M2L9k.txt

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Let's immediately go to the high field limit.
The high field limit means that the electronic Zeeman
energy is much larger than the hyperfine coupling.
And that means when we treat the problem,
we first take care of the big contributions
to the Hamiltonian.
We try to solve it, if possible, exactly.
And then the weaker term can be added perturbatively.
So now we are in a situation that the Zeeman coupling
is the big term, and the hyperfine coupling
is the weaker term.
So in other words, what comes first now is the Zeeman energy.
So we are not coupling the electronic angular momentum and
the nuclear angular momentum to total angular momentum,
because this coupling is weak.
We rather say that the electronic
and the nuclear angular momentum align
with the magnetic field.
We quantize along the direction of the magnetic field.
And then later we add the hyperfine coupling
in a perturbative way.
So B0 is now quantizes J along the direction
of the magnetic field.
And therefore we use, as a good quantum number
mJ, the projection of J on the external magnetic field axis.
So this takes care of J. J and mJ are good quantum numbers.
What about I?

Dos the nuclear angular momentum and the nuclear magnetic moment
strongly couple to the magnetic field?
Well, the answer is yes, but the argument
is a little bit more subtle.
The direct coupling of the magnetic moment
of the nucleus to the magnetic field maybe weak.
So that may be smaller than the hyperfine interaction.

So then you would say, naively well, the nuclear angular
momentum should not couple to the magnetic field.
It should first be coupled through
the hyperfine interaction.
And the hyperfine interaction is I dot J.
However, J, which coupled strongly to the magnetic field,
because it couples through the Bohr magneton,
has already been coupled to the magnetic field.
So therefore the hyperfine interaction, which is I
dot J, is now modified, because J couples
to the magnetic field, which means we have to project it
onto the magnetic field axis.
So therefore the nucleus now experiences
an electronic magnetic moment, and electronic angular
momentum, which has really been coupled to the z-axis.
And therefore, the hyperfine interaction
is also coupling the nuclear angular momentum to the z-axis.

So therefore the result is that it is now
this indirect coupling.
You couple the electron angular momentum to the z-axis,
and the electron angular momentum
couples the nuclear angular momentum to the z-axis.
So we are now - this quantizes the nuclear angular momentum
along the z-axis, which means that n sub I
becomes a good quantum number.
Anyway, maybe the result is even simpler than the explanation.
Our Zeeman Hamiltonian now simply means
that we have an external magnetic field,
and the electron couples to the magnetic field.
So what matters is the projection mJ.
The same happens for the nuclear magnetic moment.
And now we have to add the hyperfine interaction, which
was originally I dot J. But since I and J are projected
on the z-axis, what is really left over only mI and mJ.
I could have gotten this expression immediately
by just telling you J and I no longer couple to F.
This is destroyed by a strong, magnetic field.
And the good quantum numbers are J and I, and their projection,
mJ and mI.
And then just writing down the expectation value
of the Hamiltonian in this basis
would have immediately given me this result.
But I wanted to give you sort of the more
mechanistic explanation what's going on inside the atom and what leads to this result