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Suppose we have a Hamiltonian that depends on various real parameters. When tuning the values of these parameters, the energy eigenvalues will often avoid crossing each other. Why?

Is there a physically intuitive justification for level repulsion and avoided crossings? It would be nice to see a general argument.

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Questions are expected to be clear and to show some research effort. Downvoting. –  Ben Crowell Apr 8 '13 at 3:20
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For a mathematically inclined person the exposition on this link might be intuitive www2.maths.ox.ac.uk/chebfun/examples/linalg/html/… –  anna v Apr 8 '13 at 4:35
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@BenCrowell, I think the question is quite clear and direct. I don't see why it should necessarily demonstrate research effort. –  Siva Apr 8 '13 at 6:08
    
The question (v1) is essentially a duplicate of physics.stackexchange.com/q/32041/2451 –  Qmechanic Apr 8 '13 at 8:45
    
@Qmechanic Actually, the question you linked is what inspired my question. Perhaps I should have added more detail in my question, but I was thinking Adiabatic Theorem, not perturbation theory. I will edit my question correspondingly. –  ChickenGod Apr 8 '13 at 10:59

1 Answer 1

Consider what happens if there is a crossing. A crossing would imply a degeneracy in the system. A degeneracy would imply a symmetry. It would be unnatural for a perturbation to introduce a symmetry into a system, and so the eigenvalues cannot cross generically, but can under special circumstances.

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