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PROFESSOR: All right,
shall we get started?
00:00:28.350 --> 00:00:35.030
So, today-- well, before I
get started-started-- so,
00:00:35.030 --> 00:00:37.066
let me open up to questions.
00:00:37.066 --> 00:00:39.856
Do y'all have questions
from the last lecture,
00:00:39.856 --> 00:00:41.480
where we finished
off angular momentum?
00:00:45.860 --> 00:00:47.995
Or really anything
up to the last exam?
00:00:51.849 --> 00:00:53.340
Yeah?
00:00:53.340 --> 00:00:56.340
AUDIENCE: So, what exactly
happens with the half l states?
00:00:56.340 --> 00:00:57.310
PROFESSOR: Ha, ha, ha!
00:00:57.310 --> 00:00:58.890
What happens with
the half l states?
00:00:58.890 --> 00:01:00.124
OK, great question!
00:01:00.124 --> 00:01:02.540
So, we're gonna talk about
that in some detail in a couple
00:01:02.540 --> 00:01:05.960
of weeks, but let me
give you a quick preview.
00:01:05.960 --> 00:01:10.920
So, remember that when we
studied the commutation
00:01:10.920 --> 00:01:18.880
relations, Lx, Ly
is i h bar Lz .
00:01:18.880 --> 00:01:22.029
Without using the representation
in terms of derivatives,
00:01:22.029 --> 00:01:23.820
with respect to a
coordinate, without using
00:01:23.820 --> 00:01:27.780
the representations, in terms
of translations and rotations
00:01:27.780 --> 00:01:29.250
along the sphere, right?
00:01:29.250 --> 00:01:31.330
When we just used the
commutation relations,
00:01:31.330 --> 00:01:33.430
and nothing else,
what we found was
00:01:33.430 --> 00:01:38.850
that the states corresponding
to these guys, came in a tower,
00:01:38.850 --> 00:01:41.480
with either one state--
corresponding to little l
00:01:41.480 --> 00:01:43.396
equals 0-- or two
states-- with l
00:01:43.396 --> 00:01:47.010
equals 1/2-- or three
states-- with little l equals
00:01:47.010 --> 00:01:52.740
1-- or four states-- with l
equals 3/2-- and so on, and so
00:01:52.740 --> 00:01:53.880
forth.
00:01:53.880 --> 00:01:56.140
And we quickly
deduced that it is
00:01:56.140 --> 00:02:01.650
impossible to represent the
half integer states with a wave
00:02:01.650 --> 00:02:03.450
function which
represents a probability
00:02:03.450 --> 00:02:05.099
distribution on a sphere.
00:02:05.099 --> 00:02:06.640
We observed that
that was impossible.
00:02:06.640 --> 00:02:09.020
And the reason is, if
you did so, then when
00:02:09.020 --> 00:02:10.639
you take that wave
function, if you
00:02:10.639 --> 00:02:14.080
rotate by 2pi--
in any direction--
00:02:14.080 --> 00:02:16.130
if you rotate by 2pi the
wave function comes back
00:02:16.130 --> 00:02:17.710
to minus itself.
00:02:17.710 --> 00:02:20.050
But the wave function
has to be equal to itself
00:02:20.050 --> 00:02:20.910
at that same point.
00:02:20.910 --> 00:02:22.120
The value of the wave
function at some point,
00:02:22.120 --> 00:02:23.930
is equal to the wave
function at some point.
00:02:23.930 --> 00:02:24.790
That means the value
of the wave function
00:02:24.790 --> 00:02:26.560
must be equal to minus itself.
00:02:26.560 --> 00:02:28.000
That means it must be zero0.
00:02:28.000 --> 00:02:29.550
So, you can't write
a wave function--
00:02:29.550 --> 00:02:31.864
which is a probability
distribution on a sphere--
00:02:31.864 --> 00:02:34.030
if the wave function has
to be equal to minus itself
00:02:34.030 --> 00:02:35.950
at any given point.
00:02:35.950 --> 00:02:38.150
So, this is a strange thing.
00:02:38.150 --> 00:02:42.270
And we sort of said, well, look,
these are some other beasts.
00:02:42.270 --> 00:02:45.190
But the question is,
look, these furnish
00:02:45.190 --> 00:02:50.000
perfectly reasonable
towers of states respecting
00:02:50.000 --> 00:02:51.750
these commutation relations.
00:02:51.750 --> 00:02:53.477
So, are they just wrong?
00:02:53.477 --> 00:02:54.560
Are they just meaningless?
00:02:54.560 --> 00:02:57.810
And what we're going to
discover is the following--
00:02:57.810 --> 00:03:00.429
and this is really gonna go
back to the very first lecture,
00:03:00.429 --> 00:03:01.970
and so, we'll do
this in more detail,
00:03:01.970 --> 00:03:03.761
but I'm going to quickly
tell you-- imagine
00:03:03.761 --> 00:03:08.750
take a magnet, a
little, tiny bar magnet.
00:03:08.750 --> 00:03:11.570
In fact, well, imagine you
take a little bar magnet
00:03:11.570 --> 00:03:16.560
with some little
magnetization, and you send it
00:03:16.560 --> 00:03:21.760
through a region that has a
gradient for magnetic field.
00:03:21.760 --> 00:03:24.080
If there's a gradient-- so
you know that a magnet wants
00:03:24.080 --> 00:03:26.190
to anti-align with
the nearby magnet,
00:03:26.190 --> 00:03:28.590
north-south wants to
go to south-north.
00:03:28.590 --> 00:03:30.490
So, you can't put a
force on the magnet,
00:03:30.490 --> 00:03:33.160
but if you have a gradient
of a magnetic field,
00:03:33.160 --> 00:03:37.810
then one end a dipole--
one end of your magnet--
00:03:37.810 --> 00:03:41.230
can feel a stronger effective
torque then the other guy.
00:03:41.230 --> 00:03:42.740
And you can get a net force.
00:03:45.794 --> 00:03:46.960
So, you can get a net force.
00:03:46.960 --> 00:03:48.626
The important thing
here, is that if you
00:03:48.626 --> 00:03:55.550
have a magnetic field which
has a gradient, so that you've
00:03:55.550 --> 00:03:58.090
got some large B, here, and
some smaller B, here, then
00:03:58.090 --> 00:03:59.110
you can get a force.
00:03:59.110 --> 00:04:05.260
And that force is going
to be proportional to how
00:04:05.260 --> 00:04:06.350
big your magnet is.
00:04:06.350 --> 00:04:07.933
But it's also going
to be proportional
00:04:07.933 --> 00:04:10.100
to the magnetic field.
00:04:10.100 --> 00:04:15.070
And if the force is proportional
to the strength of your magnet,
00:04:15.070 --> 00:04:18.540
then how far-- if you send
this magnet through a region,
00:04:18.540 --> 00:04:20.790
it'll get deflected in one
direction or the other--
00:04:20.790 --> 00:04:23.340
and how far it gets
deflected is determined
00:04:23.340 --> 00:04:25.444
by how big of a magnet
you sent through.
00:04:25.444 --> 00:04:27.360
You send in a bigger
magnet, it deflects more.
00:04:27.360 --> 00:04:28.810
Everyone cool with that?
00:04:28.810 --> 00:04:31.930
OK, here's a funny thing.
00:04:31.930 --> 00:04:33.740
So, that's fact one.
00:04:33.740 --> 00:04:36.780
Fact two, suppose I
have a system which
00:04:36.780 --> 00:04:41.310
is a charged particle
moving in a circular orbit.
00:04:41.310 --> 00:04:41.810
OK?
00:04:41.810 --> 00:04:45.880
A charged particle moving
in a circular orbit.
00:04:45.880 --> 00:04:48.810
Or better yet, well,
better yet, imaging
00:04:48.810 --> 00:04:52.080
you have a sphere--
this is a better model--
00:04:52.080 --> 00:04:55.080
imagine you have a sphere of
uniform charge distribution.
00:04:55.080 --> 00:04:55.760
OK?
00:04:55.760 --> 00:04:59.160
A little gelatinous sphere of
uniform charge distribution,
00:04:59.160 --> 00:05:02.050
and you make it rotate, OK?
00:05:02.050 --> 00:05:06.120
So, that's charged, that's
moving, forming a current.
00:05:06.120 --> 00:05:08.110
And that current
generates a magnetic field
00:05:08.110 --> 00:05:10.160
along the axis of
rotation, right?
00:05:10.160 --> 00:05:11.610
Right hand rule.
00:05:11.610 --> 00:05:14.510
So, if you have a charged
sphere, and it's rotating,
00:05:14.510 --> 00:05:16.342
you get a magnetic moment.
00:05:16.342 --> 00:05:17.800
And how big is the
magnetic moment,
00:05:17.800 --> 00:05:19.216
it's proportional
to the rotation,
00:05:19.216 --> 00:05:21.860
to the angular momentum, OK?
00:05:21.860 --> 00:05:26.100
So, you determine that,
for a charged sphere here
00:05:26.100 --> 00:05:28.710
which is rotating
with angular momentum,
00:05:28.710 --> 00:05:32.440
let's say, l, has a
magnetic moment which
00:05:32.440 --> 00:05:34.190
is proportional to l.
00:05:37.118 --> 00:05:39.374
OK?
00:05:39.374 --> 00:05:40.540
So, let's put this together.
00:05:40.540 --> 00:05:41.970
Imagine we take
a charged sphere,
00:05:41.970 --> 00:05:43.928
we send it rotating with
same angular momentum,
00:05:43.928 --> 00:05:45.460
we send it through
a field gradient,
00:05:45.460 --> 00:05:46.710
a gradient for magnetic field.
00:05:46.710 --> 00:05:49.900
What we'll see is we can
measure that angular momentum
00:05:49.900 --> 00:05:51.804
by measuring the deflection.
00:05:51.804 --> 00:05:53.470
Because the bigger
the angular momentum,
00:05:53.470 --> 00:05:54.590
the bigger the magnetic
moment, but the
00:05:54.590 --> 00:05:56.980
bigger the magnetic moment,
the bigger the deflection.
00:05:56.980 --> 00:05:58.392
Cool?
00:05:58.392 --> 00:05:59.850
So, now here's the
cool experiment.
00:06:03.450 --> 00:06:06.060
Take an electron.
00:06:06.060 --> 00:06:08.095
And electron has some charge.
00:06:08.095 --> 00:06:09.470
Is it a little,
point-like thing?
00:06:09.470 --> 00:06:10.420
Is it a little sphere?
00:06:10.420 --> 00:06:15.120
Is it, you know-- Let's not
ask that question just yet.
00:06:15.120 --> 00:06:15.829
It's an electron.
00:06:15.829 --> 00:06:18.370
The thing you get by ripping a
negative charge off a hydrogen
00:06:18.370 --> 00:06:18.930
atom.
00:06:18.930 --> 00:06:20.070
So, take your
electron and send it
00:06:20.070 --> 00:06:21.130
through a magnetic
field gradient.
00:06:21.130 --> 00:06:22.046
Why would you do this?
00:06:22.046 --> 00:06:24.177
Because you want to
measure the angular
00:06:24.177 --> 00:06:25.260
momentum of this electron.
00:06:25.260 --> 00:06:27.635
You want to see whether the
electron is a little rotating
00:06:27.635 --> 00:06:28.620
thing or not.
00:06:28.620 --> 00:06:30.970
So, you send it through this
magnetic field gradient,
00:06:30.970 --> 00:06:33.030
and if it gets
deflected, you will
00:06:33.030 --> 00:06:35.090
have measured the
magnetic moment.
00:06:35.090 --> 00:06:36.965
And if you have measured
the magnetic moment,
00:06:36.965 --> 00:06:39.090
you'll have measured
the angular momentum.
00:06:39.090 --> 00:06:39.830
OK?
00:06:39.830 --> 00:06:43.332
Here's the funny thing, if
the electron weren't rotating,
00:06:43.332 --> 00:06:45.040
it would just go
straight through, right?
00:06:45.040 --> 00:06:46.456
It would have no
angular momentum,
00:06:46.456 --> 00:06:48.850
and it would have
no magnetic moment,
00:06:48.850 --> 00:06:51.130
and thus it would not reflect.
00:06:51.130 --> 00:06:51.660
Yeah?
00:06:51.660 --> 00:06:54.020
If it's rotating,
it's gonna deflect.
00:06:54.020 --> 00:06:56.266
Here's the experiment we do.
00:06:56.266 --> 00:06:57.765
And here's the
experimental results.
00:06:57.765 --> 00:07:00.100
The experimental results
are every electron
00:07:00.100 --> 00:07:02.765
that gets sent through bends.
00:07:02.765 --> 00:07:06.080
And it either bends
up a fixed amount,
00:07:06.080 --> 00:07:07.910
or it bends down a fixed amount.
00:07:07.910 --> 00:07:09.813
It never bends more,
it never bends less,
00:07:09.813 --> 00:07:11.730
and it certainly
never been zero.
00:07:11.730 --> 00:07:14.720
In fact, it always makes
two spots on the screen.
00:07:17.828 --> 00:07:19.250
OK?
00:07:19.250 --> 00:07:20.210
Always makes two spots.
00:07:20.210 --> 00:07:22.024
It never hits the middle.
00:07:22.024 --> 00:07:23.690
No matter how you
build this experiment,
00:07:23.690 --> 00:07:25.970
no matter how you rotate
it, no matter what you do,
00:07:25.970 --> 00:07:28.150
it always hits one of two spots.
00:07:28.150 --> 00:07:30.800
What that tells you is,
the angular momentum--
00:07:30.800 --> 00:07:33.690
rather the magnetic moment--
can only take one of two values.
00:07:33.690 --> 00:07:36.400
But the angular momentum is just
some geometric constant times
00:07:36.400 --> 00:07:37.290
the angular momentum.
00:07:37.290 --> 00:07:38.706
So, the angular
momentum must take
00:07:38.706 --> 00:07:42.450
one of two possible values.
00:07:42.450 --> 00:07:44.020
Everyone cool with that?
00:07:44.020 --> 00:07:47.460
So, from this experiment--
glorified as the Stern Gerlach
00:07:47.460 --> 00:07:48.880
Experiment-- from
this experiment,
00:07:48.880 --> 00:07:52.770
we discover that the
angular momentum, Lz,
00:07:52.770 --> 00:07:53.936
takes one of two values.
00:07:53.936 --> 00:07:55.310
L, along whatever
direction we're
00:07:55.310 --> 00:08:00.280
measuring-- but let's say in
the z direction-- Lz takes
00:08:00.280 --> 00:08:07.310
one of two values, plus
some constant and, you know,
00:08:07.310 --> 00:08:11.466
plus h bar upon 2, or
minus h bar upon 2.
00:08:11.466 --> 00:08:13.390
And you just do
this measurement.
00:08:13.390 --> 00:08:15.440
But what this tells
us is, which state?
00:08:15.440 --> 00:08:16.290
Which tower?
00:08:16.290 --> 00:08:23.100
Which set of states describe
an electron in this apparatus?
00:08:23.100 --> 00:08:25.530
L equals 1/2.
00:08:25.530 --> 00:08:27.720
But wait, we started
off by talking
00:08:27.720 --> 00:08:30.250
about the rotation
of a charged sphere,
00:08:30.250 --> 00:08:32.669
and deducing that the magnetic
moment must be proportional
00:08:32.669 --> 00:08:33.429
to the angular momentum.
00:08:33.429 --> 00:08:34.880
And what we've
just discovered is
00:08:34.880 --> 00:08:38.650
that this angular momentum-- the
only sensible angular momentum,
00:08:38.650 --> 00:08:42.100
here-- is the two
state tower, which
00:08:42.100 --> 00:08:46.920
can't be represented in terms
of rotations on a sphere.
00:08:46.920 --> 00:08:48.395
Yeah?
00:08:48.395 --> 00:08:50.020
What we've learned
from this experiment
00:08:50.020 --> 00:08:53.290
is that electrons carry a
form of angular momentum,
00:08:53.290 --> 00:08:55.480
demonstrably.
00:08:55.480 --> 00:08:59.920
Which is one of these angular
momentum 1/2 states, which
00:08:59.920 --> 00:09:01.544
never doesn't rotate, right?
00:09:01.544 --> 00:09:03.210
It always carries
some angular momentum.
00:09:03.210 --> 00:09:04.834
However, it can't be
expressed in terms
00:09:04.834 --> 00:09:08.231
of rotation of some
spherical electron.
00:09:08.231 --> 00:09:09.730
It has nothing to
do with rotations.
00:09:09.730 --> 00:09:12.345
If it did, we'd get
this nonsensical thing
00:09:12.345 --> 00:09:14.600
of the wave function
identically vanishes.
00:09:14.600 --> 00:09:16.700
So, there's some other
form of angular momentum--
00:09:16.700 --> 00:09:21.000
a totally different form of
angular momentum-- at least
00:09:21.000 --> 00:09:22.380
for electrons.
00:09:22.380 --> 00:09:25.240
Which, again, has the
magnetic moment proportional
00:09:25.240 --> 00:09:27.630
to this angular momentum
with some coefficient, which
00:09:27.630 --> 00:09:28.213
I'll call mu0.
00:09:31.580 --> 00:09:35.010
But I don't want to call
it L, because L we usually
00:09:35.010 --> 00:09:36.575
use for rotational
angular momentum.
00:09:36.575 --> 00:09:38.700
This is a different form
of angular momentum, which
00:09:38.700 --> 00:09:41.160
is purely half integer,
and we call that spin.
00:09:44.570 --> 00:09:48.540
And the spin satisfies
exactly the same commutation
00:09:48.540 --> 00:09:54.280
relations-- it's a vector-- Sx
with Sy is equal to ih bar Sz.
00:09:56.817 --> 00:09:59.150
So, it's like an angular
momentum in every possible way,
00:09:59.150 --> 00:10:01.250
except it cannot be represented.
00:10:01.250 --> 00:10:05.500
Sz does not have
any representation,
00:10:05.500 --> 00:10:08.700
in terms of h bar
upon i [INAUDIBLE].
00:10:08.700 --> 00:10:11.810
It is not related to a rotation.
00:10:11.810 --> 00:10:14.200
It's an intrinsic form
of angular momentum.
00:10:14.200 --> 00:10:16.070
An electron just has it.
00:10:16.070 --> 00:10:18.490
So, at this point, you
ask me, look, what do you
00:10:18.490 --> 00:10:20.430
mean an electron just has it?
00:10:20.430 --> 00:10:21.870
And my answer to
that question is,
00:10:21.870 --> 00:10:24.330
if you send an electron through
a Stern Gerlach Apparatus,
00:10:24.330 --> 00:10:25.663
it always hits one of two spots.
00:10:29.160 --> 00:10:30.630
And that's it, right?
00:10:30.630 --> 00:10:32.315
It's an experimental fact.
00:10:32.315 --> 00:10:34.440
And this is how we describe
that experimental fact.
00:10:34.440 --> 00:10:38.577
And the legacy of these
little L equals 1/2 states,
00:10:38.577 --> 00:10:40.660
is that they represent an
internal form of angular
00:10:40.660 --> 00:10:42.993
momentum that only exists
quantum mechanically, that you
00:10:42.993 --> 00:10:46.382
would have never
noticed classically.
00:10:46.382 --> 00:10:47.840
That was a very
long answer to what
00:10:47.840 --> 00:10:49.280
was initially a simple question.
00:10:49.280 --> 00:10:51.330
But we'll come back and
do this in more detail,
00:10:51.330 --> 00:10:52.080
this was just a quick intro.
00:10:52.080 --> 00:10:52.590
Yeah?
00:10:52.590 --> 00:10:54.158
AUDIENCE: So, for
L equals 3/2, does
00:10:54.158 --> 00:10:55.750
that mean that there's
4 values of spins?
00:10:55.750 --> 00:10:56.430
PROFESSOR: Yeah,
that means there's
00:10:56.430 --> 00:10:57.410
[? 4 ?] values of spins.
00:10:57.410 --> 00:10:58.909
And so there are
plenty of particles
00:10:58.909 --> 00:11:00.810
in the real world that
have L equals 3/2.
00:11:00.810 --> 00:11:03.580
They're not fundamental
particles, as far as we know.
00:11:03.580 --> 00:11:06.440
There are particles a nuclear
physics that carry spin 3/2.
00:11:06.440 --> 00:11:09.190
There are all sorts of
nuclei that carry spin 3/2,
00:11:09.190 --> 00:11:11.600
but we don't know of a
fundamental particle.
00:11:11.600 --> 00:11:13.480
If super symmetry
is true, then there
00:11:13.480 --> 00:11:15.990
must be a particle
called a gravitino, which
00:11:15.990 --> 00:11:21.090
would be fundamental, and would
have spin 3/2, and four states,
00:11:21.090 --> 00:11:24.150
but that hasn't
been observed, yet.
00:11:26.810 --> 00:11:29.333
Other questions?
00:11:29.333 --> 00:11:30.776
AUDIENCE: Was the
[? latter of ?]
00:11:30.776 --> 00:11:34.784
seemingly nonsensical states
discovered first, and then
00:11:34.784 --> 00:11:38.170
the experiment explain it,
or was it the experiment--
00:11:38.170 --> 00:11:39.100
PROFESSOR: Oh, no!
00:11:39.100 --> 00:11:39.860
Oh, that's a great question.
00:11:39.860 --> 00:11:41.140
We'll come back the
that at end of today.
00:11:41.140 --> 00:11:43.210
So today, we're gonna do
hydrogen, among other things.
00:11:43.210 --> 00:11:45.290
Although, I've taken so
long talking about this,
00:11:45.290 --> 00:11:47.654
we might be a little slow.
00:11:47.654 --> 00:11:49.320
We'll talk about that
a little more when
00:11:49.320 --> 00:11:52.860
we talk about hydrogen, but
it was observed and deduced
00:11:52.860 --> 00:11:56.260
from experiment before
it was understood
00:11:56.260 --> 00:11:57.960
that there was such
a physical quantity.
00:11:57.960 --> 00:12:01.340
However, the observation that
this commutation relation
00:12:01.340 --> 00:12:03.460
led to towers of
states with this
00:12:03.460 --> 00:12:05.316
pre-existed as a
mathematical statement.
00:12:05.316 --> 00:12:06.940
So, that was a
mathematical observation
00:12:06.940 --> 00:12:10.440
from long previously, and it
has a beautiful algebraic story,
00:12:10.440 --> 00:12:12.220
and all sort of
nice things, but it
00:12:12.220 --> 00:12:14.270
hadn't been connected
to the physics.
00:12:14.270 --> 00:12:15.990
And so, the observation
that the electron
00:12:15.990 --> 00:12:18.073
must carry some intrinsic
form of angular momentum
00:12:18.073 --> 00:12:20.930
with one of two values,
neither of which is 0,
00:12:20.930 --> 00:12:25.000
was actually an
experimental observation--
00:12:25.000 --> 00:12:28.640
quasi-experimental observation--
long before it was understood
00:12:28.640 --> 00:12:30.089
exactly how to
connect this stuff.
00:12:30.089 --> 00:12:31.130
AUDIENCE: So it wasn't--?
00:12:31.130 --> 00:12:32.920
PROFESSOR: I shouldn't say
long, it was like within months,
00:12:32.920 --> 00:12:33.300
but whatever.
00:12:33.300 --> 00:12:33.680
Sorry.
00:12:33.680 --> 00:12:36.013
AUDIENCE: The intent of the
experiment wasn't to solve--
00:12:36.013 --> 00:12:36.860
AUDIENCE: No, no.
00:12:36.860 --> 00:12:40.750
The experiment was this--
there are the spectrum-- Well,
00:12:40.750 --> 00:12:42.890
I'll tell you what the
experiment was in a minute.
00:12:42.890 --> 00:12:44.225
OK, yeah?
00:12:44.225 --> 00:12:48.960
AUDIENCE: [INAUDIBLE] Z has
to be plus or minus 1/2.
00:12:48.960 --> 00:12:51.390
What fixes the direction
in the Z direction?
00:12:51.390 --> 00:12:52.382
PROFESSOR: Excellent.
00:12:52.382 --> 00:12:54.030
In this experiment,
the thing that
00:12:54.030 --> 00:12:55.787
fixed the fact that
I was probing Lz
00:12:55.787 --> 00:12:57.370
is that I made the
magnetic field have
00:12:57.370 --> 00:12:59.190
a gradient in the Z direction.
00:12:59.190 --> 00:13:01.890
So, what I was sensitive to,
since the force is actually
00:13:01.890 --> 00:13:05.750
proportionally to mu dot
B-- or, really, mu dot
00:13:05.750 --> 00:13:10.510
the gradient of B,
so, we'll do this
00:13:10.510 --> 00:13:13.545
in more detail later-- the
direction of the gradient
00:13:13.545 --> 00:13:15.670
selects out which component
of the angular momentum
00:13:15.670 --> 00:13:16.190
we're looking at.
00:13:16.190 --> 00:13:18.000
So, in this experiment, I'm
measuring the angular momentum
00:13:18.000 --> 00:13:19.810
along this axis-- which
for fun, I'll call Z,
00:13:19.810 --> 00:13:21.476
I could've called it
X-- what I discover
00:13:21.476 --> 00:13:23.140
is the angular momentum
along this axis
00:13:23.140 --> 00:13:24.560
must take one of two values.
00:13:24.560 --> 00:13:27.640
But, the universe is
rotationally invariant.
00:13:27.640 --> 00:13:29.300
So, it can't possibly
matter whether I
00:13:29.300 --> 00:13:31.240
had done the experiment
in this direction,
00:13:31.240 --> 00:13:33.614
or done the experiment in this
direction, what that tells
00:13:33.614 --> 00:13:35.710
you is, in any direction
if I measure the angular
00:13:35.710 --> 00:13:38.132
momentum of the electron
along that direction,
00:13:38.132 --> 00:13:40.131
I will discover that it
takes one of two values.
00:13:43.290 --> 00:13:46.800
This is also true of
the L equals 1 states.
00:13:46.800 --> 00:13:48.510
Lz takes one of three values.
00:13:48.510 --> 00:13:49.360
What about Lx?
00:13:49.360 --> 00:13:54.040
Lx also takes one of three
values, those three values.
00:13:54.040 --> 00:13:56.936
Is is every system in a
state corresponding to one
00:13:56.936 --> 00:13:58.060
of those particular values?
00:13:58.060 --> 00:14:00.000
No, it could be in
a superposition.
00:14:00.000 --> 00:14:03.540
But the eigenvalues, are
these three eigenvalues,
00:14:03.540 --> 00:14:07.460
regardless of whether
it's Lx, or Ly, or Lz.
00:14:07.460 --> 00:14:09.111
OK, it's a good thing
to meditate upon.
00:14:12.097 --> 00:14:12.680
Anything else?
00:14:12.680 --> 00:14:13.180
One more.
00:14:13.180 --> 00:14:15.061
Yeah?
00:14:15.061 --> 00:14:19.840
AUDIENCE: [INAUDIBLE] the
last problem [INAUDIBLE].
00:14:19.840 --> 00:14:20.740
PROFESSOR: Indeed.
00:14:20.740 --> 00:14:21.090
Indeed.
00:14:21.090 --> 00:14:21.590
OK.
00:14:21.590 --> 00:14:24.775
Since some people haven't taken
the-- there will be a conflict
00:14:24.775 --> 00:14:30.108
exam later today, so I'm not
going to discuss the exam yet.
00:14:30.108 --> 00:14:34.440
But, very good observation,
and not an accident.
00:14:34.440 --> 00:14:37.590
OK, so, today we launch into 3D.
00:14:37.590 --> 00:14:41.260
We ditch our
tricked-out tricycle,
00:14:41.260 --> 00:14:43.740
and we're gonna talk about
real, physical systems in three
00:14:43.740 --> 00:14:44.470
dimensions.
00:14:44.470 --> 00:14:47.420
And as we'll discover, it's
basically the same as in one
00:14:47.420 --> 00:14:50.120
dimension, we just have to
write down more symbols.
00:14:50.120 --> 00:14:52.070
But the content is all the same.
00:14:52.070 --> 00:14:53.790
So, this will make
obvious the reason
00:14:53.790 --> 00:14:56.430
we worked with 1D
up until now, which
00:14:56.430 --> 00:14:57.990
is that there's
not a heck of a lot
00:14:57.990 --> 00:15:01.230
more to be gained for
the basic principles,
00:15:01.230 --> 00:15:03.650
but it's a lot more knowing
to write down the expressions.
00:15:03.650 --> 00:15:04.570
So, the first thing
I wanted to do
00:15:04.570 --> 00:15:06.400
is write down the
Laplacian in three
00:15:06.400 --> 00:15:12.310
dimensions in
spherical coordinates--
00:15:12.310 --> 00:15:15.930
And that is a beautiful abuse
of notation-- in spherical
00:15:15.930 --> 00:15:17.890
coordinates.
00:15:17.890 --> 00:15:19.590
And I want to note
a couple of things.
00:15:19.590 --> 00:15:21.420
So, first off, this
Laplacian, this
00:15:21.420 --> 00:15:27.650
can be written in the following
form, 1 over r dr r quantity
00:15:27.650 --> 00:15:30.164
squared.
00:15:30.164 --> 00:15:32.330
OK, that's going to be very
useful for us-- trust me
00:15:32.330 --> 00:15:39.710
on this one-- this is also
known as 1 over r dr squared r.
00:15:39.710 --> 00:15:43.090
And this, if you look
back at your notes,
00:15:43.090 --> 00:15:46.280
this is nothing other
than L squared--
00:15:46.280 --> 00:15:48.920
except for the factor of h bar
upon i-- but if it's squared,
00:15:48.920 --> 00:15:51.210
it's minus 1 upon h bar squared.
00:15:54.500 --> 00:15:57.480
OK, so this horrible
angular derivative,
00:15:57.480 --> 00:15:59.030
is nothing but L squared.
00:16:02.606 --> 00:16:05.340
OK, and you should remember
the [? dd ?] thetas,
00:16:05.340 --> 00:16:07.474
and there are these
funny sines and cosines.
00:16:07.474 --> 00:16:09.140
But just go back and
compare your notes.
00:16:09.140 --> 00:16:13.220
So, this is an observation
that the Laplacian
00:16:13.220 --> 00:16:15.200
in three dimensions and
spherical coordinates
00:16:15.200 --> 00:16:16.930
takes this simple form.
00:16:16.930 --> 00:16:18.780
A simple radial
derivative, which
00:16:18.780 --> 00:16:22.050
is two terms if you write it
out linearly in this fashion,
00:16:22.050 --> 00:16:24.560
and one term if you
write it this way,
00:16:24.560 --> 00:16:26.550
which is going to turn
out to be useful for us.
00:16:26.550 --> 00:16:29.180
And the angular part
can be written as 1
00:16:29.180 --> 00:16:34.190
over r squared, times the
angular momentum squared
00:16:34.190 --> 00:16:37.666
with a minus 1
over h bar squared.
00:16:37.666 --> 00:16:38.165
OK?
00:16:41.290 --> 00:16:43.830
So, in just to check,
remember that Lz
00:16:43.830 --> 00:16:49.330
is equal to h bar upon i d phi.
00:16:49.330 --> 00:16:53.610
So, Lz squared is going to be
equal to minus h bar squared
00:16:53.610 --> 00:16:54.622
d phi squared.
00:16:54.622 --> 00:16:57.080
And you can see that that's
one contribution to this beast.
00:17:01.570 --> 00:17:04.177
But, actually, let me-- I'm
gonna commit a capital sin
00:17:04.177 --> 00:17:06.010
and erase what I just
wrote, because I don't
00:17:06.010 --> 00:17:10.280
want it to distract you-- OK.
00:17:10.280 --> 00:17:14.040
So, with that
useful observation,
00:17:14.040 --> 00:17:16.450
I want to think about
central potentials.
00:17:16.450 --> 00:17:19.210
I want to think about systems
in 3D, which are spherically
00:17:19.210 --> 00:17:21.750
symmetric, because this is
going to be a particularly
00:17:21.750 --> 00:17:23.540
simple class of
systems, and it's also
00:17:23.540 --> 00:17:24.804
particularly physical.
00:17:24.804 --> 00:17:26.470
Simple things like a
harmonic oscillator
00:17:26.470 --> 00:17:29.011
In three dimensions, which we
solved in Cartesian coordinates
00:17:29.011 --> 00:17:30.480
earlier, we're
gonna solve later,
00:17:30.480 --> 00:17:31.790
in spherical coordinates.
00:17:31.790 --> 00:17:35.240
Things like the isotropic
harmonic oscillator, things
00:17:35.240 --> 00:17:39.840
like hydrogen, where the system
is rotationally independent,
00:17:39.840 --> 00:17:44.100
the force of the potential only
depends on the radial distance,
00:17:44.100 --> 00:17:45.770
all share a bunch of
common properties,
00:17:45.770 --> 00:17:46.950
and I want to explore those.
00:17:46.950 --> 00:17:49.960
And along the way, we'll solve
a toy model for hydrogen.
00:17:49.960 --> 00:17:54.400
So, the energy for this
is p squared upon 2m,
00:17:54.400 --> 00:17:56.400
plus a potential, which
is a function only
00:17:56.400 --> 00:17:59.500
of the radial distance.
00:17:59.500 --> 00:18:08.426
But now, p squared is equal
to minus h bar squared
00:18:08.426 --> 00:18:09.550
times the gradient squared.
00:18:12.330 --> 00:18:18.120
But this is gonna be equal to,
from the first term, minus h
00:18:18.120 --> 00:18:29.860
bar squared-- let me just
write this out-- times
00:18:29.860 --> 00:18:33.550
r dr squared r.
00:18:33.550 --> 00:18:36.589
And then from this term, plus
minus h bar squared times minus
00:18:36.589 --> 00:18:38.755
1 over h bar squared
[? to L squared ?] [? over ?] r
00:18:38.755 --> 00:18:42.595
squared, plus L
squared over r squared.
00:18:48.380 --> 00:18:51.080
So, the energy can be
written in a nice form.
00:18:51.080 --> 00:18:55.110
This is minus h bar
squared, 1 upon r dr
00:18:55.110 --> 00:18:59.435
squared r-- whoops,
sorry-- upon 2m,
00:18:59.435 --> 00:19:01.050
because it's p squared upon 2m.
00:19:01.050 --> 00:19:04.480
And from the second term, L
squared over r squared upon 2m
00:19:04.480 --> 00:19:13.935
plus 1 over 2mr squared
L squared plus u of r.
00:19:13.935 --> 00:19:18.400
OK, and this is the energy
operator when the system is
00:19:18.400 --> 00:19:21.456
rotational invariant in
spherical coordinates.
00:19:21.456 --> 00:19:21.955
Questions?
00:19:26.254 --> 00:19:26.754
Yeah?
00:19:26.754 --> 00:19:29.994
AUDIENCE: [INAUDIBLE] is
that an equals sign or minus?
00:19:29.994 --> 00:19:30.660
PROFESSOR: This?
00:19:30.660 --> 00:19:31.350
AUDIENCE: Yeah.
00:19:31.350 --> 00:19:32.350
PROFESSOR: Oh, that's
an equals sign.
00:19:32.350 --> 00:19:32.880
So, sorry.
00:19:32.880 --> 00:19:34.010
This is just quick algebra.
00:19:34.010 --> 00:19:35.440
So, it's useful to know it.
00:19:35.440 --> 00:19:39.294
So, consider the following
thing, 1 over r dr r.
00:19:39.294 --> 00:19:41.085
Why would you ever care
about such a thing?
00:19:41.085 --> 00:19:43.260
Well, let's square it.
00:19:43.260 --> 00:19:44.557
OK, because I did there.
00:19:44.557 --> 00:19:45.640
So, what is this equal to?
00:19:45.640 --> 00:19:49.320
Well, this is 1 over r dr r.
00:19:49.320 --> 00:19:52.180
1 over r dr r.
00:19:52.180 --> 00:19:53.580
These guys cancel, right?
00:19:53.580 --> 00:19:59.560
1 over r times dr. So, this is
equal to 1 over r dr squared r.
00:19:59.560 --> 00:20:04.459
But, why is this equal to dr
squared plus 2 over r times dr?
00:20:04.459 --> 00:20:06.000
And the answer is,
they're operators.
00:20:06.000 --> 00:20:08.470
And so, you should ask
how they act on functions.
00:20:08.470 --> 00:20:10.220
So, let's ask how
they act on function.
00:20:10.220 --> 00:20:15.510
So, dr squared plus 2 over
r dr times a function--
00:20:15.510 --> 00:20:18.540
acting as a function-- is
equal to f prime prime--
00:20:18.540 --> 00:20:24.400
if this is a function of
r-- plus 2 over r f prime.
00:20:24.400 --> 00:20:31.960
On the other hand, 1 over r dr
squared r, acting on f of r,
00:20:31.960 --> 00:20:36.096
well, these derivatives can
hit either the r of the f.
00:20:36.096 --> 00:20:38.990
So, there's going to be a term
where both derivatives hit
00:20:38.990 --> 00:20:42.270
f, in which case the rs cancel,
and I get f prime prime.
00:20:42.270 --> 00:20:44.550
There's gonna be two terms
where one of the d's hits
00:20:44.550 --> 00:20:47.210
this, one of the d's hits this,
then there's the other term.
00:20:47.210 --> 00:20:48.770
So, there're two
terms of that form.
00:20:48.770 --> 00:20:51.530
On d hits the r and gives
me one, one d hits the f
00:20:51.530 --> 00:20:52.640
and gives me f prime.
00:20:52.640 --> 00:20:58.150
And then there's an overall 1
over r plus 2 over r f prime.
00:20:58.150 --> 00:21:00.520
And then there's a term
were two d's hit the r,
00:21:00.520 --> 00:21:03.720
but if two d's hit
the r, that's 0.
00:21:03.720 --> 00:21:04.430
So, that's it.
00:21:04.430 --> 00:21:06.650
So, these guys are
equal to each other.
00:21:06.650 --> 00:21:09.230
So, why is this a
particularly useful form?
00:21:09.230 --> 00:21:10.690
We'll see that in just a minute.
00:21:10.690 --> 00:21:13.090
So, I'm cheating a little
bit by just writing this
00:21:13.090 --> 00:21:14.680
out and saying, this is
going to be a useful form.
00:21:14.680 --> 00:21:16.060
But trust me, it's going
to be a useful form.
00:21:16.060 --> 00:21:16.560
Yeah?
00:21:16.560 --> 00:21:21.272
AUDIENCE: Do we need to find d
squared [INAUDIBLE] dr squared
00:21:21.272 --> 00:21:21.772
r.
00:21:21.772 --> 00:21:23.210
Isn't that supposed
to be 1 over r?
00:21:23.210 --> 00:21:24.043
PROFESSOR: Oh shoot!
00:21:24.043 --> 00:21:26.291
Yes, that's supposed to
be one of our-- Thank you.
00:21:26.291 --> 00:21:26.790
Thank you!
00:21:26.790 --> 00:21:28.490
Yes, over r.
00:21:28.490 --> 00:21:29.340
Thank you.
00:21:29.340 --> 00:21:31.300
Yes, thank you for
that typo correction.
00:21:31.300 --> 00:21:31.800
Excellent.
00:21:34.970 --> 00:21:36.870
Thanks OK.
00:21:45.204 --> 00:21:46.620
So, anytime we
have a system which
00:21:46.620 --> 00:21:48.780
is rotationally invariant--
whose potential is rotationally
00:21:48.780 --> 00:21:50.238
invariant-- we can
write the energy
00:21:50.238 --> 00:21:52.290
operator in this fashion.
00:21:52.290 --> 00:21:54.160
And now, you see
something really lovely,
00:21:54.160 --> 00:21:56.640
which is that this
only depends on r,
00:21:56.640 --> 00:21:58.400
this only depends
on r, this depends
00:21:58.400 --> 00:21:59.900
on the angular
coordinates, but only
00:21:59.900 --> 00:22:03.610
insofar as it
depends on L squared.
00:22:03.610 --> 00:22:07.190
So, if we want to find
the eigenfunctions of E,
00:22:07.190 --> 00:22:09.000
our life is going to
be a lot easier if we
00:22:09.000 --> 00:22:14.611
work in eigenfunctions
of L. Because that's
00:22:14.611 --> 00:22:16.860
gonna make this one [? Ex ?]
on an eigenfunction of L,
00:22:16.860 --> 00:22:18.700
this is just going
to become a constant.
00:22:18.700 --> 00:22:20.949
So, now you have to answer
the question, well, can we?
00:22:20.949 --> 00:22:24.450
Can we find functions which are
eigenfunctions of E and of L,
00:22:24.450 --> 00:22:25.560
simultaneously?
00:22:25.560 --> 00:22:27.040
And so, the answer
to that question
00:22:27.040 --> 00:22:28.940
is, well, compute
the commutator.
00:22:28.940 --> 00:22:30.550
So, do these guys commute?
00:22:30.550 --> 00:22:33.305
In particular, of L squared.
00:22:33.305 --> 00:22:35.180
And, well, does L commute
with the derivative
00:22:35.180 --> 00:22:36.430
with respect to r, L squared?
00:22:41.330 --> 00:22:43.770
Yeah, because L only depends
on angular derivatives.
00:22:43.770 --> 00:22:45.440
It doesn't have any rs in it.
00:22:45.440 --> 00:22:49.860
And the rs don't care about
the angular variables,
00:22:49.860 --> 00:22:51.240
so they commute.
00:22:51.240 --> 00:22:52.704
What about with this term?
00:22:52.704 --> 00:22:54.620
Well, L squared trivially
commutes with itself
00:22:54.620 --> 00:22:56.070
and, again, r doesn't matter.
00:22:56.070 --> 00:22:58.490
And ditto, r and
L squared commute.
00:22:58.490 --> 00:22:59.297
So, this is 0.
00:22:59.297 --> 00:22:59.880
These commute.
00:22:59.880 --> 00:23:04.430
So, we can find
common eigenbasis.
00:23:04.430 --> 00:23:11.894
We can find a basis
of functions which
00:23:11.894 --> 00:23:13.810
are eigenfunctions both
of E and of L squared.
00:23:17.117 --> 00:23:18.200
So, now we use separation.
00:23:20.960 --> 00:23:23.220
In particular, if we
want to find a function--
00:23:23.220 --> 00:23:24.970
an eigenfunction-- of
the energy operator,
00:23:24.970 --> 00:23:33.040
E phi E is equal
to E phi E, it's
00:23:33.040 --> 00:23:36.280
going to simplify our
lives if we also let phi
00:23:36.280 --> 00:23:39.191
be an eigenfunction
of the L squared.
00:23:39.191 --> 00:23:41.190
But we know what the
eigenfunctions of L squared
00:23:41.190 --> 00:23:41.690
are.
00:23:41.690 --> 00:23:45.060
E phi E is equal to--
let me write this--
00:23:45.060 --> 00:23:54.440
of r will then be equal
to little phi of r
00:23:54.440 --> 00:23:57.890
times yLm of theta and phi.
00:24:02.030 --> 00:24:06.870
Now, quickly, because these
are the eigenfunctions of the L
00:24:06.870 --> 00:24:08.150
squared operator.
00:24:08.150 --> 00:24:11.812
Quick, is little l an
integer or a half integer?
00:24:11.812 --> 00:24:13.050
AUDIENCE: [MURMURS] Integer.
00:24:13.050 --> 00:24:13.940
PROFESSOR: Why?
00:24:13.940 --> 00:24:14.732
AUDIENCE: [MURMURS]
00:24:14.732 --> 00:24:17.315
PROFESSOR: Yeah, because we're
working with rotational angular
00:24:17.315 --> 00:24:18.380
momentum, right?
00:24:18.380 --> 00:24:21.180
And it only makes sense to talk
about integer values of little
00:24:21.180 --> 00:24:24.170
l when we have gradients on
a sphere-- when we're talking
00:24:24.170 --> 00:24:27.044
about rotations-- on a
spherical coordinates, OK?
00:24:27.044 --> 00:24:28.460
So, little l has
to be an integer.
00:24:31.980 --> 00:24:35.960
And from this point forward in
the class, any time I write l,
00:24:35.960 --> 00:24:39.960
I'll be talking about the
rotational angular momentum
00:24:39.960 --> 00:24:42.100
corresponding to integer values.
00:24:42.100 --> 00:24:44.330
And when I'm talking about
the half integer values,
00:24:44.330 --> 00:24:49.210
I'll write down s, OK?
00:24:49.210 --> 00:24:53.430
So, let's use this
separation of variables.
00:24:53.430 --> 00:24:55.070
And what does that give us?
00:24:55.070 --> 00:24:58.650
Well, l squared
acting on yLm gives us
00:24:58.650 --> 00:25:01.390
h bar squared lL plus 1.
00:25:01.390 --> 00:25:05.606
So, this tells us that
E, acting on phi E,
00:25:05.606 --> 00:25:06.980
takes a particularly
simple form.
00:25:06.980 --> 00:25:12.920
If phi E is proportional
to a spherical harmonic,
00:25:12.920 --> 00:25:19.860
then this is gonna take the form
minus h bar squared upon 2m 1
00:25:19.860 --> 00:25:28.190
over r dr squared r plus
1 over 2mr squared l
00:25:28.190 --> 00:25:30.550
squared-- but l squared
acting on the yLm
00:25:30.550 --> 00:25:35.840
gives us-- h bar
squared lL plus 1, which
00:25:35.840 --> 00:25:53.620
is just a constant over r
squared plus u of r phi E.
00:25:53.620 --> 00:25:54.480
Question?
00:25:54.480 --> 00:26:00.724
AUDIENCE: Yeah.
[INAUDIBLE] yLm1 and yLm2?
00:26:00.724 --> 00:26:01.640
PROFESSOR: Absolutely.
00:26:01.640 --> 00:26:03.850
So, can we consider
superpositions of these guys?
00:26:03.850 --> 00:26:05.090
Absolutely, we can.
00:26:05.090 --> 00:26:07.660
However, we're using separation.
00:26:07.660 --> 00:26:09.320
So, we're gonna look
at a single term,
00:26:09.320 --> 00:26:12.270
and then after
constructing solutions
00:26:12.270 --> 00:26:16.020
with a single
eigenfunction of L squared,
00:26:16.020 --> 00:26:19.340
we can then write down
arbitrary superposition of them,
00:26:19.340 --> 00:26:21.730
and generate a complete
basis of states.
00:26:21.730 --> 00:26:25.600
General statement about
separation of variables.
00:26:25.600 --> 00:26:27.850
Other questions?
00:26:27.850 --> 00:26:29.160
OK.
00:26:29.160 --> 00:26:34.810
So, here's the resulting
energy eigenvalue equation.
00:26:34.810 --> 00:26:36.419
But notice that it's
now, really nice.
00:26:36.419 --> 00:26:37.710
This is purely a function of r.
00:26:37.710 --> 00:26:40.560
We've removed all of
the angular dependence
00:26:40.560 --> 00:26:42.390
by making this
proportional to yLm.
00:26:42.390 --> 00:26:46.127
So, this has a little phi yLm,
and this has a little phi yLm,
00:26:46.127 --> 00:26:47.710
and nothing depends
on the little phi.
00:26:50.302 --> 00:26:52.593
Nothing depends on the yLm--
on the angular variables--
00:26:52.593 --> 00:26:53.702
I can make this phi of r.
00:27:00.540 --> 00:27:03.150
And if I want to make this the
energy eigenvalue equation,
00:27:03.150 --> 00:27:05.760
instead of just the action
of the energy operator,
00:27:05.760 --> 00:27:08.440
that is now my energy
eigenvalue equation.
00:27:08.440 --> 00:27:13.667
This is the result of acting on
phi with the energy operator,
00:27:13.667 --> 00:27:15.083
and this is the
energy eigenvalue.
00:27:19.840 --> 00:27:21.460
Cool?
00:27:21.460 --> 00:27:25.070
So, the upside here is that when
we have a central potential,
00:27:25.070 --> 00:27:27.320
when the system is
rotationally invariant,
00:27:27.320 --> 00:27:30.840
the potential energy is
invariant under rotations,
00:27:30.840 --> 00:27:33.880
then the energy commutes with
the angular momentum squared.
00:27:33.880 --> 00:27:36.440
And so, we can find
common eigenfunctions.
00:27:36.440 --> 00:27:39.694
When we use separation
of variable,
00:27:39.694 --> 00:27:41.360
the resulting energy
eigenvalue equation
00:27:41.360 --> 00:27:47.115
becomes nothing but a 1D energy
eigenvalue equation, right?
00:27:47.115 --> 00:27:48.240
This is just a 1D equation.
00:27:48.240 --> 00:27:49.750
Now, you might look at this
and say, well, it's not quite
00:27:49.750 --> 00:27:51.960
a 1D equation, because if
this were a 1D equation,
00:27:51.960 --> 00:27:54.620
we wouldn't have this funny
1 over r, and this funny r,
00:27:54.620 --> 00:27:55.120
right?
00:27:55.120 --> 00:27:57.650
It's not exactly what
we would have got.
00:27:57.650 --> 00:28:01.310
It's got the minus h bar
squareds upon 2m-- whoops,
00:28:01.310 --> 00:28:05.820
and there's, yeah, OK-- it's got
this funny h bar squareds upon
00:28:05.820 --> 00:28:08.010
2m, and it's got these
1 over-- or sorry,--
00:28:08.010 --> 00:28:09.900
it's got the correct h
bar squareds upon 2m,
00:28:09.900 --> 00:28:11.390
but it's got this
funny r and 1 over r.
00:28:11.390 --> 00:28:12.473
So, let's get rid of that.
00:28:12.473 --> 00:28:15.330
Let's just quickly dispense
with that funny set of r.
00:28:15.330 --> 00:28:17.490
And this comes back
to the sneaky trick
00:28:17.490 --> 00:28:21.930
I was referring to earlier,
of writing this expression.
00:28:21.930 --> 00:28:23.340
So, rather than
writing this out,
00:28:23.340 --> 00:28:25.048
it's convenient to
write it in this form.
00:28:25.048 --> 00:28:25.700
Let's see why.
00:28:28.290 --> 00:28:35.180
So, if we have the E phi of r
is equal to minus h bar squared
00:28:35.180 --> 00:28:43.580
upon 2m, 1 over r d squared
r r, plus-- and now,
00:28:43.580 --> 00:28:47.210
what I'm gonna write is-- look,
this is our potential, u of r.
00:28:47.210 --> 00:28:50.720
This is some silly,
radial-dependent thing.
00:28:50.720 --> 00:28:52.730
I'm gonna write these
two terms together,
00:28:52.730 --> 00:28:54.360
rather than writing them over,
and over, and over again, I'm
00:28:54.360 --> 00:28:56.830
going to write them together,
and call them V effective.
00:28:56.830 --> 00:29:08.040
Plus V effective of r, where V
effective is just these guys,
00:29:08.040 --> 00:29:08.540
V effective.
00:29:11.260 --> 00:29:13.500
Which has a contribution
from the original potential,
00:29:13.500 --> 00:29:15.560
and from the angular
momentum, which,
00:29:15.560 --> 00:29:17.950
notice the sign is
plus 1 over r squared.
00:29:17.950 --> 00:29:20.820
So, the potential gets really
large as you get to the origin.
00:29:24.480 --> 00:29:25.100
Phi of r.
00:29:28.200 --> 00:29:30.450
So, this r is annoying, and
this 1 over r is annoying,
00:29:30.450 --> 00:29:32.440
but there's a nice
way to get rid of it.
00:29:32.440 --> 00:29:35.810
Let phi of r-- well, this r,
we want to get rid of-- so,
00:29:35.810 --> 00:29:42.190
let phi of r equals
1 over r u of r.
00:29:42.190 --> 00:29:46.600
OK, then 1 over r squared--
or sorry, 1 over r-- dr
00:29:46.600 --> 00:29:53.020
squared r phi is equal to 1
over r dr squared r times 1
00:29:53.020 --> 00:29:56.030
over r times u, which is just u.
00:29:56.030 --> 00:29:59.636
But meanwhile, V on
phi is equal to-- well,
00:29:59.636 --> 00:30:01.010
V doesn't have
any r derivatives,
00:30:01.010 --> 00:30:06.030
it's just a function-- so, V
of phi is just 1 over r V on u.
00:30:06.030 --> 00:30:10.000
So, this equation
becomes E on u,
00:30:10.000 --> 00:30:11.920
because this also
picks up a 1 over r,
00:30:11.920 --> 00:30:17.950
is equal to minus h bar
squared upon 2m dr squared
00:30:17.950 --> 00:30:26.440
plus V effective of r u of r.
00:30:26.440 --> 00:30:30.890
And this is exactly the
energy eigenvalue equation
00:30:30.890 --> 00:30:34.750
for a 1D problem with
the following potential.
00:30:34.750 --> 00:30:41.380
The potential, V effective of r,
does the following two things--
00:30:41.380 --> 00:30:48.650
whoops, don't want to
draw it that way-- suppose
00:30:48.650 --> 00:30:52.200
we have a potential which
is the Coulomb potential.
00:30:52.200 --> 00:30:56.290
So, let's say, u is equal
to minus E squared upon r.
00:30:58.899 --> 00:30:59.690
Just as an example.
00:31:04.440 --> 00:31:06.850
So, here's r, here
is V effective.
00:31:06.850 --> 00:31:13.590
So, u first, so
there's u-- u of r,
00:31:13.590 --> 00:31:19.440
so let me draw this-- V
has another term, which
00:31:19.440 --> 00:31:25.990
is h bar squared lL
plus 1 over 2mr squared.
00:31:25.990 --> 00:31:27.470
This is for any
given value of l.
00:31:27.470 --> 00:31:30.700
This is a constant over r
squared, with a plus sign.
00:31:30.700 --> 00:31:32.724
So, that's something
that looks like this.
00:31:32.724 --> 00:31:34.140
This is falling
off like 1 over r,
00:31:34.140 --> 00:31:36.560
this is falling off
like 1 over r squared.
00:31:36.560 --> 00:31:38.950
So, it falls off more rapidly.
00:31:38.950 --> 00:31:41.045
And finally, can r be negative?
00:31:44.130 --> 00:31:44.630
No.
00:31:44.630 --> 00:31:47.070
It's defined from 0 to infinity.
00:31:47.070 --> 00:31:50.690
So, that's like having
an infinite potential
00:31:50.690 --> 00:31:52.530
for negative r.
00:31:52.530 --> 00:31:54.130
So, our effective
potential is the sum
00:31:54.130 --> 00:31:56.092
of these contributions--
wish I had
00:31:56.092 --> 00:32:00.996
colored chalk-- the sum
of these contributions
00:32:00.996 --> 00:32:02.120
is going to look like this.
00:32:02.120 --> 00:32:05.590
So, that's my V effective.
00:32:05.590 --> 00:32:13.360
This is my Ll plus 1 [INAUDIBLE]
squared over 2mr squared.
00:32:13.360 --> 00:32:16.920
And this is my u of r.
00:32:16.920 --> 00:32:17.910
Question?
00:32:17.910 --> 00:32:19.304
AUDIENCE: [INAUDIBLE].
00:32:19.304 --> 00:32:19.970
PROFESSOR: Good.
00:32:19.970 --> 00:32:23.713
OK, so this is u of r,
the original potential.
00:32:23.713 --> 00:32:24.254
AUDIENCE: OK.
00:32:24.254 --> 00:32:25.110
PROFESSOR: OK?
00:32:25.110 --> 00:32:30.030
This is 1 over L squared-- or
sorry-- lL 1 over 2mr squared.
00:32:33.250 --> 00:32:34.166
AUDIENCE: [INAUDIBLE].
00:32:36.857 --> 00:32:37.690
PROFESSOR: Oh shoot!
00:32:37.690 --> 00:32:38.090
Oh, I'm sorry!
00:32:38.090 --> 00:32:38.890
I'm terribly sorry!
00:32:38.890 --> 00:32:40.490
I've abused the
notation terribly.
00:32:40.490 --> 00:32:41.280
Let's-- Oh!
00:32:41.280 --> 00:32:44.320
This is-- Crap!
00:32:44.320 --> 00:32:46.930
Sorry.
00:32:46.930 --> 00:32:48.180
This is standard notation.
00:32:48.180 --> 00:32:51.340
And in text, when I
write this by hand,
00:32:51.340 --> 00:32:55.170
the potential is a big
U, and the wave function
00:32:55.170 --> 00:32:55.865
is a little u.
00:32:55.865 --> 00:32:57.455
So, let this be a little u.
00:32:57.455 --> 00:32:59.880
OK, this is my little
u and so, now I'm
00:32:59.880 --> 00:33:01.630
gonna have to-- oh
jeez, this is horrible,
00:33:01.630 --> 00:33:04.940
sorry-- this is the
potential, capital U
00:33:04.940 --> 00:33:07.110
with a bar underneath it.
00:33:07.110 --> 00:33:08.990
OK, seriously, so
there's capital U
00:33:08.990 --> 00:33:10.780
with a bar underneath it.
00:33:10.780 --> 00:33:13.230
And here's V, which is
gonna make my life easier,
00:33:13.230 --> 00:33:16.239
and this is the capital U
with the bar underneath it.
00:33:16.239 --> 00:33:17.780
Capital U with the
bar underneath it.
00:33:17.780 --> 00:33:19.450
Oh, I'm really sorry,
I did not realize
00:33:19.450 --> 00:33:21.600
how confusing that would be.
00:33:21.600 --> 00:33:23.960
OK, is everyone happy with that?
00:33:23.960 --> 00:33:24.791
Yeah?
00:33:24.791 --> 00:33:26.835
AUDIENCE: [INAUDIBLE].
00:33:26.835 --> 00:33:27.710
PROFESSOR: Which one?
00:33:27.710 --> 00:33:28.584
AUDIENCE: Middle.
00:33:28.584 --> 00:33:29.460
Middle.
00:33:29.460 --> 00:33:30.241
PROFESSOR: Middle.
00:33:30.241 --> 00:33:31.665
AUDIENCE: Up, up.
00:33:31.665 --> 00:33:32.165
Right there!
00:33:32.165 --> 00:33:32.665
Up!
00:33:32.665 --> 00:33:33.610
There.
00:33:33.610 --> 00:33:34.420
PROFESSOR: Where?
00:33:34.420 --> 00:33:37.288
AUDIENCE: To the right.
[CHATTER] Near the eraser mark.
00:33:37.288 --> 00:33:39.200
[LAUGHTER]
00:33:39.200 --> 00:33:41.560
PROFESSOR: So, these
are the wave function.
00:33:41.560 --> 00:33:42.840
AUDIENCE: I know.
00:33:42.840 --> 00:33:44.670
PROFESSOR: That's
the wave function.
00:33:44.670 --> 00:33:48.519
That is V.
00:33:48.519 --> 00:33:51.417
AUDIENCE: [CHATTER]
00:33:51.417 --> 00:33:54.464
PROFESSOR: Wait, if I
erased, how can I correct it?
00:33:54.464 --> 00:34:04.605
AUDIENCE: [CHATTER] There!
00:34:04.605 --> 00:34:06.730
PROFESSOR: Excellent, so
the thing that isn't here,
00:34:06.730 --> 00:34:08.410
would have a bar under it.
00:34:08.410 --> 00:34:10.151
Oh, oh, oh, oh, sorry!
00:34:10.151 --> 00:34:13.040
Ah!
00:34:13.040 --> 00:34:17.020
You wouldn't think
it would be so hard.
00:34:17.020 --> 00:34:17.560
OK, good.
00:34:17.560 --> 00:34:19.684
And this is not [? related ?]
to the wave function.
00:34:19.684 --> 00:34:20.719
OK, god, oh!
00:34:20.719 --> 00:34:23.260
That's horrible!
00:34:23.260 --> 00:34:27.090
Sorry guys, that
notation is not obvious.
00:34:27.090 --> 00:34:29.449
My apologies.
00:34:29.449 --> 00:34:31.580
Oh, there's a better
way to do this.
00:34:31.580 --> 00:34:33.800
OK, here's the better
way to do this.
00:34:33.800 --> 00:34:35.729
Instead of calling the
potential-- I'm sorry,
00:34:35.729 --> 00:34:37.770
your notes are getting
destroyed now-- so instead
00:34:37.770 --> 00:34:41.934
of calling potential capital U,
let's just call this V. Yeah.
00:34:41.934 --> 00:34:43.429
AUDIENCE: [LAUGHTER] No!
00:34:43.429 --> 00:34:45.520
PROFESSOR: And then
we have V effective.
00:34:45.520 --> 00:34:46.020
No, no.
00:34:46.020 --> 00:34:46.320
This is good.
00:34:46.320 --> 00:34:46.650
This is good.
00:34:46.650 --> 00:34:47.858
We can be careful about this.
00:34:47.858 --> 00:34:49.630
So, this is V. This
is V effective,
00:34:49.630 --> 00:34:52.270
which has V plus the
angular momentum term.
00:34:52.270 --> 00:34:54.159
Oh, good Lord!
00:34:54.159 --> 00:34:55.280
This is V effective.
00:34:55.280 --> 00:35:00.096
This is V. V phi [INAUDIBLE]
U. Good, this is V.
00:35:00.096 --> 00:35:02.634
AUDIENCE: [INAUDIBLE]
There's no U--
00:35:02.634 --> 00:35:04.050
PROFESSOR: There's
no U underline,
00:35:04.050 --> 00:35:09.310
it's now just V, V effective.
00:35:09.310 --> 00:35:10.220
Oh!
00:35:10.220 --> 00:35:11.670
Good Lord!
00:35:11.670 --> 00:35:13.160
OK, wow!
00:35:13.160 --> 00:35:14.747
That was an
unnecessary confusion.
00:35:14.747 --> 00:35:15.580
AUDIENCE: Top right.
00:35:15.580 --> 00:35:16.835
PROFESSOR: Top right.
00:35:16.835 --> 00:35:19.617
AUDIENCE: There is no bar.
00:35:19.617 --> 00:35:20.450
[? PROFESSOR: Mu. ?]
00:35:23.042 --> 00:35:24.910
AUDIENCE: Is that
V or V effective?
00:35:24.910 --> 00:35:26.846
PROFESSOR: That's V.
Although, it would've
00:35:26.846 --> 00:35:28.220
been just as true
as V effective.
00:35:28.220 --> 00:35:29.400
So, we can write V effective.
00:35:29.400 --> 00:35:30.191
It's true for both.
00:35:33.206 --> 00:35:34.980
Because it's just
a function of r.
00:35:34.980 --> 00:35:36.380
Oh, for the love of God!
00:35:36.380 --> 00:35:38.090
OK.
00:35:38.090 --> 00:35:40.440
Let's check our sanity,
and walk through the logic.
00:35:40.440 --> 00:35:43.480
So, the logic here is, we
have some potential, it's
00:35:43.480 --> 00:35:45.405
a function only of r, yeah?
00:35:45.405 --> 00:35:47.780
As a consequence, since it
doesn't care about the angles,
00:35:47.780 --> 00:35:50.113
we can write things in terms
of the spherical harmonics,
00:35:50.113 --> 00:35:52.037
we can do separation
of variables.
00:35:52.037 --> 00:35:53.620
Here's the energy
eigenvalue equation.
00:35:53.620 --> 00:35:57.479
We discover that because we're
working in spherical harmonics,
00:35:57.479 --> 00:35:59.770
the angular momentum term
becomes just a function of r,
00:35:59.770 --> 00:36:02.080
with no other coefficients.
00:36:02.080 --> 00:36:04.500
So, now we have a function
of r plus the potential V,
00:36:04.500 --> 00:36:07.310
this looks like an effective
potential, V effective,
00:36:07.310 --> 00:36:10.020
which is the sum
of these two terms.
00:36:10.020 --> 00:36:11.290
So, there's that equation.
00:36:11.290 --> 00:36:13.972
On the other hand,
this is tantalizingly
00:36:13.972 --> 00:36:16.180
close to but not quite the
energy eigenvalue equation
00:36:16.180 --> 00:36:18.870
for a 1D problem with this
potential, V effective.
00:36:18.870 --> 00:36:21.510
To make it obvious that
it's, in fact, a 1D problem,
00:36:21.510 --> 00:36:25.490
we do a change of variables,
phi goes to 1 over ru,
00:36:25.490 --> 00:36:28.450
and then 1 upon r d
squared r phi becomes 1
00:36:28.450 --> 00:36:30.720
over r d squared u,
and V effective phi
00:36:30.720 --> 00:36:33.620
becomes 1 over r V effective u.
00:36:33.620 --> 00:36:36.340
Plugging that together, gives us
this energy eigenvalue equation
00:36:36.340 --> 00:36:41.080
for u, the effective wave
function, which is 1d problem.
00:36:41.080 --> 00:36:42.660
So, we can use all
of our intuition
00:36:42.660 --> 00:36:44.660
and all of our machinery
to solve this problem.
00:36:44.660 --> 00:36:46.160
And now we have to
ask, what exactly
00:36:46.160 --> 00:36:47.630
is the effective potential?
00:36:47.630 --> 00:36:49.800
And the effective potential
has three contributions.
00:36:49.800 --> 00:36:54.321
First, it has the
original V, secondly, it
00:36:54.321 --> 00:36:55.820
has the angular
momentum term, which
00:36:55.820 --> 00:36:58.440
is a constant over r
squared-- and here is
00:36:58.440 --> 00:37:02.480
that, constant over r
squared-- and the sum of these
00:37:02.480 --> 00:37:03.502
is the effective.
00:37:03.502 --> 00:37:05.710
And this guy dominates
because it's 1 over r squared.
00:37:05.710 --> 00:37:08.280
This dominates at small r,
and this dominates at large r
00:37:08.280 --> 00:37:10.000
if it's 1 over r.
00:37:10.000 --> 00:37:12.750
So, we get an
effective potential--
00:37:12.750 --> 00:37:17.870
that I'll check-- there's
the effective potential.
00:37:21.950 --> 00:37:24.050
And finally, the
third fact is that r
00:37:24.050 --> 00:37:26.269
must be strictly positive,
so as a 1D problem,
00:37:26.269 --> 00:37:28.060
that means it can't be
negative, it's gotta
00:37:28.060 --> 00:37:29.830
have an infinite
potential on the left.
00:37:40.010 --> 00:37:44.170
So, as an example, let's go
ahead and think more carefully
00:37:44.170 --> 00:37:47.680
about specifically this problem,
about this Coulomb potential,
00:37:47.680 --> 00:37:49.050
and this 1D effective potential.
00:37:49.050 --> 00:37:49.883
AUDIENCE: Professor?
00:37:49.883 --> 00:37:50.736
PROFESSOR: Yeah?
00:37:50.736 --> 00:37:51.652
AUDIENCE: [INAUDIBLE]?
00:37:51.652 --> 00:37:52.560
PROFESSOR: Yes?
00:37:52.560 --> 00:37:54.674
AUDIENCE: Where does
the 1 over r go?
00:37:54.674 --> 00:37:55.340
PROFESSOR: Good.
00:37:55.340 --> 00:37:59.660
So, remember the ddr
squared term gave us a 1
00:37:59.660 --> 00:38:01.196
over r out front.
00:38:01.196 --> 00:38:03.320
So, from this term, there
should be 1 over r, here.
00:38:03.320 --> 00:38:05.590
From this term, there
should also be a 1 over r.
00:38:05.590 --> 00:38:07.589
And from here, there
should be a 1 over r.
00:38:07.589 --> 00:38:08.130
AUDIENCE: Ah!
00:38:08.130 --> 00:38:10.171
PROFESSOR: So, then I'm
gonna cancel the 1 over r
00:38:10.171 --> 00:38:13.100
by multiplying the
whole equation by r.
00:38:13.100 --> 00:38:13.940
Yeah?
00:38:13.940 --> 00:38:14.900
Sneaky, sneaky.
00:38:14.900 --> 00:38:18.450
So, any time you see-- any
time, this is a general lesson--
00:38:18.450 --> 00:38:20.780
anytime you see a
differential equation that
00:38:20.780 --> 00:38:23.450
has this form-- two
derivatives, plus 1
00:38:23.450 --> 00:38:26.000
over r a derivative--
you know you
00:38:26.000 --> 00:38:28.060
can play some game like this.
00:38:28.060 --> 00:38:31.400
If you see this, declare in your
mind a brief moment of triumph,
00:38:31.400 --> 00:38:33.110
because you know what
technique to use.
00:38:33.110 --> 00:38:35.520
You can do this sort of
rescaling by a power of r.
00:38:35.520 --> 00:38:38.040
And more generally, if you have
a differential equation that
00:38:38.040 --> 00:38:39.727
looks like-- let
me do this here--
00:38:39.727 --> 00:38:42.060
if you have a differential
equation that looks something
00:38:42.060 --> 00:38:46.010
like a derivative with respect
to r plus a constant over r
00:38:46.010 --> 00:38:50.160
times phi, you know
how to solve this.
00:38:50.160 --> 00:38:55.160
Let me say plus
dot, dot, dot phi.
00:38:55.160 --> 00:38:58.790
You know how to solve this
because ddr plus c over r
00:38:58.790 --> 00:39:02.350
means that phi, if there were
nothing else, equals zero.
00:39:02.350 --> 00:39:05.210
If there were no other terms
here, then this would say,
00:39:05.210 --> 00:39:06.717
ddr plus c over r
is phi, that means
00:39:06.717 --> 00:39:08.800
when you take a derivative
it's like dividing by r
00:39:08.800 --> 00:39:09.830
and multiplying by c.
00:39:09.830 --> 00:39:17.100
That means that phi goes
like r to the minus c, right?
00:39:17.100 --> 00:39:19.100
But if phi goes like
r to the minus c,
00:39:19.100 --> 00:39:21.870
that's not the exact
solution to the equation,
00:39:21.870 --> 00:39:26.880
but I can write phi is equal
to r to the minus c times u.
00:39:26.880 --> 00:39:30.730
And then this equation
becomes ddr plus dot, dot,
00:39:30.730 --> 00:39:34.320
dot u equals zero.
00:39:34.320 --> 00:39:34.980
OK?
00:39:34.980 --> 00:39:37.585
Very useful little
trick-- not really
00:39:37.585 --> 00:39:41.347
a trick, It's just observation--
and this is the second order
00:39:41.347 --> 00:39:42.430
version of the same thing.
00:39:42.430 --> 00:39:44.054
Very useful things
to have in your back
00:39:44.054 --> 00:39:47.220
pocket for moments of need.
00:39:47.220 --> 00:39:48.790
OK?
00:39:48.790 --> 00:39:56.459
So, let's pick up with this guy.
00:39:56.459 --> 00:39:58.250
So, let me give you a
little name for this.
00:39:58.250 --> 00:40:00.333
So, this term that comes
from the angular momentum
00:40:00.333 --> 00:40:03.060
[? bit, ?] this originally came
from the kinetic energy, right?
00:40:03.060 --> 00:40:04.640
It came from the
L squared over r,
00:40:04.640 --> 00:40:06.570
which was from the
gradient squared energy.
00:40:06.570 --> 00:40:07.990
This is a kinetic energy term.
00:40:07.990 --> 00:40:09.690
Why is there a
kinetic energy term?
00:40:09.690 --> 00:40:12.220
Well, what this is telling
you is that if you have some
00:40:12.220 --> 00:40:15.040
angular momentum-- if little
l is not equal to 0---
00:40:15.040 --> 00:40:17.420
then as you get closer
and closer to the origin,
00:40:17.420 --> 00:40:19.510
the potential energy is
getting very, very large.
00:40:19.510 --> 00:40:20.290
And this should make sense.
00:40:20.290 --> 00:40:22.206
If you're spinning, and
you pull in your arms,
00:40:22.206 --> 00:40:23.386
you have to do work, right?
00:40:23.386 --> 00:40:25.460
You have to pull those guys in.
00:40:25.460 --> 00:40:26.039
You speed up.
00:40:26.039 --> 00:40:27.580
You're increasing
your kinetic energy
00:40:27.580 --> 00:40:29.930
due to conservation of
angular momentum, right?
00:40:29.930 --> 00:40:31.430
If you have
rotationally invariance,
00:40:31.430 --> 00:40:35.180
as you bring in your hand you're
increasing the kinetic energy.
00:40:35.180 --> 00:40:37.620
And so, this angular
momentum barrier
00:40:37.620 --> 00:40:39.030
is just an expression of that.
00:40:39.030 --> 00:40:41.742
It's just saying
that as you come
00:40:41.742 --> 00:40:43.825
to smaller and smaller
radius, holding the angular
00:40:43.825 --> 00:40:46.800
momentum fixed, your velocity--
your angular velocity-- must
00:40:46.800 --> 00:40:49.496
increase-- your kinetic
energy must increase--
00:40:49.496 --> 00:40:51.120
and we're calling
that a potential term
00:40:51.120 --> 00:40:52.985
just because we can.
00:40:52.985 --> 00:40:56.036
Because we've worked with
definite angular momentum, OK?
00:40:56.036 --> 00:40:58.410
You should have done this in
classical mechanics as well.
00:41:02.442 --> 00:41:04.650
Well, you should have done
it in classical mechanics.
00:41:04.650 --> 00:41:06.650
So, this is called the
angular momentum barrier.
00:41:08.650 --> 00:41:14.300
Quick question, classically,
if you take a charged particle
00:41:14.300 --> 00:41:17.790
around in a Coulomb potential,
classically that system decays,
00:41:17.790 --> 00:41:18.290
right?
00:41:18.290 --> 00:41:20.070
Irradiates away energy.
00:41:20.070 --> 00:41:23.870
Does the angular momentum
barrier save us from decaying?
00:41:28.140 --> 00:41:29.565
Is that why hydrogen is stable?
00:41:37.100 --> 00:41:39.720
No one wants to
stake a claim here?
00:41:39.720 --> 00:41:41.890
Is hydrogen stable
because of conversation
00:41:41.890 --> 00:41:43.067
of angular momentum?
00:41:43.067 --> 00:41:44.380
AUDIENCE: No.
00:41:44.380 --> 00:41:45.130
PROFESSOR: No.
00:41:45.130 --> 00:41:46.390
Absolutely not, right?
00:41:46.390 --> 00:41:48.170
So, first off, in your
first problems set,
00:41:48.170 --> 00:41:50.080
when you did that
calculation, that particle
00:41:50.080 --> 00:41:52.560
had angular momentum.
00:41:52.560 --> 00:41:54.630
So, and if can radiate
that away through
00:41:54.630 --> 00:41:56.190
electromagnetic interactions.
00:41:56.190 --> 00:41:58.049
So, that didn't save us.
00:41:58.049 --> 00:41:59.340
Angular momentum won't save us.
00:41:59.340 --> 00:42:01.360
Another way to say this
is that we can construct--
00:42:01.360 --> 00:42:02.960
and we just explicitly
see-- we can construct
00:42:02.960 --> 00:42:04.870
a state with which
has little l equals 0.
00:42:04.870 --> 00:42:06.619
In which case the
angular momentum barrier
00:42:06.619 --> 00:42:09.870
is 0 over r squared,
because there's nothing.
00:42:09.870 --> 00:42:15.030
Angular momentum barrier's not
what keeps you from decaying.
00:42:15.030 --> 00:42:18.930
And the reason is that the
electron can radiate away
00:42:18.930 --> 00:42:21.620
energy and angular
momentum, and so l
00:42:21.620 --> 00:42:24.660
will decrease and decrease,
and can still fall down.
00:42:24.660 --> 00:42:27.680
So, we still need
a reason for why
00:42:27.680 --> 00:42:32.284
the hydrogen system, quantum
mechanically, is stable.
00:42:32.284 --> 00:42:34.950
[? Why do ?] [? things exist? ?]
So, let's answer that question.
00:42:38.440 --> 00:42:40.810
So, what I want to do
now is, I want to solve--
00:42:40.810 --> 00:42:43.280
do I really want to do it
that way?-- well, actually,
00:42:43.280 --> 00:42:47.080
before we do, let's consider
some last, general conditions.
00:42:47.080 --> 00:42:55.730
General facts for
central potentials.
00:42:55.730 --> 00:42:58.230
So, let's look at some general
facts for central potentials.
00:43:01.930 --> 00:43:06.680
So, the first is, regardless of
what the [? bare ?] potential
00:43:06.680 --> 00:43:10.200
was, just due to the
angular momentum barrier,
00:43:10.200 --> 00:43:13.190
we have this 1 over r squared
behavior near the origin.
00:43:15.810 --> 00:43:20.340
So, we can look at
this, we can ask, look,
00:43:20.340 --> 00:43:22.830
what are the boundary
conditions at the origin?
00:43:22.830 --> 00:43:26.680
What must be true of u
of r near the origin?
00:43:26.680 --> 00:43:31.790
Near u of r-- or
sorry, near r goes
00:43:31.790 --> 00:43:42.520
to zero-- what must
be true of u of r?
00:43:49.420 --> 00:43:51.414
So, the right way to
ask this question is not
00:43:51.414 --> 00:43:54.080
to look at this u of r, which is
not actually the wave function,
00:43:54.080 --> 00:43:56.650
but to look at the
actual wave function,
00:43:56.650 --> 00:44:01.750
phi sub E, which goes near r
equals 0, like u of r over r.
00:44:05.590 --> 00:44:06.910
So, what should be true of u?
00:44:10.450 --> 00:44:11.900
Can u diverge?
00:44:11.900 --> 00:44:13.335
Is that physical?
00:44:13.335 --> 00:44:14.560
Does u have to vanish?
00:44:14.560 --> 00:44:17.164
Can it take a constant value?
00:44:17.164 --> 00:44:18.830
So, I've given you a
hint by telling you
00:44:18.830 --> 00:44:20.590
that I want to think about there
being an infinite potential,
00:44:20.590 --> 00:44:21.075
but why?
00:44:21.075 --> 00:44:22.491
Why is that the
right thing to do?
00:44:28.050 --> 00:44:31.435
Well, imagine u of r went to a
constant value near the origin.
00:44:31.435 --> 00:44:33.560
If u of r goes to a constant
value near the origin,
00:44:33.560 --> 00:44:36.962
then the wave function
diverges near the origin.
00:44:36.962 --> 00:44:41.860
That's maybe not so bad, maybe
it has a 1 over r singularity.
00:44:41.860 --> 00:44:44.980
It's not totally obvious
that that's horrible.
00:44:44.980 --> 00:44:47.480
What's so bad about having
a 1 over r behavior?
00:44:47.480 --> 00:44:51.555
So, suppose u goes
to a constant.
00:44:59.700 --> 00:45:03.870
So, phi goes to constant over r.
00:45:03.870 --> 00:45:05.090
What's so bad about this?
00:45:14.690 --> 00:45:17.850
So, let's look back
at the kinetic energy.
00:45:17.850 --> 00:45:19.811
P is equal t--
the kinetic energy
00:45:19.811 --> 00:45:22.310
is gonna be minus h bar squared
p squared-- so the energy is
00:45:22.310 --> 00:45:26.260
going to go like, p
squared over 2md squared.
00:45:26.260 --> 00:45:30.460
But here's an important fact, d
squared-- the Laplacian-- of 1
00:45:30.460 --> 00:45:34.380
over r, well, it's
easy to see what
00:45:34.380 --> 00:45:36.720
this is at a general point.
00:45:36.720 --> 00:45:41.810
At a general point,
d squared has a term
00:45:41.810 --> 00:45:45.920
that looks like 1
over rd squared r r.
00:45:45.920 --> 00:45:55.880
So, 1 over rd squared
r on 1 over r.
00:45:55.880 --> 00:45:58.115
Well, r times 1 over
r, that's just 1.
00:46:00.910 --> 00:46:03.890
And this is 0, right?
00:46:03.890 --> 00:46:07.970
So, the gradient squared
of 1 over r, is 0.
00:46:07.970 --> 00:46:12.930
Except, can that possibly
be true at r equals 0?
00:46:12.930 --> 00:46:16.360
No, because what's the
second derivative at 0?
00:46:16.360 --> 00:46:19.230
As you approach the
origin from any direction,
00:46:19.230 --> 00:46:24.330
the function is going like 1
over r, OK, so it's growing,
00:46:24.330 --> 00:46:26.110
but it's growing
in every direction.
00:46:26.110 --> 00:46:30.220
So, what's its first
derivative at the origin?
00:46:30.220 --> 00:46:33.879
It's actually
ill-defined, because it
00:46:33.879 --> 00:46:35.420
depends on the
direction you come in.
00:46:35.420 --> 00:46:37.669
The first direction coming
in this way, the derivative
00:46:37.669 --> 00:46:40.250
looks like it's becoming
this, from this direction it's
00:46:40.250 --> 00:46:42.895
becoming this, it's
actually badly divergent.
00:46:42.895 --> 00:46:44.270
So, what's the
second derivative?
00:46:44.270 --> 00:46:45.360
Well, the second
derivative has to go
00:46:45.360 --> 00:46:46.770
as you go across
this point, it's
00:46:46.770 --> 00:46:48.740
telling you how the
first derivative changes.
00:46:48.740 --> 00:46:51.220
But it changes from plus
infinity in this direction,
00:46:51.220 --> 00:46:53.160
to plus infinity
in this direction.
00:46:53.160 --> 00:46:55.250
That's badly singular.
00:46:55.250 --> 00:46:58.140
So, this can't possibly be true,
what I just wrote down here.
00:46:58.140 --> 00:47:02.692
And, in fact, d squared
on 1 over r-- and this
00:47:02.692 --> 00:47:07.450
is a very good exercise
for recitation--
00:47:07.450 --> 00:47:10.340
is equal to delta of r.
00:47:12.910 --> 00:47:18.050
It's 0-- it's clearly 0 for r0
equals 0--- but at the origin,
00:47:18.050 --> 00:47:18.679
it's divergent.
00:47:18.679 --> 00:47:20.220
And it's divergent
in exactly the way
00:47:20.220 --> 00:47:22.986
you need to get
the delta function.
00:47:22.986 --> 00:47:26.070
OK, which is pretty awesome.
00:47:26.070 --> 00:47:29.225
So, what that tells us is that
if we have a wave function that
00:47:29.225 --> 00:47:34.930
goes like 1 over r, then the
energy contribution-- energy
00:47:34.930 --> 00:47:36.640
acting on this wave
function-- gives us
00:47:36.640 --> 00:47:38.404
a delta function at the origin.
00:47:38.404 --> 00:47:40.070
So, unless you have
the potential, which
00:47:40.070 --> 00:47:42.992
is a delta function
at the origin,
00:47:42.992 --> 00:47:44.200
nothing will cancel this off.
00:47:44.200 --> 00:47:48.964
You can't possibly satisfy the
energy eigenvalue equation.
00:47:48.964 --> 00:47:56.790
So, u of r must go
to 0 at r goes to 0.
00:47:59.330 --> 00:48:02.060
Because if it goes to a
constant-- any constant--
00:48:02.060 --> 00:48:06.545
we've got a bad divergence
in the energy, yeah?
00:48:06.545 --> 00:48:08.390
In particular, if we
calculate the energy,
00:48:08.390 --> 00:48:12.254
we'll discover that the
energy is badly divergent.
00:48:12.254 --> 00:48:17.650
It does become divergent if
we don't have u going to 0.
00:48:17.650 --> 00:48:21.930
So, notice, by the way, as a
side note, that since phi goes
00:48:21.930 --> 00:48:23.840
like, phi is equal
to u over r, that
00:48:23.840 --> 00:48:25.820
means that phi
goes to a constant.
00:48:31.680 --> 00:48:33.430
This is good, because
what this is telling
00:48:33.430 --> 00:48:36.980
us is that the wave function--
So, truly, u is vanishing,
00:48:36.980 --> 00:48:38.630
but the probability
density, which
00:48:38.630 --> 00:48:42.970
is the wave function squared,
doesn't have to vanish.
00:48:42.970 --> 00:48:44.370
That's about the
derivative of u,
00:48:44.370 --> 00:48:46.578
as you approach the origin
from [? Lucatau's ?] Rule.
00:48:49.480 --> 00:48:53.890
So, this is the first general
fact about central potential.
00:49:01.520 --> 00:49:07.185
So, the next one-- and this
is really fun one-- good Lord!
00:49:07.185 --> 00:49:12.082
Is that, note---
sorry, two more--
00:49:12.082 --> 00:49:22.350
the energy depends
on l but not on m.
00:49:30.650 --> 00:49:34.290
Just explicitly, in the
energy eigenvalue equation,
00:49:34.290 --> 00:49:37.655
we have the angular
momentum showing up int
00:49:37.655 --> 00:49:39.412
the effective
potential, little l.
00:49:39.412 --> 00:49:42.570
But little m appears absolutely
nowhere except in our choice
00:49:42.570 --> 00:49:43.630
of spherical harmonic.
00:49:43.630 --> 00:49:45.588
For any different m--
and this was pointing out
00:49:45.588 --> 00:49:48.660
before-- for any different m, we
would've got the same equation.
00:49:48.660 --> 00:49:51.400
And that means that the energy
eigenvalue can depend on l,
00:49:51.400 --> 00:49:55.510
but it can't depend on m, right?
00:49:55.510 --> 00:50:03.415
So, that means for each m in
the allowed possible values, l,
00:50:03.415 --> 00:50:09.180
l minus 1, [? i ?]
minus l-- and this
00:50:09.180 --> 00:50:14.060
is 2l plus 1 possible
values-- for each
00:50:14.060 --> 00:50:15.970
of these m's, the
energy is the same.
00:50:20.230 --> 00:50:24.600
And I'll call this E sub
l, because the energy
00:50:24.600 --> 00:50:26.840
can depend on l.
00:50:26.840 --> 00:50:28.350
Why?
00:50:28.350 --> 00:50:32.807
The degeneracy of E sub
L is equal to 2l plus 1.
00:50:47.380 --> 00:50:48.745
Why?
00:50:48.745 --> 00:50:50.160
Why do we have this degeneracy?
00:50:53.698 --> 00:50:55.359
AUDIENCE: [INAUDIBLE].
00:50:55.359 --> 00:50:56.400
PROFESSOR: Yeah, exactly.
00:50:56.400 --> 00:51:00.180
We get the degeneracies when
we have symmetries, right?
00:51:00.180 --> 00:51:02.534
When we have a symmetry,
we get a degeneracy.
00:51:02.534 --> 00:51:03.950
And so, here we
have a degeneracy.
00:51:03.950 --> 00:51:07.060
And this degeneracy isn't
fixed by rotational invariance.
00:51:07.060 --> 00:51:09.040
And why is this the right thing?
00:51:09.040 --> 00:51:10.470
Rotational symmetry?
00:51:10.470 --> 00:51:12.500
So why, did this give
us this degeneracy?
00:51:12.500 --> 00:51:15.020
But what the rotational
degeneracy is saying is,
00:51:15.020 --> 00:51:17.470
look, if you've got some
total angular momentum,
00:51:17.470 --> 00:51:20.780
the energy can't possibly depend
on whether most of it's in Z,
00:51:20.780 --> 00:51:23.886
or most of it's in X,
or most of it's and Y.
00:51:23.886 --> 00:51:25.260
It can't possibly
depend on that,
00:51:25.260 --> 00:51:26.430
but that's what
m is telling you.
00:51:26.430 --> 00:51:27.888
M is just telling
you what fraction
00:51:27.888 --> 00:51:31.580
is contained in a
particular direction.
00:51:31.580 --> 00:51:33.820
So, rotational symmetry
immediately tells you this.
00:51:33.820 --> 00:51:35.870
But there's a nice way to phrase
this, which of the following,
00:51:35.870 --> 00:51:37.250
look, what is
rotational symmetry?
00:51:37.250 --> 00:51:38.750
Rotational symmetry
is the statement
00:51:38.750 --> 00:51:40.840
that the energy doesn't
care about rotations.
00:51:40.840 --> 00:51:48.930
And in particular, it must
commute with Lx, and with Ly,
00:51:48.930 --> 00:51:49.550
and with Lz.
00:51:54.264 --> 00:51:55.680
So, this is
rotationally symmetry.
00:52:00.122 --> 00:52:02.080
And I'm going to interpret
these in a nice way.
00:52:04.730 --> 00:52:10.590
So, this guy tells me I can
find common eigenfunctions.
00:52:13.720 --> 00:52:22.830
And, more to the point, a full
common eigenbasis of E and Lz.
00:52:22.830 --> 00:52:25.590
Can I also find a common
eigenbasis of ELz and Ly?
00:52:29.980 --> 00:52:32.560
Are there common
eigenvectors of ELz and Ly?
00:52:32.560 --> 00:52:36.897
AUDIENCE: [CHATTER]
00:52:36.897 --> 00:52:37.480
PROFESSOR: No.
00:52:37.480 --> 00:52:39.355
Are there common
eigenfunctions of Lz and Ly?
00:52:39.355 --> 00:52:40.130
AUDIENCE: No.
00:52:40.130 --> 00:52:41.610
PROFESSOR: No, because
they don't commute, right?
00:52:41.610 --> 00:52:42.550
E commutes with each of these.
00:52:42.550 --> 00:52:43.870
OK, so, I'm just
going to say, I'm
00:52:43.870 --> 00:52:45.530
gonna pick a common
eigenbasis of E and Lz--
00:52:45.530 --> 00:52:47.613
but I could've picked Lx,
or I could've picked Ly,
00:52:47.613 --> 00:52:49.740
I'm just picking Lz because
that's our convention--
00:52:49.740 --> 00:52:51.722
but what do these two--
Once I've chosen this--
00:52:51.722 --> 00:52:53.930
I'm gonna work with a common
eigenbasis of E and Lz--
00:52:53.930 --> 00:52:55.700
what do these two
commutators tell me?
00:52:55.700 --> 00:52:58.150
These two commutators
tell me that E commutes
00:52:58.150 --> 00:53:02.250
with Lx plus iLy and Lx
minus iLy, L plus/minus.
00:53:06.142 --> 00:53:08.600
So, this tells you that if you
have an eigenfunctions of E,
00:53:08.600 --> 00:53:10.410
and you act with a
raising operator,
00:53:10.410 --> 00:53:13.520
you get another
eigenfunction of E.
00:53:13.520 --> 00:53:15.700
And thus, we get our
2L plus 1 degeneracy,
00:53:15.700 --> 00:53:17.950
because we can walk up and
down the tower using L plus
00:53:17.950 --> 00:53:19.730
and L minus.
00:53:19.730 --> 00:53:21.595
Cool?
00:53:21.595 --> 00:53:23.020
OK.
00:53:23.020 --> 00:53:30.160
So, this is a nice example
that when you have a symmetry
00:53:30.160 --> 00:53:32.860
you get a degeneracy,
and vice versa.
00:53:36.010 --> 00:53:36.510
OK.
00:53:39.470 --> 00:53:44.057
So, let's do some examples of
using these central potentials.
00:53:44.057 --> 00:53:44.890
AUDIENCE: Professor?
00:53:44.890 --> 00:53:45.646
PROFESSOR: Yeah
00:53:45.646 --> 00:53:46.562
AUDIENCE: [INAUDIBLE]?
00:53:52.380 --> 00:53:55.190
PROFESSOR: It's 0.
00:53:55.190 --> 00:53:57.437
So, E with Lx is 0.
00:53:57.437 --> 00:53:59.520
E with Ly-- So, are you
happy with that statement?
00:53:59.520 --> 00:54:00.705
That E with Lx is 0?
00:54:00.705 --> 00:54:01.330
AUDIENCE: Yeah.
00:54:01.330 --> 00:54:01.997
PROFESSOR: Yeah.
00:54:01.997 --> 00:54:02.496
Good.
00:54:02.496 --> 00:54:04.640
OK, and so this 0 because
this is just Lx plus iOi.
00:54:04.640 --> 00:54:06.550
So, E with Lx is
0, and E with Ly
00:54:06.550 --> 00:54:09.224
is 0, so E commutes
with these guys.
00:54:09.224 --> 00:54:10.640
And so, this is
like the statement
00:54:10.640 --> 00:54:15.912
that L squared with L
plus/minus equals 0.
00:54:15.912 --> 00:54:17.710
AUDIENCE: [INAUDIBLE].
00:54:17.710 --> 00:54:18.850
PROFESSOR: Cool.
00:54:18.850 --> 00:54:21.170
OK, so, let's do some examples.
00:54:21.170 --> 00:54:23.890
So, the first example
is gonna be-- actually,
00:54:23.890 --> 00:54:27.480
I'm going to skip this spherical
well example-- because it's
00:54:27.480 --> 00:54:31.750
just not that interesting,
but it's in the notes,
00:54:31.750 --> 00:54:33.625
and you really
need to look at it.
00:54:33.625 --> 00:54:35.000
Oh hell, yes, I'm
going to do it.
00:54:35.000 --> 00:54:36.130
OK, so, the spherical well.
00:54:38.429 --> 00:54:40.220
So, I'm going to do it
in an abridged form,
00:54:40.220 --> 00:54:43.151
and maybe it's a good
thing for recitation.
00:54:43.151 --> 00:54:45.490
AUDIENCE: Professor?
00:54:45.490 --> 00:54:47.370
PROFESSOR: Thank you
recitation leader.
00:54:47.370 --> 00:54:50.390
So, in this spherical
well, what's the potential?
00:54:50.390 --> 00:54:52.830
So, here's v of r.
00:54:52.830 --> 00:54:56.830
Not U bar, and not V
effective, just v or r.
00:54:56.830 --> 00:55:01.120
And the potential is going to
be this, so, here's r equals 0.
00:55:01.120 --> 00:55:04.180
And if it's a spherical infinite
well, then I'm gonna say,
00:55:04.180 --> 00:55:09.030
the potential is infinite
outside of some distance, l.
00:55:09.030 --> 00:55:09.530
OK?
00:55:12.070 --> 00:55:13.540
And it's 0 inside.
00:55:16.650 --> 00:55:19.930
So, what does this give us?
00:55:19.930 --> 00:55:22.394
Well, in order to
solve the system,
00:55:22.394 --> 00:55:23.810
we know that the
first thing we do
00:55:23.810 --> 00:55:26.610
is we separate out
with yLms, and then
00:55:26.610 --> 00:55:29.260
we re-scale by 1 over r
to get the function of u,
00:55:29.260 --> 00:55:33.340
and we get this
equation, which is E on u
00:55:33.340 --> 00:55:40.070
is equal to minus h bar
squared upon 2m dr squared
00:55:40.070 --> 00:55:44.910
and plus v effective--
well, plus [? lL ?] plus 1--
00:55:44.910 --> 00:55:48.445
over r squared with a
2m and an h bar squared.
00:55:52.250 --> 00:55:56.247
And the potential is 0, inside.
00:55:56.247 --> 00:55:57.330
So we can just write this.
00:56:02.700 --> 00:56:05.520
So, if you just--
let me pull out
00:56:05.520 --> 00:56:12.180
the h bar squareds over 2m--
it becomes minus [INAUDIBLE]
00:56:12.180 --> 00:56:13.200
plus 1 over r squared.
00:56:18.090 --> 00:56:20.160
So, this is not a terrible
differential equation.
00:56:20.160 --> 00:56:23.470
And one can do some
good work to solve it,
00:56:23.470 --> 00:56:25.259
but it's a harder
differential equation
00:56:25.259 --> 00:56:27.300
than I want to spend the
time to study right now,
00:56:27.300 --> 00:56:31.920
so I'm just going to consider
the case-- special case-- when
00:56:31.920 --> 00:56:36.580
there's zero angular
momentum, little l equals 0.
00:56:36.580 --> 00:56:39.320
So, in the special
case of a l equals 0,
00:56:39.320 --> 00:56:42.410
E-- and I should
call this u sub l--
00:56:42.410 --> 00:56:46.930
Eu sub 0 is equal to
h bar squared upon 2m.
00:56:46.930 --> 00:56:49.430
And now, this term is gone--
the angular momentum barrier is
00:56:49.430 --> 00:56:51.450
gone-- because there's
no angular momentum,
00:56:51.450 --> 00:56:55.600
dr squared ul.
00:56:55.600 --> 00:56:58.840
Which can be written
succinctly as ul-- or sorry,
00:56:58.840 --> 00:57:03.240
u0-- prime prime, because
this is only a function of r.
00:57:03.240 --> 00:57:05.290
So now, this is a
ridiculously easy equation.
00:57:05.290 --> 00:57:07.410
We know how to solve
this equation, right?
00:57:07.410 --> 00:57:10.360
This is saying that the
energy, a constant, times u
00:57:10.360 --> 00:57:15.150
is two derivatives
times this constant.
00:57:15.150 --> 00:57:24.970
So, u0 can be written as a
cosine of kx-- or sorry-- kr
00:57:24.970 --> 00:57:35.310
plus b sine of kr, where h
bar squared k squared upon 2m
00:57:35.310 --> 00:57:39.460
equals E. And I should
really call this E sub 0,
00:57:39.460 --> 00:57:42.740
because it could depend
on little l, here.
00:57:45.390 --> 00:57:47.000
So, there's our
momentary solution,
00:57:47.000 --> 00:57:49.416
however, we have to satisfy
our boundary conditions, which
00:57:49.416 --> 00:57:51.590
is that it's gotta
vanish at the origin,
00:57:51.590 --> 00:57:54.600
but it's also gotta
vanish at the wall.
00:57:54.600 --> 00:57:59.360
So, the boundary
conditions, u of 0
00:57:59.360 --> 00:58:06.786
equals 0 tells us that a must
be equal to 0, and u of l
00:58:06.786 --> 00:58:11.840
equals 0 tells us that, well,
if this is 0, we've just got B,
00:58:11.840 --> 00:58:15.110
but sine of kr evaluated
at l, which is sine of kl,
00:58:15.110 --> 00:58:16.390
must be equal to 0.
00:58:16.390 --> 00:58:23.250
So, kl must be the 0 of
sine, must be n pi over--
00:58:23.250 --> 00:58:27.290
must be equal to n
pi, a multiple of pi.
00:58:27.290 --> 00:58:30.240
And so, this tells you
what the energy is.
00:58:30.240 --> 00:58:32.930
So, this is just
like the 1D system.
00:58:32.930 --> 00:58:35.840
It's just exactly like
when the 1D system.
00:58:35.840 --> 00:58:38.090
So now, to finally
close this off.
00:58:38.090 --> 00:58:43.360
What does this tell you
that the eigenfunctions are?
00:58:46.440 --> 00:58:48.800
And let me do that here.
00:58:48.800 --> 00:58:57.830
So, therefore, the wave
function phi sub E0 of r theta
00:58:57.830 --> 00:59:09.640
and phi-- oh god, oh jesus,
this is so much easier
00:59:09.640 --> 00:59:12.210
in [INAUDIBLE] so,
phi [INAUDIBLE]
00:59:12.210 --> 00:59:16.062
0 of r theta and
phi is equal to y0m.
00:59:18.975 --> 00:59:21.990
But what must m be?
00:59:21.990 --> 00:59:25.115
0, because m goes from
plus L to minus L, 0.
00:59:31.334 --> 00:59:32.750
I'm just [INAUDIBLE]
the argument.
00:59:32.750 --> 00:59:38.732
Y00 times, not u of r,
times 1 over r times u.
00:59:38.732 --> 00:59:42.860
1 over r times u of r.
00:59:42.860 --> 00:59:47.220
But u of r is a constant
times sine of kr.
00:59:51.080 --> 00:59:59.810
Sine of kr, but k is equal
to n pi over L. N pi over Lr.
00:59:59.810 --> 01:00:00.510
And what's Y00?
01:00:03.350 --> 01:00:04.170
It's a constant.
01:00:04.170 --> 01:00:06.940
And so, there's an overall
normalization constant,
01:00:06.940 --> 01:00:08.780
that I'll call n.
01:00:08.780 --> 01:00:13.180
OK, so, we get that
our wave function
01:00:13.180 --> 01:00:16.900
is 1 over r times
sine of n pi over Lr.
01:00:16.900 --> 01:00:18.930
So, this looks bad.
01:00:18.930 --> 01:00:19.945
There's a 1 over r.
01:00:19.945 --> 01:00:21.035
Why is this not bad?
01:00:24.184 --> 01:00:25.850
At the origin, why
is this not something
01:00:25.850 --> 01:00:26.850
I should worry about it?
01:00:26.850 --> 01:00:27.996
AUDIENCE: [MURMURS]
01:00:27.996 --> 01:00:30.340
PROFESSOR: Yeah, because
sine is linear, first of all,
01:00:30.340 --> 01:00:31.215
[INAUDIBLE] argument.
01:00:31.215 --> 01:00:33.500
So, this goes like,
n pi over L times r.
01:00:33.500 --> 01:00:35.110
That r cancels the 1 over r.
01:00:35.110 --> 01:00:37.950
So, near the origin, this
goes like a constant.
01:00:37.950 --> 01:00:38.870
Yeah?
01:00:38.870 --> 01:00:45.090
So, u has to 0, but the
wave function doesn't.
01:00:45.090 --> 01:00:46.500
Cool?
01:00:46.500 --> 01:00:47.000
OK.
01:00:47.000 --> 01:00:50.510
So, this is a very
nice more general story
01:00:50.510 --> 01:00:58.420
for larger L, which I hope
you see in the recitation.
01:00:58.420 --> 01:01:00.100
OK.
01:01:00.100 --> 01:01:02.860
Questions on the spherical well?
01:01:02.860 --> 01:01:05.089
The whole point
here-- Oh, yeah, go.
01:01:05.089 --> 01:01:07.255
AUDIENCE: What do [INAUDIBLE]
generally [INAUDIBLE]?
01:01:17.550 --> 01:01:18.681
PROFESSOR: That's true.
01:01:18.681 --> 01:01:19.180
So, good.
01:01:19.180 --> 01:01:20.420
So, let me rephrase
the question,
01:01:20.420 --> 01:01:22.128
and tell me if this
is the same question.
01:01:22.128 --> 01:01:23.190
So, this is strange.
01:01:23.190 --> 01:01:25.510
There's nothing special
about the origin.
01:01:25.510 --> 01:01:27.990
So, why do I have
a 0 at the origin?
01:01:27.990 --> 01:01:28.865
Is that the question?
01:01:28.865 --> 01:01:29.490
AUDIENCE: Yeah.
01:01:29.490 --> 01:01:30.110
PROFESSOR: OK.
01:01:30.110 --> 01:01:30.520
It's true.
01:01:30.520 --> 01:01:32.228
There's nothing special
about the origin,
01:01:32.228 --> 01:01:33.582
except for two things.
01:01:33.582 --> 01:01:35.290
One thing that's
special about the origin
01:01:35.290 --> 01:01:36.340
is we're working
in a system which
01:01:36.340 --> 01:01:37.423
has a rotational symmetry.
01:01:37.423 --> 01:01:39.720
But rotational symmetry
is rotational symmetry
01:01:39.720 --> 01:01:41.480
around some particular point.
01:01:41.480 --> 01:01:43.640
So, there's always a special
central point anytime
01:01:43.640 --> 01:01:44.570
you have a rotational symmetry.
01:01:44.570 --> 01:01:46.153
It's the point fixed
by the rotations.
01:01:46.153 --> 01:01:49.180
So, actually, the origin
is a special point here.
01:01:49.180 --> 01:01:52.740
Second, saying that
little u has a 0
01:01:52.740 --> 01:01:56.320
is not the same as saying that
the wave function has a 0.
01:01:56.320 --> 01:01:59.390
Little u has a 0, but it
gets multiplied by 1 over r.
01:01:59.390 --> 01:02:02.440
So, the wave function, in
fact, is non-zero, there.
01:02:02.440 --> 01:02:04.902
So, the physical thing is
the probability distribution,
01:02:04.902 --> 01:02:07.110
which is the [? norm ?]
squared of the wave function.
01:02:07.110 --> 01:02:10.450
And it doesn't have
a 0 at the origin.
01:02:10.450 --> 01:02:12.237
Does that satisfy?
01:02:12.237 --> 01:02:12.820
AUDIENCE: Yes.
01:02:12.820 --> 01:02:13.403
PROFESSOR: OK.
01:02:13.403 --> 01:02:16.860
So, the origin is special when
you have a central potential.
01:02:16.860 --> 01:02:18.990
That's where the
proton is, right?
01:02:18.990 --> 01:02:19.840
Right, OK.
01:02:19.840 --> 01:02:22.970
So, there is something
special about the origin.
01:02:22.970 --> 01:02:25.630
Wow, that was a really
[? anti-caplarian ?] sort
01:02:25.630 --> 01:02:26.130
of argument.
01:02:30.410 --> 01:02:32.637
OK, so that's where
the proton-- so, there
01:02:32.637 --> 01:02:34.220
is something special
about the origin,
01:02:34.220 --> 01:02:38.870
and the wave function doesn't
vanish there, even if u does.
01:02:38.870 --> 01:02:42.310
It may vanish there, but it
doesn't necessarily have to.
01:02:42.310 --> 01:02:44.160
And we'll see that in a minute.
01:02:44.160 --> 01:02:45.940
Other questions?
01:02:45.940 --> 01:02:46.658
Yeah?
01:02:46.658 --> 01:02:48.570
AUDIENCE: So, what again, what's
the reasoning for saying that
01:02:48.570 --> 01:02:50.960
the u of r has to vanish at
[? 0 instead ?] [? of L? ?]
01:02:50.960 --> 01:02:51.626
PROFESSOR: Good.
01:02:51.626 --> 01:02:54.610
The reason that u of r had
to vanish at the origin
01:02:54.610 --> 01:02:56.540
is that if it doesn't
vanish at the origin,
01:02:56.540 --> 01:03:00.200
then the wave function
diverges-- whoops,
01:03:00.200 --> 01:03:06.610
phi goes to constant--
if u doesn't go to 0,
01:03:06.610 --> 01:03:09.020
if it goes to any
constant, non-zero,
01:03:09.020 --> 01:03:10.714
then the wave function diverges.
01:03:10.714 --> 01:03:12.130
And if we calculate
the energy, we
01:03:12.130 --> 01:03:14.170
get a delta function
at the origin.
01:03:14.170 --> 01:03:15.670
So, there's an
infinite contribution
01:03:15.670 --> 01:03:16.670
of energy at the origin.
01:03:16.670 --> 01:03:17.650
That's not physical.
01:03:17.650 --> 01:03:19.760
So, in order to get a
sensible wave function
01:03:19.760 --> 01:03:23.454
with finite energies, we need
to have the u vanishes, because
01:03:23.454 --> 01:03:24.120
of the 1 over u.
01:03:24.120 --> 01:03:26.203
And the reason that we
said it had to vanish at l,
01:03:26.203 --> 01:03:29.062
was because I was considering
this spherical well-- spherical
01:03:29.062 --> 01:03:30.770
infinite well-- where
a particle is stuck
01:03:30.770 --> 01:03:33.860
inside a region of
radius, capital L,
01:03:33.860 --> 01:03:35.860
and that's just what
I mean by saying
01:03:35.860 --> 01:03:37.126
I have an infinite potential.
01:03:37.126 --> 01:03:38.000
AUDIENCE: OK, thanks.
01:03:38.000 --> 01:03:38.666
PROFESSOR: Cool?
01:03:38.666 --> 01:03:39.250
Yeah.
01:03:39.250 --> 01:03:40.630
Others?
01:03:40.630 --> 01:03:41.870
OK.
01:03:41.870 --> 01:03:44.690
So, with all that
done, we can now
01:03:44.690 --> 01:03:47.860
do the hydrogen-- or
the Coulomb-- potential.
01:03:47.860 --> 01:03:51.560
And I want to emphasize that
we often use the following
01:03:51.560 --> 01:03:53.810
words when-- people often
use the following words when
01:03:53.810 --> 01:03:55.670
solving this
problem-- we will now
01:03:55.670 --> 01:03:58.610
solve the problem of hydrogen.
01:03:58.610 --> 01:03:59.990
This is false.
01:03:59.990 --> 01:04:02.900
I am not about to solve for
you the problem of hydrogen.
01:04:02.900 --> 01:04:07.000
I am going to construct for
you a nice toy model, which
01:04:07.000 --> 01:04:10.570
turns out to be an excellent
first pass at explaining
01:04:10.570 --> 01:04:14.360
the properties observed in
hydrogen gases, their emission
01:04:14.360 --> 01:04:15.660
spectra, and their physics.
01:04:15.660 --> 01:04:16.710
This is a model.
01:04:16.710 --> 01:04:18.170
It is a bad model.
01:04:18.170 --> 01:04:20.020
It doesn't fit the data.
01:04:20.020 --> 01:04:21.270
But it's pretty good.
01:04:21.270 --> 01:04:23.000
And we'll be able
to improve it later.
01:04:23.000 --> 01:04:24.710
OK?
01:04:24.710 --> 01:04:27.820
So, it is the solution
of the Coulomb potential.
01:04:27.820 --> 01:04:29.320
And what I want to
emphasize to you,
01:04:29.320 --> 01:04:31.000
I cannot say this
strongly enough,
01:04:31.000 --> 01:04:36.280
physics is a process of building
models that do a good job
01:04:36.280 --> 01:04:36.864
of predicting.
01:04:36.864 --> 01:04:38.863
And the better their
predictions, the better the
01:04:38.863 --> 01:04:39.450
model.
01:04:39.450 --> 01:04:40.790
But they're all wrong.
01:04:40.790 --> 01:04:44.390
Every single model you ever
get from physics is wrong.
01:04:44.390 --> 01:04:47.040
There are just some that
are less stupidly wrong.
01:04:47.040 --> 01:04:49.660
Some are a better
approximation to the data, OK?
01:04:49.660 --> 01:04:51.410
This is not hydrogen.
01:04:51.410 --> 01:04:53.380
This is going to be our
first pass at hydrogen.
01:04:53.380 --> 01:04:56.180
It's the Coulomb potential.
01:04:56.180 --> 01:05:01.700
And the Coulomb potential,
V of r, is equal to minus e
01:05:01.700 --> 01:05:04.475
squared over r.
01:05:04.475 --> 01:05:05.850
This is what you
would get if you
01:05:05.850 --> 01:05:11.740
had a classical particle with
infinite mass and charge plus
01:05:11.740 --> 01:05:12.410
b.
01:05:12.410 --> 01:05:13.910
And then another
particle over here,
01:05:13.910 --> 01:05:17.569
with mass, little m,
and charge, minus e.
01:05:17.569 --> 01:05:19.110
And you didn't pay
too much attention
01:05:19.110 --> 01:05:23.810
to things like relativity,
or spin, or, you know,
01:05:23.810 --> 01:05:24.700
lots of other things.
01:05:24.700 --> 01:05:26.450
And you have no
background magnetic field,
01:05:26.450 --> 01:05:28.330
or electric field,
and anything else.
01:05:28.330 --> 01:05:30.389
And if these are point
particles, and-- All
01:05:30.389 --> 01:05:32.180
of those things are
false that I just said.
01:05:32.180 --> 01:05:33.596
But if all those
things were true,
01:05:33.596 --> 01:05:37.370
in that imaginary universe, this
would be the salient problem
01:05:37.370 --> 01:05:37.870
to solve.
01:05:37.870 --> 01:05:39.109
So, let's solve it.
01:05:39.109 --> 01:05:40.650
Now, are all those
things that I said
01:05:40.650 --> 01:05:43.240
that were false-- the
proton's a point particle,
01:05:43.240 --> 01:05:45.250
the proton is
infinitely massive,
01:05:45.250 --> 01:05:48.660
there's no spin-- are those
preposterously stupid?
01:05:48.660 --> 01:05:49.650
AUDIENCE: No.
01:05:49.650 --> 01:05:51.610
PROFESSOR: No, they're
excellent approximations
01:05:51.610 --> 01:05:52.770
in a lot of situations.
01:05:52.770 --> 01:05:55.010
So, they're not crazy wrong.
01:05:55.010 --> 01:05:58.036
They're just not
exactly correct.
01:05:58.036 --> 01:05:59.410
I want to keep
this in your mind.
01:05:59.410 --> 01:06:02.220
These are gonna be good
models, but they're not exact.
01:06:02.220 --> 01:06:03.761
So, we're not solving
hydrogen, we're
01:06:03.761 --> 01:06:07.680
gonna solve this idealized
Coulomb potential problem.
01:06:07.680 --> 01:06:08.940
OK, so let's solve it.
01:06:08.940 --> 01:06:11.610
So, if V is minus
e over r squared,
01:06:11.610 --> 01:06:15.740
then the equation for the
rescaled wave function, u,
01:06:15.740 --> 01:06:21.570
becomes minus h bar squared
upon 2mu prime prime
01:06:21.570 --> 01:06:24.770
of r plus the effective
potential, which
01:06:24.770 --> 01:06:29.910
is h bar squared
upon 2mll plus 1
01:06:29.910 --> 01:06:41.670
over r squared minus e squared
over r u is equal to e sub l u.
01:06:41.670 --> 01:06:43.540
So, there's the equation
we want to solve.
01:06:43.540 --> 01:06:45.500
We've already used
separation of variables,
01:06:45.500 --> 01:06:46.916
and we know that
the wave function
01:06:46.916 --> 01:06:49.474
is this little u times
1 over r times yLm,
01:06:49.474 --> 01:06:50.390
for some l and some m.
01:06:53.020 --> 01:06:55.340
So, the first thing we
should do any time you're
01:06:55.340 --> 01:06:57.881
solving an interesting problem,
the first thing you should do
01:06:57.881 --> 01:06:59.740
is do dimensional analysis.
01:06:59.740 --> 01:07:02.330
And if you do dimensional
analysis, the units of e
01:07:02.330 --> 01:07:04.720
squared-- well, this
is easy-- e squared
01:07:04.720 --> 01:07:06.850
must be an energy
times a length.
01:07:06.850 --> 01:07:10.350
So, this is an energy
times a length.
01:07:10.350 --> 01:07:13.150
Also known as p
squared l, momentum
01:07:13.150 --> 01:07:16.532
squared over 2m, 2 times
the mass times the length.
01:07:16.532 --> 01:07:18.740
It's useful to put things
in terms of mass, momentum,
01:07:18.740 --> 01:07:20.614
and lengths, because
you can cancel them out.
01:07:20.614 --> 01:07:24.987
H bar has units of p times l.
01:07:24.987 --> 01:07:26.820
And what's the only
other parameter we have?
01:07:26.820 --> 01:07:30.980
We have the mass, which
has units of mass.
01:07:30.980 --> 01:07:31.630
OK.
01:07:31.630 --> 01:07:34.540
And so, from this, we can
build two nice quantities.
01:07:34.540 --> 01:07:36.015
The first, is we can build r0.
01:07:36.015 --> 01:07:40.190
We can build something
with units of a radius.
01:07:40.190 --> 01:07:42.530
And I'm going to choose the
factors of 2 judiciously, h
01:07:42.530 --> 01:07:47.444
bar squared over 2me squared--
whoops, e squared-- so,
01:07:47.444 --> 01:07:49.360
let's just make sure
this has the right units.
01:07:49.360 --> 01:07:52.190
E squared has units of
energy times the length,
01:07:52.190 --> 01:07:55.270
but h bar squared
over 2m has units
01:07:55.270 --> 01:07:58.770
of p squared l squared
over 2m, so that
01:07:58.770 --> 01:08:04.385
has units of energy
times the length squared.
01:08:04.385 --> 01:08:06.760
So, length squared over length,
this has units of length,
01:08:06.760 --> 01:08:08.240
so this is good.
01:08:08.240 --> 01:08:11.215
So, there's a parameter
that has units of length.
01:08:11.215 --> 01:08:12.840
And from this, it's
easy to see that we
01:08:12.840 --> 01:08:15.048
can build a characteristic
energy by taking e squared
01:08:15.048 --> 01:08:17.176
and dividing it by
this length scale.
01:08:17.176 --> 01:08:19.050
And so then, the
energy, which I'll
01:08:19.050 --> 01:08:24.080
call e0, which is equal
to e squared over r0,
01:08:24.080 --> 01:08:29.400
is equal to 2me to the
4th over h bar squared.
01:08:34.350 --> 01:08:37.029
So, before we do anything else,
without solving any problems,
01:08:37.029 --> 01:08:39.065
we immediately can do
a couple of things.
01:08:39.065 --> 01:08:41.439
The first is, if you take the
system and I ask you, look,
01:08:41.439 --> 01:08:42.130
what do you expect?
01:08:42.130 --> 01:08:43.505
If this is a
quantum mechanical--
01:08:43.505 --> 01:08:46.350
a 1d problem in quantum
mechanics-- with a potential,
01:08:46.350 --> 01:08:49.769
and we know something about 1D
quantum mechanical problems--
01:08:49.769 --> 01:08:51.310
I guess, this guy--
we know something
01:08:51.310 --> 01:08:52.851
about 1D quantum
mechanical problems.
01:08:52.851 --> 01:08:55.680
Which is that the ground
state has what energy?
01:08:55.680 --> 01:08:56.582
Some finite energy.
01:08:56.582 --> 01:08:58.290
It doesn't have infinite
negative energy.
01:08:58.290 --> 01:09:00.600
It's got some finite energy.
01:09:00.600 --> 01:09:03.160
What do you expect to be
roughly the ground state energy
01:09:03.160 --> 01:09:06.060
of this system?
01:09:06.060 --> 01:09:06.970
AUDIENCE: [MURMURING]
01:09:06.970 --> 01:09:07.510
PROFESSOR: Yeah.
01:09:07.510 --> 01:09:08.010
Right.
01:09:08.010 --> 01:09:09.090
Roughly minus e0.
01:09:09.090 --> 01:09:10.802
That seems like a
pretty good guess.
01:09:10.802 --> 01:09:12.510
It's the only dimensional
sensible thing.
01:09:12.510 --> 01:09:14.540
Maybe we're off by factors of 2.
01:09:14.540 --> 01:09:18.705
But, maybe it's minus e0.
01:09:18.705 --> 01:09:20.330
So, that's a good
guess, a first thing,
01:09:20.330 --> 01:09:22.890
before we do any calculation.
01:09:22.890 --> 01:09:28.180
And if you actually take mu e
to the 4th over h bar squared,
01:09:28.180 --> 01:09:31.219
this is off by,
unfortunately, a factor of 4.
01:09:31.219 --> 01:09:36.040
This is equal to 4 times
the binding energy, which
01:09:36.040 --> 01:09:39.106
is also called the
Rydberg constant.
01:09:39.106 --> 01:09:41.349
Wanna make sure I get
my factors of two right.
01:09:41.349 --> 01:09:42.874
Yep, I'm off by a factor of 4.
01:09:42.874 --> 01:09:45.180
I'm off by a factor
of 4 from what
01:09:45.180 --> 01:09:51.310
we'll call the Rydberg
energy, which is 13.6 eV.
01:09:51.310 --> 01:09:54.160
And this is observed
binding energy of hydrogen.
01:09:54.160 --> 01:09:56.970
So, before we do anything,
before we solve any equation,
01:09:56.970 --> 01:10:02.570
we have a fabulous estimate of
the binding energy of hydrogen,
01:10:02.570 --> 01:10:03.730
right?
01:10:03.730 --> 01:10:05.790
All the work we're
about to do is
01:10:05.790 --> 01:10:09.070
gonna be to deal with
this factor of 4, right?
01:10:09.070 --> 01:10:11.025
Which, I mean, is
important, but I just
01:10:11.025 --> 01:10:12.650
want to emphasize
how much you get just
01:10:12.650 --> 01:10:14.110
from doing dimensional analysis.
01:10:14.110 --> 01:10:16.515
Immediately upon knowing the
rules of quantum mechanics,
01:10:16.515 --> 01:10:18.640
knowing that this is the
equation you should solve,
01:10:18.640 --> 01:10:20.140
without ever touching
that equation,
01:10:20.140 --> 01:10:22.730
just dimensional analysis
gives you this answer.
01:10:22.730 --> 01:10:23.546
OK?
01:10:23.546 --> 01:10:24.295
Which is fabulous.
01:10:28.020 --> 01:10:31.060
So, with that motivation,
let's solve this problem.
01:10:31.060 --> 01:10:33.730
Oh, by the way,
what do you think
01:10:33.730 --> 01:10:37.330
r0 is a good approximation to?
01:10:37.330 --> 01:10:38.740
Well, it's a length scale.
01:10:38.740 --> 01:10:39.884
AUDIENCE: [INAUDIBLE].
01:10:39.884 --> 01:10:40.550
PROFESSOR: Yeah!
01:10:40.550 --> 01:10:42.633
It's probably something
like the expectation value
01:10:42.633 --> 01:10:44.820
of the radius-- or maybe
of the radius squared--
01:10:44.820 --> 01:10:47.236
because the expectation value
of the radius is probably 0.
01:10:49.660 --> 01:10:52.950
OK, so, let's solve this system.
01:10:52.950 --> 01:10:55.960
And at this point, I'm not
gonna actually solve out
01:10:55.960 --> 01:10:57.820
the differential
equation in detail.
01:10:57.820 --> 01:10:59.980
I'm just gonna tell you
how the solution goes,
01:10:59.980 --> 01:11:04.540
because solving it is a sort
of involved undertaking.
01:11:04.540 --> 01:11:07.630
And so, here's the first thing,
so we look at this equation.
01:11:07.630 --> 01:11:10.310
So, we had this differential
equation-- this guy--
01:11:10.310 --> 01:11:13.150
and we want to solve it.
01:11:13.150 --> 01:11:16.520
So, think back to the
harmonic oscillator
01:11:16.520 --> 01:11:19.910
when we did the brute force
method of solving the hydrogen
01:11:19.910 --> 01:11:21.445
system, OK?
01:11:21.445 --> 01:11:26.830
When we did the brute force
method-- she sells seashells--
01:11:26.830 --> 01:11:30.430
when the brute force method
of solving, what did we do?
01:11:30.430 --> 01:11:33.920
We first did, we did
asymptotic analysis.
01:11:33.920 --> 01:11:36.690
We extracted the
overall asymptotic form,
01:11:36.690 --> 01:11:38.742
at infinity and at
the origin, to get
01:11:38.742 --> 01:11:40.450
a nice regular
differential equation that
01:11:40.450 --> 01:11:42.340
didn't have any
funny singularities,
01:11:42.340 --> 01:11:44.950
and then we did a
series approximation.
01:11:44.950 --> 01:11:46.180
OK?
01:11:46.180 --> 01:11:48.189
Now, do most
differential equations
01:11:48.189 --> 01:11:49.730
have a simple closed
form expression?
01:11:49.730 --> 01:11:50.725
A solution?
01:11:50.725 --> 01:11:53.350
No, most differential equations
of some, maybe if you're lucky,
01:11:53.350 --> 01:11:55.940
it's a special function that
people have studied in detail,
01:11:55.940 --> 01:11:58.148
but most don't have a simple
solution like a Gaussian
01:11:58.148 --> 01:11:59.670
or a power large, or something.
01:11:59.670 --> 01:12:01.941
Most of them just have
some complicated solution.
01:12:01.941 --> 01:12:04.440
This is one of those miraculous
differential equations where
01:12:04.440 --> 01:12:06.773
we can actually exactly write
down the solution by doing
01:12:06.773 --> 01:12:09.510
the series approximation,
having done asymptotic analysis.
01:12:13.190 --> 01:12:15.790
So, the first thing when doing
dimensional analysis too, let's
01:12:15.790 --> 01:12:17.645
make everything dimensionless.
01:12:20.650 --> 01:12:23.834
OK, and it's easy to see what
the right thing to do is.
01:12:23.834 --> 01:12:26.500
Take r and make it dimensionless
by pulling out a factor of rho,
01:12:26.500 --> 01:12:27.540
or of r0.
01:12:27.540 --> 01:12:29.790
So, I'll pick our new variable
is gonna be called rho,
01:12:29.790 --> 01:12:32.420
this is dimensionless.
01:12:32.420 --> 01:12:34.630
And the second thing is I
want to take the energy,
01:12:34.630 --> 01:12:37.600
and I will write it
as minus e0, times
01:12:37.600 --> 01:12:39.720
some dimensionless
energy, epsilon.
01:12:39.720 --> 01:12:43.015
So, these guys are my
dimensionless variables.
01:12:43.015 --> 01:12:45.390
And when you go through and
do that, the equation you get
01:12:45.390 --> 01:12:55.590
is minus d rho squared plus l l
plus 1 over rho squared minus 1
01:12:55.590 --> 01:13:01.670
over rho plus epsilon
u is equal to 0.
01:13:01.670 --> 01:13:03.420
So, the form of this
differential equation
01:13:03.420 --> 01:13:05.450
is, OK, it's not
different in any deep way,
01:13:05.450 --> 01:13:06.955
but it's a little bit easier.
01:13:06.955 --> 01:13:09.455
This is gonna be the easier way
to deal with this, because I
01:13:09.455 --> 01:13:11.840
don't have to deal with
any stupid constant.
01:13:11.840 --> 01:13:14.450
And so now, let's do
the brute force thing.
01:13:14.450 --> 01:13:16.955
Three, asymptotic analysis.
01:13:25.050 --> 01:13:27.220
And here, I'm just going
to write down the answers.
01:13:27.220 --> 01:13:28.791
And the reason is,
first off, this
01:13:28.791 --> 01:13:30.790
is something you should
either do in recitation,
01:13:30.790 --> 01:13:33.030
or see-- go through--
on your own,
01:13:33.030 --> 01:13:35.650
but this is just the mathematics
of solving a differential
01:13:35.650 --> 01:13:36.150
equation.
01:13:36.150 --> 01:13:37.610
This is not the important part.
01:13:37.610 --> 01:13:41.429
So, when rho goes to infinity,
which terms dominate?
01:13:41.429 --> 01:13:42.970
Well, this is not
terribly important.
01:13:42.970 --> 01:13:44.700
This is not terribly important.
01:13:44.700 --> 01:13:46.030
That term is gonna dominate.
01:13:49.470 --> 01:13:54.389
And if we get that d rho squared
plus u, rho goes to infinity,
01:13:54.389 --> 01:13:55.430
these two terms dominate.
01:13:55.430 --> 01:13:57.304
Well, two derivatives
is a constant.
01:13:57.304 --> 01:13:58.720
You know what those
solutions look
01:13:58.720 --> 01:14:00.310
like, they look
like exponentials,
01:14:00.310 --> 01:14:02.250
with the exponential
being brute--
01:14:02.250 --> 01:14:06.830
with the power-- the exponent,
sorry, being root epsilon.
01:14:06.830 --> 01:14:11.550
So, u is going to go like e to
the minus square root epsilon
01:14:11.550 --> 01:14:12.070
rho.
01:14:12.070 --> 01:14:13.820
For normalize-ability,
I picked the minus,
01:14:13.820 --> 01:14:16.195
I could've picked the plus,
that would've been divergent.
01:14:17.990 --> 01:14:22.120
So, as rho goes to
0, what happens?
01:14:22.120 --> 01:14:24.530
Well, as rho goes to 0,
this is insignificant.
01:14:24.530 --> 01:14:26.400
And this totally
dominates over this guy.
01:14:29.800 --> 01:14:31.722
On the other hand,
if l is equal to 0,
01:14:31.722 --> 01:14:33.430
then this is the only
term that survives,
01:14:33.430 --> 01:14:35.680
so we'd better make sure
that that behaves gracefully.
01:14:35.680 --> 01:14:37.980
As rho goes to 0,
asymptotic analysis
01:14:37.980 --> 01:14:40.750
is gonna tell us
that u goes like rho.
01:14:40.750 --> 01:14:43.090
Well, two derivatives, we
pulled down a rho squared,
01:14:43.090 --> 01:14:45.975
and so two derivatives in
this guy, we pulled down an l,
01:14:45.975 --> 01:14:46.797
then an l plus 1.
01:14:46.797 --> 01:14:48.630
So, this should go like
rho to the l plus 1.
01:14:54.800 --> 01:14:56.749
There's also another term.
01:14:56.749 --> 01:14:59.290
So, in the same way that there
were two solutions to this guy
01:14:59.290 --> 01:15:02.160
asymptotically-- one growing,
one decreasing-- here,
01:15:02.160 --> 01:15:04.410
there's another solution,
which is rho to the minus l.
01:15:04.410 --> 01:15:06.410
That also does it, because
we get minus l, then
01:15:06.410 --> 01:15:09.932
minus minus l minus 1, which
gives us the plus l l plus 1.
01:15:09.932 --> 01:15:11.890
But that is also badly
diversion at the origin,
01:15:11.890 --> 01:15:14.090
it goes like 1 over 0 to the l.
01:15:14.090 --> 01:15:14.820
That's bad.
01:15:14.820 --> 01:15:15.945
So, these are my solutions.
01:15:18.750 --> 01:15:20.760
So, this tells us, having
done this in analysis,
01:15:20.760 --> 01:15:24.150
we should write that u is
equal to rho to the l plus
01:15:24.150 --> 01:15:28.170
1 times e to the
minus root epsilon
01:15:28.170 --> 01:15:30.690
rho times some
remaining function,
01:15:30.690 --> 01:15:35.672
which I'll call v, little
v. Little v of rho,
01:15:35.672 --> 01:15:37.130
and this,
asymptotically, should go
01:15:37.130 --> 01:15:40.730
to a constant near the
origin and something that
01:15:40.730 --> 01:15:46.560
vanishes slower than an
exponential at infinity.
01:15:46.560 --> 01:15:50.285
So then, we take this and
we do our series expansion.
01:15:54.759 --> 01:15:56.550
So, we take that
expression, we plug it in.
01:15:56.550 --> 01:15:59.970
At that point, all we're doing
is a change of variables.
01:15:59.970 --> 01:16:02.690
We plug it in, and we get
a resulting differential
01:16:02.690 --> 01:16:03.190
equation.
01:16:05.810 --> 01:16:13.670
Rho v prime prime plus 2 1
plus l minus root epsilon rho
01:16:13.670 --> 01:16:24.437
v prime plus 1 minus 2 root
epsilon l plus 1 v equals 0.
01:16:24.437 --> 01:16:26.395
So, this is the resulting
differential equation
01:16:26.395 --> 01:16:29.840
for the little v guy.
01:16:29.840 --> 01:16:33.400
And we do a series expansion.
01:16:33.400 --> 01:16:39.285
V is equal to sum
over, sum from j
01:16:39.285 --> 01:16:46.260
equals 0 to infinity,
of a sub j rho to the j.
01:16:46.260 --> 01:16:49.600
Plug this guy in here, just
like in the case of the harmonic
01:16:49.600 --> 01:16:52.850
oscillator equation, and
get a series expansion.
01:16:52.850 --> 01:16:57.010
Now, OK, let me write
it out this way.
01:17:02.670 --> 01:17:04.470
And the series expansion
has a solution,
01:17:04.470 --> 01:17:05.646
which is a sub j plus 1.
01:17:05.646 --> 01:17:07.520
And this is, actually,
kind of a fun process.
01:17:07.520 --> 01:17:11.824
So, if you, you know, like
quick little calculations,
01:17:11.824 --> 01:17:13.240
this is a sweet
little calculation
01:17:13.240 --> 01:17:14.530
to take this expression.
01:17:14.530 --> 01:17:17.510
Plug it in and derive
this recursion relation,
01:17:17.510 --> 01:17:23.100
which is root-- or 2 root--
epsilon times j plus l
01:17:23.100 --> 01:17:33.340
plus 1 minus 1 over j plus
1 j plus 2 l plus 2 aj.
01:17:38.570 --> 01:17:40.990
So, here's our series expansion.
01:17:40.990 --> 01:17:49.130
And in order for
this terminate, we
01:17:49.130 --> 01:17:54.120
must have that some aj
max plus 1 is equal to 0.
01:17:54.120 --> 01:17:56.210
So, one of these guys
must eventually vanish.
01:17:56.210 --> 01:17:58.260
And the only thing's that's
changing is little j.
01:17:58.260 --> 01:18:01.700
So, what that tells us is
that for some maximum value
01:18:01.700 --> 01:18:05.750
of little j, root 2 epsilon
times j maximum plus l plus 1
01:18:05.750 --> 01:18:07.700
is equal to minus 1.
01:18:07.700 --> 01:18:10.140
But that gives us a
relationship between overall j
01:18:10.140 --> 01:18:13.074
max, little l, and the energy.
01:18:13.074 --> 01:18:14.740
And if you go through,
what you discover
01:18:14.740 --> 01:18:21.460
is that the energy is equal
to 1 over 4 n squared, where
01:18:21.460 --> 01:18:27.370
n is equal to j
max plus l plus 1.
01:18:30.010 --> 01:18:35.700
And what this tells is
that the energy is labeled
01:18:35.700 --> 01:18:40.010
by an integer, n, and an
integer, l, and an integer,
01:18:40.010 --> 01:18:42.450
m-- these are from the
spherical harmonics,
01:18:42.450 --> 01:18:44.640
and n came from the
series expansion--
01:18:44.640 --> 01:18:51.895
and it's equal to minus e0 over
4 n squared, independent of l
01:18:51.895 --> 01:18:52.395
and m.
01:18:58.680 --> 01:19:01.310
And so, by solving the
differential equation exactly,
01:19:01.310 --> 01:19:05.400
which in this case we kind of
amazingly can, what we discover
01:19:05.400 --> 01:19:09.420
is that the energy eigenvalues
are, indeed, exactly 1/4 of e0.
01:19:12.229 --> 01:19:14.020
And they're spaced with
a 1 over n squared,
01:19:14.020 --> 01:19:15.040
which does two things.
01:19:15.040 --> 01:19:17.150
Not only does that
explain-- so, let's think
01:19:17.150 --> 01:19:19.790
about the consequence of this
very briefly-- not only does
01:19:19.790 --> 01:19:25.440
that explain the minus 13.6
eV, not only does that explain
01:19:25.440 --> 01:19:28.650
the binding energy of
hydrogen as is observed,
01:19:28.650 --> 01:19:29.460
that it does more.
01:19:29.460 --> 01:19:31.960
Remember in the very beginning
one of the experimental facts
01:19:31.960 --> 01:19:33.584
we wanted to explain
about the universe
01:19:33.584 --> 01:19:43.320
was that the spectrum
of light of hydrogen
01:19:43.320 --> 01:19:47.530
went like 30 over 4 n squared.
01:19:51.550 --> 01:19:53.040
This was the Rydberg relation.
01:19:56.400 --> 01:19:58.600
And now we see explicitly.
01:19:58.600 --> 01:20:00.730
So, we've solved
for that expansion.
01:20:00.730 --> 01:20:02.180
But there's a real puzzle here.
01:20:05.350 --> 01:20:07.512
Purely on very
general grounds, we
01:20:07.512 --> 01:20:09.970
derived earlier that when you
have a rotationally invariant
01:20:09.970 --> 01:20:11.664
potential-- a
central potential--
01:20:11.664 --> 01:20:13.080
every energy should
be degenerate,
01:20:13.080 --> 01:20:14.860
with degeneracy 2l plus 1.
01:20:14.860 --> 01:20:18.530
It can depend on l, but it
must be independent of m.
01:20:18.530 --> 01:20:20.190
But here, we've
discovered-- first off,
01:20:20.190 --> 01:20:21.840
we've fit a nice bit
of experimental data,
01:20:21.840 --> 01:20:24.048
but we've discovered the
energy is, in fact, not just
01:20:24.048 --> 01:20:27.370
independent of m, but it's
independent of l, too.
01:20:27.370 --> 01:20:28.530
Why?
01:20:28.530 --> 01:20:33.260
What symmetry is explaining
this extra degeneracy?
01:20:33.260 --> 01:20:35.720
We'll pick that up next time.