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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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  • $\begingroup$ The question (v1) is essentially a duplicate of physics.stackexchange.com/q/32041/2451 $\endgroup$
    – Qmechanic ♦
    Commented Apr 8, 2013 at 8:45
  • $\begingroup$ @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. $\endgroup$
    – ChickenGod
    Commented Apr 8, 2013 at 10:59
  • $\begingroup$ Concerning the Adiabatic Theorem, see also e.g. Wikipedia. $\endgroup$
    – Qmechanic ♦
    Commented Apr 8, 2013 at 11:17

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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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    $\begingroup$ Only upto accidental degeneracies. For e.g. it happens in H-atom due to circular symmetric Coulomb potential V(r) $\endgroup$
    – L.K.
    Commented Apr 16, 2019 at 14:29
  • $\begingroup$ Can you show it implies a symmetry? How does the symmetry act? $\endgroup$
    – Kvothe
    Commented Jun 29, 2022 at 16:58

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