Spin-orbit interaction and bremsstrahlung

[wc2351]

Hi,

It never occurred to me before, but when you derive the spin-orbit Hamiltonian as a perturbation to the hydrogen Hamiltonian, you imagine your electron orbiting around the nucleus, and of course that's not the correct picture because of the electromagnetic radiation that leaks out. So I guess a rigorous derivation of the spin-orbit Hamiltonian has to be done at the level of Dirac eq or QED...but I haven't learned either of them so could someone describe how it goes?

 

[Jano L.]

This is an interesting question. The only good derivation I came across seems to be from the Dirac equation and the term itself is relativistic, containing factor 1/c^2. There is some explanation of this in the book by Davydov, Quantum Mechanics, p. 250.

I think that a term similar to the LS term could be derived also in classical theory, if one assumes that the electron is a charged spinning sphere or something which has extended charge distribution and revolves around the nucleus. Because the electron is extended, it should feel varying electric field of the nucleus and this can perhaps lead to a coupling between the orbital and inner rotation.

 

[DrDu]

There is a nice derivation of the SO coupling in the book

Grundzüge der Quantentheorie. Mit exemplarischen Anwendungen
Werner R. Theis (Autor)

which considers the equation of motion of a spin in it's non-inertial rest frame.
Especially interesting is how the non-commutativity of the Lorentz transformation gives an additional contribution to the SO coupling.
In German though.

 

[Bill_K]

This is a very interesting question, and primarily from the point of view of human psychology. More about that in a moment.

The original poster is correct, the only valid explanation for spin-orbit coupling comes from the Dirac equation. A full derivation of the term can be found for example in "Relativistic Wave Mechanics" by Corinaldesi and Strocchi, Chap VIII, Non-Relativistic Limit of the Dirac Equation. What's especially interesting is not so much that it's a relativistic effect (Of course it is! Electron spin is already a relativistic effect) but that it's an effect which is second order. The first order nonrelativistic approximation to the Dirac equation is the Schrödinger-Pauli equation. The energy from the spin-orbit term is a small correction to this, smaller than the first-order atomic binding energy by a factor of e²/ħc, the fine structure constant. (Which of course is why we call it that.)

Not only is it a relativistic effect, the derivation is quite mathematical. So people start saying to themselves, "Wait, isn't there a simpler approach? Some way we can understand this intuitively?" Well no, there isn't. It just has to be swallowed. But this is where the psychology comes in.

Of course, Physics is full of models, in which you simplify things by making an assumption. But to mean anything the assumption must be an approximation, i.e. it can't just be blatantly false, it must be valid in some limit. But sometimes we forget that, and make assumptions which are so intuitively appealing that they take on a life of their own, even though they contradict physics. Examples are the hole model of antiparticles, the vector model of angular momentum, and so on.

One of these ideas was in the first suggestion above, that an electron is in some ways like a classical extended object, a charged sphere, rotating on its axis perhaps. Another one, more common in the spin-orbit context, is to imagine that the electron stands still and the nucleus revolves about it. The nucleus then produces a B field which the electron spin reacts to, causing the spin-orbit term. Now if you know any physics at all, you know that it is not valid to take over the Dirac equation to a noninertial frame. Nevertheless, this idea is in many textbooks, and has been handed down from generation to generation of students because it is so intuitively appealing. Accompanying this is usually mention of an additional correction due to the Thomas precession.

[By the way, non-commutativity of Lorentz transformations is an equally pseudo way of deriving the Thomas precession, but that's another topic!]

Any such scheme (I can't call it a model) will result in an answer that's pretty close, maybe off by a factor of 2 or 3, but otherwise very satisfying. This reinforces the feeling that we now "understand" the effect much better. Why is this? Because by dimensionality arguments, almost anything that produces the right units can be at most off by a small numerical factor, even if what it's based on is complete horseradish.

 

[Jano L.]

...the only valid explanation for spin-orbit coupling comes from the Dirac equation.

Bill, do you mean that other explanations are not known or that they cannot exist?
I wouldn't be so harsh against models - after all, ultimately everything is a model, even Dirac's equation.

 

[DrDu]

Bill, I find this statement also astonishing. After all, spin orbit coupling is well defined also for classical systems (with spin meaning only angular momentum in the rest frame) and the classical derivation should hold true in that limit.
I am also not sure what we learn from a series expansion of the Dirac equation in terms of c. The Dirac equation is an operator equation for free particles and as such quite useless as it does not describe real world electrons.
On the other hand solutions of bound state problems in QED are tremendously complicated and rely on ad hoc approximations (e.g. which diagrams to retain in Bethe Salpeter equation or alternative formulations).
Of course that is only my impression from someone who has never delved more deeply into that matter.
Strocchi is a mathematical physicist and it may be that there is really some valid approximation to derive the spin orbit coupling up to c squared from first principles.

 

[Bill_K]

Jano, As I said, models are a good thing! The development of physics has been based on models. But not every ad hoc explanation deserves to be called one.

Generally a model is an approximation that is valid in some limited domain. "The Earth is flat" is wrong, but provides a useful model. The idea that "an eclipse is a dragon's bite" is not useful in any way. "The nucleus is a point" is a useful approximation. "The electron is a spinning sphere" is not. (a) because it has no valid consequences and (b) there is no way to extend the idea consistently with special relativity. The "positrons are holes in the Fermi sea" suggestion violated charge conjugation, and so on.

As you point out, DrDu, the electron is correctly described by QED, with field operators. But a useful approximation is to use the Dirac equation as an equation for a C-number field. In fact this approach is typically used as the starting point for bound state calculations in QED - one seeks the spinor wavefunction solutions and energy levels in the Coulomb field of the nucleus.

A further approximation is the nonrelativistic limit of the Dirac equation, the Schrödinger-Pauli equation, i.e. the Schrödinger equation for a Pauli 2-spinor. You can trace through the analysis in the book I mentioned, and see that the spin-orbit term arises from the commutator of two sigma matrices, which is not very illuminating. And so the wish for a more intuitive explanation is certainly well motivated.

But I have never seen one that had any valid physical basis, and I seriously doubt there is one. They are all of the "dragon bites moon" variety.

 

[wc2351]

Thank you all for your detailed replies. I'll try to check out some of the suggested references once my final exams are over.

 

[DrDu]

Thinking about it, maybe it is easier to consider spin orbit coupling in scattering. In pure form it should be responsible for the scattering of a neutron from a charged spinless boson, but also make a contribution to e.g. Moeller scattering of electrons from electrons.

 

[dextercioby]

[...] Another one, more common in the spin-orbit context, is to imagine that the electron stands still and the nucleus revolves about it. The nucleus then produces a B field which the electron spin reacts to, causing the spin-orbit term. Now if you know any physics at all, you know that it is not valid to take over the Dirac equation to a noninertial frame. Nevertheless, this idea is in many textbooks, and has been handed down from generation to generation of students because it is so intuitively appealing. [...]

The idea you speak of is usually present in atomic physics books, or in QM books when discussing applications of perturbation theory. Incidentally or not, this "picture" with the nucleus in motion generating a magnetic field to which the electron's spin magnetic moment "couples" is correct from the perspective of classical physics, just as assuming the classical Hamiltonian for the nonrelativistc H-atom is: KE_nucleus+KE_electron+static Coulomb potential energy. In exactly the same manner, both models are <stamped> with the (more or less correct) quantization rules leading us from classical mechanics to subchapters or sections of quantum mechanics. The spin vector becomes an operator, the orbital angular momentum likewise, etc. So I <swallow> this <idea> in the same way I swallow the Schrödinger equation for the harmonic oscillator in 1 spatial dimension.

No one ever mentions Dirac equation in the same paragraph/chapter with <noninertial frames of reference>. It would be absurd. So no worries for a confusion. Someone correct if I'm wrong, but the Dirac's equation for the H-atom's not fully solved, so bringing noninertial frames of reference into quantum mechanics done with Dirac's equation is wild, even wrong.

The bottom line should be: the spin-orbit interaction is purely a quantum mechanical and specially relativistic effect, just as the electron's spin value of 1/2 has the deepest root in Galilei invariant quantum mechanics and not in Dirac's theory.

P.S. In curved 4D spacetime, though, one can incorporate Dirac spinors withing the realms of quantum field theory.

 

[DrDu]

No one ever mentions Dirac equation in the same paragraph/chapter with <noninertial frames of reference>. It would be absurd. So no worries for a confusion. Someone correct if I'm wrong, but the Dirac's equation for the H-atom's not fully solved, so bringing noninertial frames of reference into quantum mechanics done with Dirac's equation is wild, even wrong.

I am not sure about that. Transforming the Dirac equation to a classical non-intertial frame is not a big deal.
What is more complicated is transforming to a frame that depends on the quantum coordinates of the electron. However I think this is exactly the content of the Foldy Wouthuysen transformation.
It can be seen that the alpha matrix is the velocity operator in the Dirac representation. The FW trafo aims at eliminating alpha from the equation, which in turn corresponds to a transformation to the rest frame of the electron.

 

[Bill_K]

The exact solution of the Dirac equation in a central Coulomb potential is discussed "in all the old familiar places", e.g. Messiah vol II, and Bjorken and Drell vol I. An important conclusion drawn from this is that even these exact relativistic solutions are still twofold degenerate, (2S_½ and 2P_½ for example) and the Lamb shift is required to completely split the levels.

One reason it makes no sense to talk about the Dirac wavefunction in a noninertial frame is that it would require a global accelerating coordinate system, which is not a well-defined concept. A noninertial frame can be used to describe a point particle but not a field.

 

[dextercioby]

The Dirac's equation for the H atom is solved in the infinite nuclear mass approximation, thus it's not a valid description of the H atom. I was referring the so called 2 centric Dirac equation which would suit the H atom, as it has a (pointlike) nucleus and a(n). (pointlike) electron

 

[DrDu]

One reason it makes no sense to talk about the Dirac wavefunction in a noninertial frame is that it would require a global accelerating coordinate system, which is not a well-defined concept. A noninertial frame can be used to describe a point particle but not a field.

Hm, I don't see a problem in transforming e.g. to a rotating coordinate system.

 

[Bill_K]

The Dirac's equation for the H atom is solved in the infinite nuclear mass approximation, thus it's not a valid description of the H atom

This gets crazier and crazier. Can you say, "center of mass coordinates and reduced mass"?

 

[Bill_K]

I don't see a problem in transforming e.g. to a rotating coordinate system.

A global rotating coordinate system is not a valid concept in relativity since at large distances the tangential velocity exceeds c.

 

[DrDu]

A global rotating coordinate system is not a valid concept in relativity since at large distances the tangential velocity exceeds c.

I don't see a problem with this. The spot of a lighthouse will also move with superluminal speed at large enough distances. Yet this does not mean that relativity is violated.
Anandan and Suzuki consider the Dirac equation in a rotating system:
http://arxiv.org/abs/quant-ph/0305081
And here for a semi-classical wavepacket a FW trafo is shown to lead to the usual SO coupling in the rest frame:
http://www.google.de/url?sa=t&rct=j...sg=AFQjCNFvvrYcdgV4-K_T7F5cQPgNdjf2Vg&cad=rja

 

[dextercioby]

This gets crazier and crazier. Can you say, "center of mass coordinates and reduced mass"?

I can, but with no use. The trick of separating the dynamics (COM virtual particle vs. reduced-mass virtual particle) in the 2 body problem in the context of classical mechanics (the trick works and leads to correct quantum results) doesn't work for the Dirac equation as explained by W. Greiner in his book <Relativistic Quantum Mechanics. Wave Equations>, page 250 in the 3. edition. So my statement you quoted is accurate.