Why are Diagonal Elements of a Time-Dependent Perturbed Hamiltonian Often Zero?

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SUMMARY

The diagonal elements of a time-dependent perturbed Hamiltonian are often zero because the perturbation primarily serves to mix the eigenstates of the unperturbed Hamiltonian. While it is possible for these diagonal elements to be non-zero due to actual physical perturbations that shift energy levels, it is generally more practical to consider perturbations that only mix states. In such scenarios, the diagonal elements are effectively incorporated into the unperturbed Hamiltonian.

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Morten
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Why does a time-dependent perturbed Hamiltonian commonly have diagonal elements equal to zero?
 
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Because the perturbation is what is going to mix the eigenstates of the unperturbed Hamiltonian. But the diagonal elements could be non-zero: an actual physical perturbation could also shift the energy levels. Most often, it is easier to work with a perturbation that only mixes the states, so I would move the diagonal elements to the "unperturbed" Hamiltonian in such a case.
 
Thanks!
 

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