Lamb shift in hydrogen atoms

Question

Has a Lamb shift been observed in any atom other than hydrogen?

I am highly confused. The Lamb shift makes the P_1/2 state lower than the S_1/2 state. Is Lamb shift observed in other atoms? If not, then why?

Answer 1

Hydrogen is the only neutral atom for which we have an analytic solution to the Schrodinger equation, so it is the only atom for which we have an exact expression for the energies of the ${}^2S_{1/2}$ and ${}^2P_{1/2}$ states. And the Dirac equation tells us the energies of the two states should be the same. The Lamb shift is the observation that the energies are actually slightly different.

For any atom with more than one electron the SE cannot be solved analytically and we have to resort to a numerical solution. This means we have no reason to suppose that the two states should have the same energy, and indeed they don't.

So the Lamb shift only applies to the hydrogen atom because it is only for hydrogen that we expect the ${}^2S_{1/2}$ and ${}^2P_{1/2}$ to have equal energies. For all other atoms there will be corrections to the states due to vacuum fluctuations, but the two states already have different energies so the corrections just change the difference very slightly, and this has never been seen as important enough to have its own name.

Answer 2

This expands slightly the answer by John Rennie.

The term "Lamb shift" ordinarily is understood to mean the case of hydrogen, and this has been the most important case historically. Nonetheless the QED contribution being referred to (i.e. an effect in addition to what can be obtained from Dirac equation employed to do single-particle quantum theory, which is a somewhat limited calculational approach) is present in all electronic states of all atoms.

In most non-hydrogen cases, as John Rennie says, the Lamb shift is just a contribution to an energy level difference which is already non-zero before bringing in the QED corrections. So it is only in hydrogen that it furnishes a split between levels which would otherwise be degenerate. However, experimental efforts to probe quantum electrodynamics (QED) have included effort to detect the equivalent shift in other cases. In particular, one goes to highly-charged ions. In such systems the QED effects are enhanced and one can test the strong-field region of QED. Here the QED calculations get harder. Of course if the highly charged ion being measured has just one electron remaining then it is a hydrogen-like case, but not quite the same because the nucleus will be larger which may complicate matters. More generally, whether there is one electron or many, the Lamb shift is large enough in such systems to contribute a significant effect which can be measured.

Here are a couple of papers on the Lamb shift in hydrogen-like uranium (the first that popped up in a search just now):

https://www.osti.gov/etdeweb/biblio/7263593

https://iopscience.iop.org/article/10.1088/0031-8949/1988/T22/045

Answer 3

Has a Lamb shift been observed in any atom other than hydrogen?

Strictly speaking, the answer is "yes", even if one defines the Lamb shift as narrowly as @JohnRennie seems to do. For example, people studied (theoretically and experimentally) the Lamb shift in deuterium, tritium, and muonic atoms of isotopes of hydrogen.