# Does the Lamb shift occur only in $n=2$?

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 $$\frac{1}{2}$$?

• well, you can only hope to measure it for degenerate states, otherwise the shift is too small to be detected unless you work really hard Sep 8, 2021 at 15:13
• I believe your questions are all answered in this post: physics.stackexchange.com/questions/429690/… Sep 8, 2021 at 20:43

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.