M2L9g.txt

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So our next topic is now atoms in external magnetic fields,
and the first chapter is on fine structure and the Lande'
g-factor.
But maybe more colloquially, atoms in external fields
means that we add one more vector to the mix.
In fine structure, we had orbital angular momentum
and spin angular momentum, and we
discussed how spin orbit coupling eventually couples
L and S to J, and so on.
But now we extend the game by one more vector, B,
an external magnetic field.
And it really becomes sort of a player in the game,
because you know that if you have spin orbit coupling,
we use the vector model that L and S couple and precess
around the axis of J, the total angular momentum.
So the game we played when we couple angular momentum is
that angular momentum couple, they
precess around an axis that involves new quantum
numbers and so on.
But now, we can add a new axis, a quantization
axis with a magnetic field, and then
angular momenta will precess around the magnetic field.
And there maybe actually a conflict,
that for strong magnetic fields, the precession
is different than at weak magnetic fields.
And this is what we want to discuss now.
So it is a game of L, S, and B. So what
we are adding with magnetic fields is
we are adding one term to the Hamiltonian, which
is the Zeeman term where we have an external magnetic field
and we couple with a magnetic moment.
So the one thing which makes it interesting
when we discuss fine structure is the following,
that we have actually two components to the angular
momentum of the atom, which is the spin
and the orbital angular momentum.
And the two have different g-factors.
So it's not that angular momentum,
that the magnetic moment is just proportional to the angular
momentum, if the angular momentum comes from spin,
it has a different weight than when the angular momentum comes
from orbital angular momentum.
And this is what ...  we want to now understand.
We want to understand, we
want to determine what is the magnetic moment
of the atom when the angular momentum has
two different sources.
And what we will find is we will find that there's
a Lande' g-factor, which is sometimes 2, which is sometimes
1, or which it sometimes has a value in between, depending
how S and L are arranged with respect to each other.
So our Hamilitonian is the Hamiltonian for the atom.
We know that we have fine structure, we discuss that,
which couples L and S. And now, we
have the--doesn't want to move--
we have a magnetic moment due to the spin,
and due to the orbital angular momentum,
And that couples to the external magnetic field B.
So in other words, if you had no fine structure coupling,
if S and L would not be coupled, the answer
would be very simple.
S just couples to the magnetic field,
gives the Zeeman effect, with a g-factor of 2,
and L couples and gives it-- and couples to the magnetic field
with a g-factor of 1.
But now the two are coupled with respect to each other,
and if you have LS coupling, the projection of S and L
on the z-axis is not a good quantum number anymore.
So therefore we have two different terms,
which are diagonal in two different bases.
And that's what we want to discuss.
OK, so the g-factor of orbital angular momentum is 1,
the g-factor of the spin is 2, or if you
want to include the leading correction due to qed
is the fine structure constant over 2p-- over 2pi.
OK, we did discuss that the fine structure can
be related as the Zeeman energy of the spin
in a magnetic field, which is created
by the electron due to it's motion.
Or if you take the frame of the electron,
the electron sees the proton orbiting
around that creates a magnetic field,
and this magnetic field couples to this spin.
So therefore, we can associate fine structure
with an internal magnetic field inside the atom.

And this internal magnetic field is rather large,
it's on the order of one Tesla.
So therefore, for our discussion of the Lande' g-factor
and fine structure in applied magnetic fields,
we will assume that we are in the weak field limit, where
the fine structure term, the first term,
is much larger than the Zeeman term.
Of course, if you use very strong magnets,
you can go to the high field case,
but I will discuss explicitly the transition from weak field
to high field for hyper-fine structure.
And the phenomenon for fine structure
is completely analogous, it just happens at much higher fields.
So anyway, I will discuss the high field case
and the transition to the high field case
with a much more relevant example
of hyper-fine structure.
And you can immediately apply to fine structure if you want.
So if I wanted to calculate-- if i
want to solve the problem-- calculate the Lande' g-factor,
I could directly calculate just one matrix element
and it would be done.
So all I want to know is, what is the Zeeman
energy-- the Zeeman energy divided by the magnetic field
is a magnetic moment.
And I can-- and since I assume that I'm in the weak field
limit, I can simply use the quantum numbers S, L, and S
and L coupled to J. And the magnetic quantum number is MJ.

So by simply calculating this expectation value, I'm done,
and I've solved the problem.

However, I want to do the derivation using the vector
model, because it provides some additional insight.
So in the vector model, we have L and S. L and S
coupled to J to the total angular momentum.
So L-- and in the vector model, we
assume that L and S rapidly precess around J,
and therefore the only thing which matters
are the projetion-- only the projections of L and S
on the J-axis are important.
So you can say if you have a rapid precession of L and S
around J, the transverse components rapidly average
out and do not contribute.
So therefore, our Zeeman Hamilitonian
has to be rewritten in the following way.
The Zeeman Hamiltonian was the magnetic moment
times the external magnetic field, with a minus sign.
But what matters are only-- what matters is the projection
on the J-direction so we do
the projection in this way.
And also, in the end what matters
is since the magnetic moments are aligned with J,
it is now the scalar product
of the magnetic field with J.
So in the vector model, we calculate the Zeeman energies
in that way.
But just to mention that if you don't like the vector
model and the assumption of rapid precession,
just take this matrix element, it's exactly the same result.
In other words, I give you an intuitive picture
what is inside those matrix elements.
OK, so let's evaluate that.
Let me factor out the Bohr magneton
and we have J squared taking 1 of each bracket.
And now, the magnetic moment, assuming
that the g-factor of the spin is 2,
the magnetic moment is the Bohr magneton times L,
the g-factor of L is 1, plus S, but the g-factor of S is 2.
So this is now the magnetic moment accounting
for the two different g-factors we projected on the j-axis.
And the second bracket, B.J becomes the value
of the magnetic field.
We assume that the magnetic field points
in the z direction, so therefore it
is the z component of the total angular momentum.

So let me collect the simple terms.
Now L plus 2S can be written, because L plus S is
J, can be written as J plus S. So now we have the product of J
with J, which gives us J squared,
and then we need the product of S and J. And as usual,
we can get an expression for that
by using the summation of angular momentum,
and if we square it on the right hand side
we have the scalar product of J and S,
but we have now the scalar product of J and S
expressed by L squared J squared S squared.
J squared plus S squared minus L squared divided by J squared.
We're just one line away from the final result.
Jz is a good quantum number, its mJ,
the projection of the total angular momentum on the z-axis.
And on the bracket here is now the result, the famous result,
for the Lande' g-factor.
So we have J squared over J squared which gives us 1,
and then I simply put in the quantum numbers
for J squared S squared L squared, which
is J times J plus 1, plus S times S plus 1,
minus L times L plus 1.
And we divide by 2 times J times J plus 1.
The Zeeman structure in a magnetic field is
the Bohr magneton times the magnetic field times
the angular momentum in the J direction,
but then we multiply with the g-factor.

If we don't have spin, these are two
limiting-- these are now limiting cases.
if you do not have spin, then the only ingredients
the only contribution to angular momentum
is orbital angular momentum, and we have a g-factor of 1.
So the Lande' g-factor simply becomes gL.
In the case we don't have angular momentum,
and you can just evaluate this expression for L equals 0,
you find indeed that the g-factor is 2.
But it can have different values depending
on the-- it depends on the atomic structure.
So that's all I want to say about fine structure in a magnetic field.