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The shift is due to interactions between atomic electrons and vacuum state fluctuations, so I would think that there would be an associated energy shift for all states, and not just s-states.

Also, is there any physical relationship between the Lamb shift and the Darwin correction for hydrogen? The explanation for the Darwin term is usually "Zitterbewegung", which is often in turn justified by the same processes contributing to the Lamb shift.

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The s-states are the only orbitals that do not vanish at the origin. This is where the coulomb field of the proton is strong enough to produce measurable vacuum polarization or Lamb shift. Hence the Lamb shift only affects s-states.

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