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Let's think about a system that has a two-fold degeneracy for some given energy level. That is, two states $\psi_{a}$ and $\psi_{b}$, both of which correspond to energy $E_{0}$. An example would be a spin-1/2 particle with a Hamiltonian that is spin-independent. Now imagine that when we apply a perturbation, H', to the system, the degeneracy breaks ...