WEBVTT

00:00:00.000 --> 00:00:03.600
PROFESSOR: Let me discuss, for
a second, effective potential.

00:00:13.090 --> 00:00:19.560
And-- actually, let me ask you
a question to get you started.

00:00:19.560 --> 00:00:23.830
We're going to look at high n.

00:00:23.830 --> 00:00:26.920
N-- so n's, say, 100.

00:00:26.920 --> 00:00:35.620
And then l will go
from 0 up to 99.

00:00:35.620 --> 00:00:38.080
Very high l's.

00:00:38.080 --> 00:00:41.330
What's going to be
different from them?

00:00:41.330 --> 00:00:45.530
Well here is one
possibility, and that's

00:00:45.530 --> 00:00:48.360
a possibility that happens.

00:00:48.360 --> 00:00:53.660
The orbits are going to go
from being circular to being

00:00:53.660 --> 00:00:59.150
elliptical as you change l.

00:00:59.150 --> 00:01:00.830
And I ask you--

00:01:00.830 --> 00:01:04.340
which l would correspond
to circular orbit,

00:01:04.340 --> 00:01:10.800
and which l would correspond
to the most elliptical orbit?

00:01:10.800 --> 00:01:16.210
So let's take a poll/
so most circular orbit--

00:01:22.952 --> 00:01:29.620
is it l equals 0 or l equals 99?

00:01:29.620 --> 00:01:34.885
Who votes for l equals 0
for the most circular orbit?

00:01:38.374 --> 00:01:43.280
I have about half [INAUDIBLE],
a little bit more than half.

00:01:43.280 --> 00:01:44.180
OK.

00:01:44.180 --> 00:01:45.853
Who votes for l equals 99?

00:01:50.130 --> 00:01:53.250
[INAUDIBLE]

00:01:53.250 --> 00:01:54.620
Which is the most [INAUDIBLE]?

00:01:54.620 --> 00:01:58.560
OK, this is funny,
because we all

00:01:58.560 --> 00:02:03.150
associate most
circular with l equals

00:02:03.150 --> 00:02:05.370
0, spherically symmetric.

00:02:05.370 --> 00:02:06.420
But it's wrong.

00:02:06.420 --> 00:02:07.679
This is the most circular.

00:02:16.094 --> 00:02:18.590
AUDIENCE: What if l equals 100?

00:02:18.590 --> 00:02:23.260
PROFESSOR: You cannot
reach l equal 100.

00:02:23.260 --> 00:02:31.620
Actually-- and this
is most elliptical.

00:02:35.420 --> 00:02:37.700
and you can have the
intuition for that.

00:02:37.700 --> 00:02:43.770
Let's avoid the calculation
and be intuitive.

00:02:43.770 --> 00:02:50.430
And the intuition it is kind
of interesting, because we

00:02:50.430 --> 00:02:51.870
said at effective potential.

00:02:51.870 --> 00:02:54.120
And indeed, I want to discuss--

00:02:54.120 --> 00:02:55.850
that's what I want
to understand.

00:02:55.850 --> 00:02:57.640
Effective potential.

00:02:57.640 --> 00:03:01.980
So maybe I'll draw this
line a little lower.

00:03:01.980 --> 00:03:04.320
And what is the
effective potential?

00:03:04.320 --> 00:03:08.460
We've written it
already many times.

00:03:08.460 --> 00:03:15.810
It's h squared l times l
plus 1 over 2mr squared

00:03:15.810 --> 00:03:19.020
minus e squared over r.

00:03:19.020 --> 00:03:22.170
So the minus e squared
over r is here.

00:03:27.530 --> 00:03:30.305
Maybe I should draw it in a
way that gives me more room.

00:03:34.020 --> 00:03:34.561
Like that.

00:03:37.210 --> 00:03:39.290
OK.

00:03:39.290 --> 00:03:45.180
And then there is the energy
level that I'm looking at.

00:03:45.180 --> 00:03:48.760
It's some high energy level.

00:03:48.760 --> 00:03:51.250
So I'll put it near
here, not too near

00:03:51.250 --> 00:03:52.900
that I don't see
it, but near here.

00:03:52.900 --> 00:03:57.450
Here is-- my en is here.

00:04:00.960 --> 00:04:01.770
OK.

00:04:01.770 --> 00:04:05.440
So the answer is
actually already there.

00:04:05.440 --> 00:04:08.460
Not too clearly, but here it is.

00:04:08.460 --> 00:04:13.310
What does a particle
do in this potential?

00:04:13.310 --> 00:04:17.079
It goes with lots of kinetic
energy, very fast here,

00:04:17.079 --> 00:04:20.050
it goes from here to here.

00:04:20.050 --> 00:04:22.200
Here to here.

00:04:22.200 --> 00:04:25.300
But what does that mean?

00:04:25.300 --> 00:04:33.570
That the radius is changing
from r to some other value of r.

00:04:33.570 --> 00:04:38.910
From r equals 0 to
some other value of r.

00:04:38.910 --> 00:04:43.890
So what looks like an orbit?

00:04:43.890 --> 00:04:47.580
If you have a planet
going around the sun,

00:04:47.580 --> 00:04:53.710
and at some point it
seems to reach the sun,

00:04:53.710 --> 00:04:55.750
and at some point it goes far--

00:04:55.750 --> 00:04:57.850
here is the sun--

00:04:57.850 --> 00:05:03.560
then the orbit
must be like this.

00:05:03.560 --> 00:05:05.375
That's likely to be the orbit.

00:05:08.060 --> 00:05:12.440
On the other hand, as
soon s is equal to 1,

00:05:12.440 --> 00:05:15.720
you have the part
of the contributions

00:05:15.720 --> 00:05:17.240
that affect the potential.

00:05:17.240 --> 00:05:21.410
You have a 1 over
r square like this.

00:05:21.410 --> 00:05:23.600
And then the total
potential, which

00:05:23.600 --> 00:05:26.930
is the original
term plus that, will

00:05:26.930 --> 00:05:33.180
have something that looks
like this, and go like that.

00:05:33.180 --> 00:05:38.450
And then this orbit is going to
go from some small value of r,

00:05:38.450 --> 00:05:41.250
but different from
0, to another one.

00:05:41.250 --> 00:05:44.180
So those are all
elliptical orbits,

00:05:44.180 --> 00:05:46.480
because the radius
is not constant.

00:05:49.320 --> 00:05:51.720
And then this orbit,
it goes like that.

00:05:51.720 --> 00:05:54.160
Now look at it--

00:05:54.160 --> 00:05:58.930
if you have some
elliptical orbit-- so then

00:05:58.930 --> 00:06:02.530
after that time, it just
becomes a little more

00:06:02.530 --> 00:06:07.570
like this, in which the minimum
radius for, say l equal 1

00:06:07.570 --> 00:06:11.680
is here, and the maximum
radius has been reduced.

00:06:11.680 --> 00:06:19.060
l equal 2, l equal 3, they
all produce different ones.

00:06:19.060 --> 00:06:21.490
My graph is not perfect.

00:06:21.490 --> 00:06:23.910
Different ones.

00:06:23.910 --> 00:06:24.770
Different ones.

00:06:24.770 --> 00:06:28.160
And finally, one
that just touches it.

00:06:31.070 --> 00:06:33.620
And that one, the
radius is fixed.

00:06:33.620 --> 00:06:35.690
The particle is
moving like that.

00:06:35.690 --> 00:06:38.550
And that is your
spherical orbit.

00:06:38.550 --> 00:06:42.740
And if you increase l a
little more, no more solution.

00:06:42.740 --> 00:06:46.910
So it's the top
l that produces--

00:06:46.910 --> 00:06:52.400
this is already almost circular,
and this is perfectly circular.

00:06:52.400 --> 00:06:59.360
So then it goes like this, and
then eventually goes like this,

00:06:59.360 --> 00:07:03.170
and finally at the
end it goes like that.

00:07:03.170 --> 00:07:07.820
And it's circular
at the last case.

00:07:07.820 --> 00:07:12.930
For the most l, you finally
get your circular orbit.

00:07:12.930 --> 00:07:17.300
And also the intuition is clear.

00:07:17.300 --> 00:07:19.940
If you have a
circular orbit, you

00:07:19.940 --> 00:07:22.670
have lots of angular momentum--

00:07:22.670 --> 00:07:25.970
r cross p is very good.

00:07:25.970 --> 00:07:29.870
When two vectors are
orthogonal, the cross-product

00:07:29.870 --> 00:07:33.200
of angular momentum is large.

00:07:33.200 --> 00:07:35.550
But look here.

00:07:35.550 --> 00:07:39.470
Here is r, for the
shortest orbit,

00:07:39.470 --> 00:07:41.570
and p is in that direction.

00:07:41.570 --> 00:07:43.760
R cross p almost like that.

00:07:43.760 --> 00:07:47.630
This is a very low
angular momentum orbit.

00:07:51.600 --> 00:07:55.860
So l equals 0 is the
most elliptical orbit.

00:07:55.860 --> 00:08:01.290
L equal very high is
the most circular orbit.

00:08:01.290 --> 00:08:02.850
There is one thing--

00:08:02.850 --> 00:08:06.840
a calculation-- maybe I'll
use 10 minutes next time--

00:08:06.840 --> 00:08:12.080
is to show that there's
two turning points here--

00:08:12.080 --> 00:08:15.490
an r plus and an r minus.

00:08:15.490 --> 00:08:19.990
So there is an r plus for
one solution, and an r minus,

00:08:19.990 --> 00:08:21.670
and it goes like that.

00:08:21.670 --> 00:08:24.880
So the r pluses
and the r minuses

00:08:24.880 --> 00:08:28.860
corresponds to an ellipse that--

00:08:28.860 --> 00:08:30.210
here is the center.

00:08:30.210 --> 00:08:32.000
That's where the proton sits.

00:08:32.000 --> 00:08:37.630
And you go r plus out is the
maximum, and r minus is here.

00:08:37.630 --> 00:08:41.530
So the two turning points
that define the maximum r

00:08:41.530 --> 00:08:44.650
and minimum r tell
you about the lips

00:08:44.650 --> 00:08:47.650
on these r plus plus r minus.

00:08:47.650 --> 00:08:52.850
And when you find those critical
points and do the calculation,

00:08:52.850 --> 00:08:57.670
you in fact can show
that r plus plus r minus

00:08:57.670 --> 00:09:01.428
is indeed equal
to n squared a 0.

00:09:04.056 --> 00:09:09.500
I think that number
comes out exactly there.

00:09:09.500 --> 00:09:12.230
R plus plus r minus over 2.

00:09:12.230 --> 00:09:15.480
The average is exactly that.

00:09:15.480 --> 00:09:20.360
So the roots, the two roots,
are such that their average

00:09:20.360 --> 00:09:21.650
is always the same.

00:09:21.650 --> 00:09:23.690
So this is not--

00:09:23.690 --> 00:09:26.330
I'm not sure I get it
right with my graph,

00:09:26.330 --> 00:09:27.920
but it should be
that if you take

00:09:27.920 --> 00:09:31.280
the average of that
intersection points,

00:09:31.280 --> 00:09:34.570
they're always at
the same point.

00:09:34.570 --> 00:09:41.930
And so that's how it
happens under an analog in--

00:09:41.930 --> 00:09:43.730
so you get the intuition here.

00:09:43.730 --> 00:09:46.880
Here are your orbits.

00:09:46.880 --> 00:09:49.790
And these are
equal energy orbits

00:09:49.790 --> 00:09:53.990
of different eccentricity
as you go in there.

00:09:53.990 --> 00:09:59.480
And even Kepler knew that,
because Kepler figured out

00:09:59.480 --> 00:10:03.800
that the time that it takes
the planet to go around

00:10:03.800 --> 00:10:10.220
depends only on the sum of these
two things, the major diameter.

00:10:10.220 --> 00:10:14.010
So all these
solutions over here--

00:10:14.010 --> 00:10:15.710
if they were planets--

00:10:15.710 --> 00:10:20.220
would take the same time
to go around the sun.

00:10:20.220 --> 00:10:22.610
So they all, in
quantum mechanics,

00:10:22.610 --> 00:10:26.420
have the same energy, and
they're therefore degenerate.

00:10:26.420 --> 00:10:29.630
So the mystery of the
quadratic potential

00:10:29.630 --> 00:10:33.200
that makes all of these
periods to be the same

00:10:33.200 --> 00:10:37.790
is the mystery that is making
the energies the same in all

00:10:37.790 --> 00:10:39.530
of these orbits.

00:10:39.530 --> 00:10:42.710
So we'll finalize
this story next time

00:10:42.710 --> 00:10:45.760
and do some other fun things.