M2L9n.txt

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So let me show you the weak field, the strong field limit,
and do a graphic interpolation.
What I've chosen as an example is the case
of a doublet S_1/2 ground state and a nuclear angular
momentum of 3/2.
And examples for that is sodium and rubidium 87.
So it's moved up by 3/4, the other state
is moved down by 5/4.
And since F equals 1, has three components,
F equals 2 has 5 components, the center of mass is preserved.
We have calculated the g-factor for those states.
And the g-factor tells us what is this structure
in weak magnetic fields.
So this is the weak magnetic field solution.
At high magnetic field-- well you
know what you have at high magnetic field.
You have a single electron, which
can be spin up and spin down.
So if you have an electron which has spin up and spin down,
it pretty much is linear Zeeman shift for spin down
and linear Zeeman shift for spin up.
So therefore, the energy levels will evolve like this.

So another words, we have 8 levels.
We have the structure at weak magnetic fields.
At high magnetic fields of course, we also have 8 levels,
but they pretty much group into spin up of the electrons,
spin down of the electrons.
And then for this, a smaller hyperfine structure
on top of it, because now the nuclear spin
can have various orientations.
And I equals 3/2 state has four orientations, so
therefore electron spin up, and electron spin down.
We obtain 4 sub levels.
And if you ask how did I connect it?
What I've connected, the quantum numbers
are such that what is here, the states are labeled
by mI and mS. And here they're labeled by mF,
but mF equals mI plus mS. So this
is how you correlate the states in the high field case,
to the states in the low field case.
Here, we have mJ equals 1/2.
Here we have the electron spin minus 1/2.

These 4 levels are now four different quantum numbers
for the nuclear angular momentum,
which are minus 3/2, minus 1/2, plus 1/2, and plus 3/2.
And this is what I meant by avoided crossing.
You at some point I think, draw it yourself
put the quantum numbers on it.
Look at it, and you learn a lot by doing it.
But what you realize also when you
solve the Hamiltonian, that this structure can
be explained the following way.
You will always find you have states which are stretched,
where there is only one state which has the maximum angular
momentum, these are called the stretched states.
And then the other states you always
will find two states which have the same total mF.
And those two states avoid each other.
So you can say, just pointing on two states,
those two states, let's just assume the have same mF,
they undergo an avoided crossing.
And that's exactly what you get out
of the diagonalization of a 2 by 2 matrix.
So this whole diagram can be understood
by you have stretched states which form a 1 by 1 matrix.
There is no recoupling taking place.
And then you have three pairs of states.
You have three pairs of states which form 2 by 2 matrix.
And in each pair, if you would now focus on it,
you really see the avoided crossing, typical for a 2 by 2 matrix.