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00:00:01,170 --> 00:00:08,270
PROFESSOR: OK, so this is the
picture you've studied so far.

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And now we will
consider the next thing.

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So basis states.

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What's going to happen?

5
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We have to deal
with basis states.

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00:00:30,090 --> 00:00:32,830
We have electron spin.

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We have electron
angular momentum.

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What is the total
angular momentum?

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The sum of them.

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00:00:40,430 --> 00:00:42,910
We will have to deal with that.

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So we will have to deal with
addition of angular momentum

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00:00:46,200 --> 00:00:47,730
in the hydrogen atom.

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It's awfully important.

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That's the key of the matter.

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So let's see what we have.

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Let's look at our basis states.

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So now the spin, the
most important thing,

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we take it very seriously.

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Recall that when you
have angular momentum--

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in general, we use j and m,
and those are the two quantum

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numbers, j squared, the
angular momentum eigenvalues,

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h squared, l times--

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j times j plus 1.

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And jc over h bar
has eigenvalues, m.

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That's a notation
for angular momentum.

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So we have the electron spin.

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So what is the notation.

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It's sms, and it's
always equal to 1/2,

29
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because the electron
always has been 1/2.

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The electron will have many
orbital angular momentum, 0, 1,

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3, but spin, it
only has been 1/2.

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So s is always 1/2.

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ms can be plus minus 1/2.

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So that's two states.

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For orbital electron,
orbital, we have l and m.

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Those are the names, l and
m is the quantum number.

37
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So how do we define
the uncoupled basis?

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The uncoupled basis
is a set of states

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that enumerates the whole
spectrum of the hydrogen atom,

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using those quantum numbers to
distinguish all those states.

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So the uncoupled basis are
those states that we have there.

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Uncoupled basis are
all these states,

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and they're described by n, and
the principal quantum number,

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l and m.

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This is the orbital.

46
00:03:36,680 --> 00:03:40,520
And you could say,
well, s and ms,

47
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that would be a
correct thing to do.

48
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But as we said, s is always
So copying and copying again,

49
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something that is
always the same value

50
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and doesn't have
any new information

51
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is not worth it, so people
don't include the s.

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00:03:59,616 --> 00:04:04,420
And we put ms.

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And that's electron spin along
the z direction, electron sz.

54
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And it takes values
plus minus 1/2.

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So this is our uncoupled basis.

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For any electron
in that table, you

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need to know uniquely
that electron state.

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You need to give me
all these numbers.

59
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You have to tell me
where I am horizontally,

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after that, where I am
vertically, I'm sorry,

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for m, where I am horizontally
for l, within the multiplate,

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which is my value of m.

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And once you've done that,
you should tell me up or down.

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So all those numbers
are important.

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They're one to
one correspondence

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to the basis state.

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But now, let's do
the coupled basis.

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That's where things
begin to get interesting.

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Coupled basis.

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So we'll consider the
total angular momentum

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j, which is l plus s.

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When we add angular
momentum, we basically say,

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you know, you have states
that are representations

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of orbital angular
momentum and spin.

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But I want you to express
those as eigenstates

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of the total angular momentum.

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That's all adding
angular momentum means.

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It's recognizing that we
want to re-express our basis

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states in terms of eigenstates
of the total angular momentum.

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It's all you do.

81
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So what do we do then?

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We have an l multiplate.

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What does an l multiple mean?

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That's a set of
vectors, a vector space.

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And we tensor it with
a spin multiplate,

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and the result is equal
to the sum of j multiplate

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because your states are nothing
else but tensor products

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of these things, even
though we never wrote it

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in the way of tensor product.

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Basically, the wave
function has some expression

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having to do with
l and m in here

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and has some value of the spin.

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So a given state has
all these properties.

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A given state lives
in the tensor product,

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and we want to write it
the sum of j multiplate.

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So I want to say a couple
more things about this.

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When we do addition
of angular momentum,

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what happens to the
quantum numbers?

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That's, again,
also a thing that's

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sometimes a little subtle,
and I want to emphasize it.

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So an l multiple it is
described by l and m.

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A spin multiplate is
defined by s and ms.

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But when you have a j
mulitplate, you have j and j--

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how do they call it?

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jm.

106
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I think they call it.

107
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Yeah, jm.

108
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But actually, you
have a little more.

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I claim that the states--

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so this is what I'm
meaning my this.

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These states have two quantum
numbers you can specify.

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You cannot specify any more.

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This is the z component of l,
and you cannot specify the x

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or the y component.

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Here is all you can
specify of these states.

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You certainly can
specify the value of j,

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and the value of jm, the m
component, the z component

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of j, but can you specify more?

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And the answer is yes.

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You can specify a little more.

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You can actually specify
here all the states in here

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are eigenstates of l,
of s, of j, and of jm.

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So you add two more.

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This may sound a little
funny, but every state here

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had the same l eigenvalues.

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They were, for
example l equal 3.

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So all the states
have l equal 3.

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All the states are eigenstates
of l squared, eigenstates.

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All these states here are
l squared eigenstates,

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and all the states here
are s squared eigenstates

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with the same eigenvalues.

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So if you multiply them,
and you rearrange them,

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because that's all this
addition of angular momentum

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is is just rearranging
the states,

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you still have that all
the states that are here

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have the same l squared
and the same s squared.

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So l and s are good
quantum numbers here.

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The things that are not good
quantum numbers are m and ms

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are not good.

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They are not good quantum
numbers of this state.

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They are not
eigenstates of m or ms.

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So this is something you've
learned with addition

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of angular momentum.

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If this is a little fuzzy,
it will be important

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that you review it and make
sure this becomes clear.

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There will be [? stuffed ?]
recitation about these things,

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and we'll do more with it.

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But now, what is the application
in the hydrogen atom for this?

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Well, our multiplates
are multiplates

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of [INAUDIBLE]
value of l that are

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being tensored with spin 1/2.

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Orbital l, and
when you have this,

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you know that the answer is
l plus 1/2 plus l minus 1/2.

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Those are the two values of j,
j max, and j min in this case.

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So in the hydrogen
atom, the notation

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is that this is the l state
with j equal l plus 1/2.

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This is the j value plus the
l state with l minus 1/2.

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So this is the capital L
we were mentioning before,

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in this case L of l.

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And our notation is evolving.

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This is the spectroscopic
notation in which we

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will describe states by an nLj.

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So that's the
spectroscopic notation.

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You put the principal quantum
number, the capital L,

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that is for L equals 0, put
an s, a p, a d, and the number

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which is the value of j here.

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So let's look at
our spectrum again.

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We have to do that.

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It's an important thing.

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So we'll do three cases
here, n equal 1, n equals 2,

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and n equals 3.

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We have l equals 0, l
equal 1, l equals 2.

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And that's s, p, and
D. OK, let's begin.

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This state, ground state,
what is the ground state?

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It's l equals 0, tensor with
1/2 for the ground state.

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This is just the state 1/2.

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00:13:31,760 --> 00:13:35,870
So it will be a state
with j equal 1/2.

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00:13:35,870 --> 00:13:44,080
So it should be written as
n, which is a 1, the capital

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00:13:44,080 --> 00:13:47,120
L, which is 0.

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00:13:47,120 --> 00:13:51,250
So it's s and the j value 1/2.

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That's the state.

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00:13:52,060 --> 00:13:55,630
1s 1/2.

183
00:13:55,630 --> 00:13:57,820
Go here.

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00:13:57,820 --> 00:14:00,080
Well, that's still l equals 0.

185
00:14:00,080 --> 00:14:07,620
So this product source is
2s 1/2, and here is 3s 1/2.

186
00:14:10,270 --> 00:14:13,300
That's the name
of the multiplates

187
00:14:13,300 --> 00:14:15,190
in the coupled basis.

188
00:14:15,190 --> 00:14:16,480
What do we have here?

189
00:14:22,140 --> 00:14:25,550
We have l equals 1.

190
00:14:25,550 --> 00:14:32,660
So with l equals 1,
we have 1 tensor 1/2.

191
00:14:32,660 --> 00:14:38,090
So we will get j
equals 3/2 and 1/2.

192
00:14:38,090 --> 00:14:40,765
So we must have 2.

193
00:14:43,800 --> 00:14:45,360
What is the value of l?

194
00:14:45,360 --> 00:14:49,000
Remember, the value
of l is preserved.

195
00:14:49,000 --> 00:14:54,240
So if this was l equals 1,
after you do the tensor product,

196
00:14:54,240 --> 00:14:58,120
l is still a good
quantum number.

197
00:14:58,120 --> 00:15:07,290
So you have 2p 3/2, and 2p 1/2.

198
00:15:09,840 --> 00:15:18,500
Here, you would have two
states, 3p 3/2 and 3p 1/2.

199
00:15:23,220 --> 00:15:26,790
Finally, here l
equals 2, you have

200
00:15:26,790 --> 00:15:33,640
2 tensor 1/2 is 5/2 plus 3/2.

201
00:15:33,640 --> 00:15:43,140
So you would have
3D 5/2 and 3D 3/2.

202
00:15:43,140 --> 00:15:45,850
Those are your states.

203
00:15:45,850 --> 00:15:47,640
This is the notation
we will use.

204
00:15:47,640 --> 00:15:52,890
So what is the uncouple
basis at the end of the day?

205
00:15:52,890 --> 00:15:55,620
The uncoupled basis is n.

206
00:15:55,620 --> 00:15:58,720
You still have n.

207
00:15:58,720 --> 00:16:00,940
What is the coupled basis now?

208
00:16:00,940 --> 00:16:03,550
We have the uncoupled
basis being this.

209
00:16:06,810 --> 00:16:11,910
The coupled basis is still n.

210
00:16:11,910 --> 00:16:13,730
Still l.

211
00:16:13,730 --> 00:16:16,240
We said l carries through.

212
00:16:16,240 --> 00:16:20,200
Still s would carry through,
but we don't have it there.

213
00:16:20,200 --> 00:16:22,720
Now I cannot put m.

214
00:16:22,720 --> 00:16:27,360
m is no good when [INAUDIBLE],,
but I have j and jm.

215
00:16:33,210 --> 00:16:40,665
So these are your coupled
basis states, coupled basis.

216
00:16:43,250 --> 00:16:47,570
And it's represented in
this notation, the n,

217
00:16:47,570 --> 00:16:51,860
the little l is representative
for whatever letter is here,

218
00:16:51,860 --> 00:16:55,550
the j is here, and well,
the jm is not said,

219
00:16:55,550 --> 00:17:00,210
but that's a multiplate.

220
00:17:00,210 --> 00:17:04,470
OK, so we've rewritten the
stating the couple basis,

221
00:17:04,470 --> 00:17:07,710
because we will need
those states in order

222
00:17:07,710 --> 00:17:11,700
to do perturbations theory.