A comment I wrote on an American Journal of Physics paper from last April has now been published by the journal and appears on their site:
(It's pretty far down the page, the third item in the Notes and Discussions section, and followed by the response by the authors of the original paper.)
The comment is also posted for unrestricted viewing on arxiv.org: http://arxiv.org/abs/1005.3841.
Unfortunately, there's no way to view the response to the comment, by the authors of the original paper, other than by looking in the AJP which requires a subscription. If I may paraphrase, basically, the authors agree with all of my claims except for the last, and go on to show that the empirically-determined result for the spin-orbit coupling can continue to be obtained even after inclusion of hidden momentum as required according to the modern textbooks.
It was a surprise to me when they did this, I have to admit. So I tried to reproduce their result, but there aren't very many details in their response, so when they said they included the hidden momentum in the Bohr postulate that orbital angular momentum is quantized, I interpreted it differently than they meant. I literally put the hidden angular momentum in the angular momentum that is quantized, whereas as I found out empirically and eventually confirmed through correspondence, the comment response doesn't assume this, it only includes the hidden momentum in the equation for the electron velocity. How I did it is presented in the arxiv paper linked to below.
So then when the hidden momentum is included in this way the result no longer agrees with experiment, as I expected it wouldn't. At that point I thought the response to my comment as simply in error. When we figured out through correspondence that the assumptions of their and my analyses were different, then I decided to submit my analysis as a separate paper, but only after I went further with it by linking in the so-called "hidden energy," that must accompany the hidden momentum.
It is only very simple relativistic covariance arguments that are required to prove that existence of hidden momentum implies existence of hidden energy. When I put in the hidden energy then surprisingly to me, the experimental result was recovered in the conventional general method of analysis that includes the Thomas precession. So this is saying that if we re-examine the whole semiclassical atom spin-orbit coupling analysis using the electrodynamic equations according to the modern electrodynamics textbooks (e.g., Jackson 3rd edition but not Jackson 2nd edition, which doesn't include hidden momentum in the equation of translational motion of a magnetic dipole), the expected result is obtained and involves in an essential way the Thomas precession that contributes a necessary factor of one-half. However I was surprised by this also, because I expected to get the correct result only by not invoking the Thomas precession, because I don't think the Thomas precession can affect the energy as argued originally by Thomas and retained in textbooks to the present day.
For a couple of reasons I think the Thomas precession as only a kinematical effect should not reduce the spin-orbit coupling energy. The most powerful argument I have for this belief is that if the T.P. does affect the energy as usually thought, then it will contradict the energy value one will obtain if the energy to invert the orbit is calculated, as opposed to the energy to invert the spin, as is usually done. But, these are two equally-valid ways to compute the same quantity, and simply must obtain the same result. So I think there is more to be learned here and intend to revisit the problem at some point in the not-too-distant future.
As I have already mentioned, and wanted to mention especially to anyone who comes here on account of seeing the comment in AJP, I have written a response to the response to my comment and posted it on arxiv (here: http://arxiv.org/abs/1009.0495 ). I also submitted the latter to AJP but it has been rejected, although two out of three referee reports were quite positive and unambiguously recommended publication. The third was equally unambiguous in the other direction and AJP is a very conservative journal, of course, so it's not surprising the paper has been rejected on account of one out of three reviews being negative. The reviews are posted in the previous posting here. I will respond to the negative review eventually but it will take a while. I will take the referee's suggestion to see what the BMT equation has to say about it. I don't think he will turn out to be correct (that I double-counted the spin-orbit orientational potential energy somehow) but I will give it my best to prove or disprove whether it's true. But it is not my top priority to do this right now so it will have to wait.
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