consider a two fold degeneracy such that H Psi_a = E Psi_a and H Psi_b = E Psi_b and <Psi_a | Psi_b> = 0 All of the above are the unperturbed states, Hamiltonian and eigenvalue. Notice the two states share the eigen value E. Form the linear combination of the two states Psi = a * Psi_a + b * Psi_b Clearly H Psi = E Psi and the eigenvalue of the linear combination is still the value E. Now a perturbation to the hamiltonian H will split the energy level into two values E+ and E- and two states Psi+ and Psi-. Finally the question: I am told that if the perturbation is "dialed down to zero" then Psi+ and Psi- reduce to two different linear combinations of Psi_a and Psi_b. What does this mean? Shouldn't the states be simply Psi_a and Psi_b? Why would there be two unique linear combinations instead of just the two eigenstates themselves?