WEBVTT

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PROFESSOR: So we'll do the
relativistic corrections.

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And all the corrections
that I'll do today,

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I'll skip the easy but sometimes
a little tedious algebra.

00:00:12.860 --> 00:00:14.550
It's not very tedious.

00:00:14.550 --> 00:00:17.470
Nothing that is pages
and pages of algebra.

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It's lines of algebra.

00:00:19.650 --> 00:00:23.070
But why would I
do it in lecture?

00:00:23.070 --> 00:00:25.620
No point for that.

00:00:25.620 --> 00:00:30.240
So let's see what we can do.

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This is the
relativistic correction,

00:00:33.000 --> 00:00:34.500
the minus p squared.

00:00:34.500 --> 00:00:38.320
So could we write this for
the relativistic correction?

00:00:38.320 --> 00:00:42.390
We're going to do first order
correction, relativistic,

00:00:42.390 --> 00:00:46.188
of the levels n l ml.

00:00:50.780 --> 00:00:52.190
Let's put a question mark.

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Minus 1 over 8 m cubed c squared
psi n l ml p to the fourth.

00:01:12.450 --> 00:01:17.160
Now recall that p to the
fourth, the way it was given,

00:01:17.160 --> 00:01:20.685
is really p squared
times p squared.

00:01:24.020 --> 00:01:27.480
You have four things that
have to be multiplied.

00:01:27.480 --> 00:01:31.590
So it's not px to the fourth
plus py to the fourth plus

00:01:31.590 --> 00:01:36.870
pz to the fourth is px
squared plus py squared

00:01:36.870 --> 00:01:41.950
plus pz squared, all
squared, just in case

00:01:41.950 --> 00:01:44.430
there's an ambiguity.

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That seems reasonable.

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The first order
corrections should

00:01:48.210 --> 00:01:53.810
be found by taking the
states and finding this.

00:01:53.810 --> 00:01:56.684
But there is a
big question mark.

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And this kind of
question is going

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to come up every time you
think about these things.

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This formula, where
I said the shift

00:02:07.450 --> 00:02:11.920
of the energy of this state
is that state evaluated here,

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applies for nondegenerate
perturbation theory.

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And if the hydrogen
atom is anything,

00:02:19.820 --> 00:02:22.670
it's a system with a
lot of degeneracies.

00:02:22.670 --> 00:02:29.510
So why can I use that,
or can I use that?

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We have the hydrogen atom.

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I just deleted it here.

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So here, if you have n,
for degeneracies you fix n.

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For degeneracies, you
fix some value of n.

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And now you have the
degeneracies between

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the various l's, for each
l between the various m's.

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A gigantic amount of degeneracy.

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Who allows me to do that?

00:03:03.770 --> 00:03:10.480
I'm supposed to take that
level three has nine states,

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remember?

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n square states.

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Well, we should do a 9 by 9
matrix here and calculate this.

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Nine sounds awful.

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We don't want to do
that so we better think.

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So this is the situation
you find yourself.

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Technically speaking,
this is a problem

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in the degenerate
perturbation theory.

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We should do that.

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And you better think about
this every time you face this

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problem because sometimes you
can get away without doing

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the degenerate analysis,
but sometimes you can't.

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Yes?

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AUDIENCE: It's like
rotationally symmetric.

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So you can mix terms with
different [INAUDIBLE]..

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PROFESSOR: OK.

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So you're saying, basically,
that this thing in this basis--

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so we have nine states here.

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n equal 3.

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In these nine states,
it doesn't mix them.

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So this is diagonal here.

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And what one is
claiming by doing

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that is that this
is a good basis,

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that delta H is
already diagonal there.

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And don't worry, we can do it.

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In fact, that is true.

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And the argument goes like that.

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We know that p to the
fourth, the perturbation,

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commutes with l squared.

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We'll discuss it a little more.

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And p to the fourth
commutes with lz as well.

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So these are two claims.

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Very important claims.

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Remember, we had a
remark that I told you

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few times few lectures ago.

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Very important.

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If you have a
Hermitian operator that

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commutes with your
perturbation for which

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the states of your
bases are eigenstates

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with different eigenvalues,
then the basis is good.

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So here it is.

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l squared commutes
with p to the fourth.

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Why?

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Because, in fact, p
to the fourth commutes

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with any angular momentum
because p to the fourth

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is p squared times p squared.

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And p squared is
rotational invariant.

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p squared commutes with any l.

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If that's not
obvious intuitively,

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which it should become
something you trust--

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this is rotational invariant.
p squared dot product doesn't

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depend on rotation.

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If you have a p and you square
it or you have a rotated p

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and you square
it, it's the same.

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So p to the fourth
commutes with any component

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of angular momentum.

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So these two are written
like great facts,

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but the basic fact is that
p squared with any li is 0.

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And all this follows from here.

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But this is a
Hermitian operator.

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This is a Hermitian operator.

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And the various states,
when you have fixed n,

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you can have different l's.

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But when you have
different l's, there

00:06:42.475 --> 00:06:45.950
are different
eigenvalues of l squared.

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So in those cases, the
matrix element will vanish.

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When you have the same l's
but different m's, these

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are different eigenvalues of lz.

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So the matrix element
should also vanish.

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So this establishes rigorously
that that perturbation,

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p to the fourth, is
diagonal in that subspace.

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So the subspace relevant
here is this whole thing.

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And in this subspace,
it's completely diagonal.

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Good.

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So generally, this kind of point
is not emphasized too much.

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But it's, in fact, the most
important and more interesting

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and more difficult point
in this calculations.

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We'll have one more
thing to say about this.

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But let's continue with this.

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I'll say the following.

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We use the Hermiticity
of p squared

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to move one p squared
to the other side.

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So Enl ml 1 is equal to
minus 1 over 8m cubed

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c squared p squared psi
nlm p squared psi nml--

00:08:26.917 --> 00:08:27.417
nlm.

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OK.

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We move this p to the fourth.

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It was p squared
times p squared.

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One p squared is Hermitian.

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We move it here.

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And then, instead of calculating
a billion derivatives here,

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you use the fact that p squared
over 2m plus v of r on the wave

00:08:53.330 --> 00:08:58.370
function is equal to the energy
of that wave function that

00:08:58.370 --> 00:09:00.650
depends on n times
the wave function.

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These are eigenstates.

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So p squared-- we don't
want to take derivatives,

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and those expectation values can
be replaced by a simpler thing.

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P squared on psi is just
2m En minus v of r psi.

00:09:31.120 --> 00:09:44.700
So Enlm 1 is equal to minus
1 over 8m cubed c squared.

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Here we have, well, the m's.

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Two m's are out, so we'll
put a 2 and an mc squared.

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Yep.

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En minus v of r psi En
minus v of r psi nlm nlm.

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OK.

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We got it to the point
where I think you can all

00:10:20.080 --> 00:10:22.990
agree this is doable.

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Why?

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Because, again, this
term is Hermitian,

00:10:27.460 --> 00:10:29.230
so you can put it
to the other side.

00:10:29.230 --> 00:10:32.800
And you'll have terms in which
you compute the expectation

00:10:32.800 --> 00:10:36.150
value on this
state of E squared.

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E squared is a number, so
it goes out, times 1, easy.

00:10:40.540 --> 00:10:44.840
En cross terms with vr
is the expectation of v

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of r in this state,
is the expectation

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of 1 over r in a state.

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That's easy.

00:10:50.140 --> 00:10:52.930
It comes from the
Virial theorem.

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Then you'll have the expectation
of v squared in a state,

00:10:56.830 --> 00:11:00.520
and that's the expectation of
1 over r squared in a state.

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You've also done it.

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So yes, getting all together,
getting the factors right

00:11:07.570 --> 00:11:13.360
would take you 15 minutes
or 20 minutes or whatever.

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But the answer is already clear.

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So let's write the answer.

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And the answer is that
Enl ml 1 relativistic is

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minus 1/8 alpha to the fourth.

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That's our very
recognizable factor.

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mc squared 4n over
l plus 1/2 minus 3.

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Now, fine structure
is something all of us

00:12:07.210 --> 00:12:09.860
must do at least
once in our life.

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So I do encourage you to
read the notes carefully

00:12:13.300 --> 00:12:15.160
and just do it.

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Just become familiar with it.

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It's a very nice subject,
and it's something

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you should understand.

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So here, again, I have to
do a comment about basis,

00:12:31.460 --> 00:12:35.080
and those comments keep
coming because it's

00:12:35.080 --> 00:12:36.760
an important subject.

00:12:36.760 --> 00:12:42.790
And I want to emphasize it.

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So what is the reason?

00:12:43.950 --> 00:12:46.630
The reason I wanted to comment
is because in a second,

00:12:46.630 --> 00:12:49.720
I'm going to do the
spin orbit term.

00:12:49.720 --> 00:12:55.800
And in that case, I would like
to work with a coupled basis.

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Here, I'm working with
the uncoupled basis.

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And really, this thing
is the expectation value

00:13:03.730 --> 00:13:24.130
of Hl relativistic in nl ml ms
nl ml ms. This is really that.

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I wrote psi nl ml, so you should
trust the first three labels.

00:13:33.350 --> 00:13:36.170
And ms goes for the ride.

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It's this spin.

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The operator you're putting
here, delta H relativistic,

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has nothing to do with spin.

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Could not change the
spin of the states.

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This has to be diagonal in spin.

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So this number you've
computed is nothing else

00:13:53.690 --> 00:14:00.920
than this overlap in
the uncoupled basis.

00:14:00.920 --> 00:14:11.460
So this calculation was
uncoupled basis matrix element.

00:14:15.180 --> 00:14:16.950
And we saw that it's diagonal.

00:14:21.980 --> 00:14:24.800
In fact, this whole
thing is nothing

00:14:24.800 --> 00:14:28.250
but the function of n and l.

00:14:31.110 --> 00:14:32.230
n and l.

00:14:32.230 --> 00:14:45.860
And independent
of ml and ms. OK.

00:14:45.860 --> 00:14:47.210
That's what we've calculated.

00:14:47.210 --> 00:14:48.965
So here is the question.

00:14:51.800 --> 00:15:05.190
We could consider this in
the coupled bases nlj mj.

00:15:08.160 --> 00:15:13.235
nlj mj.

00:15:21.120 --> 00:15:26.490
And the question is, do I
have to recalculate this

00:15:26.490 --> 00:15:28.155
in the coupled basis or not?

00:15:32.470 --> 00:15:37.290
And here is an argument that I
don't have to recalculate it.

00:15:37.290 --> 00:15:42.560
So I'm going to claim that
this is really equal to that.

00:15:42.560 --> 00:15:43.290
Just the same.

00:15:45.950 --> 00:15:48.520
It's the kind of thing that
makes you a little uneasy,

00:15:48.520 --> 00:15:50.700
but bear with me.

00:15:50.700 --> 00:15:53.310
Why should it be the same?

00:15:53.310 --> 00:16:01.080
Think of this as fixed
n and l because this

00:16:01.080 --> 00:16:02.790
depends on n and l.

00:16:02.790 --> 00:16:09.620
If we have the
hydrogen atom here,

00:16:09.620 --> 00:16:17.660
you'd take one of these
elements, one of these states--

00:16:17.660 --> 00:16:21.740
this is a fixed n, fixed l.

00:16:21.740 --> 00:16:25.861
And we're looking
at fixed n, fixed l.

00:16:25.861 --> 00:16:26.360
Yes.

00:16:26.360 --> 00:16:31.730
There are lots of states here
that have different ml and ms.

00:16:31.730 --> 00:16:37.070
But the answer doesn't depend
on ml and ms. In this basis,

00:16:37.070 --> 00:16:40.920
we are also looking at that
subspace, that multiplet,

00:16:40.920 --> 00:16:42.710
nl fixed.

00:16:42.710 --> 00:16:48.170
And they have reorganized
the states with j and mj.

00:16:48.170 --> 00:16:54.830
In fact, with two values of
j and several values of mj.

00:16:54.830 --> 00:16:57.860
But at the end of the
day, the coupled basis

00:16:57.860 --> 00:17:01.250
is another way to describe
these states coming

00:17:01.250 --> 00:17:05.750
from tensoring the l
multiplet with a spin 1/2.

00:17:05.750 --> 00:17:08.450
So it gives you two multiplets,
but they are the same states.

00:17:10.980 --> 00:17:17.460
So the fact that
every state here

00:17:17.460 --> 00:17:21.660
is some linear combination
of states in the uncoupled

00:17:21.660 --> 00:17:27.210
bases with different values of
ml and ms that add up to mj.

00:17:27.210 --> 00:17:30.280
But this answer doesn't
depend on ml and ms.

00:17:30.280 --> 00:17:33.480
So whatever linear
combination you need,

00:17:33.480 --> 00:17:36.330
it doesn't change because
the answer doesn't

00:17:36.330 --> 00:17:44.680
depend on ml and ms. So this
must be the same as that.

00:17:44.680 --> 00:17:46.390
I'll give another argument.

00:17:46.390 --> 00:17:50.950
Maybe a little more abstract,
but clearer perhaps.

00:17:50.950 --> 00:17:52.780
Think of this.

00:17:52.780 --> 00:17:54.640
So this can be--

00:17:54.640 --> 00:17:59.230
in the notes, I explain that by
changing basis and explaining

00:17:59.230 --> 00:18:02.170
why exactly
everything works out.

00:18:02.170 --> 00:18:04.870
But there is no need
for that argument

00:18:04.870 --> 00:18:07.600
if you think a little
more abstractly.

00:18:07.600 --> 00:18:09.460
Think of this subspace.

00:18:09.460 --> 00:18:14.200
Because with fixed n and
l, we have this subspace.

00:18:14.200 --> 00:18:20.720
In this subspace,
the uncoupled basis

00:18:20.720 --> 00:18:23.700
makes the perturbation diagonal.

00:18:23.700 --> 00:18:28.520
But more than diagonal, it makes
the perturbation proportional

00:18:28.520 --> 00:18:34.010
to the unit matrix, because
every eigenvalue is the same.

00:18:34.010 --> 00:18:37.780
Because in this subspace,
n and l is fixed.

00:18:37.780 --> 00:18:41.450
And yes, m and ms
change, but the answer

00:18:41.450 --> 00:18:43.100
doesn't depend on that.

00:18:43.100 --> 00:18:48.410
So this matrix, delta
H, in this subspace

00:18:48.410 --> 00:18:52.730
is proportional to
the unit matrix.

00:18:52.730 --> 00:18:55.820
And when a matrix is
proportional to the unit

00:18:55.820 --> 00:18:59.030
matrix, it is
proportional to the unit

00:18:59.030 --> 00:19:02.390
matrix in any orthogonal basis.

00:19:02.390 --> 00:19:05.990
A unit matrix
doesn't get rotated.

00:19:05.990 --> 00:19:09.440
So it should be a unit
matrix here as well,

00:19:09.440 --> 00:19:11.820
and it should be
the same matrix.

00:19:11.820 --> 00:19:14.290
So this is the same pair.