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Could the Lamb shift be used in $n=3,4,5,...$? Or does it only work with $n=2$? And does it work for values of $j$ other than $\frac{1}{2}$?

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Yes, there will be a Lamb shift contribution to more highly excited states, and the origin is much the same as for the famous $n=2$ case. However, the size of the shift scales like $n^{-3}$, so that it becomes smaller for increasingly energetic states. For example, in hydrogen there is a $3S_{1/2}$-$3P_{1/2}$ splitting but it is approximately 313.5 MHz, as compared to the 1058 MHz splitting for the $n=2$ levels.

Here's a paper reporting a measurement of the Lamb shift in the hydrogen $n=3$ system and finding a splitting of about 314.8 MHz, very close to that predicted based on the approximate $n^{-3}$ scaling.

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