Something incorrect in the physics textbooks

[Eggs Ackley]

At least that's how it seems to me. I invite attempts to disabuse me of my notion.

What I think is obviously incorrect is the contention that the Thomas precession can account for the so-called spin-orbit coupling anomaly of atomic physics. The anomaly is between the magnitude of atomic emission spectral line splitting predicted prior to L. H. Thomas's 1927 explanation, and that measured, which differed by a factor of two. Thomas showed how special relativity requires an accelerating reference frame to rotate relative to an inertial frame if the former is also moving transverse to the acceleration relative to the latter. The rotation is now known as the Thomas precession, and I don't dispute its existence. What I dispute is the idea that because the T.P. reduces the rate of electron spin precession as observed in the inertial laboratory frame, it must also reduce the spin-orbit interaction energy accordingly. That it would is a very appealing idea, I would agree, but unfortunately (or not as the case may be), it does not lead to a consistent description when the implications are followed to their conclusion. Therefore we must conclude, the T.P. causes a violation of the usual relationship (i.e., one of equality) between applied torque and rate of change of vector angular momentum, rather than that it accounts for the spin-orbit coupling anomaly. I will now explain why.

The consistency problem arises when we consider that the spectral line splitting corresponds to a difference in the energy required to (considering atomic hydrogen most simply) separate the bound electron from the proton, depending on the relative spin and orbit orientation. The "up" or "down" orientation of the spin relative to the orbital angular momentum causes a small increase or decrease in the binding force that's due primarily to the electrostatic (or Coulomb) attraction between the electron and proton. We can calculate the work required to separate the two particles by considering either particle fixed and moving the other against its attractive force to the other. Consistency requires that both calculations must yield the same result. But, they cannot yield the same result if the Thomas precession halves the rate of electron precession and also halves the amount of torque it takes to re-orient the electron (assuming the electron has a g-factor of two), and the description was already consistent in the absence of the Thomas precession. (It's easy to confirm that the latter is true.)

Another way to see this is to observe that the Thomas precession doesn't have any effect on the rate of precession of the orbit. The precession frequency of the orbit depends only on the magnetic field strength due to the electron intrinsic magnetic moment, which isn't affected by the T.P. So, if the T.P. halves the torque required to re-orient the electron spin, the amount of work done in flipping the spin is half that required to flip the orbit, if they were equal in the absence of T.P., which they are. (If they weren't then macroscopic systems such as interacting solenoids would violate angular momentum conservation.)

It's easy to see also from this argument that the T.P. must result in nonconservation of the total angular momentum. However, it turns out that this nonconservation is relatively benign in the case of a g-factor of two, since the total magnetic moment remains fixed in spite of it.

[Dancing David]

Um, I will wait but electrons don't orbit the nucleus.

ETA:

It looks like there are two seperate components to the idea the spin of a particle (which is most like QM) and the other about orbiting macroscopic body.

Electron spin is a QM term unrelated to the other.

But I could be wrong:
http://www.ece.ucsb.edu/faculty/Kroemer/pubs/13_04Thomas.pdf

It appears to be about movement through an electric field

Historically, this discrepancy provided a major puzzle,1
until it was pointed out by Thomas2 that this argument overlooks
a second relativistic effect that is less widely known,
but is of the same order of magnitude: An electric field with
a component perpendicular to the electron velocity causes an
additional acceleration of the electron perpendicular to its
instantaneous velocity, leading to a curved electron trajectory.
In essence, the electron moves in a rotating frame of
reference, implying an additional precession of the electron,
called the Thomas precession.

[Eggs Ackley]

Um, I will wait but electrons don't orbit the nucleus.

ETA:

It looks like there are two seperate components to the idea the spin of a particle (which is most like QM) and the other about orbiting macroscopic body.

Electron spin is a QM term unrelated to the other.

But I could be wrong:
http://www.ece.ucsb.edu/faculty/Kroemer/pubs/13_04Thomas.pdf

It appears to be about movement through an electric field


Even in some quantum mechanical interpretations it's fair to say the electron orbits the nucleus, if the orbital angular momentum is nonzero. In any case, it doesn't really matter. When there is orbital angular momentum, there's an orbital magnetic moment, and spin-orbit coupling.

That Kroemer paper is not one I would reference for a standard and introductory overview. A standard modern reference for calculating the spin-orbit coupling including the Thomas precession would be Jackson's Classical Electrodynamics. I can't link you to that. Here is a paper I like that's very close to getting at the issue I raise, it just doesn't go far enough: http://www.hep.princeton.edu/~mcdonald/examples/EM/munoz_ajp_69_554_01.pdf . Importantly, Munoz observes that the "hidden momentum" must be included to make sense of things.

The original paper by Thomas is available here (you will have to scroll down to it, it's the 1927 one): http://home.tiscali.nl/physis/HistoricPaper/

There's an algebra mistake in it. His equation 4.122 assumes the electron g-factor is two (that's the modern equivalent of his lambda = e/(mc) ), however his statement on page 16, in the second paragraph from the end of section 6, that "it would be natural to put 2 I w = Omega and still obtain (6.71), (6.72)" does not hold up. Setting (the spin vector) Omega = 2 I w is equivalent to setting g=1, which invalidates his use of 4.122 in deriving his 6.71. When you work it through correctly it gives a different result than the case he shows the details of, which is based on his 5.1. But 5.1 isn't the correct modern form for the equation of translational motion of a magnetic dipole, because it omits the hidden momentum (see Jackson or the references in Munoz). The correct form is now known to be the one at the start of his penultimate paragraph of section 6, which he draws the mistaken conclusion about. So, the rest of the paper showing how the T.P. can obtain the proper spin-orbit coupling is mooted because his stated need for secular (i.e., orbit-averaged) conservation of the total angular momentum is not met in that case. So there is no basis after all for claiming that the T.P. accounts for the spin-orbit coupling anomaly. This should not be that controversial, since it is obtained properly using the Dirac equation, and there is no mention of the T.P. in that connection. However the T.P. explanation is still in a lot of textbooks and I guess is still taught at the undergraduate level.

[Eggs Ackley]

Something fortuitous from my point of view has happened. American Journal of Physics published a paper just last month, which I just became aware of last week, that revisits the calculation of the Thomas factor once again and reaffirms Thomas' result. Here's a link:

http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000078000004000428000001&idtype=cvips&gifs=yes&ref=no

Here's the abstract (all you can see without an institutional license, anyhow):


On the classical analysis of spin-orbit coupling in hydrogenlike atoms

Edit: I forgot to mention that the missing term is approximately the same size as the spin-orbit coupling magnitude the authors calculate, so adding it in reverts the calculated value to the old anomalous one. I have a speculation I will post on my blog for how the anomaly might nonetheless be resolved classically. When I post it, I hope some real physicist can tell me whether it is consistent with what the Dirac theory says.

American Journal of Physics -- April 2010 -- Volume 78, Issue 4, pp. 428-432
Issue Date: April 2010 ABSTRACTREFERENCES (14)Buy This PDF (US$28)



We reanalyze the usual classical derivation of spin-orbit coupling in hydrogenlike atoms. We point out the presence of an additional force exerted on a spinning electron due to the appearance of its electric dipole moment in the rest frame of the nucleus. This force has been ignored, although its inclusion in the electron's equation of motion influences the energy level of an orbiting electron on an equal footing with other effects in the usual analysis of spin-orbit coupling. A fortuitous cancellation between two terms leaves the overall energy level unaffected, which explains in part why this effect has been overlooked. An account of this effect in the Bohr model produces the usual expression for the spin-orbit coupling but with different radii of the electron's orbit for different spatial orientations of the electron's spin. This result is in qualitative agreement with the solution of the Dirac–Coulomb equation for hydrogenlike atoms.




Even though the paper directly contradicts me, it's good because when I looked at the title my concern was that it would reveal what only I seem to recognize to be the case, and without mentioning me. But it completely overlooks the issue. That means I get to write a comment.

It's a fine paper, otherwise. The authors are simply unaware of the existence of hidden momentum, and they didn't do the further calculations to discover that omitting the hidden momentum, linear momentum is not conserved. Anyhow their Equation (26) is directly contrary to the direction of Jackson 3rd edition to add the hidden momentum (m x E)/c to Newton's equation of motion (i.e. F = dP/dt). This is not in the first or second editions, however. If you don't add it in you get an answer that agrees with Thomas for the "effective" force on the electron (and more importantly, the resulting binding energy), but this force is not equal and opposite the force on the proton.

The nice thing here is that this has saved me a lot of work, essentially that I would have to write a paper like that one just to set up the problem. But they have set it up for me very nicely, probably far better than I could have done as an amateur. I get to come along with a two-paragraph comment saying what they missed and that the result must contradict Thomas that the Thomas precession accounts for the spin-orbit coupling anomaly, as is in many many physics textbooks.

I wrote the comment up two days ago and submitted it, and the editor said yesterday he sent it to the authors, and that they can contact me directly. If we can't sort it out amongst ourselves, then he will call in more reviewers, he said.

BTW I started a little alternative-physics blog. Nobody hardly has looked at it yet. There isn't much there yet but I have several more posts drafted. But there are (mostly) open links to many classic papers many of which pre-date the LANL/Cornell preprint archive and so may not be easy to find otherwise. The way they're posted (which is not by me), they don't get found easily by search engines. Like, for example Eliezer, "The Interaction of Electrons in an Electromagnetic Field", or Bhabha, "General Theory of Spinning Particles in a Maxwell Field". These are mainstream papers. There are no papers from alternative journals, but some of them have only been published on arxiv, to my knowledge.

http://quantumskeptic.blogspot.com/

[Eggs Ackley]

Oops, that edit paragraph was supposed to be at the bottom, and certainly not inside the quote. I'm not sure how I managed to do that. Now the edit window appears to have closed.

[Eggs Ackley]

My comment is now up on arxiv: http://arxiv.org/abs/1005.3841

[Eggs Ackley]

This has turned out to be a very interesting experience. It turned out that the authors agreed with part of my comment but not all. Also, where I said adding the hidden momentum doubles the non-Coulomb component of the binding force, the authors did the calculation and got that it was half, not twice. Then, I looked at my own analysis that's been posted on arxiv for over a year, and it agreed with what they said, not with what I said. So then I asked for a do-over, and the editor and the authors said ok. So then, after my comment was revised and agreed to, it was accepted for publication, but I told the editor I still did not believe their response, and that maybe he should hold off until I can analyze it further. They said that even though the non-Coulomb binding force is halved it doesn't ultimately affect the spin-orbit coupling magnitude, which I found very difficult to believe. But, I could not find any sign errors or other algebra errors and their argument was almost convincing me, but yesterday I think I figured out what they omitted that will change result in accordance with my expectations. So now I have proposed that they look into it and I would just as soon let them make their own corrections, if they agree. If not then I will write it up and send it in. That will only be two do-overs, well under the standard limit of three.

Also, in another shameless plug for my blog, I want to point out for anyone who's interested in how Thomas precession can cause angular momentum nonconservation, but who doesn't want to wade through all the (elementary but voluminous) math of my arxiv paper, I have a math-free description using a toy model of two superconducting coil magnets mutually precessing in space here: http://quantumskeptic.blogspot.com/2010/05/two-faces-of-one-coin.html

[Eggs Ackley]

In case anyone is interested, I can report that my comment appeared in AJP this month: http://ajp.aapt.org/resource/1/ajpias/v78/i12

It's down the page in the notes and comments section.

Unfortunately a subscription is required to read it in place. The comment as accepted is posted on arxiv, though, here: http://arxiv.org/abs/1005.3841 , but then, sorry there's no access to the authors' response. Basically they agree with all of my claims except the last. Their response shows how they can get the same result in spite of incorporating the hidden momentum as required, which I admit was a surprise to me. I argue that their analysis in their response is not covariant, in my response to their response, here: http://arxiv.org/abs/1009.0495 , and then show that the expected (empirical) result can be recovered in a relativistically-covariant fashion if the so-called "hidden energy" is included. This has some interesting implications, and seems to be consistent with the Dirac equatiuon.

The journal would not print a response to a response (not surprisingly) so I submitted it as a separate paper. It got two favorable reviews and one pretty highly unfavorable, and the journal rejected it. The reviews can be read on my blog if desired (http://quantumskeptic.blogspot.com/2010/11/reviews-and-decision-in-today.html). Eventually I will follow up and investigate the claims of the negative review, which I am skeptical of, but I also think there's more to be understood there. In particular, there is still a problem of inconsistency of the coupling calculation, if the Thomas precession is invoked, as I had to do to get the expected result. I wanted to get the right answer without invoking the T.P., and was very disappointed when I was unable to do so. AJP would probably not have ever published that, anyhow.

I have a couple of other projects in front of it though.

[quarky]

Correctness X time = uncertainty.

Certainly.