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PROFESSOR: Good morning.

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Today's lecture will
deal with Zeeman effect.

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And then we'll get started with
a semi-classical approximation.

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So Zeeman in effect
is the last topic

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we do with respect
to the hydrogen

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atom and the corrections
of perturbation theory

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and with WKB or the
semi-classical approximation,

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we begin a new chapter in 806.

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Zeeman effect.

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So this is an
effect having to do

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with an atom in
a magnetic field.

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It was discovered by a Dutch
physicist, Peter Zeeman,

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who lived from 1865 to
1943, who was a Dutch.

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The work was
actually done in 1896

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at a time where there was
very little idea of quantum

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mechanics to be, and
he got a Nobel Prize

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for in the year 1902.

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So what he discovered
was the spectral lines

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seemed to split in the
presence of a magnetic field.

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It's an old result,
therefore, 100 years old.

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Its explanation
and understanding

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took about a couple of decades,
because you couldn't do it

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without quantum mechanics.

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So nobody could
quite figure out what

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had happened, but was very,
very important in its time.

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It still remains very important.

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People use the Zeeman
effect all the time.

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In fact, it's used
nowadays in studies

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of astrophysics, studies
of the sun, the sunspots.

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You know, there are
places in the sun,

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where the temperature
is a little lower,

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and that's places where the
magnetic field lines in the sun

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sort of breakout from the
interior to the exterior.

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And it's interesting,
because the sun produces

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this cycle of
solar spots, and it

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happens, because the sun
doesn't rotate uniformly,

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has a different rotation
speed in the equator,

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faster than in the poles, so
the magnetic field lines that

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go from north to south in
the sun get tangled up,

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and they start breaking up
and doing all these things.

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So people want to know
what is the magnetic field

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in the solar sunspot.

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And in fact, they observe
the weak magnetic fields away

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from the solar
sunspots, and then they

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see the spectrum of an atom.

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And suddenly, as you move
inside the spot, the field,

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the spectrum, the lines split.

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And they can measure
the magnetic fields

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very accurately.

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They're of the order of 3,000
Gauss, 2,000, 3,000 Gauss.

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And it's pretty
interesting work.

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So this Zeeman effect
remains very important.

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So what is the Zeeman effect?

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We have Zeeman effect here.

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We have the magnetic field
interacting with the electron,

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and the electron now has
two magnetic moments,

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a magnetic moment associated
with the orbital motion.

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This looks completely
like the classical formula

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of the magnetic moment due to a
particle that goes in circles.

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It produces a current,
and that current

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is proportional to the angular
momentum of the rotating

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particle.

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And of course, there
is the magnetic moment

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due to the spin that
has a factor of 2 here.

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This g factor, we've
discussed before.

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S.

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So H Zeeman, you put an
external magnetic field

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into the atom, a constant
uniform external magnetic

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field, and you have
a minus mu dot b,

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so you have a interaction
mu l plus mu s,

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dot B. So this is e over 2mc.

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That's the typical factor here,
and a recognizable l plus 2s.

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So it's not l plus s.

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It's l plus 2s times b.

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And many times, we think
of B, align the axis

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so that it is in
the z direction.

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So this turns out to be e
over 2mc lz plus 2sz times

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B. So this is the
Zeeman Hamiltonian.

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But this is part of
a story of an atom.

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So if we want to think of
the hydrogen atom properly,

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we must consider and reconsider
what was the Hamiltonian there.

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And we had an H for
the hydrogen atom.

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That was an H 0.

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That was the familiar
one, p squared

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over 2m minus e squared over r.

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Then we had a fine
structure Hamiltonian,

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delta H, fine structure.

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Let's put fs for fine structure.

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Those were the relativistic
terms, the Darwin term,

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and the spin orbit coupling.

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The three of them
constituted what we call

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defined structure Hamiltonian.

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And now we have a Zeeman effect.

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I probably should
go delta H Zeeman,

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because it's an addition here
to the term we had before.

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It refers to what I
call just H Zeeman.

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But in the context
of the hydrogen atom,

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we should call it
delta H Zeeman.

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And now, we have to rethink.

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And the reason the Zeeman
effect is non-trivial for us,

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and it's a very interesting and
somewhat challenging example

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of what we have to do
in perturbation theory,

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is that we cannot forget
about the fine structure.

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So if it would be just this,
it would be kind of simple.

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But we have the whole thing.

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So we have to make an
approximation sometimes.

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And we're going to consider
two interesting cases,

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the very weak Zeeman effect and
the very strong Zeeman effect.

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I will not consider the
intermediate Zeeman effect,

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not because it's
not interesting,

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but because there's very little
you can do to simplify it,

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and to think about it.

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You basically have to
go ahead and diagonalize

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the large matrix.

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So while it's important, and
if your life depended on this,

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and your research dependent
on this, you would do it.

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For us, we have a lot to
learn from the weak case

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and the strong case,
lots of concepts,

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and we'll leave the
intermediate one for later.

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So how can we decide how
to treat these terms?

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Well, we should look
physically at what's happening.

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We have a magnetic field,
an external magnetic field.

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But the fine structure
constant taught you

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that there is something like
an internal magnetic field

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in the atom.

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It's that magnetic field
responsible for spin orbit

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coupling is that magnetic field
that the electron sees when

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it's going around the proton.

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There's a static electric field,
but whenever you are in motion,

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a static electric
field in the lab

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also has a magnetic
field by relativity,

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and you do see a magnetic field.

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You could also imagine it
as you are the electron,

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and the problem is
going around you,

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and creates at the center of
the loop, a magnetic field.

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In any case, you've looked
at that magnetic field.

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And we can call it the
internal magnetic field

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due to spin orbit.

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As one exercise in
the homework, you've

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been asked to estimate the
value of that internal magnetic

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field.

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Is it a Gauss, 1,000
Gauss, 10,000 Gauss?

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How much it is for
a typical level?

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So there's a number
here that this interest.

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And now we can decide
whether we have a weak Zeeman

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effect or a strong
Zeeman effect,

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by looking how your external
magnetic field compares

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with this little magnetic field.

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So we'll have a weak
Zeeman effect, A. Zeeman.

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When B is much smaller
than B internal.

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And therefore, the
effects of Zeeman

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is going to be smaller
than fine structure.

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So really, when you
look at this line,

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and you have the
first two terms,

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you should think
of these two terms

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as your new known Hamiltonian.

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And the Zeeman term,
at its perturbation.

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Yes.

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You've solved for
the fine structure

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coupling and the shift.

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So this is the known thing.

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These are going to be the known
states with known energies.

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You only know them the first
order, but you know them.

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And then the Zeeman
effect will be

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a perturbation theory on this.

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This is the weak Zeeman effect.

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How about the strong
Zeeman effect?

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So B will be strong Zeeman.

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This time, the magnetic
field associated

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with the Zeeman effect, the
external magnetic field,

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is much smaller than
the magnetic field

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responsible for
spin orbit coupling.

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And this time, what
are we going to do?

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Well, we will take
the Hamiltonian.

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It will be H 0.

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The strong Zeeman effect
means this Zeeman Hamiltonian

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is a lot more important than
the spin orbit coupling.

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So we'll add delta
H Zeeman here.

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And we hope this will be
our known Hamiltonian,

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because anyway, the Zeeman
stuff is much stronger now

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than fine structure.

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So this should be
our non-Hamiltonian.

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You should complain,
no, this is not known.

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I haven't done Zeeman,
but we'll look at it.

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And once we have our
known Hamiltonian here,

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spin orbit has to be rethought.

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Fine structure has to be
rethought as a perturbation.