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M2L9j.txt
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M2L9j.txt
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#
# File: content-mit-8-421-2x-subtitles/M2L9j.txt
#
# Captions for 8.421x module
#
# This file has 113 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
The next step is now hyperfine structure.
So we're adding one more vector to the game.
So we add angular momentum of the nucleus.
So now the game we play is not only L and S. We have I and B.
So it's the game of the four vectors
and eventually how they precess around each other.
And that gives rise to the structure of hyperfine levels
in the external magnetic field.
We assume that L and S have coupled to J.
So we have actually the coupling of J, I,
but now we have an external quantization axis
and Zeeman energies due to the external magnetic field.
And of course, in hyperfine structure, we discussed that.
I and J-- I and J are no longer conserved angular momenta
because they couple to a total angular momentum, which is F.
So our Hamiltonian is the sort of Hamiltonian
without any hyperfine and fine structure.
Then we have the hyperfine coupling, which couples I
and J with the product, I dot J. And then we
have an external magnetic field, which
couples to the magnetic moment of the electron.
And this maybe a smaller term, but we
can easily carry it with us.
There's also a coupling with the magnetic moment of the nucleus.
So in this case, because the hyperfine structure
is smaller than the fine structure,
I want to discuss both the weak field and the strong field case
because magnetic fields of a few hundred gauss
may actually take you to the high field limit.
So I want to discuss both the low field and the high field
limit.
The low field limit implies
that the Zeeman energies are much smaller
than the hyperfine splittings.
And the sort of way how we describe the system
is that J and I couple.
So J, which is responsible for the magnetic moment,
precesses around F, the total angular momentum.
But then the total angular momentum
processes around the magnetic field.
So in other words, you assume that the coupling between J
and I is so strong, if they couple to F,
the magnetic field is not breaking up
the coupling between J and I. J and I together form F.
And this hyperfine state, it's magnetic moment,
precesses around B. So this is sort of the picture
and you have to get used to it.
J precesses around F, and F precesses
around the external magnetic field, B naught.
But again, if you don't like the precession model,
just calculate the quantum mechanical energy levels,
diagonalize the Hamiltonian.
The answer is Identical.
So the Zeeman Hamiltonian couples to the magnetic field.
And we have two contributions to the magnetic moment,
the electron and the nucleus.
And in the weak field limit we use a treatment
which is almost completely analogous
to the treatment we use when we derive the Lande g-factor.
So in the weak field limit we can treat the Zeeman Hamiltonian
perturbation theory, and it's exactly analogous
when we added a weak magnetic field to the fine structure.
So in the vector model, we have the coupling of J and I to F.
And the relevant term in the Hamiltonian
is we have the magnetic moment of the electron, which
is proportional to J, and it couples to B.
So this relevant term-- and this is fully analogous to what
I did five or ten minutes ago, has
to be replaced due to the presence of the nuclear angular
momentum, we have to project everything
on the axis of the total angular momentum, F.
So therefore, the Zeeman Hamiltonian,
it had the contribution to the magnetic moment
due to the electron and due to the nucleus.
This is proportional to J, but now we
have to project it onto F. And similarly, the magnetic moment
of the nucleus is proportional to I.
But what matters is the projection on F.
And since I've factored out the Bohr magneton,
the magnetic moment of the nucleus
is proportional to the nuclear magneton.
I have to account for the ratio.
And what matters now is the projection of F and B naught.
So therefore, collecting all of the terms,
we have the Bohr magneton, which
is setting the scale of the interaction.
The last term is the magnetic field.
But the projection of F onto the magnetic field
gives us the MF quantum number.
And all the rest is called the G factor
of the hyperfine structure.
And the G factor of the hyperfine structure
is-- let me just simply find and neglect
the small contribution-- it's a thousand times smaller--
of the nuclear magnetic moment.
But if you want, you can easily include it.
With this approximation, the G factor
of the hyperfine structure is this.
So it's proportionate to the Lande g-factor we just derived.
And then using exactly the same thing,
you have J dot F. You can express it now
by the quantum numbers of F square, J square, and I square.
You find the final result, what are
the G factors of the hyperfine levels,
of the hyperfine states.
So this is the hyperfine structure
of atoms in weak magnetic fields.