# Fine and hyperfine structure

1. Dec 25, 2007

### ehrenfest

1. The problem statement, all variables and given/known data
I am trying to see if I understand these corrections to the hydrogen atom:

To explain the fine structure correction, we go to the electrons frame of reference, in which the proton is moving with velocity v. This produces a magnetic field according to Maxwell's equations the points perpendicular to the plane of the protons "orbit". This magnetic field interacts with the electrons magnetic moment, so that the electron lines up its magnetic moment antiparallel to the magnetic field.

Why is it only the spin magnetic moment of the electron that interacts with the magnetic field generated by the proton?

EDIT: The answer is that the electron does not have any orbital angular momentum in its own rest frame. In the rest frame of the electron, the proton acquires the orbital angular momentum that belonged to the electron in the proton's rest frame.

The interaction energy gets added to the Hamiltonian as $$H_1 = \mu_s\cdot\vec{B}$$. This gets multiplied by the Thomas correction of 1/2. Then we use perturbation theory to calculate the first-order energy perturbation which depends on the principal quantum number, total angular momentum, the orbital angular momentum. When we then add the relativistic correction, the dependence on the orbital angular momentum cancels out.

This is also called the spin-orbit interaction because it is an interaction between the "orbital" angular momentum of the proton in the electrons frame of reference and the spin angular momentum of the electron.

The hyperfine correction arises from the spin magnetic moment of the proton which interacts with creates a magnetic field as the proton "orbits" the electron in the electrons frame of reference.

This is also called the spin-spin interaction because it is an interaction between the spin angular momentum of the proton and the spin-angular momentum of the electron.

If the proton had spin 0 instead of spin 1/2, then the hyperfine structure would not occur but the fine structure would still exist, right?

2. Relevant equations

3. The attempt at a solution

Last edited: Dec 25, 2007
2. Dec 26, 2007

### Gokul43201

Staff Emeritus
First off, you must bear in mind that this is only a hand waving argument (or so it seems to me). The electron doesn't really "orbit" the proton. And even if it did, you would be transforming between non-inertial reference frames, so there's no good reason to naively expect the numbers to come out right. But they do come out approximately right! But even within this argument, there is no need to invoke orbital motion for the hyperfine interaction. You get a magnetic field directly from the spin (magnetic moment), without any need for orbiting.

Last edited: Dec 26, 2007
3. Dec 26, 2007

### ehrenfest

Yes, my mistake.