileo invariance.