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PROFESSOR: All right.
00:00:26.380 --> 00:00:28.050
Welcome, everyone.
00:00:28.050 --> 00:00:29.710
Hi.
00:00:29.710 --> 00:00:32.850
So today we're going
to pick up where
00:00:32.850 --> 00:00:36.430
we left off last time in our
study of angular momentum
00:00:36.430 --> 00:00:38.865
and rotations in
quantum mechanics.
00:00:38.865 --> 00:00:40.240
Before I get
started, let me open
00:00:40.240 --> 00:00:43.520
up for questions, pragmatic
and physics related.
00:00:48.350 --> 00:00:49.786
Yeah?
00:00:49.786 --> 00:00:52.744
AUDIENCE: When we were solving
for the 3D harmonic oscillator
00:00:52.744 --> 00:00:55.866
we solved for the energy
eigenfunction that
00:00:55.866 --> 00:00:59.153
was a product of phi
x, phi y, and phi z.
00:00:59.153 --> 00:01:02.604
We made an assumption
that phi e was
00:01:02.604 --> 00:01:05.972
equal to phi sub n
of x plus that sum.
00:01:05.972 --> 00:01:08.433
How did you get
from [INAUDIBLE]??
00:01:08.433 --> 00:01:09.100
PROFESSOR: Good.
00:01:09.100 --> 00:01:11.100
So what we had was that
we had that the energy--
00:01:11.100 --> 00:01:15.590
we wanted to find the energy
of the 3D harmonic oscillator.
00:01:15.590 --> 00:01:18.345
And we wanted to find the energy
eigenfunctions and eigenvalues.
00:01:18.345 --> 00:01:19.970
And they way we did
this was by saying,
00:01:19.970 --> 00:01:22.540
look, the energy of the 3D
harmonic oscillator, which
00:01:22.540 --> 00:01:24.730
I can think of as a
function of x and px
00:01:24.730 --> 00:01:28.480
and y and py and z and
pz, has this nice form.
00:01:28.480 --> 00:01:31.540
We could write it as
the energy operator
00:01:31.540 --> 00:01:35.080
purely in terms of x,
p squared x upon 2m
00:01:35.080 --> 00:01:37.710
plus m omega squared
upon 2x squared.
00:01:37.710 --> 00:01:42.500
Plus-- so this is a single
1D harmonic oscillator energy
00:01:42.500 --> 00:01:44.150
operator in the x direction.
00:01:44.150 --> 00:01:48.730
Plus E 1D in the y
direction, plus a harmonic
00:01:48.730 --> 00:01:52.510
oscillator energy 1D
in the z direction.
00:01:52.510 --> 00:01:54.110
So that was the
first observation.
00:01:54.110 --> 00:01:58.430
And then we said that given that
this splits in this fashion,
00:01:58.430 --> 00:02:03.230
I'm going to write my
energy eigenfunction, phi
00:02:03.230 --> 00:02:08.710
of x, y, and z in separated
form as a product.
00:02:08.710 --> 00:02:16.620
Phi x of x, phi y of
y, and phi z of z.
00:02:16.620 --> 00:02:19.830
And we used this and deduced
that in order for this
00:02:19.830 --> 00:02:22.280
to be an eigenfunction of
the 3D harmonic oscillator
00:02:22.280 --> 00:02:25.140
it must be true that
5x of x was itself
00:02:25.140 --> 00:02:28.520
an eigenfunction of the
x harmonic oscillator
00:02:28.520 --> 00:02:32.716
equals with some energy
epsilon, which I will
00:02:32.716 --> 00:02:35.870
call epsilon sub x, phi sub x.
00:02:35.870 --> 00:02:37.450
And ditto for y and z.
00:02:37.450 --> 00:02:40.770
And if this is true, if phi sub
x is an eigenfunction of the 1D
00:02:40.770 --> 00:02:43.780
harmonic oscillator, then
this is an eigenfunction
00:02:43.780 --> 00:02:45.272
of the 3D harmonic oscillator.
00:02:45.272 --> 00:02:46.730
But we know what
the eigenfunctions
00:02:46.730 --> 00:02:47.850
are of the harmonic oscillator.
00:02:47.850 --> 00:02:49.490
The eigenfunctions
of the 1D harmonic
00:02:49.490 --> 00:02:51.130
oscillator we gave a name.
00:02:51.130 --> 00:02:53.790
We call them pi sub n.
00:02:53.790 --> 00:02:56.850
So what we then said
was, look, if this
00:02:56.850 --> 00:02:59.790
is an eigenfunction of
the 1D harmonic oscillator
00:02:59.790 --> 00:03:04.930
in the x direction, then
it's labeled by an n.
00:03:04.930 --> 00:03:06.900
And if this one is
in the y direction,
00:03:06.900 --> 00:03:08.150
it's labeled by an l.
00:03:08.150 --> 00:03:10.630
And if this one is in the z
direction it's labeled by a z.
00:03:10.630 --> 00:03:11.980
But on top of that we know more.
00:03:11.980 --> 00:03:13.970
We know what these
energy eigenvalues are.
00:03:13.970 --> 00:03:16.270
The energy eigenvalues
corresponding to these guys
00:03:16.270 --> 00:03:19.750
are if this is
phi n, this is En.
00:03:19.750 --> 00:03:23.460
I can simply write this as En
for the 1D harmonic oscillator.
00:03:23.460 --> 00:03:27.570
And taking this, using the fact
that they're 1D eigenfunctions
00:03:27.570 --> 00:03:29.320
and plugging it into
the energy eigenvalue
00:03:29.320 --> 00:03:31.040
equation for the 3D
harmonic oscillator
00:03:31.040 --> 00:03:33.510
tells us that the energy
eigenvalues for the 3D case
00:03:33.510 --> 00:03:40.680
are of the form E1d
x plus E1d in the y
00:03:40.680 --> 00:03:44.750
plus E1d in the z, which is
equal to, since these were
00:03:44.750 --> 00:03:50.600
the same frequency, h bar
omega times n plus l plus m,
00:03:50.600 --> 00:03:53.820
from each of them
a 1/2, so plus 3/2.
00:03:53.820 --> 00:03:54.320
Cool?
00:03:54.320 --> 00:03:55.460
That answer your question?
00:03:55.460 --> 00:03:56.310
Excellent.
00:03:56.310 --> 00:03:57.541
Other questions?
00:03:57.541 --> 00:03:59.505
Yeah?
00:03:59.505 --> 00:04:01.469
AUDIENCE: Energy
and angular momentum
00:04:01.469 --> 00:04:03.257
have to be related
somehow, right?
00:04:03.257 --> 00:04:03.924
PROFESSOR: Yeah.
00:04:03.924 --> 00:04:05.397
AUDIENCE: I mean, of course.
00:04:05.397 --> 00:04:06.390
Because it's both.
00:04:06.390 --> 00:04:07.140
PROFESSOR: Indeed.
00:04:07.140 --> 00:04:09.340
AUDIENCE: [INAUDIBLE].
00:04:09.340 --> 00:04:13.741
The thing is, we have a ladder.
00:04:13.741 --> 00:04:14.829
Is there limits on them?
00:04:14.829 --> 00:04:16.079
PROFESSOR: Very good question.
00:04:16.079 --> 00:04:17.420
So this is where we're
going to pick up.
00:04:17.420 --> 00:04:18.380
Let me rephrase this.
00:04:18.380 --> 00:04:19.588
This is really two questions.
00:04:19.588 --> 00:04:20.523
Question number one.
00:04:20.523 --> 00:04:22.690
Look, there should be a
relationship between angular
00:04:22.690 --> 00:04:23.780
momentum and energy.
00:04:23.780 --> 00:04:25.697
But we're just talking
about angular momentum.
00:04:25.697 --> 00:04:26.510
Why?
00:04:26.510 --> 00:04:28.410
Second question, look,
we've got a ladder.
00:04:28.410 --> 00:04:29.910
But is the ladder infinite?
00:04:29.910 --> 00:04:31.280
So let me come back to
the second question.
00:04:31.280 --> 00:04:32.850
That's going to be the
beginning of the lecture.
00:04:32.850 --> 00:04:34.900
On the first question,
yes angular momentum
00:04:34.900 --> 00:04:37.108
is going to play a role when
we calculate the energy.
00:04:37.108 --> 00:04:38.723
But two quick things to note.
00:04:38.723 --> 00:04:40.140
First off, consider
a system which
00:04:40.140 --> 00:04:42.637
is spherically symmetric,
rotationally invariant.
00:04:42.637 --> 00:04:44.970
That means that the energy
doesn't depend on a rotation.
00:04:44.970 --> 00:04:47.137
If I rotate the system I
haven't changed the energy.
00:04:47.137 --> 00:04:50.250
So if the system is
rotationally invariant,
00:04:50.250 --> 00:04:52.550
that's going to imply some
constraints on the energy
00:04:52.550 --> 00:04:55.390
eigenvalues and how they
depend on the angular momentum,
00:04:55.390 --> 00:04:56.880
as we discussed last time.
00:04:56.880 --> 00:04:58.422
Let me say that
slightly differently.
00:05:00.630 --> 00:05:03.620
When we talk about the free
particle, 1D free particle--
00:05:03.620 --> 00:05:05.250
we've talked about
this one to death.
00:05:05.250 --> 00:05:06.850
Take the 1D free particle.
00:05:06.850 --> 00:05:08.870
We can write the
energy eigenfunctions
00:05:08.870 --> 00:05:11.750
as momentum eigenfunctions,
because the momentum commutes
00:05:11.750 --> 00:05:13.765
with the energy.
00:05:13.765 --> 00:05:15.723
And so the way the
eigenfunctions of the energy
00:05:15.723 --> 00:05:18.240
operate are indeed e to
iKX there are plane waves,
00:05:18.240 --> 00:05:20.980
they're eigenfunctions of the
momentum operator as well.
00:05:20.980 --> 00:05:23.085
Similarly, when we
talk about a 3D system
00:05:23.085 --> 00:05:25.210
it's going to be useful in
talking about the energy
00:05:25.210 --> 00:05:29.690
eigenvalues to know a basis of
eigenfunctions of the angular
00:05:29.690 --> 00:05:30.600
momentum operator.
00:05:30.600 --> 00:05:32.320
Knowing the angular
momentum operator
00:05:32.320 --> 00:05:35.713
is going to allow us to
write energy eigenfunctions
00:05:35.713 --> 00:05:37.130
in a natural way
and a simply way,
00:05:37.130 --> 00:05:39.530
in the same way that knowing
the momentum operator allowed
00:05:39.530 --> 00:05:40.905
us to write energy
eigenfunctions
00:05:40.905 --> 00:05:43.700
in a simple way in the 1D case.
00:05:43.700 --> 00:05:44.717
That make sense?
00:05:44.717 --> 00:05:47.300
We're going to have a glorified
version of the Fourier theorum
00:05:47.300 --> 00:05:49.090
where instead of
something over e to iKX,
00:05:49.090 --> 00:05:50.150
we're going to have something
over angular momentum
00:05:50.150 --> 00:05:51.072
eigenstates.
00:05:51.072 --> 00:05:53.130
And those are called
the spherical harmonics.
00:05:53.130 --> 00:05:55.750
And they are the analog
of Fourier expansion
00:05:55.750 --> 00:05:56.912
for this year.
00:05:56.912 --> 00:05:57.620
But you're right.
00:05:57.620 --> 00:05:58.960
We're going to have
to understand how
00:05:58.960 --> 00:06:00.252
that interacts with the energy.
00:06:00.252 --> 00:06:02.160
And that'll be the topic
of the next lecture.
00:06:02.160 --> 00:06:04.160
We're going to finish up
angular momentum today.
00:06:04.160 --> 00:06:06.950
Other questions?
00:06:06.950 --> 00:06:11.360
OK, so from last time, these
are the commutation relations
00:06:11.360 --> 00:06:13.450
which we partially
derived in lecture
00:06:13.450 --> 00:06:16.630
and which you will be
driving on your problem set.
00:06:16.630 --> 00:06:18.140
It's a really good exercise.
00:06:18.140 --> 00:06:19.230
Commit these to memory.
00:06:19.230 --> 00:06:20.840
They're your friends.
00:06:20.840 --> 00:06:23.130
Key thing here to keep in mind.
00:06:23.130 --> 00:06:26.270
h bar has units of angular
momentum, so this makes sense.
00:06:26.270 --> 00:06:29.270
Angular momentum, angular
momentum, angular momentum.
00:06:29.270 --> 00:06:34.220
So when you see an h
bar in this setting,
00:06:34.220 --> 00:06:37.520
its job, in some sense, is to
make everything dimensionally
00:06:37.520 --> 00:06:39.330
sensible.
00:06:39.330 --> 00:06:42.545
So the important things here are
that lx and ly do not commute.
00:06:42.545 --> 00:06:44.840
They commute to lz.
00:06:44.840 --> 00:06:47.070
Can you have a state with
definite angular momentum
00:06:47.070 --> 00:06:49.445
in the x direction and definite
angular momentum in the y
00:06:49.445 --> 00:06:50.622
direction simultaneously?
00:06:50.622 --> 00:06:52.330
No, because of this
commutation relation.
00:06:52.330 --> 00:06:55.760
It would have to vanish.
00:06:55.760 --> 00:06:58.065
This is, say, the x component
of the angular momentum.
00:06:58.065 --> 00:07:00.190
Can you have a state with
definite angular momentum
00:07:00.190 --> 00:07:03.850
in the x direction and total
angular momentum all squared?
00:07:03.850 --> 00:07:04.380
Yes.
00:07:04.380 --> 00:07:04.880
OK, great.
00:07:04.880 --> 00:07:06.570
That's going to be
important for us.
00:07:06.570 --> 00:07:08.410
So in order to construct
the eigenfunctions
00:07:08.410 --> 00:07:09.993
it turns out to be
useful to construct
00:07:09.993 --> 00:07:12.140
these so-called raising
and lowering operators,
00:07:12.140 --> 00:07:15.167
which are Lx plus iLy.
00:07:15.167 --> 00:07:16.750
They have a couple
of nice properties.
00:07:16.750 --> 00:07:19.260
The first is, since these are
built up out of Lx and Ly,
00:07:19.260 --> 00:07:20.802
both of which commute
with L squared,
00:07:20.802 --> 00:07:24.280
the L plus minuses
commute with L squared.
00:07:24.280 --> 00:07:25.370
So these guys commute.
00:07:25.370 --> 00:07:29.990
So if we have an
eigenfunction of L squared,
00:07:29.990 --> 00:07:34.700
acting with L plus does
not change its eigenvalue.
00:07:34.700 --> 00:07:37.130
Similarly, L plus
commuting with Lz
00:07:37.130 --> 00:07:42.360
gives us h bar L plus or minus,
with a plus or minus out front.
00:07:42.360 --> 00:07:47.570
This is just like the raising
and lowering operators
00:07:47.570 --> 00:07:49.060
for the harmonic oscillator.
00:07:49.060 --> 00:07:52.850
But instead of the energy we
have the angular momentum.
00:07:52.850 --> 00:07:57.260
So this is going to tell us
that the angular momentum
00:07:57.260 --> 00:07:59.970
eigenvalues, the
eigenvalues of Lz,
00:07:59.970 --> 00:08:02.330
are shifted by
plus or minus h bar
00:08:02.330 --> 00:08:04.140
when we raise or
lower with L plus or L
00:08:04.140 --> 00:08:07.200
minus, just like the
energy was shifted--
00:08:07.200 --> 00:08:12.130
for the 1d harmonic oscillator
the energy was shifted,
00:08:12.130 --> 00:08:15.990
plus h bar omega a
dagger, was shifted
00:08:15.990 --> 00:08:17.800
by h bar omega when
we acted with a plus
00:08:17.800 --> 00:08:18.860
on an energy eigenstate.
00:08:18.860 --> 00:08:19.370
Same thing.
00:08:22.580 --> 00:08:24.930
Questions on the commutators
before we get going?
00:08:24.930 --> 00:08:27.090
In some sense, we're
going to just to just take
00:08:27.090 --> 00:08:28.757
advantage of these
commutation relations
00:08:28.757 --> 00:08:30.820
and explore their
consequences today.
00:08:30.820 --> 00:08:40.360
So our goal is going to be
to build the eigenfunctions
00:08:40.360 --> 00:08:47.347
and eigenvalues of the
angular momentum operators,
00:08:47.347 --> 00:08:49.680
and in particular of the most
angular momentum operators
00:08:49.680 --> 00:08:51.980
we [INAUDIBLE] complete set
of commuting observables,
00:08:51.980 --> 00:08:53.310
L squared and Lz.
00:08:53.310 --> 00:08:54.810
You might complain,
look, why Lz?
00:08:54.810 --> 00:08:55.310
Whoops.
00:08:55.310 --> 00:08:56.550
I don't mean a commutator.
00:08:56.550 --> 00:08:57.710
I mean the set.
00:08:57.710 --> 00:08:59.480
You might say, why Lz?
00:08:59.480 --> 00:09:00.620
Why not Lx?
00:09:00.620 --> 00:09:03.247
And if you call this
the z direction then
00:09:03.247 --> 00:09:04.830
I will simply choose
a new basis where
00:09:04.830 --> 00:09:06.380
this is called the x direction.
00:09:06.380 --> 00:09:08.110
So it makes no
difference whatsoever.
00:09:08.110 --> 00:09:08.950
It's just a name.
00:09:08.950 --> 00:09:10.490
The reason we're
going to choose Lz
00:09:10.490 --> 00:09:13.480
is because that coordinate
system plays nicely
00:09:13.480 --> 00:09:16.710
with spherical coordinates,
just the conventional choice
00:09:16.710 --> 00:09:19.260
of spherical coordinates where
theta equals 0 is the up axis.
00:09:19.260 --> 00:09:20.760
But there's nothing
deep about that.
00:09:20.760 --> 00:09:24.670
We could have
taken any of these.
00:09:24.670 --> 00:09:26.220
OK.
00:09:26.220 --> 00:09:27.740
So this is our goal.
00:09:27.740 --> 00:09:28.670
So let's get started.
00:09:28.670 --> 00:09:32.902
So first, because of these
commutation relations
00:09:32.902 --> 00:09:34.360
and in particular
this one, we know
00:09:34.360 --> 00:09:36.485
that we can find common
eigenfunctions of L squared
00:09:36.485 --> 00:09:37.480
and Lz.
00:09:37.480 --> 00:09:41.610
Let us call those common
eigenfunctions by a name,
00:09:41.610 --> 00:09:47.720
Y sub lm such that--
so let these guys
00:09:47.720 --> 00:09:56.460
be the common eigenfunctions
of L squared and Lz.
00:09:56.460 --> 00:10:02.120
I.e., L squared Ylm is equal
to-- well first off, units.
00:10:02.120 --> 00:10:04.220
This has units of
angular momentum squared,
00:10:04.220 --> 00:10:05.040
h bar squared.
00:10:05.040 --> 00:10:06.590
So that got rid of the units.
00:10:06.590 --> 00:10:08.490
And we want our eigenvalue, lm.
00:10:08.490 --> 00:10:11.495
And because I know
the answer, I'm
00:10:11.495 --> 00:10:13.120
going to give--
instead of calling this
00:10:13.120 --> 00:10:16.210
a random dimensionless number,
which would be the eigenvalue,
00:10:16.210 --> 00:10:19.950
I'm going to call it a very
specific thing, l, l plus 1.
00:10:19.950 --> 00:10:21.880
This is a slightly
grotesque thing to do,
00:10:21.880 --> 00:10:24.120
but it will make the
algebra much easier.
00:10:24.120 --> 00:10:28.080
So similarly, so that's
what the little l is.
00:10:28.080 --> 00:10:30.080
Little l is labeling the
eigenvalue of L squared
00:10:30.080 --> 00:10:31.747
and the actual value
of that eigenvalue.
00:10:31.747 --> 00:10:33.668
I'm just calling h
bar squared ll plus 1.
00:10:33.668 --> 00:10:35.460
That doesn't tell you
anything interesting.
00:10:35.460 --> 00:10:37.430
This was already a
real positive number.
00:10:37.430 --> 00:10:40.660
So this could have been any
real positive number as well,
00:10:40.660 --> 00:10:43.260
by tuning L.
00:10:43.260 --> 00:10:48.140
Similarly, Lz Ylm-- I want
this to be an eigenfunction--
00:10:48.140 --> 00:10:49.795
this has units of
angular momentum.
00:10:49.795 --> 00:10:50.670
So I'll put an h bar.
00:10:50.670 --> 00:10:52.640
Now we have a dimensionless
coefficient Ylm.
00:10:52.640 --> 00:10:55.870
And I'll simply call that m.
00:10:55.870 --> 00:10:57.690
So if you will, these
are the definitions
00:10:57.690 --> 00:11:02.440
of the symbols m and little l.
00:11:02.440 --> 00:11:05.600
And Ylm are just the names I'm
giving to the angular momentum
00:11:05.600 --> 00:11:06.820
eigenfunctions.
00:11:06.820 --> 00:11:07.830
Cool?
00:11:07.830 --> 00:11:09.330
So I haven't actually
done anything.
00:11:09.330 --> 00:11:11.120
I've just told you that
these are the eigenfunctions.
00:11:11.120 --> 00:11:13.493
What we want to know is what
properties do they have,
00:11:13.493 --> 00:11:15.910
and what are the actual allowed
values of the eigenvalues.
00:11:18.680 --> 00:11:30.470
So the two key things to note
are first that L plus and minus
00:11:30.470 --> 00:11:32.160
leave the eigenvalue
of L squared alone.
00:11:32.160 --> 00:11:33.077
So they leave l alone.
00:11:36.350 --> 00:11:40.890
So this is the statement
that L squared on L plus Ylm
00:11:40.890 --> 00:11:45.230
is equal to L plus
on L squared Ylm
00:11:45.230 --> 00:11:50.180
is equal to h bar
squared ll plus 1,
00:11:50.180 --> 00:11:56.760
the eigenvalue of L squared
acting on Ylm, L plus Ylm.
00:11:56.760 --> 00:12:00.120
And so L plus Ylm is just as
much an eigenfunction of L
00:12:00.120 --> 00:12:05.960
squared as Ylm was itself,
with the same eigenvalue.
00:12:05.960 --> 00:12:06.850
Two.
00:12:06.850 --> 00:12:07.932
So that came from--
00:12:10.710 --> 00:12:14.200
where are we--- this
commutation relation.
00:12:14.200 --> 00:12:23.620
OK, so similarly, L plus
minus raise or lower m by one.
00:12:26.420 --> 00:12:29.290
And the way to see that is to
do exactly the same computation,
00:12:29.290 --> 00:12:37.530
Lz on L plus, for example,
Ylm is equal to L plus--
00:12:37.530 --> 00:12:39.750
now we can write
the commutator--
00:12:39.750 --> 00:12:50.110
Lz, L plus, plus L plus Lz Ylm.
00:12:50.110 --> 00:12:53.160
But the commutator of Lz
with L plus we already have,
00:12:53.160 --> 00:12:54.820
is plus h bar L plus.
00:12:54.820 --> 00:12:58.340
So this is equal
to h bar L plus.
00:12:58.340 --> 00:13:01.210
And from this term, L plus
Lz of Ylm, Lz acting on Ylm
00:13:01.210 --> 00:13:08.360
gives us h bar m, plus h bar
m L plus Ylm is equal to,
00:13:08.360 --> 00:13:11.800
pulling this out, this is h bar
times m plus 1 times L plus.
00:13:11.800 --> 00:13:16.810
h bar m plus 1 L plus Ylm.
00:13:16.810 --> 00:13:21.390
So L plus has raised
the eigenvalue m by one.
00:13:21.390 --> 00:13:24.270
This state, what we get by
acting on Ylm with the raising
00:13:24.270 --> 00:13:26.987
operator is a thing
with m greater by one.
00:13:26.987 --> 00:13:28.820
And that came from the
commutation relation.
00:13:28.820 --> 00:13:30.530
And if we had done the
same thing with minus,
00:13:30.530 --> 00:13:33.060
if you go through the minus
signs, it just gives us this.
00:13:36.660 --> 00:13:39.460
What does that tell us?
00:13:39.460 --> 00:13:41.460
What this tells us is we
get a ladder of states.
00:13:44.773 --> 00:13:46.190
Let's look at what
they look like.
00:13:46.190 --> 00:13:50.110
Each ladder, for a given value
of L, if you raise with L plus
00:13:50.110 --> 00:13:52.180
and lower with L minus,
you don't change L
00:13:52.180 --> 00:13:53.670
but you do change m.
00:13:53.670 --> 00:13:56.890
So we get ladders that are
labeled by L. So for example,
00:13:56.890 --> 00:13:59.910
if I have some value L1,
this is going to give me
00:13:59.910 --> 00:14:03.330
some state labeled by m.
00:14:03.330 --> 00:14:04.790
Let me put the m to the side.
00:14:04.790 --> 00:14:08.590
I can raise it to get
m plus 1 by L plus.
00:14:08.590 --> 00:14:13.930
And I can lower it to
get m minus 1 by L minus.
00:14:13.930 --> 00:14:15.855
So I got a tower.
00:14:18.500 --> 00:14:21.240
And if we have another value,
a different value of L,
00:14:21.240 --> 00:14:25.370
I'll call it L2, we
got another tower.
00:14:25.370 --> 00:14:28.050
You get m, m plus 1, m plus
2, dot, dot, dot, minus
00:14:28.050 --> 00:14:29.890
1, dot, dot, dot.
00:14:29.890 --> 00:14:34.270
So we have separated towers with
different values of L squared.
00:14:34.270 --> 00:14:37.310
And within each tower we can
raise and lower by L plus,
00:14:37.310 --> 00:14:38.110
skipping by one.
00:14:41.840 --> 00:14:43.830
OK, questions?
00:14:43.830 --> 00:14:47.260
So that was basically the
end of the last lecture,
00:14:47.260 --> 00:14:48.710
said slightly differently.
00:14:52.500 --> 00:14:54.767
Now here's the question.
00:14:54.767 --> 00:14:56.850
So this is the question
that a student asked right
00:14:56.850 --> 00:14:59.102
at the beginning.
00:14:59.102 --> 00:15:00.060
Is this tower infinite?
00:15:05.225 --> 00:15:05.850
Or does it end?
00:15:10.890 --> 00:15:12.140
So I pose to you the question.
00:15:12.140 --> 00:15:15.410
Is the tower infinite,
or does it end?
00:15:15.410 --> 00:15:16.010
And why?
00:15:19.209 --> 00:15:21.040
AUDIENCE: [INAUDIBLE] direction.
00:15:21.040 --> 00:15:23.498
PROFESSOR: OK, so it's tempting
to say it's infinite in one
00:15:23.498 --> 00:15:25.290
direction, because--?
00:15:25.290 --> 00:15:28.020
AUDIENCE: There are no bounds
to the angular momentum.
00:15:28.020 --> 00:15:29.520
PROFESSOR: OK, so
it's because there
00:15:29.520 --> 00:15:31.800
are no bounds to the angular
momentum one can have.
00:15:31.800 --> 00:15:32.467
That's tempting.
00:15:34.950 --> 00:15:37.557
AUDIENCE: But at
the same time, when
00:15:37.557 --> 00:15:40.735
you act a raising or
lowering operator on L,
00:15:40.735 --> 00:15:44.550
the eigenvalue of L
squared remains the same.
00:15:44.550 --> 00:15:48.327
So then you can't
raise the z [INAUDIBLE]
00:15:48.327 --> 00:15:50.410
of the angular momentum
above the actual momentum.
00:15:50.410 --> 00:15:50.950
PROFESSOR: Thank you.
00:15:50.950 --> 00:15:51.520
Exactly.
00:15:51.520 --> 00:15:52.320
So here's the statement.
00:15:52.320 --> 00:15:53.153
Let me restate that.
00:15:53.153 --> 00:15:54.380
That's exactly right.
00:15:54.380 --> 00:15:57.450
So look, L squared is the
eigenvalue of the total angular
00:15:57.450 --> 00:15:57.950
momentum.
00:15:57.950 --> 00:16:00.870
Roughly speaking, it's giving
you precisely in the state Ylm
00:16:00.870 --> 00:16:03.730
it tells you the expected value
of the total angular momentum,
00:16:03.730 --> 00:16:04.900
L squared.
00:16:04.900 --> 00:16:06.510
That's some number.
00:16:06.510 --> 00:16:08.890
Now, if you act with L
plus you keep increasing
00:16:08.890 --> 00:16:10.525
the expected value of Lz.
00:16:10.525 --> 00:16:12.692
But if you keep increasing
it and keep increasing it
00:16:12.692 --> 00:16:16.010
and keep increasing
it, Lz will eventually
00:16:16.010 --> 00:16:19.505
get much larger than the
square root of L squared.
00:16:19.505 --> 00:16:21.990
That probably isn't true.
00:16:21.990 --> 00:16:23.502
That sounds wrong.
00:16:23.502 --> 00:16:24.460
That was the statement.
00:16:24.460 --> 00:16:24.960
Excellent.
00:16:24.960 --> 00:16:25.720
Exactly right.
00:16:25.720 --> 00:16:27.400
So let's make that precise.
00:16:27.400 --> 00:16:28.930
So is the tower infinite?
00:16:28.930 --> 00:16:29.540
No.
00:16:29.540 --> 00:16:31.810
It's probably not, for
precisely that reason.
00:16:31.810 --> 00:16:32.935
So let's make that precise.
00:16:32.935 --> 00:16:34.560
So here's the way
we're going to do it.
00:16:34.560 --> 00:16:35.980
This is a useful
trick in general.
00:16:35.980 --> 00:16:37.940
This will outlive
angular momentum
00:16:37.940 --> 00:16:41.400
and be a useful trick throughout
quantum mechanics for you.
00:16:41.400 --> 00:16:43.090
I used it in a paper once.
00:16:43.090 --> 00:16:45.630
So here's the nice observation.
00:16:45.630 --> 00:16:47.365
Suppose it's true
that the tower ends.
00:16:47.365 --> 00:16:49.365
Just like for the raising
and lowering operators
00:16:49.365 --> 00:16:51.852
for the harmonic oscillator
in one dimension,
00:16:51.852 --> 00:16:53.560
that tells us that in
order for the power
00:16:53.560 --> 00:16:57.210
to end that state must
be 0 once we raise it.
00:16:57.210 --> 00:17:02.360
The last state, so Yl, and I'll
call this m plus, must be 0.
00:17:02.360 --> 00:17:03.740
There must be a max.
00:17:03.740 --> 00:17:06.150
Oh, sorry.
00:17:06.150 --> 00:17:08.480
Let me actually, before
I walk through exactly
00:17:08.480 --> 00:17:09.859
this statement--
00:17:09.859 --> 00:17:11.859
So let me make,
first, this, the no,
00:17:11.859 --> 00:17:13.560
slightly more
obvious and precise.
00:17:13.560 --> 00:17:16.420
So let's turn that argument
into a precise statement.
00:17:16.420 --> 00:17:18.420
L squared is equal to,
just from the definition,
00:17:18.420 --> 00:17:22.780
Lx squared plus Ly
squared plus Lz squared.
00:17:22.780 --> 00:17:24.630
Now let's take the
expectation value
00:17:24.630 --> 00:17:29.940
in this state Ylm of both
sides of this equation.
00:17:29.940 --> 00:17:34.130
So on the left-hand side we
get h bar squared ll plus 1.
00:17:34.130 --> 00:17:36.770
And on the right-hand side we
get the expectation value of Lx
00:17:36.770 --> 00:17:40.470
squared plus the expectation
value of Ly squared
00:17:40.470 --> 00:17:42.653
plus the expectation
value of Lz squared.
00:17:42.653 --> 00:17:44.820
But we know that the
expectation value of Lz squared
00:17:44.820 --> 00:17:46.520
is h bar squared, m squared.
00:17:46.520 --> 00:17:49.850
But the expectation of
Lx squared and Ly squared
00:17:49.850 --> 00:17:52.070
are strictly positive.
00:17:52.070 --> 00:17:54.410
Because this can
be written as a sum
00:17:54.410 --> 00:17:57.932
over all possible
eigenvalues of Lx squared,
00:17:57.932 --> 00:17:59.890
which is the square of
the possible eigenvalues
00:17:59.890 --> 00:18:03.170
of Lx times the
probability distribution.
00:18:03.170 --> 00:18:05.570
That's a sum of positive,
strictly positive [INAUDIBLE]..
00:18:05.570 --> 00:18:07.830
These are positive [INAUDIBLE].
00:18:07.830 --> 00:18:10.580
So ll plus 1 is
equal to positive
00:18:10.580 --> 00:18:12.700
plus positive plus h
bar squared m squared.
00:18:12.700 --> 00:18:14.283
In particular that
tells you that it's
00:18:14.283 --> 00:18:18.690
greater than or equal to
h bar squared m squared.
00:18:18.690 --> 00:18:20.620
Maybe these are 0.
00:18:20.620 --> 00:18:22.350
So the least it can be is--
00:18:22.350 --> 00:18:24.270
so the most m
squared can possibly
00:18:24.270 --> 00:18:26.340
be is the square
root of L plus 1.
00:18:26.340 --> 00:18:28.310
Or the most m can be.
00:18:28.310 --> 00:18:30.120
So m is bounded.
00:18:30.120 --> 00:18:33.800
There must be a maximum m,
and there must be a minimum m.
00:18:33.800 --> 00:18:34.800
Because this is squared.
00:18:34.800 --> 00:18:36.160
The sign doesn't matter.
00:18:36.160 --> 00:18:37.160
Everyone cool with that?
00:18:39.920 --> 00:18:42.770
OK, so let's turn this into,
now, a precise argument.
00:18:42.770 --> 00:18:44.710
What are the values
of m plus and m minus?
00:18:44.710 --> 00:18:47.770
What is the top of the
value of each tower?
00:18:47.770 --> 00:18:50.388
Probably it's going to
depend on total L, right?
00:18:50.388 --> 00:18:52.930
So it's going to depend on each
value of L. Let's check that.
00:18:52.930 --> 00:18:54.030
So here's the nice trick.
00:18:54.030 --> 00:18:57.300
Suppose we really do
have a maximum m plus.
00:18:57.300 --> 00:19:01.190
That means that if I try to
raise the state Ylm plus,
00:19:01.190 --> 00:19:05.420
I should get the state
0, which ends the tower.
00:19:05.420 --> 00:19:07.120
So suppose this is true.
00:19:07.120 --> 00:19:10.210
In that case, in particular
here's the nice trick,
00:19:10.210 --> 00:19:12.830
L plus Ylm, if we
take its magnitude,
00:19:12.830 --> 00:19:15.450
if we take the magnitude of
this state, the state is 0.
00:19:15.450 --> 00:19:16.450
It's the zero function.
00:19:16.450 --> 00:19:18.690
So what's its magnitude?
00:19:18.690 --> 00:19:19.360
Zero.
00:19:19.360 --> 00:19:22.370
You might not think that's all
that impressive an observation.
00:19:22.370 --> 00:19:24.660
OK, but note what this is.
00:19:24.660 --> 00:19:26.125
We know how to work with this.
00:19:26.125 --> 00:19:27.625
And in particular
this is equal to--
00:19:27.625 --> 00:19:30.860
I should have done
this, dot, dot, dot--
00:19:30.860 --> 00:19:33.140
I'm now going to use
the Hermitian adjoint
00:19:33.140 --> 00:19:35.110
and pull this over to
the right-hand side.
00:19:35.110 --> 00:19:36.970
The adjoint of L
plus is L minus.
00:19:36.970 --> 00:19:43.330
So this gives us Ylm,
L minus, L plus Ylm.
00:19:43.330 --> 00:19:45.517
This also doesn't look like
much of an improvement,
00:19:45.517 --> 00:19:47.600
until you notice from the
definition of L plus and
00:19:47.600 --> 00:19:50.140
L minus that what's
L minus L plus?
00:19:50.140 --> 00:19:53.870
Well, L minus L plus, we're
going to get an Lx squared.
00:19:53.870 --> 00:20:01.470
So Ylm, we get an Lx
squared plus an Ly squared.
00:20:01.470 --> 00:20:04.890
I'm going to get my sign
right, if I'm not careful.
00:20:04.890 --> 00:20:06.560
Plus-- well, let's just do it.
00:20:06.560 --> 00:20:08.240
So we have L minus L plus.
00:20:08.240 --> 00:20:12.770
So we're going to get a
Lx iLy minus i, iLyLx.
00:20:12.770 --> 00:20:15.410
So plus i commutator
of Lx with Ly.
00:20:18.930 --> 00:20:19.980
Good.
00:20:19.980 --> 00:20:23.050
So that's progress, Ylm.
00:20:23.050 --> 00:20:25.820
But it's still not progress,
because the natural operators
00:20:25.820 --> 00:20:28.900
with which to act on the
Ylm's are L squared and Lz.
00:20:28.900 --> 00:20:31.350
So can we put this in the
form L squared and Lz?
00:20:31.350 --> 00:20:31.850
Sure.
00:20:31.850 --> 00:20:35.970
This is L squared
minus Lz squared.
00:20:35.970 --> 00:20:41.790
Ylm L squared minus Lz squared.
00:20:41.790 --> 00:20:47.430
And this is i h bar Lz
times i is minus h bar Lz.
00:20:47.430 --> 00:20:51.270
So minus h bar Lz, Ylm.
00:20:53.920 --> 00:20:56.750
And this must be equal to 0.
00:20:56.750 --> 00:20:58.510
But this is equal to--
00:20:58.510 --> 00:21:04.160
from L squared we get h
bar squared, ll plus 1.
00:21:04.160 --> 00:21:08.100
From the Lz squared we get
minus m squared h bar squared.
00:21:08.100 --> 00:21:10.530
And from here we get
minus h bar, h bar m,
00:21:10.530 --> 00:21:13.395
so minus m each bar squared.
00:21:13.395 --> 00:21:14.770
Notice that the
units worked out.
00:21:14.770 --> 00:21:17.970
And all of this was
multiplying Ylm, Ylm.
00:21:17.970 --> 00:21:20.850
But Ylm, Ylm, if it's a properly
normalized eigenstate, which
00:21:20.850 --> 00:21:24.310
is what we were assuming at
the beginning, is just 1.
00:21:24.310 --> 00:21:27.310
So we get this times 1 is 0.
00:21:27.310 --> 00:21:28.080
Aha.
00:21:28.080 --> 00:21:29.730
And notice that in all
of this, this was m plus.
00:21:29.730 --> 00:21:31.313
We were assuming,
we were working here
00:21:31.313 --> 00:21:35.740
with the assumption that L plus
annihilated this top state.
00:21:35.740 --> 00:21:38.488
And so grouping this together,
this says therefore h bar
00:21:38.488 --> 00:21:40.780
squared times-- pulling out
a common h bar squared from
00:21:40.780 --> 00:21:41.620
of all this--
00:21:41.620 --> 00:21:52.740
times l, l plus 1 minus m plus,
m plus, plus 1 is equal to 0.
00:21:52.740 --> 00:21:57.120
And this tells us that
m plus is equal to l.
00:21:57.120 --> 00:22:01.870
And if you want to be
strict, put a plus sign.
00:22:01.870 --> 00:22:03.160
Everyone cool with that?
00:22:03.160 --> 00:22:04.480
This is a very useful trick.
00:22:04.480 --> 00:22:06.130
If you know something
is 0 as function,
00:22:06.130 --> 00:22:07.220
you know its norm is 0.
00:22:07.220 --> 00:22:09.500
And now you can use things
like Hermitian adjoints.
00:22:09.500 --> 00:22:10.300
Very, very useful.
00:22:13.400 --> 00:22:14.925
OK questions about that?
00:22:14.925 --> 00:22:16.410
Yeah.
00:22:16.410 --> 00:22:20.755
AUDIENCE: [INAUDIBLE]
00:22:20.755 --> 00:22:21.630
PROFESSOR: Excellent.
00:22:21.630 --> 00:22:23.463
Where this came from
is I was just literally
00:22:23.463 --> 00:22:26.600
taking L plus and L minus,
taking the definitions,
00:22:26.600 --> 00:22:27.680
and plugging them in.
00:22:27.680 --> 00:22:30.320
So if I have L minus
L plus, L minus
00:22:30.320 --> 00:22:31.820
is, just to write
it out explicitly,
00:22:31.820 --> 00:22:33.090
L minus is equal to Lx.
00:22:33.090 --> 00:22:34.990
Because the [INAUDIBLE]
are observables,
00:22:34.990 --> 00:22:37.115
and so they're Hermitians,
so they're self-adjoint,
00:22:37.115 --> 00:22:38.470
the minus iLy.
00:22:38.470 --> 00:22:41.270
So L minus L plus
gives me an LxLx.
00:22:41.270 --> 00:22:43.020
It gives me an LxiLy.
00:22:43.020 --> 00:22:45.020
It gives you a minus iLyLx.
00:22:45.020 --> 00:22:49.370
And it gives me a minus ii,
which is plus 1 Ly squared.
00:22:49.370 --> 00:22:50.140
Cool?
00:22:50.140 --> 00:22:51.400
Excellent.
00:22:51.400 --> 00:22:53.650
I'm in a state of
serious desperation here.
00:22:58.720 --> 00:22:59.290
Good.
00:22:59.290 --> 00:23:00.468
Other questions?
00:23:00.468 --> 00:23:01.782
Yeah.
00:23:01.782 --> 00:23:04.640
AUDIENCE: [INAUDIBLE] How do
you get from that to that?
00:23:04.640 --> 00:23:05.640
PROFESSOR: This to here?
00:23:05.640 --> 00:23:07.010
OK, good.
00:23:07.010 --> 00:23:08.463
Sorry, I jumped a step here.
00:23:08.463 --> 00:23:09.880
So here what I
said is, look, I've
00:23:09.880 --> 00:23:12.210
got this nice set of
operators acting on Ylm.
00:23:12.210 --> 00:23:15.240
I know how each of these
operators acts on Ylm.
00:23:15.240 --> 00:23:19.060
Lz gives me an h bar m,
minus some h bar squared.
00:23:19.060 --> 00:23:20.360
Lz squared, L squared.
00:23:20.360 --> 00:23:23.290
And then overall that was a just
some number times Ylm, which
00:23:23.290 --> 00:23:24.950
I can pull out of
the inner product.
00:23:24.950 --> 00:23:25.450
Cool?
00:23:25.450 --> 00:23:26.498
Yeah?
00:23:26.498 --> 00:23:28.513
AUDIENCE: [INAUDIBLE]
00:23:28.513 --> 00:23:29.180
PROFESSOR: Good.
00:23:29.180 --> 00:23:30.750
This subscript plus
meant that, look,
00:23:30.750 --> 00:23:32.250
there was a maximum value of m.
00:23:32.250 --> 00:23:34.990
So m squared had to be
less than or equal to this.
00:23:34.990 --> 00:23:37.630
There's a maximum value
and a minimum value.
00:23:37.630 --> 00:23:38.630
Maybe they're different.
00:23:38.630 --> 00:23:39.172
I don't know.
00:23:39.172 --> 00:23:42.470
I'm just going to be
open-minded about that.
00:23:42.470 --> 00:23:42.970
Others?
00:23:42.970 --> 00:23:43.850
Yeah?
00:23:43.850 --> 00:23:45.975
AUDIENCE: Will that mean,
then, that you can't ever
00:23:45.975 --> 00:23:48.758
have all the angular
momentum [INAUDIBLE]??
00:23:48.758 --> 00:23:50.300
PROFESSOR: Yeah,
awesome observation.
00:23:50.300 --> 00:23:51.490
Exactly.
00:23:51.490 --> 00:23:53.282
We were going to get
there in a little bit.
00:23:53.282 --> 00:23:53.782
I like that.
00:23:53.782 --> 00:23:54.700
That's exactly right.
00:23:57.920 --> 00:24:00.100
Let me go through a couple
more steps, and then I'll
00:24:00.100 --> 00:24:01.400
come back to that observation.
00:24:01.400 --> 00:24:02.655
And I promise I will say so.
00:24:05.200 --> 00:24:07.560
So if we go through
exactly a similar argument,
00:24:07.560 --> 00:24:10.710
let's see what happens if
we did L minus on Ylm minus,
00:24:10.710 --> 00:24:12.330
just to walk through the logic.
00:24:12.330 --> 00:24:15.660
So if this were-- you lower
the lowest one, you get 0.
00:24:15.660 --> 00:24:18.170
Then we'd get the same
story, L minus L minus.
00:24:18.170 --> 00:24:21.800
When we take the adjoint
we'd get L plus, L minus.
00:24:21.800 --> 00:24:23.200
And L plus L minus,
what changes?
00:24:23.200 --> 00:24:24.742
The only thing that's
going to change
00:24:24.742 --> 00:24:27.790
is that you get this
commutator the other direction.
00:24:27.790 --> 00:24:29.720
And there are minus
signs in various places.
00:24:29.720 --> 00:24:33.470
The upshot of which is that
m minus is equal to minus l.
00:24:35.990 --> 00:24:37.240
So this is quite parsimonious.
00:24:37.240 --> 00:24:38.440
It's symmetric.
00:24:38.440 --> 00:24:40.118
If you take z to
minus z, if you switch
00:24:40.118 --> 00:24:42.660
the sign of the angular momentum
you get the same thing back.
00:24:42.660 --> 00:24:43.940
That's satisfying, perhaps.
00:24:48.580 --> 00:24:50.790
But it's way more than that.
00:24:50.790 --> 00:24:53.460
This tells us a lot about
the possible eigenvalues,
00:24:53.460 --> 00:24:55.330
in the following way.
00:24:55.330 --> 00:24:56.980
Look back at our towers.
00:24:56.980 --> 00:25:01.242
In our towers we have that
the angular momentum is
00:25:01.242 --> 00:25:02.450
raised and lowered by L plus.
00:25:02.450 --> 00:25:05.450
So the Lz angular momentum is
raised and lowered by L plus.
00:25:05.450 --> 00:25:06.970
l remains the same.
00:25:06.970 --> 00:25:10.230
But there's a maximum
state now, which is plus l.
00:25:10.230 --> 00:25:12.450
And there's a minimum
stay in here, l1.
00:25:12.450 --> 00:25:14.527
And here there's a
minimum state, minus l1.
00:25:14.527 --> 00:25:16.110
Similarly for the
other tower, there's
00:25:16.110 --> 00:25:20.000
a minimum state minus l2
and a maximum state, l2.
00:25:20.000 --> 00:25:23.127
So how big each tower is depends
on the total angular momentum.
00:25:23.127 --> 00:25:24.460
That kind of makes sense, right?
00:25:24.460 --> 00:25:26.150
If you've got more
angular momentum
00:25:26.150 --> 00:25:28.080
and you can only step
Lz by one, you've
00:25:28.080 --> 00:25:30.535
got more room to move
with a large value of l
00:25:30.535 --> 00:25:32.500
than with a small value of l.
00:25:32.500 --> 00:25:33.000
OK.
00:25:35.600 --> 00:25:38.370
So what does that tell
us about the values of m?
00:25:38.370 --> 00:25:47.530
Notice that m spans the
values from its minimal value,
00:25:47.530 --> 00:25:53.220
minus l to l in integer
steps, in unit steps.
00:25:58.140 --> 00:26:04.895
So if you think about m
is l, then m is l minus 1.
00:26:04.895 --> 00:26:09.490
And if I keep lowering I get
down to m is minus l plus 1.
00:26:09.490 --> 00:26:11.976
And then m is equal to minus l.
00:26:15.240 --> 00:26:18.435
So the difference in the Lz
eigenvalue between these guys,
00:26:18.435 --> 00:26:19.310
the difference is 2l.
00:26:23.070 --> 00:26:24.930
But the number of
unit steps in here
00:26:24.930 --> 00:26:30.230
is one fewer, because one,
dot, dot, dot 2l minus 1.
00:26:30.230 --> 00:26:33.090
So this is some
integer, which is
00:26:33.090 --> 00:26:35.330
the number of states minus 1.
00:26:35.330 --> 00:26:40.940
So if these are n states, and
I'll call this N sub l states,
00:26:40.940 --> 00:26:44.570
then this difference twice
l is N sub l minus 1,
00:26:44.570 --> 00:26:45.590
because it's unit steps.
00:26:45.590 --> 00:26:47.440
Cool?
00:26:47.440 --> 00:26:51.690
So for example, if there
were two states, so N is 2,
00:26:51.690 --> 00:26:54.530
2l is equal to--
00:26:54.530 --> 00:26:58.575
well, that's 1,
which is 2 minus 1,
00:26:58.575 --> 00:27:00.492
the number of states
minus 1, also known as 1.
00:27:00.492 --> 00:27:02.700
So l is 1/2.
00:27:02.700 --> 00:27:04.420
And more generally
we find that l
00:27:04.420 --> 00:27:06.107
must be, in order
for this process
00:27:06.107 --> 00:27:07.690
to make sense, in
order for the m plus
00:27:07.690 --> 00:27:11.380
and the m minus to
match up, we need
00:27:11.380 --> 00:27:17.860
that l is of the form an
integer N sub l minus 1 upon 2.
00:27:17.860 --> 00:27:20.670
So this tells us that
where Nl is an integer--
00:27:20.670 --> 00:27:23.780
Nl is just the number
of states in this tower,
00:27:23.780 --> 00:27:25.580
and it's a strictly
positive integer.
00:27:25.580 --> 00:27:27.122
If you have zero
states in the tower,
00:27:27.122 --> 00:27:29.440
then that's not
very interesting.
00:27:29.440 --> 00:27:40.500
So this tells us that l is
an integer a half integer,
00:27:40.500 --> 00:27:42.810
but nothing else.
00:27:42.810 --> 00:27:43.310
Cool?
00:27:49.106 --> 00:27:52.940
In particular, if
it's a half integer,
00:27:52.940 --> 00:27:59.837
that means 1/2, 3/2, 5/2, if
it's an integer or it can be 0.
00:27:59.837 --> 00:28:01.420
There's nothing wrong
with it being 0.
00:28:01.420 --> 00:28:03.376
But then 1 on.
00:28:07.750 --> 00:28:12.010
So that means we can plot our
system in the following way.
00:28:12.010 --> 00:28:22.320
If we have l equals 0, how
many states do we have?
00:28:22.320 --> 00:28:24.690
If little l is 0,
how many states are?
00:28:24.690 --> 00:28:28.990
What's the largest value of Lz?
00:28:28.990 --> 00:28:32.230
What's the largest allowed
value of Lz or of m
00:28:32.230 --> 00:28:34.160
if little l is equal to 0?
00:28:34.160 --> 00:28:34.660
AUDIENCE: 0.
00:28:34.660 --> 00:28:35.202
PROFESSOR: 0.
00:28:35.202 --> 00:28:37.350
Because it goes from
plus 0 to minus 0.
00:28:37.350 --> 00:28:38.760
So that's pretty much 0.
00:28:38.760 --> 00:28:43.930
So we have a single state with
m equals 0 and l equals 0.
00:28:43.930 --> 00:28:46.470
If l is equal to-- what's
the next possible value?--
00:28:46.470 --> 00:28:52.130
1/2, then m can be
either 1/2 or we lower it
00:28:52.130 --> 00:28:53.740
by 1, which is minus 1/2.
00:28:53.740 --> 00:28:57.460
So there are two
states, m equals
00:28:57.460 --> 00:28:59.400
1/2 and m equals minus 1/2.
00:29:02.030 --> 00:29:03.040
So there's one power.
00:29:03.040 --> 00:29:04.040
It's a very short tower.
00:29:04.040 --> 00:29:07.400
This is the shortest
possible tower.
00:29:07.400 --> 00:29:11.670
Then we also have the state l
equals 1, which has m equals 0.
00:29:11.670 --> 00:29:15.050
It has m equals 1.
00:29:15.050 --> 00:29:18.120
And it has m equals minus 1.
00:29:18.120 --> 00:29:20.900
And that's it.
00:29:20.900 --> 00:29:22.030
And so on and so forth.
00:29:25.570 --> 00:29:33.190
Four states for l equals 3/2,
with states 3/2, 1/2, this is m
00:29:33.190 --> 00:29:38.710
equals 1/2, m equals
minus 1/2, and minus 3/2.
00:29:38.710 --> 00:29:41.390
The values of m span from minus
l to l with integer steps.
00:29:46.480 --> 00:29:51.260
And this is possible for
all values of l which are
00:29:51.260 --> 00:30:02.920
of the form all integer
or half integer l's.
00:30:02.920 --> 00:30:05.040
So for every different
value of the total angular
00:30:05.040 --> 00:30:06.390
momentum, the total
amount of angular momentum
00:30:06.390 --> 00:30:08.300
you have, you have
a tower of states
00:30:08.300 --> 00:30:10.660
labeled by Lz's eigenvalue
in this fashion.
00:30:13.960 --> 00:30:14.460
Questions?
00:30:17.508 --> 00:30:21.510
Does anyone notice anything
troubling about these?
00:30:21.510 --> 00:30:23.458
Something physically
a little discomfiting
00:30:23.458 --> 00:30:24.250
about any of these?
00:30:27.262 --> 00:30:29.470
What does it mean to say
I'm in the state l equals 0,
00:30:29.470 --> 00:30:29.970
m equals 0?
00:30:29.970 --> 00:30:33.050
What is that telling you?
00:30:33.050 --> 00:30:34.060
Zero angular momentum.
00:30:34.060 --> 00:30:36.540
What's the expectation
value of l squared?
00:30:36.540 --> 00:30:37.040
Zero.
00:30:37.040 --> 00:30:39.400
The expectation value of Lz?
00:30:39.400 --> 00:30:44.050
And by rotational invariance,
the expected value of Lx or Ly?
00:30:44.050 --> 00:30:45.500
That thing is not rotating.
00:30:45.500 --> 00:30:47.580
There's no angular momentum.
00:30:47.580 --> 00:30:49.990
So the no angular momentum
state is one with l equals 0,
00:30:49.990 --> 00:30:50.490
m equals 0.
00:30:50.490 --> 00:30:52.840
So this is not spinning.
00:30:52.840 --> 00:30:54.590
What about this guy?
00:30:54.590 --> 00:30:56.900
L equals 1, m equals 0.
00:30:56.900 --> 00:31:00.300
Does this thing carry
angular momentum?
00:31:00.300 --> 00:31:01.965
Yeah, absolutely.
00:31:01.965 --> 00:31:03.840
So it just doesn't carry
any angular momentum
00:31:03.840 --> 00:31:04.680
in the z direction.
00:31:04.680 --> 00:31:06.660
But its total angular
momentum on average,
00:31:06.660 --> 00:31:09.180
its expected total
angular momentum
00:31:09.180 --> 00:31:13.940
is ll plus 1, which is
2, times h bar squared.
00:31:13.940 --> 00:31:18.295
So if you measure Lx and Ly
what do you expect to get?
00:31:18.295 --> 00:31:20.170
Well, if you measure Lx
squared or Ly squared
00:31:20.170 --> 00:31:22.087
you probably expect to
get something non-zero.
00:31:22.087 --> 00:31:24.260
We'll come back to
that in just a second.
00:31:24.260 --> 00:31:28.073
And what about the state
l equals 1, m equals 1?
00:31:28.073 --> 00:31:29.490
Your angular
momentum-- you've got
00:31:29.490 --> 00:31:30.850
as much angular momentum
in the z direction as you
00:31:30.850 --> 00:31:31.970
possibly can.
00:31:31.970 --> 00:31:33.890
So that definitely
carries angular momentum.
00:31:33.890 --> 00:31:35.580
So there's a state
that has no angular
00:31:35.580 --> 00:31:36.860
momentum in the z direction.
00:31:36.860 --> 00:31:38.277
And there's a state
that has some,
00:31:38.277 --> 00:31:40.210
and there's a state
that has less.
00:31:40.210 --> 00:31:41.190
That make sense.
00:31:41.190 --> 00:31:42.628
Yeah?
00:31:42.628 --> 00:31:45.048
AUDIENCE: If m equals
1 and l equals 1,
00:31:45.048 --> 00:31:47.952
that means that the angular
momentum for the z direction
00:31:47.952 --> 00:31:50.173
is the total angular momentum?
00:31:50.173 --> 00:31:50.881
PROFESSOR: Is it?
00:31:50.881 --> 00:31:54.180
AUDIENCE: And then
Lx and Ly was 0?
00:31:54.180 --> 00:31:56.998
PROFESSOR: OK, that's
an excellent question.
00:31:56.998 --> 00:31:58.540
Let me answer that
now, and then I'll
00:31:58.540 --> 00:32:00.470
come back to the point
I wanted to make.
00:32:00.470 --> 00:32:00.730
Hold on a second.
00:32:00.730 --> 00:32:02.105
Let me just answer
that question.
00:32:05.610 --> 00:32:06.270
I'll work here.
00:32:11.390 --> 00:32:14.120
So here's the crucial thing.
00:32:14.120 --> 00:32:16.860
Even in this state, so you
were asking about the state l
00:32:16.860 --> 00:32:18.110
equals 1, m equals 1.
00:32:18.110 --> 00:32:21.170
And the question that was asked,
a very good question, is look,
00:32:21.170 --> 00:32:23.930
does that mean all the angular
momentum is in the z direction,
00:32:23.930 --> 00:32:26.247
and Lx and Ly are 0?
00:32:26.247 --> 00:32:27.830
But let me just ask
this more broadly.
00:32:27.830 --> 00:32:29.830
Suppose we have a state
with angular momentum l,
00:32:29.830 --> 00:32:31.840
and m is equal to l.
00:32:31.840 --> 00:32:33.590
The most angular
momentum you can possibly
00:32:33.590 --> 00:32:35.215
have in the z direction,
same question.
00:32:35.215 --> 00:32:38.025
Is all the angular momentum
in the z direction?
00:32:38.025 --> 00:32:39.650
And what I want to
emphasize you is no,
00:32:39.650 --> 00:32:41.040
that's absolutely not the case.
00:32:41.040 --> 00:32:42.730
So two arguments for that.
00:32:42.730 --> 00:32:45.950
The first is, suppose it's true
that Lx and Ly are identically
00:32:45.950 --> 00:32:48.512
0.
00:32:48.512 --> 00:32:50.220
Can that satisfy the
uncertainty relation
00:32:50.220 --> 00:32:51.430
due to those commutators?
00:32:51.430 --> 00:32:51.930
No.
00:32:51.930 --> 00:32:53.670
There must be
uncertainty in Lx and Ly,
00:32:53.670 --> 00:32:57.540
because Lz has a non-zero
expectation value.
00:32:57.540 --> 00:32:59.873
So it can't be that Lx
squared and Ly squared
00:32:59.873 --> 00:33:01.040
have zero expectation value.
00:33:01.040 --> 00:33:03.100
But let's be more
precise about this.
00:33:03.100 --> 00:33:05.440
The expectation value of L
squared is easy to calculate.
00:33:05.440 --> 00:33:09.050
It's h bar squared l, plus 1.
00:33:09.050 --> 00:33:10.570
Because we're in the state Ylm.
00:33:10.570 --> 00:33:15.660
This is the state
Y l sub l, or ll.
00:33:15.660 --> 00:33:21.490
The expectation of Lz squared
is equal to h bar squared,
00:33:21.490 --> 00:33:22.492
m squared.
00:33:22.492 --> 00:33:23.950
And m squared now,
m is equal to l.
00:33:23.950 --> 00:33:25.880
So h bar squared, l squared.
00:33:25.880 --> 00:33:26.425
Aha.
00:33:26.425 --> 00:33:27.800
So the expected
value of l square
00:33:27.800 --> 00:33:29.425
is not the same as
Lx squared, but this
00:33:29.425 --> 00:33:33.510
is equal to the expected value
of Lx squared plus Ly squared
00:33:33.510 --> 00:33:34.340
plus Lz squared.
00:33:41.790 --> 00:33:44.260
Therefore the
expected value of Lx
00:33:44.260 --> 00:33:49.160
squared plus the expected
value of Ly squared
00:33:49.160 --> 00:33:51.498
is equal to the difference
between this and this.
00:33:51.498 --> 00:33:54.040
We just subtract this off h bar
squared ll plus 1 minus h bar
00:33:54.040 --> 00:33:57.000
squared, l squared,
h bar squared l.
00:33:57.000 --> 00:33:58.900
And by symmetry you
don't expect the symmetry
00:33:58.900 --> 00:34:00.932
to be broken between Lx and Ly.
00:34:00.932 --> 00:34:02.390
You can actually
do the calculation
00:34:02.390 --> 00:34:03.450
and not just be glib about it.
00:34:03.450 --> 00:34:05.408
But both arguments give
you the correct answer.
00:34:05.408 --> 00:34:08.110
The expectation
value of Lx squared
00:34:08.110 --> 00:34:11.960
is equal to 1/2 h bar squared l.
00:34:11.960 --> 00:34:16.690
And ditto for y, in this state.
00:34:16.690 --> 00:34:17.667
So notice two things.
00:34:17.667 --> 00:34:20.000
First off is we make the total
angular momentum little l
00:34:20.000 --> 00:34:21.719
large.
00:34:21.719 --> 00:34:24.460
The amount by which we fail
to have all the angular
00:34:24.460 --> 00:34:26.860
momentum in the z direction
is getting larger and larger.
00:34:26.860 --> 00:34:30.410
We're increasing the crappiness
of putting all the angular
00:34:30.410 --> 00:34:32.010
momentum in the z direction.
00:34:32.010 --> 00:34:37.080
However, as a ratio of the total
angular momentum divided by L
00:34:37.080 --> 00:34:40.630
squared, so this
divided by L squared,
00:34:40.630 --> 00:34:43.280
and this is h bar
squared, l, l plus 1,
00:34:43.280 --> 00:34:45.690
and in particular this
is l squared plus l,
00:34:45.690 --> 00:34:48.578
so if we took the ratio,
the rational mismatch
00:34:48.578 --> 00:34:49.870
is getting smaller and smaller.
00:34:49.870 --> 00:34:50.449
And that's good.
00:34:50.449 --> 00:34:52.280
Because as we go to very
large angular momentums
00:34:52.280 --> 00:34:53.719
where things should
start getting classical
00:34:53.719 --> 00:34:56.250
in some sense, we should get
back the familiar intuition
00:34:56.250 --> 00:34:59.290
that you can put all the angular
momentum in the z direction.
00:34:59.290 --> 00:35:00.228
Yeah?
00:35:00.228 --> 00:35:02.520
AUDIENCE: Why are we imposing
the [INAUDIBLE] condition
00:35:02.520 --> 00:35:05.440
that the expectation values of
Lx and Ly should be identical?
00:35:05.440 --> 00:35:05.600
PROFESSOR: Excellent.
00:35:05.600 --> 00:35:07.030
That's why I was saying
this is a glib argument.
00:35:07.030 --> 00:35:08.190
You don't have to impose--
00:35:08.190 --> 00:35:10.130
that needs to be done a
little bit more delicately.
00:35:10.130 --> 00:35:11.400
But we can just
directly compute this.
00:35:11.400 --> 00:35:12.830
And you do so on
your problem set.
00:35:15.610 --> 00:35:16.140
Yeah?
00:35:16.140 --> 00:35:18.250
AUDIENCE: Why
doesn't the existence
00:35:18.250 --> 00:35:21.770
of the l equals 0
eigenfunction where the angular
00:35:21.770 --> 00:35:26.095
momentum the L squared
is definitely 0
00:35:26.095 --> 00:35:27.870
violate the
uncertainty principle?
00:35:27.870 --> 00:35:28.250
PROFESSOR: Awesome.
00:35:28.250 --> 00:35:30.250
On your problem set you're
going to answer that,
00:35:30.250 --> 00:35:33.030
but let me give you
a quick preview.
00:35:33.030 --> 00:35:34.520
This is such a great response.
00:35:34.520 --> 00:35:37.140
Just Let me give
you a quick preview.
00:35:37.140 --> 00:35:38.420
So from that Lx--
00:35:38.420 --> 00:35:40.090
OK, this is a
really fun question.
00:35:40.090 --> 00:35:43.060
Let me go into it
in some detail.
00:35:43.060 --> 00:35:44.610
Wow, I'm going fast today.
00:35:44.610 --> 00:35:46.320
Am I going way too fast?
00:35:46.320 --> 00:35:47.550
No?
00:35:47.550 --> 00:35:48.110
A little?
00:35:48.110 --> 00:35:49.210
Little too fast?
00:35:49.210 --> 00:35:50.960
OK, ask me more questions
to slow me down.
00:35:50.960 --> 00:35:51.520
I'm excited.
00:35:51.520 --> 00:35:52.978
I didn't get much
sleep last night.
00:35:57.090 --> 00:35:58.840
One of the great joys
of being a physicist
00:35:58.840 --> 00:36:01.280
is working with
other physicists.
00:36:01.280 --> 00:36:03.210
So yesterday one of
my very good friends
00:36:03.210 --> 00:36:06.870
and a collaborator I really
delight in talking with
00:36:06.870 --> 00:36:07.790
came to visit.
00:36:07.790 --> 00:36:09.210
And we had a late night dinner.
00:36:09.210 --> 00:36:11.835
And this led to me late at
night, not doing my work,
00:36:11.835 --> 00:36:14.210
but reading papers about what
our conversation was about.
00:36:14.210 --> 00:36:17.552
And only then at the very end,
when I was just about to die
00:36:17.552 --> 00:36:19.760
did I write the response to
the reviewer on the paper
00:36:19.760 --> 00:36:22.135
that I was supposed to be
doing by last night's deadline.
00:36:22.135 --> 00:36:24.750
So I'm kind of tired, but
I'm in a really good mood.
00:36:28.610 --> 00:36:29.610
It's a really good job.
00:36:34.580 --> 00:36:36.330
Now I've totally lost
my train of thought.
00:36:36.330 --> 00:36:37.992
What was the question again?
00:36:37.992 --> 00:36:39.760
AUDIENCE: The existence
of the l equals
00:36:39.760 --> 00:36:43.250
0 state where the total angular
momentum is definitely--
00:36:43.250 --> 00:36:45.000
PROFESSOR: Excellent,
and the uncertainty.
00:36:45.000 --> 00:36:47.910
So the question is, why don't
we violate the uncertainty when
00:36:47.910 --> 00:36:49.530
we know that L
equals-- where am I;
00:36:49.530 --> 00:36:53.608
I just covered it-- when we know
that L equals 0 and m equals 0,
00:36:53.608 --> 00:36:55.150
doesn't that destroy
our uncertainty?
00:36:55.150 --> 00:36:57.233
Because we know that the
angular momentum Lz is 0.
00:36:57.233 --> 00:36:58.442
Angular momentum for Lx is 0.
00:36:58.442 --> 00:36:59.665
Angular momentum for Ly is 0.
00:36:59.665 --> 00:37:00.902
All of them vanish.
00:37:00.902 --> 00:37:02.860
Doesn't that violate the
uncertainty principle?
00:37:02.860 --> 00:37:04.980
So let's remind ourselves what
the form of that uncertainty
00:37:04.980 --> 00:37:05.750
relation is.
00:37:05.750 --> 00:37:08.680
The form of the uncertainty
relation following from Lx Ly
00:37:08.680 --> 00:37:10.200
is i h bar Lz.
00:37:10.200 --> 00:37:13.260
Recall the general statement.
00:37:13.260 --> 00:37:16.710
The uncertainty in A times
the uncertainty in B, squared,
00:37:16.710 --> 00:37:18.700
squared, is equal to--
00:37:18.700 --> 00:37:23.430
let me just write
it as h bar upon 2--
00:37:23.430 --> 00:37:25.290
sorry, 1/2.
00:37:25.290 --> 00:37:27.230
The absolute value of
the expectation value
00:37:27.230 --> 00:37:33.550
of the commutator,
A with B. Good lord.
00:37:33.550 --> 00:37:35.230
Dimensionally, does this work?
00:37:35.230 --> 00:37:36.090
Yes.
00:37:36.090 --> 00:37:36.590
OK, good.
00:37:36.590 --> 00:37:39.570
Because units of A, units of
B. Unites of A, units of B.
00:37:39.570 --> 00:37:41.602
Triumph.
00:37:41.602 --> 00:37:43.810
And these are going to be
quantum-mechanically small,
00:37:43.810 --> 00:37:45.660
because commutators have h bars.
00:37:45.660 --> 00:37:47.120
And commutators have h bars why?
00:37:49.850 --> 00:37:51.020
Not because God hates us.
00:37:51.020 --> 00:37:52.460
Why do commutators have h bars?
00:37:55.810 --> 00:37:58.950
What happens classically?
00:37:58.950 --> 00:38:01.670
In classical mechanics,
do things commute?
00:38:01.670 --> 00:38:02.250
Yes.
00:38:02.250 --> 00:38:04.470
Why are the h bars
and commutators
00:38:04.470 --> 00:38:06.020
physical observables?
00:38:06.020 --> 00:38:08.133
Because we exist.
00:38:08.133 --> 00:38:09.550
Because there's a
classical limit.
00:38:09.550 --> 00:38:11.630
OK so this is going to make
quantum-mechanically small,
00:38:11.630 --> 00:38:13.505
so we expect the
uncertainty relation to also
00:38:13.505 --> 00:38:15.398
be quantum-mechanically small.
00:38:15.398 --> 00:38:16.440
Just important intuition.
00:38:16.440 --> 00:38:20.730
So let's look at the specific
example of Lx, Ly, and Lz.
00:38:20.730 --> 00:38:23.290
So the uncertainty in
Lx, in any particular--
00:38:23.290 --> 00:38:24.550
remember that this is
defined as the uncertainty
00:38:24.550 --> 00:38:26.770
in a particular state psi,
in a particular state psi.
00:38:26.770 --> 00:38:29.145
And this expectation value is
taken in a particular state
00:38:29.145 --> 00:38:32.010
psi, that same state.
00:38:32.010 --> 00:38:37.470
So the uncertainty of
Lx, in some state Ylm,
00:38:37.470 --> 00:38:41.610
times the uncertainty of
Ly in that same state Ylm
00:38:41.610 --> 00:38:47.171
should be greater
than or equal to 1/2
00:38:47.171 --> 00:38:48.963
the absolute value of
the expectation value
00:38:48.963 --> 00:38:50.330
of the commutator of Lx and Ly.
00:38:50.330 --> 00:38:53.390
But the commutator of
Lx and Ly is i h bar Lz.
00:38:53.390 --> 00:38:56.140
And i, when we pull it
through this absolute value,
00:38:56.140 --> 00:38:58.710
is going to give me just 1. h
bar is going to give me h bar.
00:38:58.710 --> 00:39:04.330
So h bar upon 2, expectation
value of Lz, absolute value.
00:39:04.330 --> 00:39:06.420
Yeah?
00:39:06.420 --> 00:39:08.480
Can Lx and Ly have
zero uncertainty?
00:39:16.340 --> 00:39:18.450
When?
00:39:18.450 --> 00:39:21.197
Expectation value of Lz is 0.
00:39:21.197 --> 00:39:22.030
So that sounds good.
00:39:22.030 --> 00:39:27.080
It sounds like if Lz has
expectation value of 0,
00:39:27.080 --> 00:39:29.590
then we can have Lx
and Ly, definite.
00:39:29.590 --> 00:39:31.430
But that's bad.
00:39:31.430 --> 00:39:32.700
Really?
00:39:32.700 --> 00:39:34.460
Really, can we do that?
00:39:34.460 --> 00:39:35.790
Why not?
00:39:35.790 --> 00:39:37.690
AUDIENCE: [INAUDIBLE]
00:39:37.690 --> 00:39:40.065
[LAUGHTER]
00:39:43.532 --> 00:39:44.740
PROFESSOR: Bless you, my son.
00:39:51.060 --> 00:39:54.520
Can we have Lx and Ly both
take definite values, just
00:39:54.520 --> 00:39:57.050
because Lz?
00:39:57.050 --> 00:39:59.210
Why?
00:39:59.210 --> 00:40:03.070
What else do we have to satisfy?
00:40:03.070 --> 00:40:05.750
What other uncertainty
relations must we satisfy?
00:40:08.390 --> 00:40:10.820
There are two more.
00:40:10.820 --> 00:40:15.230
And I invite you to go look
at what those two more are
00:40:15.230 --> 00:40:19.520
and deduce that this is
only possible if Lx squared,
00:40:19.520 --> 00:40:23.110
Ly squared, and Lz squared all
have zero expectation value.
00:40:23.110 --> 00:40:26.770
In fact, I think it's just
a great question that I
00:40:26.770 --> 00:40:30.358
think it's on your problem set.
00:40:30.358 --> 00:40:31.650
So thank you for that question.
00:40:31.650 --> 00:40:32.460
It's a really good question.
00:40:32.460 --> 00:40:34.040
There was another
question in here.
00:40:34.040 --> 00:40:35.160
Yeah?
00:40:35.160 --> 00:40:40.944
AUDIENCE: Something that
you said earlier [INAUDIBLE]
00:40:40.944 --> 00:40:41.812
half integer.
00:40:41.812 --> 00:40:43.354
So were you deriving
this by counting
00:40:43.354 --> 00:40:46.728
the number of equations
and somehow asserting--
00:40:46.728 --> 00:40:47.792
Why does--?
00:40:47.792 --> 00:40:48.500
PROFESSOR: Great.
00:40:48.500 --> 00:40:50.570
So the question
is, wait, really?
00:40:50.570 --> 00:40:52.273
Why is L and integer
a half integer.
00:40:52.273 --> 00:40:53.440
That was a little too quick.
00:40:53.440 --> 00:40:55.017
Is that roughly the
right statement?
00:40:55.017 --> 00:40:56.503
AUDIENCE: [INAUDIBLE].
00:40:56.503 --> 00:40:57.170
PROFESSOR: Good.
00:40:57.170 --> 00:40:57.660
Excellent.
00:40:57.660 --> 00:40:58.827
Let me go through the logic.
00:40:58.827 --> 00:41:01.590
So the logic goes like this.
00:41:01.590 --> 00:41:02.870
I know that the Ylm's--
00:41:02.870 --> 00:41:06.640
if the Ylm's are eigenfunctions
of L squared and Lz,
00:41:06.640 --> 00:41:09.178
and I've constructed this
tower of them using the raising
00:41:09.178 --> 00:41:10.720
and lowering operators,
we've already
00:41:10.720 --> 00:41:13.300
shown that the largest
possible value of m is l
00:41:13.300 --> 00:41:15.470
and the least possible
value is minus l.
00:41:15.470 --> 00:41:17.000
And these states
must be separated
00:41:17.000 --> 00:41:19.370
by integer steps in m.
00:41:19.370 --> 00:41:19.870
OK, good.
00:41:19.870 --> 00:41:23.960
So pick a value of l,
a particular tower.
00:41:23.960 --> 00:41:27.530
And let the number of states
in that tower be N sub l.
00:41:27.530 --> 00:41:28.990
So there's N sub l of them.
00:41:28.990 --> 00:41:29.700
Great.
00:41:29.700 --> 00:41:34.355
And what's the distance
between these guys?
00:41:34.355 --> 00:41:35.980
We haven't assumed
l is an integer yet.
00:41:35.980 --> 00:41:38.610
We haven't assumed that.
00:41:38.610 --> 00:41:42.431
So this N sub l is an integer.
00:41:42.431 --> 00:41:43.848
Because it's the
number of states.
00:41:43.848 --> 00:41:45.938
And the number of
states can't be a pi.
00:41:50.313 --> 00:41:51.230
Now let's count that--
00:41:51.230 --> 00:41:53.693
that so how many
states are there?
00:41:53.693 --> 00:41:56.195
There are Nl.
00:41:56.195 --> 00:41:59.360
But if I count one,
two, three, four,
00:41:59.360 --> 00:42:02.320
the total angular
momentum down here is 2l.
00:42:04.830 --> 00:42:11.500
I had some pithy way of
giving this a fancy name.
00:42:11.500 --> 00:42:13.820
But I can't remember
what it was.
00:42:13.820 --> 00:42:15.670
So the length of
the tower in units
00:42:15.670 --> 00:42:22.140
of h bar, the height
of this tower is 2l.
00:42:22.140 --> 00:42:27.030
But the number of steps I
took in here was Nl minus 1.
00:42:27.030 --> 00:42:29.550
And that number of
steps is times 1.
00:42:29.550 --> 00:42:32.435
So we get that 2l is N minus 1.
00:42:32.435 --> 00:42:33.560
There's nothing fancy here.
00:42:33.560 --> 00:42:36.128
I'm just saying if L is 0
we go from here to here.
00:42:36.128 --> 00:42:37.170
There's just one element.
00:42:37.170 --> 00:42:39.400
So number states is 1, L is 0.
00:42:39.400 --> 00:42:46.288
So it's that same logic just
repeated for every value of L.
00:42:46.288 --> 00:42:46.955
Other questions?
00:42:52.890 --> 00:42:59.030
Coming back to this,
something on this board
00:42:59.030 --> 00:43:01.825
should cause you some
serious physical discomfort.
00:43:05.525 --> 00:43:07.650
We've talked about the l
equals 0 m equals 0 state.
00:43:07.650 --> 00:43:10.025
This is a state which has no
angular momentum whatsoever,
00:43:10.025 --> 00:43:11.490
in any direction at all.
00:43:11.490 --> 00:43:13.867
We've talked about the l
equals 1 m equals 0 state.
00:43:13.867 --> 00:43:15.950
It carries no angular
momentum in the z direction,
00:43:15.950 --> 00:43:18.540
but it presumably has non-zero
expectation value for L
00:43:18.540 --> 00:43:22.830
squared x and a Ly squared.
00:43:22.830 --> 00:43:23.980
These guys are also fine.
00:43:23.980 --> 00:43:26.980
What about these guys?
00:43:26.980 --> 00:43:29.920
AUDIENCE: [INAUDIBLE].
00:43:29.920 --> 00:43:32.290
PROFESSOR: Yes!
00:43:32.290 --> 00:43:34.180
That's disconcerting.
00:43:34.180 --> 00:43:38.034
Do I have to have angular
momentum in the z direction?
00:43:38.034 --> 00:43:40.022
AUDIENCE: [INAUDIBLE]
00:43:40.022 --> 00:43:42.010
[LAUGHTER]
00:43:45.000 --> 00:43:48.620
PROFESSOR: That should
go on a shirt somewhere.
00:43:48.620 --> 00:43:50.420
Let me ask the question
more precisely,
00:43:50.420 --> 00:43:54.080
or in a way that's a little
less threatening to me.
00:43:54.080 --> 00:43:58.420
Do you have to have angular
momentum in the z direction?
00:43:58.420 --> 00:43:59.630
I'm sorry, what's your name?
00:43:59.630 --> 00:44:02.587
Does David need to have angular
momentum in the z direction?
00:44:02.587 --> 00:44:03.170
AUDIENCE: Yes.
00:44:03.170 --> 00:44:04.710
PROFESSOR: Does this chalk
need to have angular dimension
00:44:04.710 --> 00:44:05.240
in the--
00:44:05.240 --> 00:44:06.760
well, the chalk's--
00:44:06.760 --> 00:44:10.163
OK, classically no.
00:44:10.163 --> 00:44:12.330
It can have some total
angular momentum, which is 0,
00:44:12.330 --> 00:44:14.840
and it can be rotating
not in the z direction.
00:44:14.840 --> 00:44:17.720
It can be rotating
in the zx plane.
00:44:17.720 --> 00:44:19.430
If it's rotating
in the zx plane,
00:44:19.430 --> 00:44:22.670
it's got total angular
momentum L squared as non-zero.
00:44:22.670 --> 00:44:25.410
But its angular momentum
in the z direction is 0.
00:44:25.410 --> 00:44:28.610
Its axis is exactly
along the y direction.
00:44:28.610 --> 00:44:30.360
And so it's got no
angular moment-- that's
00:44:30.360 --> 00:44:32.020
perfectly possible classically.
00:44:32.020 --> 00:44:34.690
And that's perfectly possible
when L is an integer.
00:44:34.690 --> 00:44:36.600
Similarly when L is 2.
00:44:36.600 --> 00:44:39.200
This is the particular
tower that I love the most.
00:44:39.200 --> 00:44:43.790
2, 1, 0, minus 1, minus 2.
00:44:43.790 --> 00:44:45.900
The reason I love
this the most is
00:44:45.900 --> 00:44:49.300
that it's related to gravity,
which is pretty awesome.
00:44:52.035 --> 00:44:53.160
That's a whole other story.
00:44:56.050 --> 00:44:59.760
I really shouldn't
have said that.
00:44:59.760 --> 00:45:01.740
That's only going
to confuse you.
00:45:01.740 --> 00:45:04.980
So for any integer, it's
possible to have no angular
00:45:04.980 --> 00:45:06.573
momentum in the z direction.
00:45:06.573 --> 00:45:07.990
That means it's
possible to rotate
00:45:07.990 --> 00:45:11.543
around the x-axis or the y-axis,
orthogonal to the z-axis.
00:45:11.543 --> 00:45:12.210
That make sense?
00:45:12.210 --> 00:45:15.690
But for these half integer guys,
you are inescapably spinning.
00:45:15.690 --> 00:45:18.160
There is no such thing as
a state of this guy that
00:45:18.160 --> 00:45:19.880
carries no angular momentum.
00:45:19.880 --> 00:45:23.780
Anything well described
by these quantum states
00:45:23.780 --> 00:45:26.830
is perpetually
rotating or spinning.
00:45:26.830 --> 00:45:29.380
It carries angular
momentum, we say precisely.
00:45:29.380 --> 00:45:31.797
Perpetually carries angular
momentum in the z direction.
00:45:31.797 --> 00:45:34.130
Any time you measure it, it
carries an angular momentum,
00:45:34.130 --> 00:45:36.920
either plus a half integer
or minus a half integer.
00:45:36.920 --> 00:45:40.700
But never, ever zero.
00:45:40.700 --> 00:45:43.390
AUDIENCE: But in the
classical limit where
00:45:43.390 --> 00:45:47.610
you have very large L, the
m equals one half state,
00:45:47.610 --> 00:45:49.534
or the m equals
minus a half state is
00:45:49.534 --> 00:45:51.784
going to get arbitrarily
small compared to the angular
00:45:51.784 --> 00:45:52.284
momentum.
00:45:52.284 --> 00:45:53.880
So isn't it just
like where we said,
00:45:53.880 --> 00:45:56.196
well, OK, you can never
have your angular momentum
00:45:56.196 --> 00:45:58.710
only in the z direction,
but we don't care?
00:45:58.710 --> 00:46:01.985
Because in the classical
limit it gets arbitrarily
00:46:01.985 --> 00:46:02.610
close to there.
00:46:02.610 --> 00:46:03.920
PROFESSOR: See, one of the
nice things about writing
00:46:03.920 --> 00:46:06.462
lectures like this is that you
get to leave little landlines.
00:46:06.462 --> 00:46:07.970
So this is exactly one of those.
00:46:07.970 --> 00:46:09.700
Thank you for asking
this question.
00:46:09.700 --> 00:46:11.470
Let me rephrase that question.
00:46:11.470 --> 00:46:14.250
Look, we all took
high school chemistry.
00:46:14.250 --> 00:46:16.080
We all know about spin.
00:46:16.080 --> 00:46:17.577
The nuclei have spin.
00:46:17.577 --> 00:46:18.910
They have some angular momentum.
00:46:18.910 --> 00:46:20.327
But if you build
up a lot of them,
00:46:20.327 --> 00:46:21.830
you build up a piece
of chalk, look,
00:46:21.830 --> 00:46:23.560
as we said before,
while it's true
00:46:23.560 --> 00:46:26.385
that there's some
mismatch in the angular
00:46:26.385 --> 00:46:28.510
momentum in the z direction
for some large L state,
00:46:28.510 --> 00:46:29.802
it's not only angular momentum.
00:46:29.802 --> 00:46:32.025
Some is in Lz, Lx,
and Ly as well.
00:46:32.025 --> 00:46:34.700
It's preposterously small
for a macroscopic object
00:46:34.700 --> 00:46:36.200
where L is macroscopic.
00:46:36.200 --> 00:46:38.030
It's the angular
momentum in Planck units,
00:46:38.030 --> 00:46:41.010
in units of the Planck constant.
00:46:41.010 --> 00:46:42.220
10 to the 26th--
00:46:42.220 --> 00:46:43.960
something huge.
00:46:43.960 --> 00:46:45.410
Why would we even notice?
00:46:45.410 --> 00:46:46.700
But here's the real statement.
00:46:46.700 --> 00:46:48.230
The statement isn't
just that this
00:46:48.230 --> 00:46:49.837
is true of macroscopic objects.
00:46:49.837 --> 00:46:51.420
But imagine you take
a small particle.
00:46:51.420 --> 00:46:54.500
Imagine you take a single atom.
00:46:54.500 --> 00:46:56.690
We're deep in the quantum
mechanical regime.
00:46:56.690 --> 00:46:59.220
We're not in the
classical regime.
00:46:59.220 --> 00:47:01.540
We take that single atom.
00:47:01.540 --> 00:47:03.280
And if it carries
angular momentum
00:47:03.280 --> 00:47:05.240
and it's described by
L equals 1/2 state,
00:47:05.240 --> 00:47:09.000
that atom will never, ever, ever
be measured to have its angular
00:47:09.000 --> 00:47:13.600
momentum in the z direction,
or indeed any direction, be 0.
00:47:13.600 --> 00:47:15.278
You will never,
in any direction,
00:47:15.278 --> 00:47:16.820
measure its angular
momentum to be 0.
00:47:16.820 --> 00:47:20.110
That atom perpetually
carries angular momentum.
00:47:20.110 --> 00:47:22.010
And that is weird.
00:47:22.010 --> 00:47:23.798
OK, maybe you don't
find it weird.
00:47:23.798 --> 00:47:24.340
This is good.
00:47:24.340 --> 00:47:26.673
You've grown up in an era
when that's not a weird thing.
00:47:26.673 --> 00:47:29.230
But I find this
deeply disconcerting.
00:47:29.230 --> 00:47:32.680
And you might say, look, we
never actually measure an atom.
00:47:32.680 --> 00:47:33.640
But we do.
00:47:33.640 --> 00:47:34.803
We do all the time.
00:47:34.803 --> 00:47:36.970
Because we measure things
like the spectre of light,
00:47:36.970 --> 00:47:40.960
as we'll study when we study
atoms in a week, a couple
00:47:40.960 --> 00:47:42.580
weeks because of the exam--
00:47:42.580 --> 00:47:44.650
sorry guys, there
has to be an exam--
00:47:44.650 --> 00:47:46.400
as we will find
when we study atoms
00:47:46.400 --> 00:47:50.210
in more detail,
or indeed, at all,
00:47:50.210 --> 00:47:52.140
we'll be sensitive
to the angular
00:47:52.140 --> 00:47:54.780
momentum of the
constituents of the atom.
00:47:54.780 --> 00:47:56.190
And we'll see different spectra.
00:47:56.190 --> 00:48:01.730
So it's an observable property
when you shine light on gases.
00:48:01.730 --> 00:48:05.380
This is something we
can really observe.
00:48:05.380 --> 00:48:09.390
So what this suggests
is one of two things.
00:48:09.390 --> 00:48:13.005
Either these are just
crazy and ridiculous
00:48:13.005 --> 00:48:15.305
and we should ignore
them, or there's
00:48:15.305 --> 00:48:16.930
something interesting
and intrinsically
00:48:16.930 --> 00:48:19.580
quantum-mechanical about
them that's not so familiar.
00:48:19.580 --> 00:48:21.038
And the answer is
going to turn out
00:48:21.038 --> 00:48:24.500
to be the second,
the latter of those.
00:48:24.500 --> 00:48:25.030
The ladder?
00:48:28.210 --> 00:48:31.120
That was not intentional.
00:48:31.120 --> 00:48:32.300
Maybe it was subconscious.
00:48:32.300 --> 00:48:35.300
OK, so I want to think
about some more consequences
00:48:35.300 --> 00:48:37.630
of the structure of the
Ylm's and the eigenvalues,
00:48:37.630 --> 00:48:39.698
in particular of
this tower structure.
00:48:39.698 --> 00:48:40.990
I want to understand some more.
00:48:40.990 --> 00:48:43.925
What other physics can we
extract from this story?
00:48:48.540 --> 00:48:50.290
First, a very
useful thing is just
00:48:50.290 --> 00:48:52.370
to get a picture of
these guys in your head.
00:48:52.370 --> 00:48:55.940
Let's draw the angular
momentum eigenfunctions.
00:48:58.232 --> 00:48:59.190
So what does that mean?
00:48:59.190 --> 00:49:02.350
Well first, when we talked about
the eigenfunctions of momentum,
00:49:02.350 --> 00:49:04.730
linear momentum
in one dimension,
00:49:04.730 --> 00:49:06.740
we immediately went
to the wave function.
00:49:06.740 --> 00:49:09.260
We talked about
how the amplitude
00:49:09.260 --> 00:49:11.450
to beat a particular
spot varied in space.
00:49:11.450 --> 00:49:13.820
And the amplitude was
just e to the iKX.
00:49:13.820 --> 00:49:15.660
So the amplitude
was an oscillating--
00:49:15.660 --> 00:49:17.010
the phase rotated.
00:49:17.010 --> 00:49:19.060
And the absolute value of
the probability density
00:49:19.060 --> 00:49:19.940
was completely constant.
00:49:19.940 --> 00:49:21.040
Everybody cool with that?
00:49:21.040 --> 00:49:22.537
That was the 1D plane wave.
00:49:22.537 --> 00:49:25.120
So the variables there, we had
an angular momentum eigenstate.
00:49:25.120 --> 00:49:27.913
And that's a function
of the position.
00:49:27.913 --> 00:49:29.580
Or, sorry, a linear
momentum eigenstate.
00:49:29.580 --> 00:49:30.340
And that's a function
of the position.
00:49:30.340 --> 00:49:32.110
Angular momentum
eigenstates are going
00:49:32.110 --> 00:49:35.920
to be functions of
angular position.
00:49:35.920 --> 00:49:39.770
So I want to know what these
wave functions look like, not
00:49:39.770 --> 00:49:40.658
just the eigenvalues.
00:49:40.658 --> 00:49:42.950
But I want to know what is
the wave function associated
00:49:42.950 --> 00:49:44.980
to eigenvalues
little l and little m
00:49:44.980 --> 00:49:48.910
look like, y sub lm of
the angles theta and phi.
00:49:48.910 --> 00:49:52.118
What do these guys look like?
00:49:52.118 --> 00:49:52.910
How do we get them?
00:49:54.358 --> 00:49:56.650
This is going to be your goal
for the next two minutes.
00:49:59.160 --> 00:50:03.970
So the first thing
to notice is that we
00:50:03.970 --> 00:50:06.340
know what the form of the
eigenvalues and eigenfunctions
00:50:06.340 --> 00:50:06.840
are.
00:50:06.840 --> 00:50:09.610
If we act with Lz
we get h bar m back.
00:50:09.610 --> 00:50:13.510
If we act with L squared we get
h bar squared ll plus 1 back.
00:50:13.510 --> 00:50:16.910
But we also have other
expressions for Lz and L
00:50:16.910 --> 00:50:17.410
squared.
00:50:17.410 --> 00:50:19.210
In particular-- I wrote
them down last time
00:50:19.210 --> 00:50:20.430
in spherical coordinates.
00:50:20.430 --> 00:50:23.140
So I'm going to working in the
spherical coordinates where
00:50:23.140 --> 00:50:26.440
the declination from the
vertical is an angle theta,
00:50:26.440 --> 00:50:31.510
and the angle around the
equator is an angle phi.
00:50:31.510 --> 00:50:34.523
And theta equals 0 is going
to be up in the z direction.
00:50:34.523 --> 00:50:35.940
It's just a choice
of coordinates.
00:50:35.940 --> 00:50:37.942
There's nothing deep.
00:50:37.942 --> 00:50:39.150
There's nothing even shallow.
00:50:39.150 --> 00:50:40.653
It's just definitions.
00:50:40.653 --> 00:50:42.820
So we're going to work in
the spherical coordinates.
00:50:42.820 --> 00:50:44.620
And in spherical
coordinates we observe
00:50:44.620 --> 00:50:46.370
that this angular
momentum, just following
00:50:46.370 --> 00:50:49.850
the definition from
r cross p, takes
00:50:49.850 --> 00:50:51.040
a particularly simple form.
00:50:54.580 --> 00:50:55.790
That's a typo.
00:50:55.790 --> 00:50:59.310
h bar upon i, dd phi.
00:50:59.310 --> 00:51:00.810
And instead of
writing L squared I'm
00:51:00.810 --> 00:51:02.980
going to write L plus
minus, because it's shorter
00:51:02.980 --> 00:51:05.397
and also because it's going
to turn out to be more useful.
00:51:05.397 --> 00:51:08.260
So this takes the form h
bar, e to the plus minus i
00:51:08.260 --> 00:51:19.060
phi, d theta, plus or minus
cotangent of theta, d phi.
00:51:23.652 --> 00:51:25.110
So these are the
expressions for Lz
00:51:25.110 --> 00:51:27.943
and L plus minus in
spherical coordinates,
00:51:27.943 --> 00:51:29.235
in these spherical coordinates.
00:51:34.900 --> 00:51:36.900
I want to know how Ylm
depends on theta and phi.
00:51:36.900 --> 00:51:38.358
And it's clearly
going to be easier
00:51:38.358 --> 00:51:40.660
to ask about the
Lz eigenequation.
00:51:40.660 --> 00:51:41.960
So let's look at that.
00:51:41.960 --> 00:51:48.530
Lz on Ylm gives me h--
00:51:48.530 --> 00:51:49.040
Oh yeah?
00:51:49.040 --> 00:51:51.255
AUDIENCE: Is it h or h bar?
00:51:51.255 --> 00:51:52.130
PROFESSOR: Oh, Jesus.
00:51:56.140 --> 00:51:59.580
When I write letters
by hand, which
00:51:59.580 --> 00:52:03.610
is basically when I write to
my mom, all my h's are crossed.
00:52:03.610 --> 00:52:04.283
I can't help it.
00:52:04.283 --> 00:52:05.450
So this is like the inverse.
00:52:10.550 --> 00:52:12.550
It's been a long time
since I made that mistake.
00:52:12.550 --> 00:52:14.120
It's usually the other.
00:52:14.120 --> 00:52:22.122
So Lz acting on Ylm gives
me h bar m, acting on Ylm.
00:52:22.122 --> 00:52:23.830
But that's what I get
when I act with Lz,
00:52:23.830 --> 00:52:25.955
so let's just write out
the differential equation .
00:52:25.955 --> 00:52:29.150
So h bar m Ylm where m is an
integer, or a half integer,
00:52:29.150 --> 00:52:31.660
depending on whether l is
an integer or half integer.
00:52:31.660 --> 00:52:37.000
h bar m Ylm is equal to h
bar upin i, d phi of Ylm.
00:52:39.360 --> 00:52:41.610
Using the awesome power of
division and multiplication
00:52:41.610 --> 00:52:43.860
I will divide both
sides by h bar.
00:52:43.860 --> 00:52:47.730
And I will multiply
both sides by i.
00:52:47.730 --> 00:52:53.707
And we now have the equation
for the eigenfunctions of Lz,
00:52:53.707 --> 00:52:55.290
which we actually
worked on last time.
00:52:55.290 --> 00:52:57.410
And we can solve
this very simply.
00:52:57.410 --> 00:52:59.973
This says that, remember, Ylm
is a function of theta and phi.
00:52:59.973 --> 00:53:01.890
Here we're only looking
at the phi dependence,
00:53:01.890 --> 00:53:04.015
because that's all that
showed up in this equation.
00:53:04.015 --> 00:53:06.820
So this tells us that the
eigenfunctions Ylm are
00:53:06.820 --> 00:53:12.080
of the form of theta and phi,
are of the form e to the im
00:53:12.080 --> 00:53:17.540
phi times some
remaining dependence
00:53:17.540 --> 00:53:21.080
on theta, which I'll
write as p of theta.
00:53:21.080 --> 00:53:24.510
And that p could
depend on l and m.
00:53:24.510 --> 00:53:25.010
Cool?
00:53:34.820 --> 00:53:36.980
Already, before we even
ask about that dependence
00:53:36.980 --> 00:53:41.230
on theta, the p dependence, we
learn something pretty awesome.
00:53:41.230 --> 00:53:43.880
Look at this wave function.
00:53:43.880 --> 00:53:48.450
Whatever else we know,
its dependence on phi
00:53:48.450 --> 00:53:50.500
is e to the im phi.
00:53:50.500 --> 00:53:52.260
Now, remember what phi is.
00:53:52.260 --> 00:53:54.670
Phi is the angle
around the equator.
00:53:54.670 --> 00:53:57.180
So it goes from 0 to 2 pi.
00:53:57.180 --> 00:53:59.790
And when it comes back to
2 pi it's the same point.
00:53:59.790 --> 00:54:01.250
Phi equals 0 and
phi equals pi are
00:54:01.250 --> 00:54:03.120
two names for the same point.
00:54:03.120 --> 00:54:03.620
Yes?
00:54:06.750 --> 00:54:08.520
But that should worry you.
00:54:08.520 --> 00:54:11.540
Because note that as
a consequence of this,
00:54:11.540 --> 00:54:18.630
Ylm of theta 0 is equal
to, well, whatever it is.
00:54:21.600 --> 00:54:31.120
Sorry, theta of 2 pi is equal
to e to the im 2 pi times Plm
00:54:31.120 --> 00:54:31.620
to theta.
00:54:35.370 --> 00:54:36.720
But this is equal to--
00:54:36.720 --> 00:54:39.230
oh, now we're in trouble.
00:54:39.230 --> 00:54:45.710
If m is an integer, this is
equal to e to the i integer 2
00:54:45.710 --> 00:54:50.240
pi 1.
00:54:50.240 --> 00:54:51.090
Yes, OK.
00:54:51.090 --> 00:54:52.715
You're supposed to
cheer at that point.
00:54:52.715 --> 00:54:54.548
It's like the coolest
identity in the world.
00:54:54.548 --> 00:54:58.040
So e to the i 2 pi, that's one.
00:54:58.040 --> 00:55:05.400
So if m is an integer
then this is just
00:55:05.400 --> 00:55:10.180
Plm of theta, which is also
what we get by putting in phi
00:55:10.180 --> 00:55:11.210
equals 0.
00:55:11.210 --> 00:55:17.930
So this is equal to
Ylm of theta, comma, 0.
00:55:17.930 --> 00:55:21.350
But if m is a half integer
then e to the i half
00:55:21.350 --> 00:55:26.010
integer times 2 pi is minus 1.
00:55:26.010 --> 00:55:31.650
And so that gives us
minus Ylm of theta and 0,
00:55:31.650 --> 00:55:34.010
if m is a half integer.
00:55:38.330 --> 00:55:43.730
So let me say that again,
in the same words, actually.
00:55:43.730 --> 00:55:47.760
But let me just say it again
with different emphasis.
00:55:47.760 --> 00:55:53.180
What this tells us
is that Ylm at 0,
00:55:53.180 --> 00:55:55.805
as function of the coordinates
theta and phi, Ylm--
00:55:58.390 --> 00:56:02.240
so let's take m as an integer.
00:56:02.240 --> 00:56:11.530
Ylm at theta and 0 is equal
to Ylm at theta and 0.
00:56:11.530 --> 00:56:12.630
That's good.
00:56:12.630 --> 00:56:20.310
But if Y is a half integer,
then Ylm at theta and 2 pi,
00:56:20.310 --> 00:56:23.760
which is the same point
as Ylm at theta and 0,
00:56:23.760 --> 00:56:30.000
is equal to minus
Ylm at theta and 0.
00:56:30.000 --> 00:56:31.660
That's less good.
00:56:31.660 --> 00:56:34.494
What must be true of
Ylm, of theta and 0?
00:56:34.494 --> 00:56:36.150
0.
00:56:36.150 --> 00:56:38.470
And was there anything
special about the point 0?
00:56:38.470 --> 00:56:38.970
No.
00:56:38.970 --> 00:56:40.345
I could have just
taken any point
00:56:40.345 --> 00:56:41.690
and rotated it around by pi.
00:56:41.690 --> 00:56:47.610
So this tells us that
Ylm of theta and phi
00:56:47.610 --> 00:56:51.990
is identically equal to
0 if m is a half integer.
00:56:57.410 --> 00:56:58.700
Huh.
00:56:58.700 --> 00:57:00.387
That's bad.
00:57:00.387 --> 00:57:01.970
Because what's the
probability density
00:57:01.970 --> 00:57:05.030
of being found at any
particular angular position?
00:57:05.030 --> 00:57:06.390
0.
00:57:06.390 --> 00:57:07.814
Can you normalize that state?
00:57:07.814 --> 00:57:08.603
No.
00:57:08.603 --> 00:57:10.520
That is not a state that
describes a particle.
00:57:10.520 --> 00:57:12.895
That is a state that describes
the absence of a particle.
00:57:12.895 --> 00:57:14.400
That is not what we want.
00:57:14.400 --> 00:57:20.060
So these states cannot describe
these values of l and m,
00:57:20.060 --> 00:57:22.480
which seem like perfectly
reasonable values of l and m,
00:57:22.480 --> 00:57:25.740
perfectly reasonable
eigenvalues of L squared and lm.
00:57:29.930 --> 00:57:35.570
They cannot be used to label
wave functions of physical
00:57:35.570 --> 00:57:38.670
states corresponding to
wave functions on a sphere.
00:57:38.670 --> 00:57:40.070
You can't do it.
00:57:40.070 --> 00:57:42.560
Because if you try, you find
that those wave functions
00:57:42.560 --> 00:57:44.750
identically vanish.
00:57:44.750 --> 00:57:45.840
OK?
00:57:45.840 --> 00:57:47.310
So these cannot be used.
00:57:47.310 --> 00:57:56.755
These do not describe wave
functions of a particle
00:57:56.755 --> 00:57:57.630
in quantum mechanics.
00:57:57.630 --> 00:57:58.463
They cannot be used.
00:57:58.463 --> 00:58:01.598
Those values, those towers
cannot be used to describe
00:58:01.598 --> 00:58:03.140
particles moving in
three dimensions.
00:58:06.950 --> 00:58:07.830
Questions about that?
00:58:07.830 --> 00:58:11.550
This is a slightly
subtle argument.
00:58:11.550 --> 00:58:12.050
Yeah?
00:58:12.050 --> 00:58:13.800
AUDIENCE: You said
something earlier about
00:58:13.800 --> 00:58:17.484
how atoms would never have
any zero angular momentum.
00:58:17.484 --> 00:58:20.760
And so the ones that have
no zero angular momentum,
00:58:20.760 --> 00:58:22.177
we just said they're
not possible.
00:58:22.177 --> 00:58:23.170
So [INAUDIBLE]?
00:58:23.170 --> 00:58:24.420
PROFESSOR: Excellent question.
00:58:24.420 --> 00:58:26.295
I said it slightly diff--
so the question is,
00:58:26.295 --> 00:58:28.555
look, earlier you were
saying, yeah, yeah, yeah.
00:58:28.555 --> 00:58:31.180
There are atoms in the world and
they have half integer angular
00:58:31.180 --> 00:58:31.690
momentum.
00:58:31.690 --> 00:58:33.065
And you can shine
a light on them
00:58:33.065 --> 00:58:35.620
and you can tell and stuff.
00:58:35.620 --> 00:58:38.490
But you just said
these can't exist.
00:58:38.490 --> 00:58:41.143
So how can those two
things both be true?
00:58:41.143 --> 00:58:42.310
Thank you for this question.
00:58:42.310 --> 00:58:43.393
It's a very good question.
00:58:43.393 --> 00:58:45.490
I actually said a
slightly different thing.
00:58:45.490 --> 00:58:49.660
What I said was, states where
angular momentum lm are half
00:58:49.660 --> 00:58:53.070
integers cannot be
described by a wave function
00:58:53.070 --> 00:58:54.500
of the coordinates.
00:58:54.500 --> 00:58:56.790
We're going to need some
different description.
00:58:56.790 --> 00:58:58.420
And in particular, we're going
to need a different description
00:58:58.420 --> 00:58:59.440
that does what?
00:58:59.440 --> 00:59:06.650
Well, as we take phi from 0 to
2 pi, as we rotate the system,
00:59:06.650 --> 00:59:08.750
we're going to pick
up a minus sign.
00:59:08.750 --> 00:59:11.020
So in order to describe
an object with lm
00:59:11.020 --> 00:59:13.710
being a half integer, we
can't use the wave function.
00:59:13.710 --> 00:59:17.125
We need something that is
allowed to be doubly valued.
00:59:17.125 --> 00:59:19.500
And in particular, we need
something that behaves nicely.
00:59:19.500 --> 00:59:21.830
When you rotate
around by 2 pi, we
00:59:21.830 --> 00:59:25.330
need to come back to a
minus sign, not itself.
00:59:25.330 --> 00:59:29.910
So at some object
that's not a function,
00:59:29.910 --> 00:59:31.420
it's called a spinner.
00:59:31.420 --> 00:59:35.080
So we'll talk about it later.
00:59:35.080 --> 00:59:36.880
We need some object
that does that.
00:59:36.880 --> 00:59:39.347
So there's this
classic demonstration
00:59:39.347 --> 00:59:41.680
at this point, which is
supposed to be done in a quantum
00:59:41.680 --> 00:59:42.675
mechanics class.
00:59:42.675 --> 00:59:44.050
So at this point
the lecturist is
00:59:44.050 --> 00:59:45.650
obliged to do the
following thing.
00:59:45.650 --> 00:59:48.700
You say, blah, blah, blah,
if things rotate by 2 pi
00:59:48.700 --> 00:59:50.480
they have to come
back to themselves.
00:59:50.480 --> 00:59:52.790
And then you do this.
00:59:52.790 --> 00:59:59.170
I'm going to rotate my
hand like a record by 2 pi.
00:59:59.170 --> 01:00:00.760
And it will not
come back to itself.
01:00:00.760 --> 01:00:01.260
OK?
01:00:04.250 --> 01:00:07.080
And it quite uncomfortably
has not come back to itself.
01:00:07.080 --> 01:00:08.970
But I can do a further
rotation to show you
01:00:08.970 --> 01:00:09.928
that it's a minus sign.
01:00:09.928 --> 01:00:11.640
I can do a further
the rotation by 2 pi
01:00:11.640 --> 01:00:12.973
and have it come back to itself.
01:00:15.390 --> 01:00:16.970
And I kept the axis
vertical, yeah?
01:00:16.970 --> 01:00:19.512
OK, so at this point you're all
supposed to go like, oh, yes,
01:00:19.512 --> 01:00:22.070
uh-huh, mmm.
01:00:22.070 --> 01:00:28.460
So now that we've got that
out of the way, I have an arm.
01:00:28.460 --> 01:00:30.687
So the story is a little
more complicated than that.
01:00:30.687 --> 01:00:32.270
This is actually a
fair demonstration,
01:00:32.270 --> 01:00:33.645
but it's a slightly
subtle story.
01:00:33.645 --> 01:00:36.145
If you want to understand it,
ask your recitation instructor
01:00:36.145 --> 01:00:37.383
or come to my office hours.
01:00:37.383 --> 01:00:40.100
AUDIENCE: What about USB sticks?
01:00:40.100 --> 01:00:41.342
PROFESSOR: USB sticks?
01:00:41.342 --> 01:00:43.800
AUDIENCE: You insert them here
and they don't go and insert
01:00:43.800 --> 01:00:45.045
the other way.
01:00:45.045 --> 01:00:46.905
[LAUGHTER]
01:00:48.805 --> 01:00:50.180
PROFESSOR: What
about USB sticks?
01:00:50.180 --> 01:00:51.410
You insert them this
way, they don't work.
01:00:51.410 --> 01:00:53.160
You insert them this
way, they don't work.
01:00:53.160 --> 01:00:55.450
But if you do it
again, then they do.
01:00:55.450 --> 01:00:57.410
[LAUGHTER]
01:00:59.860 --> 01:01:01.820
[APPLAUSE]
01:01:02.707 --> 01:01:03.290
PROFESSOR: OK.
01:01:07.220 --> 01:01:10.200
That's pretty good.
01:01:10.200 --> 01:01:13.245
So for the moment, as long as
we want to describe our system
01:01:13.245 --> 01:01:14.620
with a wave function
of position,
01:01:14.620 --> 01:01:16.870
which means we're thinking
about where will we find it
01:01:16.870 --> 01:01:19.230
as a function of angle, we
cannot use the half integer l
01:01:19.230 --> 01:01:20.932
or m.
01:01:20.932 --> 01:01:23.015
So if we can't use the
half interger l or m, fine.
01:01:23.015 --> 01:01:24.230
We'll just throw them
out for the moment
01:01:24.230 --> 01:01:26.050
and we'll use the
integer l and m.
01:01:26.050 --> 01:01:28.860
And let's keep going.
01:01:28.860 --> 01:01:30.500
What we need to
determine now is the P.
01:01:30.500 --> 01:01:32.000
We've determined
the phi dependence,
01:01:32.000 --> 01:01:33.430
but we need the
theta dependence.
01:01:33.430 --> 01:01:36.940
We can get the theta
dependence in a sneaky fashion.
01:01:36.940 --> 01:01:39.470
Remember the harmonic
oscillator in 1D.
01:01:39.470 --> 01:01:41.720
When we wanted to find the
ground state wave function,
01:01:41.720 --> 01:01:44.230
we could either solve the energy
eigenvalue equation, which
01:01:44.230 --> 01:01:47.380
is a second order differential
equation and kind of horrible,
01:01:47.380 --> 01:01:50.450
or we could solve the
ground state equation that
01:01:50.450 --> 01:01:53.020
said that the ground state is
annihilated by the annihilation
01:01:53.020 --> 01:01:55.458
operator, which is a first
order difference equation
01:01:55.458 --> 01:01:56.500
and much easier to solve.
01:01:56.500 --> 01:01:57.140
Yeah?
01:01:57.140 --> 01:01:58.140
Let's do the same thing.
01:01:58.140 --> 01:01:59.620
We have an
annihilation condition.
01:01:59.620 --> 01:02:04.047
If we have the top state, L
plus on Y ll is equal to 0.
01:02:04.047 --> 01:02:06.130
But L plus is a first order
differential operator.
01:02:06.130 --> 01:02:07.370
This is going to be easier.
01:02:07.370 --> 01:02:09.490
So we need to find a
solution to this equation.
01:02:09.490 --> 01:02:11.327
And do I want to
go through this?
01:02:11.327 --> 01:02:11.910
Yeah, why not.
01:02:11.910 --> 01:02:13.440
OK, so I want this
to be equal to 0.
01:02:13.440 --> 01:02:16.340
But L plus Yll is
equal to-- well,
01:02:16.340 --> 01:02:25.460
it's h bar, e to the plus i phi,
d theta, plus cotangent theta,
01:02:25.460 --> 01:02:28.560
d phi on Yll.
01:02:28.560 --> 01:02:36.390
And Yll is e to the i l
phi times Plm of theta.
01:02:36.390 --> 01:02:37.720
Cool?
01:02:37.720 --> 01:02:41.760
So dd phi on e to the il phi,
there's no phi dependence here.
01:02:41.760 --> 01:02:44.970
It's just going to
give us a vector of il.
01:02:44.970 --> 01:02:46.336
Did I get the--
01:02:46.336 --> 01:02:47.210
yeah.
01:02:47.210 --> 01:02:52.030
So that's going to give us
a plus il and no dd phi.
01:02:58.860 --> 01:03:01.500
And this e to the il
phi we can pull out.
01:03:01.500 --> 01:03:02.840
But this has to be equal to 0.
01:03:02.840 --> 01:03:10.670
So this says that 0 is equal to
d theta plus il cotangent theta
01:03:10.670 --> 01:03:13.090
P lm of theta.
01:03:15.843 --> 01:03:17.510
And this is actually
much better than it
01:03:17.510 --> 01:03:18.802
seems for the following reason.
01:03:18.802 --> 01:03:21.420
dd theta-- find the dd theta.
01:03:21.420 --> 01:03:25.350
Cotangent of theta
is cosine over sine.
01:03:25.350 --> 01:03:28.030
That's what you get if you
take the derivative of lots
01:03:28.030 --> 01:03:30.270
of sine functions--
01:03:30.270 --> 01:03:32.210
sine to the l, say.
01:03:32.210 --> 01:03:32.960
Take a derivative.
01:03:32.960 --> 01:03:35.418
You lose a power of sine and
you pick up a power of cosine.
01:03:35.418 --> 01:03:37.840
So multiplying by cotangent
gets rid of a power of sign
01:03:37.840 --> 01:03:43.450
and gives you a power of cosine,
which is a derivative of sine.
01:03:43.450 --> 01:03:45.830
So Plm, noticing the l--
01:03:45.830 --> 01:03:49.400
and I screwed up an i somewhere.
01:03:49.400 --> 01:03:52.330
I think I wanted an i up here.
01:03:52.330 --> 01:03:53.980
Let's see.
01:03:53.980 --> 01:03:54.480
i.
01:04:06.290 --> 01:04:06.790
Sorry.
01:04:10.280 --> 01:04:12.490
Yes, I want an i cotangent.
01:04:12.490 --> 01:04:14.280
OK, that's much better.
01:04:14.280 --> 01:04:16.510
So i cotangent-- good, good.
01:04:16.510 --> 01:04:20.970
And then the i squareds
give me a minus l.
01:04:20.970 --> 01:04:25.260
And this tells us that Plm,
so if this is sine to the l,
01:04:25.260 --> 01:04:29.690
then d theta gives us an l sine
to the l minus 1 cosine, which
01:04:29.690 --> 01:04:31.777
is what I get by
taking sine to the l
01:04:31.777 --> 01:04:33.610
and multiplying by
cosine, dividing by sine,
01:04:33.610 --> 01:04:35.200
and multplying by l.
01:04:35.200 --> 01:04:38.120
So this gives me Pll.
01:04:38.120 --> 01:04:40.847
This is for the
particular state ll.
01:04:40.847 --> 01:04:43.180
We're looking at the top state
and we're annhilating it.
01:04:43.180 --> 01:04:47.570
So Pll is equal to
some coefficient,
01:04:47.570 --> 01:04:52.150
so I'll just say proportional
to sine to the l of theta.
01:04:55.450 --> 01:05:02.710
So this tells us that Yll is
equal to some normalization,
01:05:02.710 --> 01:05:06.730
sub ll, just some number,
times from the phi dependence
01:05:06.730 --> 01:05:10.400
e to the il phi, and from
the theta dependence,
01:05:10.400 --> 01:05:12.640
sine to the l of theta.
01:05:16.580 --> 01:05:19.570
So this is the form.
01:05:19.570 --> 01:05:20.846
Sorry, go ahead.
01:05:20.846 --> 01:05:22.980
AUDIENCE: What's
the symbol there?
01:05:22.980 --> 01:05:25.250
PROFESSOR: Oh, this
twisted horrible thing?
01:05:25.250 --> 01:05:26.610
It's proportional to.
01:05:26.610 --> 01:05:29.930
But it was a long night.
01:05:33.350 --> 01:05:36.160
So now we have the wave function
explicitly as a function phi
01:05:36.160 --> 01:05:38.870
and as a function of theta
completely understood
01:05:38.870 --> 01:05:41.640
for the top state in any tower.
01:05:41.640 --> 01:05:43.740
This is for any L. The
top state in any tower
01:05:43.740 --> 01:05:45.070
is e to the il phi.
01:05:45.070 --> 01:05:46.050
Does that make sense?
01:05:46.050 --> 01:05:48.495
Well Lz is h bar upon id phi.
01:05:48.495 --> 01:05:51.760
So that gives us
h bar as m as l.
01:05:51.760 --> 01:05:52.400
So that's good.
01:05:52.400 --> 01:05:53.340
That's the top state.
01:05:53.340 --> 01:05:55.007
And from the sine
theta we just checked.
01:05:55.007 --> 01:05:58.490
We constructed that
this indeed has the--
01:05:58.490 --> 01:06:00.860
well, if you then check you
will find that the L squared
01:06:00.860 --> 01:06:02.818
eigenvalue, which you'll
do on the problem set,
01:06:02.818 --> 01:06:05.990
the l squared eigenvalue
is h bar squared ll plus 1.
01:06:05.990 --> 01:06:07.720
AUDIENCE: Quick question.
01:06:07.720 --> 01:06:10.530
The expression we have for
the L plus minus operators,
01:06:10.530 --> 01:06:12.970
how did we construct the
expressions for Lx and Ly?
01:06:12.970 --> 01:06:14.130
PROFESSOR: Good.
01:06:14.130 --> 01:06:15.800
It's much easier than you think.
01:06:15.800 --> 01:06:18.380
So Lx is equal to--
01:06:18.380 --> 01:06:19.500
L is r cross b, right?
01:06:19.500 --> 01:06:22.460
So this is going to
be yPz minus zPy.
01:06:28.200 --> 01:06:37.391
And this is equal to h
bar upon i, ydz minus zdy.
01:06:37.391 --> 01:06:40.010
But you know what y is
in spherical coordinates.
01:06:40.010 --> 01:06:40.990
And you know what
derivitive with respect to z
01:06:40.990 --> 01:06:41.750
is in spherical coordinates.
01:06:41.750 --> 01:06:43.810
Because you know what z is,
and you know the chain rule.
01:06:43.810 --> 01:06:45.940
So taking this and just
plugging in explicit expression
01:06:45.940 --> 01:06:48.107
for the change of variables
to spherical coordinates
01:06:48.107 --> 01:06:49.550
takes care of it.
01:06:49.550 --> 01:06:50.050
Yeah?
01:06:50.050 --> 01:06:51.855
AUDIENCE: What does the
superscript of the sine
01:06:51.855 --> 01:06:52.465
indicate?
01:06:52.465 --> 01:06:53.960
Is that sine to the power of l?
01:06:53.960 --> 01:06:54.668
PROFESSOR: Sorry.
01:06:54.668 --> 01:06:56.090
This is bad notation.
01:06:56.090 --> 01:06:59.410
It's not bad notation, it's
just not familiar notation.
01:06:59.410 --> 01:07:02.075
It's notation that is used
throughout theoretical physics.
01:07:02.075 --> 01:07:04.310
It means this, sine
theta to the l.
01:07:04.310 --> 01:07:08.090
The lth power of sine.
01:07:08.090 --> 01:07:10.250
For typesetting reasons
we often put the power
01:07:10.250 --> 01:07:12.130
before the argument.
01:07:12.130 --> 01:07:13.630
Yeah, no, it's a
very good question.
01:07:13.630 --> 01:07:15.505
Thank you for asking,
because it was unclear.
01:07:15.505 --> 01:07:16.610
I appreciate that.
01:07:16.610 --> 01:07:19.122
Other questions?
01:07:19.122 --> 01:07:22.220
AUDIENCE: [INAUDIBLE]
L hat [INAUDIBLE]??
01:07:22.220 --> 01:07:24.860
PROFESSOR: How did we come
up with the L hat plus minus?
01:07:24.860 --> 01:07:27.030
That was from this.
01:07:27.030 --> 01:07:29.645
So we know what the components
of the angular momentum
01:07:29.645 --> 01:07:31.600
are in Cartesian coordinates.
01:07:31.600 --> 01:07:33.910
And you know how,
because it's coordinates,
01:07:33.910 --> 01:07:36.640
to change variables from
Cartesian to spherical.
01:07:36.640 --> 01:07:38.290
So you just plug this in for Lx.
01:07:38.290 --> 01:07:41.200
But L plus is Lx plus
ioy, and so you just
01:07:41.200 --> 01:07:43.130
take these guys
in spherical form
01:07:43.130 --> 01:07:44.880
and add them together
with the relative i.
01:07:44.880 --> 01:07:48.008
And that gives you
that expression.
01:07:48.008 --> 01:07:49.966
AUDIENCE: Maybe I missed
this, but can you just
01:07:49.966 --> 01:07:53.173
explain the distinction
between Y sub ll and Y sub lm?
01:07:53.173 --> 01:07:54.340
PROFESSOR: Yeah, absolutely.
01:07:54.340 --> 01:08:01.756
So Y sub ll, it means Y sub
lm where m is equal to l.
01:08:01.756 --> 01:08:02.464
AUDIENCE: Oh, OK.
01:08:02.464 --> 01:08:04.780
It was just the
generic [INAUDIBLE]..
01:08:04.780 --> 01:08:07.030
PROFESSOR: It's a generic
eigenfunction of the angular
01:08:07.030 --> 01:08:10.600
momentum, with the angular
momentum in the z direction
01:08:10.600 --> 01:08:13.140
being equal to the angular
momentum in the total angular
01:08:13.140 --> 01:08:14.200
momentum.
01:08:14.200 --> 01:08:16.680
At least for these numbers.
01:08:16.680 --> 01:08:17.380
OK?
01:08:17.380 --> 01:08:19.490
Cool.
01:08:19.490 --> 01:08:22.609
Good, so now, if we know this,
how do we just as a side note--
01:08:22.609 --> 01:08:24.609
suppose we know-- well,
suppose we know this?
01:08:24.609 --> 01:08:25.170
We know this.
01:08:25.170 --> 01:08:26.760
We know what the top state
in the tower looks like.
01:08:26.760 --> 01:08:28.677
How do I get the next
state down in the tower?
01:08:28.677 --> 01:08:30.923
how do I get Yl l minus 1?
01:08:30.923 --> 01:08:31.715
AUDIENCE: Lower it.
01:08:31.715 --> 01:08:32.548
PROFESSOR: Lower it.
01:08:32.548 --> 01:08:33.130
Exactly.
01:08:33.130 --> 01:08:35.643
So this is easy, L minus on Yll.
01:08:35.643 --> 01:08:37.560
And we have to be careful
about normalization.
01:08:37.560 --> 01:08:41.262
So again, it's proportional to.
01:08:41.262 --> 01:08:41.970
But this is easy.
01:08:41.970 --> 01:08:43.260
We don't have to solve
any difference equations.
01:08:43.260 --> 01:08:44.490
We just have to
take derivatives.
01:08:44.490 --> 01:08:45.819
So it's just like
the raising operator
01:08:45.819 --> 01:08:47.040
for a harmonic oscillator.
01:08:47.040 --> 01:08:48.787
We can raise and
lower along the tower
01:08:48.787 --> 01:08:50.120
and get the right wave function.
01:08:54.920 --> 01:08:57.924
To give you some examples--
01:08:57.924 --> 01:08:59.043
yeah, let's do that here.
01:08:59.043 --> 01:09:01.340
Let me just quickly
give you a few examples
01:09:01.340 --> 01:09:05.125
of the first few
spherical harmonics.
01:09:05.125 --> 01:09:06.750
Sorry, I should give
these guys a name.
01:09:06.750 --> 01:09:08.939
These functions, Ylm
of theta and phi,
01:09:08.939 --> 01:09:10.970
they're called the
spherical harmonics.
01:09:10.970 --> 01:09:13.630
They're called these because
they solve the Laplacian
01:09:13.630 --> 01:09:16.750
equation on the sphere, which
is just the eigenvalue equation.
01:09:16.750 --> 01:09:19.500
L squared on them is equal to
a constant times those things
01:09:19.500 --> 01:09:20.000
back.
01:09:24.240 --> 01:09:28.279
Just to tabulate a couple of
examples for you concretely,
01:09:28.279 --> 01:09:31.529
consider the l equals 0 states.
01:09:31.529 --> 01:09:34.500
What are the allowed values
of m for little l equals 0?
01:09:34.500 --> 01:09:35.000
0.
01:09:35.000 --> 01:09:36.710
So Y0,0 is the only state.
01:09:36.710 --> 01:09:41.020
And if you properly normalize
it, it's 1 over root 4 pi.
01:09:41.020 --> 01:09:41.660
OK, good.
01:09:41.660 --> 01:09:43.460
what about l equals 1?
01:09:43.460 --> 01:09:50.520
Then we have Y1, 0 and
we have Y1 minus 1.
01:09:50.520 --> 01:09:52.960
And we have Y1,1.
01:09:52.960 --> 01:09:56.370
So these guys take a
particularly simple form.
01:09:56.370 --> 01:09:59.805
Root 3-- I'm not even going to
worry about the coefficient.
01:09:59.805 --> 01:10:00.680
They're in the notes.
01:10:00.680 --> 01:10:03.500
You can look them up anywhere.
01:10:03.500 --> 01:10:08.110
So first off let's
look at Y1, 1.
01:10:08.110 --> 01:10:09.000
So non-linear today.
01:10:09.000 --> 01:10:11.045
So Y1, 1, it's going to
be some normalization.
01:10:11.045 --> 01:10:12.130
And what is the form?
01:10:15.940 --> 01:10:19.490
It's just e to the il phi
sine theta to the l. l is 1.
01:10:19.490 --> 01:10:27.040
So this is some constant times
e to the i phi, sine theta.
01:10:27.040 --> 01:10:27.740
Who 1, 0?
01:10:27.740 --> 01:10:32.560
Well, it's got angular momentum
0 in the z direction, in Lz.
01:10:32.560 --> 01:10:35.540
So that means how
does it depend on phi?
01:10:35.540 --> 01:10:36.590
It doesn't.
01:10:36.590 --> 01:10:39.112
And you can easily
see that, because when
01:10:39.112 --> 01:10:42.500
we lower we get an e
to the i minus i phi.
01:10:42.500 --> 01:10:45.780
Anyway, so this gives
us a constant times
01:10:45.780 --> 01:10:47.760
no e to the i phi,
no phi dependents,
01:10:47.760 --> 01:10:50.380
and cosine of theta.
01:10:50.380 --> 01:10:51.990
And if you get a
cosine of theta,
01:10:51.990 --> 01:10:53.760
the d theta and
the cotangent d phi
01:10:53.760 --> 01:10:55.270
will give you the same thing.
01:10:55.270 --> 01:10:58.350
And Y1 minus 1 is again
a constant, times e
01:10:58.350 --> 01:10:59.790
to the minus i phi.
01:10:59.790 --> 01:11:04.530
So it's got m equals minus
1 and sine theta again.
01:11:04.530 --> 01:11:07.170
Notice a pleasing
parsimony here.
01:11:07.170 --> 01:11:10.795
The theta dependence is the
same for plus m and minus m.
01:11:13.660 --> 01:11:14.560
So what about Y2, 2?
01:11:17.740 --> 01:11:25.970
Some constant e to the i 2 phi,
sine squared theta, dot, dot,
01:11:25.970 --> 01:11:27.957
dot, Y0, 0.
01:11:27.957 --> 01:11:29.040
And here it's interesting.
01:11:29.040 --> 01:11:31.850
Here we just got one term
from taking the derivative.
01:11:31.850 --> 01:11:33.190
They both give you cosine.
01:11:33.190 --> 01:11:36.780
But now there are two ways to
act with the two derivatives.
01:11:36.780 --> 01:11:38.190
And this gives you a constant.
01:11:38.190 --> 01:11:39.440
Now what's the phi dependence?
01:11:39.440 --> 01:11:41.560
It's nothing, because
it's got m equals 0.
01:11:41.560 --> 01:11:43.250
And so the only
dependence is on theta.
01:11:43.250 --> 01:11:46.470
And we get a cos squared
whoops, there's a 3--
01:11:46.470 --> 01:11:49.150
3 cos squared theta minus 1.
01:11:49.150 --> 01:11:50.895
AUDIENCE: Do you mean Y2, 0?
01:11:50.895 --> 01:11:51.770
PROFESSOR: Oh, shoot.
01:11:51.770 --> 01:11:52.270
Thank you.
01:11:52.270 --> 01:11:53.080
Yes, I mean Y2, 0.
01:11:53.080 --> 01:11:53.980
Thank you.
01:11:53.980 --> 01:11:58.600
And then if we continue
lowering to Y2 minus 2,
01:11:58.600 --> 01:12:00.100
this is equal to,
again, a constant.
01:12:00.100 --> 01:12:02.350
And the it's going to be the
same dependence on theta,
01:12:02.350 --> 01:12:05.890
but a different dependence of
phi. e to the minus 2i phi,
01:12:05.890 --> 01:12:07.850
sine squared theta.
01:12:07.850 --> 01:12:08.350
OK?
01:12:08.350 --> 01:12:09.348
Yeah.
01:12:09.348 --> 01:12:11.340
AUDIENCE: Aren't these
not normalizable?
01:12:14.325 --> 01:12:14.950
PROFESSOR: Why?
01:12:14.950 --> 01:12:16.550
AUDIENCE: Oh, never mind.
01:12:16.550 --> 01:12:17.480
PROFESSOR: Good.
01:12:17.480 --> 01:12:18.980
So let me turn that
into a question.
01:12:18.980 --> 01:12:21.790
The question is, are
these normalizable?
01:12:21.790 --> 01:12:23.590
Yeah, so how would
we normalize them?
01:12:23.590 --> 01:12:25.320
What's the check for if
they're normalizable or not?
01:12:25.320 --> 01:12:26.270
AUDIENCE: [INAUDIBLE]
01:12:26.270 --> 01:12:27.687
PROFESSOR: Yeah,
we integrate them
01:12:27.687 --> 01:12:30.890
norm squared over a sphere.
01:12:30.890 --> 01:12:32.800
Not over a volume,
just over a sphere.
01:12:32.800 --> 01:12:35.150
Because they're only wave
functions on the sphere.
01:12:35.150 --> 01:12:36.360
We haven't dealt with
the radial function.
01:12:36.360 --> 01:12:37.485
We'll deal with that later.
01:12:37.485 --> 01:12:39.403
That will come in
the next lecture.
01:12:39.403 --> 01:12:40.070
Other questions?
01:12:40.070 --> 01:12:40.630
Yeah?
01:12:40.630 --> 01:12:43.020
AUDIENCE: Can you explain
one more time why m equals
01:12:43.020 --> 01:12:44.915
0 doesn't have [INAUDIBLE]?
01:12:44.915 --> 01:12:46.540
PROFESSOR: Yeah, why
m equals 0 doesn't
01:12:46.540 --> 01:12:49.070
have any phi dependence?
01:12:49.070 --> 01:12:51.570
If m equals 0 had
phi dependence,
01:12:51.570 --> 01:12:53.990
then we know that
the eigenvalue of Llz
01:12:53.990 --> 01:12:56.540
is what we get when we take a
derivative with respect to phi.
01:12:56.540 --> 01:12:59.110
But if the Lz
eigenvalue is 0, that
01:12:59.110 --> 01:13:01.190
means that when we act
with dd phi we get 0.
01:13:01.190 --> 01:13:03.220
That means it can't
depend on phi.
01:13:03.220 --> 01:13:05.950
Cool?
01:13:05.950 --> 01:13:07.280
Other questions?
01:13:07.280 --> 01:13:09.280
What I'm going to do next
is I'm going show you,
01:13:09.280 --> 01:13:10.863
walk you through
some of these angular
01:13:10.863 --> 01:13:14.280
momentum eigenstates,
graphically on the computer.
01:13:14.280 --> 01:13:18.700
Before we do that, any questions
about the calculation so far?
01:13:18.700 --> 01:13:19.200
OK.
01:13:30.620 --> 01:13:34.690
So this mathematical package
I'll post on the web page.
01:13:34.690 --> 01:13:37.050
And at the moment I think
it's doing the real part.
01:13:37.050 --> 01:13:38.610
So what we're looking
at now is the real part.
01:13:38.610 --> 01:13:40.402
Actually, let's look
at the absolute value.
01:13:43.350 --> 01:13:44.300
Good.
01:13:44.300 --> 01:13:50.490
So here we are, looking
at the absolute value of--
01:13:50.490 --> 01:13:54.180
that's not what I wanted to do.
01:13:54.180 --> 01:13:56.465
So what we're looking
at in this notation
01:13:56.465 --> 01:13:57.840
is some horrible
parametric plot.
01:13:57.840 --> 01:14:00.090
You don't really need to see
the mathemat-- oh, shoot.
01:14:02.330 --> 01:14:02.830
Sorry.
01:14:05.605 --> 01:14:07.480
You don't need to see
the code, particularly.
01:14:07.480 --> 01:14:10.240
So I'm not going to worry about
it, but it will be posted.
01:14:10.240 --> 01:14:13.230
So here we're looking at
the absolute value, the norm
01:14:13.230 --> 01:14:19.810
squared of the spherical
harmonic Y. And the lower
01:14:19.810 --> 01:14:22.926
eigenvalue here, lower
coordinate is the l.
01:14:22.926 --> 01:14:25.440
And the upper is m.
01:14:25.440 --> 01:14:29.210
So when l is 0, what we get--
01:14:29.210 --> 01:14:32.000
here's what this
plot is indicating.
01:14:32.000 --> 01:14:34.570
The distance away from the
origin in a particular angular
01:14:34.570 --> 01:14:37.430
direction is the absolute
value of the wave function.
01:14:37.430 --> 01:14:39.330
So the further away
from the origin
01:14:39.330 --> 01:14:42.320
the colored point you see is,
the larger the absolute value.
01:14:45.720 --> 01:14:48.560
And the color here is just to
indicate depth and position.
01:14:48.560 --> 01:14:50.110
It's not terribly meaningful.
01:14:50.110 --> 01:14:52.520
So here we see that
we get a sphere.
01:14:52.520 --> 01:14:54.330
So the probability
density or the norm
01:14:54.330 --> 01:14:57.280
squared of the wave function
of the spherical harmonic
01:14:57.280 --> 01:14:57.997
is constant.
01:14:57.997 --> 01:14:58.830
So that makes sense.
01:14:58.830 --> 01:14:59.960
It's spherically symmetric.
01:14:59.960 --> 01:15:01.400
It has no angular momentum.
01:15:01.400 --> 01:15:03.400
As we start increasing
the angular momentum,
01:15:03.400 --> 01:15:06.778
let's take the angular
momentum l is 1, m is 0 state,
01:15:06.778 --> 01:15:08.195
now something
interesting happens.
01:15:11.037 --> 01:15:12.370
The total angular momentum is 1.
01:15:12.370 --> 01:15:15.260
And we see that there
are two spheres.
01:15:15.260 --> 01:15:16.940
Let me sort of rotate this.
01:15:16.940 --> 01:15:19.700
So there are two spheres,
and there's the z-axis
01:15:19.700 --> 01:15:20.720
passing through them.
01:15:20.720 --> 01:15:23.430
And so the probability
is much larger
01:15:23.430 --> 01:15:25.980
around this lobe on the top
or the lobe on the bottom.
01:15:25.980 --> 01:15:29.070
And it's 0 on the plane.
01:15:29.070 --> 01:15:31.280
Now, that's kind
of non-intuitive
01:15:31.280 --> 01:15:37.060
if you think, well,
Lz is large, so why
01:15:37.060 --> 01:15:38.220
is it along the vertical?
01:15:38.220 --> 01:15:39.300
Why is that true?
01:15:39.300 --> 01:15:41.990
So here, this is the
Lz equals 0 state.
01:15:41.990 --> 01:15:46.550
That means it carries no angular
momentum along the z-axis.
01:15:46.550 --> 01:15:50.740
That means it's not rotating
far out in the xy plane.
01:15:50.740 --> 01:15:52.950
So your probability of
finding it in the xy plane
01:15:52.950 --> 01:15:53.550
is very small.
01:15:53.550 --> 01:15:55.450
Because if it was
rotating in the xy plane
01:15:55.450 --> 01:15:57.367
it would carry a large
angular momentum in Lz.
01:15:57.367 --> 01:15:58.110
But Lz is 0.
01:15:58.110 --> 01:16:00.560
So it can't be extended
out in the xy plane.
01:16:00.560 --> 01:16:01.580
Cool?
01:16:01.580 --> 01:16:03.656
On the other hand, it can
carry angular momentum
01:16:03.656 --> 01:16:05.063
in the x or the y direction.
01:16:05.063 --> 01:16:06.480
But if it carries
angular momentum
01:16:06.480 --> 01:16:07.990
in the x direction
for example, that
01:16:07.990 --> 01:16:09.920
means the system is
rotating around the z--
01:16:09.920 --> 01:16:10.420
sorry.
01:16:10.420 --> 01:16:12.503
If it carries angular
momentum in the x direction,
01:16:12.503 --> 01:16:15.020
it's rotating in the zy plane.
01:16:15.020 --> 01:16:17.600
So there's some
probability to find it out
01:16:17.600 --> 01:16:20.090
of the plane in y and z.
01:16:20.090 --> 01:16:22.290
But it can't be in the xy plane.
01:16:22.290 --> 01:16:24.690
Hence it's got to be
in the lobes up above.
01:16:24.690 --> 01:16:25.710
That cool?
01:16:25.710 --> 01:16:28.230
So it's very useful to develop
an intuition for this stuff
01:16:28.230 --> 01:16:30.342
if you're going to do
chemistry or crystallography
01:16:30.342 --> 01:16:31.800
or any condensed
manner of physics.
01:16:31.800 --> 01:16:32.717
It's just very useful.
01:16:32.717 --> 01:16:35.330
So I encourage you to play
with these little applets.
01:16:35.330 --> 01:16:38.490
And I'll post this
mathematics package.
01:16:38.490 --> 01:16:41.170
But let's looked at what happens
now if we crank up the angular
01:16:41.170 --> 01:16:41.670
momentum.
01:16:41.670 --> 01:16:43.897
So as we crank up the
l angular momentum,
01:16:43.897 --> 01:16:45.980
now we're getting this
sort of lobe-y thing, which
01:16:45.980 --> 01:16:49.290
looks like some sort of
'50s sci-fi apparatus.
01:16:49.290 --> 01:16:52.460
So what's going on there?
01:16:52.460 --> 01:16:54.360
This is the 2, 0 state.
01:16:54.360 --> 01:16:57.495
And the 2, 0 state has a 3
cos squared theta minus 1.
01:16:57.495 --> 01:16:58.870
But cos squared
theta, that means
01:16:58.870 --> 01:17:02.910
it's got two periods as it
goes from vertical to negative.
01:17:02.910 --> 01:17:04.630
And if you take that
and you square it,
01:17:04.630 --> 01:17:06.670
you get exactly this.
01:17:06.670 --> 01:17:07.540
OK?
01:17:07.540 --> 01:17:10.200
So they're using the cos squared
as a function of the angle
01:17:10.200 --> 01:17:12.080
of declination from vertical.
01:17:12.080 --> 01:17:16.885
And it's m equals 0, so you're
saying no dependence on phi.
01:17:16.885 --> 01:17:18.260
Of course, that's
a little cheap.
01:17:18.260 --> 01:17:19.970
Because the angular
dependence on phi
01:17:19.970 --> 01:17:21.870
is just an overall phase.
01:17:21.870 --> 01:17:24.416
So we're not going to see
it in the absolute value.
01:17:24.416 --> 01:17:25.570
Everyone agree on that?
01:17:25.570 --> 01:17:26.810
We're not going to
see the absolute--
01:17:26.810 --> 01:17:27.090
OK.
01:17:27.090 --> 01:17:28.140
So let's check that.
01:17:28.140 --> 01:17:30.880
Let's take l2 and m2.
01:17:30.880 --> 01:17:34.520
So now when l is 2 and m is
2, we just get this donut.
01:17:34.520 --> 01:17:37.470
So what that's saying is, we've
got some angular momentum.
01:17:37.470 --> 01:17:37.970
l is 2.
01:17:37.970 --> 01:17:40.390
But all the angular momentum,
almost all of it, anyway,
01:17:40.390 --> 01:17:41.932
is in the z direction.
01:17:41.932 --> 01:17:43.390
And is that what
we're seeing here?
01:17:43.390 --> 01:17:44.070
Well, yeah.
01:17:44.070 --> 01:17:46.340
It seems like it's most
likely to find the particle,
01:17:46.340 --> 01:17:49.980
the probability is greatest, out
in this donut around the plane.
01:17:49.980 --> 01:17:53.890
Now, if it were Lz is equal
to l, it would be flat.
01:17:53.890 --> 01:17:55.955
It would be a strictly
0 thickness pancake.
01:17:55.955 --> 01:17:59.750
But we have some uncertainty
in what Lx and Ly are,
01:17:59.750 --> 01:18:01.510
which is why we got
this fattened donut.
01:18:01.510 --> 01:18:02.760
Everyone cool with that?
01:18:02.760 --> 01:18:07.630
And if we crank up l and
we make-- yeah, right?
01:18:07.630 --> 01:18:10.460
So you can see that you've
got some complicated shapes.
01:18:10.460 --> 01:18:12.010
But as we crank up
l and crank up m,
01:18:12.010 --> 01:18:13.690
we just get a thinner
and thinner donut.
01:18:13.690 --> 01:18:16.065
And the fact that the donut's
getting thinner and thinner
01:18:16.065 --> 01:18:18.050
is that l over L squared
that we did earlier.
01:18:18.050 --> 01:18:18.590
Cool?
01:18:18.590 --> 01:18:20.210
It's still a donut,
but it's getting
01:18:20.210 --> 01:18:22.660
relatively thinner
and thinner, by virtue
01:18:22.660 --> 01:18:24.940
of getting wider and wider.
01:18:24.940 --> 01:18:30.060
And a last thing to show you
is let's take a look now at--
01:18:30.060 --> 01:18:32.780
in fact, let's go to
the l equals 0 state.
01:18:32.780 --> 01:18:35.620
Let's take a look
at the real part.
01:18:39.490 --> 01:18:42.240
So now we're looking
at the real part.
01:18:42.240 --> 01:18:45.990
And nothing much
changed for the Y0, 0.
01:18:45.990 --> 01:18:49.620
But for Y2, 0, well,
still not much changed.
01:18:49.620 --> 01:18:52.280
For Y2, 0 let's now--
01:18:52.280 --> 01:18:53.540
this is sort of disheartening.
01:18:53.540 --> 01:18:56.490
Nothing really has changed.
01:18:56.490 --> 01:18:59.410
Why?
01:18:59.410 --> 01:19:00.850
Because it's real, exactly.
01:19:00.850 --> 01:19:04.530
So for Y2, 0, as long as m
is equal to 0, this is real.
01:19:04.530 --> 01:19:06.210
There's no phase.
01:19:06.210 --> 01:19:08.380
The phase information
contains information
01:19:08.380 --> 01:19:10.710
about the Lz eigenvalue.
01:19:10.710 --> 01:19:19.540
So we can correct this by
changing the angular momentum.
01:19:19.540 --> 01:19:22.113
Let's-- oh shoot,
how do I do that?
01:19:22.113 --> 01:19:23.530
I can't turn it
off at the moment.
01:19:23.530 --> 01:19:24.600
OK, whatever.
01:19:24.600 --> 01:19:29.930
So here we have a
large M. And now
01:19:29.930 --> 01:19:31.750
we've got this very
funny lobe-y structure.
01:19:31.750 --> 01:19:34.250
So this is the Y2, 2, which a
minute ago looked rotationally
01:19:34.250 --> 01:19:34.873
symmetric.
01:19:34.873 --> 01:19:36.540
And now it's not
rotationally symmetric.
01:19:36.540 --> 01:19:40.340
It's this lobe-y structure,
where the lobes are-- remember,
01:19:40.340 --> 01:19:41.430
previously we had a donut.
01:19:41.430 --> 01:19:44.580
Now we have these lobes when
we look at the real part.
01:19:44.580 --> 01:19:45.850
How does that make sense?
01:19:45.850 --> 01:19:53.050
Well, we've got an
e to the i 2 phi.
01:19:53.050 --> 01:19:56.420
And if we look at the real
part, that's cosine of 2 pi.
01:19:56.420 --> 01:19:59.095
And so we're getting a cosine
function modulating the donut.
01:19:59.095 --> 01:20:01.470
And if we look at the real
part, let's do the same thing.
01:20:01.470 --> 01:20:03.550
Let's look at the
imaginary part.
01:20:06.420 --> 01:20:10.360
And the imaginary part
of Y2, 0, we get nothing.
01:20:10.360 --> 01:20:13.098
That's good, because
Y2, 0 was real.
01:20:13.098 --> 01:20:14.140
That would have been bad.
01:20:14.140 --> 01:20:16.140
But if we look at the
imaginary part of Y2, 2,
01:20:16.140 --> 01:20:19.380
we get the corresponding
lobes, the other lobes,
01:20:19.380 --> 01:20:22.473
so that cos squared
plus sine squared is 1.
01:20:22.473 --> 01:20:23.140
Play with these.
01:20:23.140 --> 01:20:24.440
Develop some intuition.
01:20:24.440 --> 01:20:26.023
They're going to be
very useful for us
01:20:26.023 --> 01:20:28.640
when we talk about hydrogen
and the structure of solids.
01:20:28.640 --> 01:20:30.640
And I will see you next week.
01:20:30.640 --> 01:20:33.390
[APPLAUSE]