M2L10a.txt

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I want to give you sort of a summary
that you see the bigger picture.
That you see beyond the details.
That what I actually taught you about atoms in magnetic field
is a paradigmatic example of quantum
physics:
what happens if you have two different terms in Hamiltonian
and you have to interpolate between one and the other.
Let's summarize as follows.
What we have is a Hamiltonian
and it has one part, 
the hyperfine interaction which depends on I dot J.
And then it has an external magnetic field part.
And what couples to the magnetic field which we assume,
is in the J direction-- in the Z direction,
are the Z components of the magnetic moment.
And the Z component of the magnetic moment
are proportional to the MJ, or MI quantum number, 
to the magnetic quantum number of the atom and the nucleus.
So in a weak field-- in a weak field
it is the hyperfine structure which dominates.
So in a weak field we first solve
for the hyperfine structure.
And then we use the eigenfunction
of the hyperfine structure.
And the eigenfunction of the hyperfine structure
have the quantum number F where J and I have coupled to F.
And then we treat the magnetic Zeeman Hamiltonian
perturbatively and that led us to the formulation of the Lande
g -factor gF.
The other case is the strong field case where
the magnetic field dominates.
Then we say-- OK.
Then we simply solve for the hyperfine structure
in the magnetic field.
I know it's one of those words in quantum physics,
or in physics, or maybe even in life.
First things first.
You should first take care of the big things,
and this is now the magnetic field.
And since the magnetic field Hamiltonian
is diagonalized when we have eigenfunctions where MJ and MI
are good quantum numbers.
This is sort of-- if you ignore the hyperfine coupling,
this is the exact diagonalization of the Zeeman term.
And then in perturbation theory we
look for the hyperfine coupling.
And well we do perturbation theory
in eigenfunctions with MI and MJ.
And that means if the I dot J term it is only the component
MI MJ which remains.
So I've given you those two cases.
Now what you should also learn here in this example
is the language which we use.
And sometimes I would say the language can be more
confusing than the equations.
What we say here is we say that the angular
momenta of the electron and of the nucleus
are coupled to the magnetic field axis.
They are quantized.
The approximate eigenstates are those
which have a specific quantum number in the Z direction
because a magnetic field points to the Z direction.
So we're saying I and J are strongly coupled to the Z-axis
by the magnetic field.
And then we treat to the coupling of I
and J with each other in perturbation theory.
Whereas, in the previous case, we
said I and J strongly couple.
When I and J strongly couple, F becomes a good quantum number.
And that means I and J both precess
around the axis of the total angular momentum, F.
And therefore, we say I and J couple to F.
And then we solve the coupling for this coupling of F
to the magnetic field in the second step.
I hope you sort of see that there's sort of two
limiting cases we take.
We can exactly diagonalize one term
and then we perturbatively add on the result
for the second term.
Of course in the age of computers
I could have simply written down for you an Hamiltonian
and say, well, it has to be-- it has
to be numerically diagonalized, what I discussed instead
were the two limiting cases.
This discussion now allows me to discuss
what happens when we go to even stronger fields.
Well when we go to even stronger fields then
we may have fields which are even
stronger than the fine structure coupling.
The coupling of the orbital angular
momentum of the electron, and the spin angular momentum to J.
And well without even any derivation, which is obvious,
you know what happens now is that each component which
provides magnetic moment.
The spin, the orbital angular momentum, and the nucleus.
The dominant term for each of them
is the coupling to the magnetic field.
So in strong magnetic fields these are the eigenstates.
The eigenstates are labeled by MI, ML, MS.
So we've taken care of the strong coupling term.
And now, in addition, we are now treating
in perturbation theory.
Some fine structure coupling, but the quantum numbers
are already distributed, ML MS. There
is a coupling between MI MS and a coupling between MI and ML.

So this is sort of the limiting cases.
But as a general illustration of quantum mechanics,
I thought that this was a nice example for an Hamiltonian,
where we have different scalar products like B times S, B dot
L, S dot L, I dot J. The question is, how do we
take care of those different parts
because they do not commute.
Of course a theorist would just say I simply diagonalize it
and that's it.
But if you want to develop intuition,
then you have to discuss the limiting cases.
And in particular the approach which
allows an intuitive understanding
is first things first.
And we treat -- we first treat the stronger terms and then
the weaker terms.
And we can do-- we can quantitatively
derive -- analytically derive expressions, for instance,
for the Lande g -factor in this vector model.
This vector model says that-- assumes, so to speak,
that a state which has an eigenfunction of MJ
rapidly precesses around the Z axis.
And this vector model actually allows
you to do easy calculations without
Clebsch-Gordan coefficient.
So the concept of the vector model
is rapid precession for transverse components
and projecting of vectors onto the axis around which we
have rapid precession.
But this is simply a tool to do calculations
without the explicit use of Clebsch-Gordan coefficients.

OK so this is what I wanted to tell you
about atoms in magnetic field.
Any questions?