Spin-Orbit Coupling

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Let me talk some Physics (before it is too late):

This is about the historically significant semi-classical paradox of spin-orbit coupling and its resolution.

Let us take the simplest hydrogen atom. It has a proton with a charge e sitting in an inertial frame called the lab. And an electron of charge e circulating around the proton.

Seen from the proton frame, there is an electric field produced by the proton which interacts with the electron charge. This leading first order term in the Hamiltonian gives the Hydrogen spectrum without the fine structure.

The fine structure is explained by the spin-orbit coupling: the electron has a tiny magnetic moment which interacts with the proton field.

Now, in the rest frame of the proton (lab) the proton produces only an electric field at the site of the electron (forget the tiny proton magnetic moment which gives the hyperfine splittings).

And this electric field can't interact with the magnetic moment of the orbiting electron.

So, they jumped to the electron frame. Here the proton circulates about the electron. The orbiting proton's charge produces a magnetic field at the site of the electron. This magnetic field interacts with the magnetic moment of the electron giving the spin-orbit term in the Hamiltonian. But if we put the experimentally observed value of the anomalous magnetic moment of the electron (taken from the Zeeman splittings) there is a factor of 2 which spoils the fine structure.

So, Thomas observed that the electron frame to which we jumped is not an inertial frame, since the electron is undergoing a centripetal acceleration.

He then showed that in such an accelerating frame there is what we now call the Thomas Precession.

If we take this second order relativistic term into account, we get a compensating factor of 2 which nicely explains the observed fine structure.

On seeing this Einstein remarked that it is wonder that a second-order relativistic correction gives a mighty factor of 2 (instead of a a decimal term).

Then came Dirac who worked in the lab frame of the proton.

In his fully quantum theory, there is no anomalous magnetic moment...the factor of 2 vanishes, giving its right value.

Also there is no need to jump frames and no need of any Thomas Precession term.

Both the factors of 2 vanish giving the right Zeeman splittings and the right fine structure marvelously.

Let us now get back to our semi-classical thing.

If something like a fine structure can be observed in the electron frame, it should also be observed in the proton frame (inertial lab).

What is it that we missed?

In the proton frame there is only its electric field. This electric field can't interact with the electron's magnetic moment as we have seen above.

But the electron is circulating about the proton. And it is a magnetic dipole.

A moving magnetic dipole creates an electric dipole, if we remember our special relativity correctly.

It is this relativistic second order electric dipole that gives the coupling with the electric field of the proton at the site of the circulating electron.

So, the spin-orbit term between the electric field of the proton and the relativistically induced electric dipole moment of the electron should give us the right fine structure without the need of jumping to the electron frame and invoking the Thomas term as was done historically.

To my knowledge, this beautiful point is discussed only in an obscure fat Russian Book titled: 'Problems in Electrodynamics' by V. V. Batygin and I. N. Toptygin. Only one copy of this excellent book was available in the CL of IIT KGP.

Why is it excellent?

Because it also has Solutions {;-}

The book is out of print.

I returned it to the CL upon my retirement to get my Library 'no-dues' for getting my pension which unfortunately I needed very badly...


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