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I don't understand the part about turning off the perturbation. What is meant by or what is he referring to when he says "upper" and "lower" states. Why must the "upper" state reduce to a combination of psi0_a and psi0_b. Why isn't it the other way--the lower state reduces to a combination of psi0_a and psi0_b. This entire paragraph is extremely confusing to me and I have no idea what he is referring to by upper and lower states, and why the lower state reduces to a different combination of orthogonal states. can someone please clarify? Everything seems arbitrary.

  • $\begingroup$ Sorry for not using LaTex, I'm not on my PC $\endgroup$ – CrypticParadigm Feb 11 '16 at 0:16
  • $\begingroup$ A convenient formalism for dealing with perturbations of degenerate eigenvalues is Kato's perturbation theory. It is presented in A. Messiah "Quantum Mechanics " volume 2 and of course in Kato's "Perturbation Theory of Linear Operators ". $\endgroup$ – Urgje Feb 11 '16 at 8:46

The assumption is that the perturbation will break the degeneracy and there will be some linear combinations of states that diagonalise the perturbation. For a two-level system, one (diagonalized) matrix element will be larger than the other. The linear combination that has the larger matrix element is the upper state and the other is the lower state. Now consider the case where the perturbing potential is multiplied by a variable lambda. As lambda approaches 0, the degeneracy returns. This is what is meant by turning the perturbation off. You still end up with the two linear combinations of the original states, and these are the states that should be used for perturbation calculations. Hence, the next step is to find the linear combinations that diagonalise the perturbation matrix.


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