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## Main Question or Discussion Point

Take the usual time-independent perturbation theory in QM for example,H'=H_0+V, a basic assumption is we can expand the new states of H' in terms of the old ones of H_0, most of the textbooks justify this assumption by reasoning that the set of eigenfunctions of Hamiltonian is complete, regardless of the old H_0 or the new H'. But I guess this assumption can't be always true, for example, you can't possibly write a free-particle wavepacket in terms of eigenfuntions of infinite square well. So when can this assumption be applied exactly?