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PROFESSOR: All right, it
is time to get started.
00:00:25.940 --> 00:00:30.750
Thanks for coming for
this cold and rainy
00:00:30.750 --> 00:00:32.540
Wednesday before Thanksgiving.
00:00:35.490 --> 00:00:40.150
Today we're supposed to talk
about the radial equation.
00:00:40.150 --> 00:00:43.710
That's our main subject today.
00:00:43.710 --> 00:00:47.640
We discussed last
time the states
00:00:47.640 --> 00:00:51.430
of angular momentum from
the abstract viewpoint,
00:00:51.430 --> 00:00:55.460
and now we make contact with
some important problems,
00:00:55.460 --> 00:01:00.390
and differential equations,
and things like that.
00:01:00.390 --> 00:01:05.730
And there's a few concepts
I want to emphasize today.
00:01:05.730 --> 00:01:09.570
And basically, the
main concept is
00:01:09.570 --> 00:01:13.480
that I want you to just become
familiar with what we would
00:01:13.480 --> 00:01:18.330
call the diagram, the key
diagram for the states
00:01:18.330 --> 00:01:23.250
of a theory, of a particle in
a three dimensional potential.
00:01:23.250 --> 00:01:27.860
I think you have to have a good
understanding of what it looks,
00:01:27.860 --> 00:01:30.700
and what is special
about it, and when
00:01:30.700 --> 00:01:32.790
it shows particular properties.
00:01:32.790 --> 00:01:37.730
So to begin with, I'll
have to do a little aside
00:01:37.730 --> 00:01:43.296
on a object that is
covered in many courses.
00:01:43.296 --> 00:01:46.050
I don't know to what
that it's covered,
00:01:46.050 --> 00:01:48.920
but it's the subject
of spherical harmonics.
00:01:48.920 --> 00:01:53.730
So we'll talk about spherical
harmonics for about 15 minutes.
00:01:53.730 --> 00:01:56.295
And then we'll do
the radial equation.
00:01:56.295 --> 00:01:59.360
And for the radial equation,
after we discuss it,
00:01:59.360 --> 00:02:01.550
we'll do three examples.
00:02:01.550 --> 00:02:05.610
And that will be the
end of today's lecture.
00:02:05.610 --> 00:02:10.830
Next time, as you come back
from the holiday next week,
00:02:10.830 --> 00:02:14.470
we are doing the addition of
angular momentum basically.
00:02:14.470 --> 00:02:19.070
And then the last week,
more examples and a few more
00:02:19.070 --> 00:02:23.420
things for emphasis to
understand it all well.
00:02:23.420 --> 00:02:26.610
All right, so in terms
of spherical harmonics,
00:02:26.610 --> 00:02:37.530
I wanted to emphasize that
our algebraic analysis led
00:02:37.530 --> 00:02:42.380
to states that we
called jm, but today I
00:02:42.380 --> 00:02:47.750
will call lm, because they
will refer to orbital angular
00:02:47.750 --> 00:02:49.180
momentum.
00:02:49.180 --> 00:02:52.860
And as you've seen in
one of your problems,
00:02:52.860 --> 00:02:55.360
orbital angular
momentum has to do
00:02:55.360 --> 00:03:01.620
with values of j,
which are integers.
00:03:01.620 --> 00:03:09.670
So half integers values of j
cannot be realized for orbital
00:03:09.670 --> 00:03:10.820
angular momentum.
00:03:10.820 --> 00:03:12.460
It's a very interesting thing.
00:03:12.460 --> 00:03:18.840
So spin states don't have wave
functions in the usual way.
00:03:18.840 --> 00:03:23.490
It's only states of
integer angular momentum
00:03:23.490 --> 00:03:26.260
that have wave functions.
00:03:26.260 --> 00:03:27.905
And those are the
spherical harmonics.
00:03:27.905 --> 00:03:32.330
So I will talk about
lm, and l, as usual,
00:03:32.330 --> 00:03:36.040
will go from 0 to infinity.
00:03:36.040 --> 00:03:40.910
And m goes from l to minus l.
00:03:43.420 --> 00:03:48.660
And you had these states, and
we said that algebraically you
00:03:48.660 --> 00:03:56.440
would have L squared equals h
squared l times l plus 1 lm.
00:03:56.440 --> 00:04:03.960
And Lz lm equal hm lm.
00:04:07.610 --> 00:04:10.440
Now basically, the
spherical harmonics
00:04:10.440 --> 00:04:17.339
are going to be wave
functions for these states.
00:04:17.339 --> 00:04:20.019
And the way we
can approach it is
00:04:20.019 --> 00:04:24.790
that we did a little bit of work
already with constructing the l
00:04:24.790 --> 00:04:27.120
squared operator.
00:04:27.120 --> 00:04:34.050
And in last lecture we derived,
starting from the fact that L
00:04:34.050 --> 00:04:39.720
is r cross p and using
x,y, and z, px, py, pz,
00:04:39.720 --> 00:04:44.270
and passing through spherical
coordinates that L squared is
00:04:44.270 --> 00:04:52.230
the operator minus h squared 1
over sine theta d d theta sine
00:04:52.230 --> 00:04:59.295
theta d d theta again plus
1 over sine squared theta d
00:04:59.295 --> 00:05:00.866
second d phi squared.
00:05:03.830 --> 00:05:10.220
And we didn't do it,
but Lz, which you know
00:05:10.220 --> 00:05:23.790
is h bar over i x d dy
minus y d dx can also
00:05:23.790 --> 00:05:28.320
be translated into
angular variables.
00:05:28.320 --> 00:05:31.820
And it has a very simple form.
00:05:31.820 --> 00:05:33.740
Also purely angular.
00:05:33.740 --> 00:05:38.550
And you can interpret it Lz is
rotations around the z-axis,
00:05:38.550 --> 00:05:40.420
so they change phi.
00:05:40.420 --> 00:05:42.530
So it will not
surprise you, if you
00:05:42.530 --> 00:05:48.840
do this exercise, that
this is h over i d d phi.
00:05:48.840 --> 00:05:52.870
And you should really check it.
00:05:52.870 --> 00:05:57.020
There's another one that
is a bit more laborious.
00:05:57.020 --> 00:06:03.630
L plus minus, remember,
is Lx plus minus i Lz.
00:06:03.630 --> 00:06:06.010
We have a big attendance today.
00:06:10.250 --> 00:06:13.900
Is equal to-- more people.
00:06:13.900 --> 00:06:30.630
h bar e to the plus minus i phi
i cosine theta over sine theta
00:06:30.630 --> 00:06:36.130
d d phi plus minus d d theta.
00:06:41.000 --> 00:06:44.010
And that takes a bit of algebra.
00:06:44.010 --> 00:06:45.300
You could do it.
00:06:45.300 --> 00:06:47.520
It's done in many books.
00:06:47.520 --> 00:06:50.040
It's probably there
in Griffith's.
00:06:50.040 --> 00:06:52.590
And these are the
representations
00:06:52.590 --> 00:06:56.840
of these operators as
differential operators that
00:06:56.840 --> 00:06:59.690
act and function
on theta and phi
00:06:59.690 --> 00:07:03.940
and don't care about radius.
00:07:03.940 --> 00:07:08.910
So in mathematical physics,
people study these things
00:07:08.910 --> 00:07:14.460
and invent these things
called spherical harmonics
00:07:14.460 --> 00:07:19.210
Ylm's of theta and phi.
00:07:19.210 --> 00:07:22.940
And the way you
could see their done
00:07:22.940 --> 00:07:28.230
is in fact, such that
this L squared viewed
00:07:28.230 --> 00:07:34.870
as this operator, differential
operator, acting on Ylm
00:07:34.870 --> 00:07:43.690
is indeed equal to h squared
l times l plus 1 Ylm.
00:07:43.690 --> 00:07:49.890
And Lz thought also as a
differential operator, the one
00:07:49.890 --> 00:07:51.930
that we've written there.
00:07:51.930 --> 00:07:58.960
On the Ylm is h bar m Ylm.
00:08:01.700 --> 00:08:06.430
So they are constructed
in this way,
00:08:06.430 --> 00:08:10.100
satisfying these equations.
00:08:10.100 --> 00:08:12.980
These are important equations
in mathematical physics,
00:08:12.980 --> 00:08:14.930
and these functions
were invented
00:08:14.930 --> 00:08:18.380
to satisfy those equations.
00:08:18.380 --> 00:08:24.130
Well, these are the properties
of those states over there.
00:08:24.130 --> 00:08:30.620
So we can think
of these functions
00:08:30.620 --> 00:08:34.429
as the wave functions
associated with those states.
00:08:34.429 --> 00:08:39.289
So that's interpretation that
is natural in quantum mechanics.
00:08:39.289 --> 00:08:42.210
And we want to think
of them like that.
00:08:42.210 --> 00:08:50.780
We want to think of the Ylm's
as the wave functions associated
00:08:50.780 --> 00:08:52.900
to the states lm.
00:08:52.900 --> 00:08:56.760
So lm.
00:08:56.760 --> 00:09:02.575
And here you would put a
position state theta phi.
00:09:08.610 --> 00:09:11.360
This is analogous to the
thing that we usually
00:09:11.360 --> 00:09:17.090
call the wave function being a
position state times the state
00:09:17.090 --> 00:09:18.590
side.
00:09:18.590 --> 00:09:25.440
So we want to think of
the Ylm's in this way
00:09:25.440 --> 00:09:29.850
as pretty much the wave
functions associated
00:09:29.850 --> 00:09:32.550
to those states.
00:09:32.550 --> 00:09:42.160
Now there is a little bit of
identities that come once you
00:09:42.160 --> 00:09:45.150
accept that this is what
you think of the Ylm's.
00:09:45.150 --> 00:09:50.480
And then the compatibility
of these equations.
00:09:50.480 --> 00:09:57.390
Top here with these ones makes
in this identification natural.
00:09:57.390 --> 00:10:01.920
Now in order to manipulate
and learn things
00:10:01.920 --> 00:10:03.950
about those spherical
harmonics the way
00:10:03.950 --> 00:10:06.220
we do things in
quantum mechanics,
00:10:06.220 --> 00:10:10.190
we think of the
completeness relation.
00:10:10.190 --> 00:10:17.450
If we have d cube x x x, this
is a completeness relation
00:10:17.450 --> 00:10:19.770
for position states.
00:10:24.500 --> 00:10:31.320
And I want to derive or
suggest a completeness relation
00:10:31.320 --> 00:10:33.760
for these theta phi states.
00:10:33.760 --> 00:10:38.900
For that, I would
pass this integral
00:10:38.900 --> 00:10:41.170
to do it in spherical
coordinates.
00:10:41.170 --> 00:10:52.290
So I would do dr rd
theta r sine theta d phi.
00:10:52.290 --> 00:10:59.100
And I would put r theta
phi position states
00:10:59.100 --> 00:11:01.050
for these things.
00:11:01.050 --> 00:11:06.180
And position states r theta phi.
00:11:06.180 --> 00:11:07.810
Still being equal to 1.
00:11:13.510 --> 00:11:21.920
And we can try to
split this thing.
00:11:21.920 --> 00:11:25.360
It's natural for us to
think of just theta phi,
00:11:25.360 --> 00:11:30.390
because these wave functions
have nothing to do with r,
00:11:30.390 --> 00:11:34.940
so I will simply do
the integrals this way.
00:11:34.940 --> 00:11:39.780
d theta sine theta d phi.
00:11:39.780 --> 00:11:45.360
And think just like a
position state in x, y, z.
00:11:45.360 --> 00:11:48.770
It's a position state in x,
in y, and in z multiplied.
00:11:48.770 --> 00:11:52.280
We'll just split these
things without trying
00:11:52.280 --> 00:11:55.790
to be too rigorous about it.
00:11:55.790 --> 00:12:00.700
Theta and phi like this.
00:12:00.700 --> 00:12:11.845
And you would have the integral
dr r squared r r equal 1.
00:12:15.800 --> 00:12:21.410
And at this point,
I want to think
00:12:21.410 --> 00:12:29.580
of this as the natural way of
setting a completeness relation
00:12:29.580 --> 00:12:32.060
for theta and phi.
00:12:32.060 --> 00:12:35.360
And this doesn't
talk to this one,
00:12:35.360 --> 00:12:40.260
so I will think of this that
in the space of theta and phi,
00:12:40.260 --> 00:12:42.740
objects that just
depend on theta and phi,
00:12:42.740 --> 00:12:45.570
this acts as a complete thing.
00:12:45.570 --> 00:12:48.220
And if objects depend
also in r, this
00:12:48.220 --> 00:12:50.580
will act as a complete thing.
00:12:50.580 --> 00:12:53.140
So I will-- I don't know.
00:12:53.140 --> 00:12:56.130
Maybe the right way
to say is postulate
00:12:56.130 --> 00:12:59.960
that we'll have a completeness
relation of this form.
00:12:59.960 --> 00:13:12.975
d theta sine theta d phi
theta phi theta phi equals 1.
00:13:23.260 --> 00:13:31.070
And then with this we can
do all kinds of things.
00:13:31.070 --> 00:13:34.230
First, this integral
is better written.
00:13:34.230 --> 00:13:42.520
This integral really represents
0 to pi d theta sine theta 0
00:13:42.520 --> 00:13:46.690
to 2 pi d phi.
00:13:46.690 --> 00:13:49.785
Now this is minus
d cosine theta.
00:13:57.530 --> 00:14:02.020
And when theta is equal
to 0, cosine theta
00:14:02.020 --> 00:14:10.280
is 1 to minus 1 integral
d phi 0 to 2 pi.
00:14:10.280 --> 00:14:18.550
So this integral, really
d theta sine theta d
00:14:18.550 --> 00:14:25.090
phi this is really the
integral from minus 1 to 1.
00:14:25.090 --> 00:14:32.955
Change that order of d cos theta
integral d phi from 0 to 2 pi.
00:14:37.650 --> 00:14:43.227
And this is called the
integral over solid angle.
00:14:43.227 --> 00:14:44.060
That's a definition.
00:14:47.710 --> 00:14:51.930
So we could write the
completeness relation
00:14:51.930 --> 00:14:57.350
in the space theta phi as
integral over solid angle theta
00:14:57.350 --> 00:15:02.800
phi theta phi equals 1.
00:15:08.280 --> 00:15:12.460
Then the key property of
the spherical harmonics,
00:15:12.460 --> 00:15:20.220
or the lm states, is
that they are orthogonal.
00:15:20.220 --> 00:15:25.870
So delta l, l prime,
delta m, m prime.
00:15:25.870 --> 00:15:28.180
So the orthogonality
are of this state
00:15:28.180 --> 00:15:31.860
is guaranteed because
Hermitian operators,
00:15:31.860 --> 00:15:36.790
different eigenvalues,
they have to be orthogonal.
00:15:36.790 --> 00:15:38.990
Eigenstates of Hermitian.
00:15:38.990 --> 00:15:41.400
Operators with
different eigenvalues.
00:15:41.400 --> 00:15:46.670
Here, you introduce a complete
set of states of theta phi.
00:15:46.670 --> 00:15:58.950
So you put l prime m prime
theta phi theta phi lm.
00:16:02.340 --> 00:16:12.850
And this is the integral over
solid angle of Yl prime m
00:16:12.850 --> 00:16:17.610
prime of theta phi star.
00:16:17.610 --> 00:16:21.400
This is in the wrong position.
00:16:21.400 --> 00:16:27.866
And here Ylm of theta
phi being equal delta l
00:16:27.866 --> 00:16:31.440
l prime delta m m prime.
00:16:37.650 --> 00:16:47.630
So this is orthogonality
of the spherical harmonics.
00:16:47.630 --> 00:16:50.570
And this is pretty
much all we need.
00:16:50.570 --> 00:16:58.690
Now there's the standard ways
of constructing these things
00:16:58.690 --> 00:17:02.860
from the quantum mechanical
sort of intuition.
00:17:02.860 --> 00:17:07.450
Basically, you can
try to first build
00:17:07.450 --> 00:17:15.425
Yll, which corresponds
to the state ll.
00:17:20.050 --> 00:17:23.390
Now the kind of
differential equations
00:17:23.390 --> 00:17:27.349
this Yll satisfies
are kind of simple.
00:17:27.349 --> 00:17:30.100
But in particular,
the most important one
00:17:30.100 --> 00:17:34.980
is that L plus kills this state.
00:17:34.980 --> 00:17:37.720
So basically you
use the condition
00:17:37.720 --> 00:17:40.920
that L plus kills this
state to find a differential
00:17:40.920 --> 00:17:46.050
equation for this, which
can be solved easily.
00:17:46.050 --> 00:17:47.850
Not a hard
differential equation.
00:17:47.850 --> 00:17:50.300
Then you find Yll.
00:17:50.300 --> 00:17:57.700
And then you can find Yll
minus 1 and all the other ones
00:17:57.700 --> 00:18:01.400
by applying the
operator L minus.
00:18:01.400 --> 00:18:03.960
The lowering operator of m.
00:18:03.960 --> 00:18:06.420
So in principle, if you
have enough patience,
00:18:06.420 --> 00:18:10.130
you can calculate all the
spherical harmonics that way.
00:18:10.130 --> 00:18:14.240
There's no obstruction.
00:18:14.240 --> 00:18:17.170
But the form is a
little messy, and if you
00:18:17.170 --> 00:18:21.640
want to find the normalizations
so that these things work out
00:18:21.640 --> 00:18:25.790
correctly, well, it takes some
work at the end of the day.
00:18:25.790 --> 00:18:30.680
So we're not going
to do that here.
00:18:30.680 --> 00:18:34.500
We'll just leave it at
that, and if we ever
00:18:34.500 --> 00:18:40.260
need some special harmonics,
we'll just hold the answers.
00:18:40.260 --> 00:18:44.250
And they are in most textbooks.
00:18:44.250 --> 00:18:46.940
So if you do need
them, well, you'll
00:18:46.940 --> 00:18:49.620
have to do with
complicated normalizations.
00:18:52.230 --> 00:18:56.190
So that's really all I wanted to
say about spherical harmonics,
00:18:56.190 --> 00:19:00.210
and we can turn then to
the real subject, which
00:19:00.210 --> 00:19:02.720
is the radial equation.
00:19:02.720 --> 00:19:03.870
So the radial equation.
00:19:12.440 --> 00:19:17.480
So we have a Hamiltonian
H equals p squared vector
00:19:17.480 --> 00:19:21.160
over 2m plus v of r.
00:19:21.160 --> 00:19:23.960
And we've seen that
this is equal to h over
00:19:23.960 --> 00:19:33.030
2m 1 over r d second dr
squared r plus 1 over 2mr
00:19:33.030 --> 00:19:37.755
squared L squared plus v of r.
00:19:40.500 --> 00:19:42.695
So this is what we're
trying to solve.
00:19:45.510 --> 00:19:48.070
And the way we
attempt to solve this
00:19:48.070 --> 00:19:49.920
is by separation of variables.
00:19:49.920 --> 00:19:54.600
So we'll try to write the wave
function, psi, characterized
00:19:54.600 --> 00:19:56.980
by three things.
00:19:56.980 --> 00:20:04.680
Its energy, the value of
l, and the value of m.
00:20:04.680 --> 00:20:08.350
And it's a function of
position, because we're
00:20:08.350 --> 00:20:12.380
trying to solve H
psi equal E psi.
00:20:12.380 --> 00:20:16.580
And that's the energy
that we want to consider.
00:20:16.580 --> 00:20:20.980
So I will write here to begin
with something that will not
00:20:20.980 --> 00:20:25.840
turn out to be exactly
right, but it's
00:20:25.840 --> 00:20:29.520
important to do
it first this way.
00:20:29.520 --> 00:20:36.700
A function of art r that
has labels E, l, and m.
00:20:36.700 --> 00:20:40.000
Because it certainly could
depend on E, could depend on l,
00:20:40.000 --> 00:20:43.100
and could depend on m,
that radial function.
00:20:43.100 --> 00:20:45.420
And then the angular
function will
00:20:45.420 --> 00:20:50.530
be the Ylm's of theta and phi.
00:20:50.530 --> 00:20:53.060
So this is the [INAUDIBLE]
sets for the equation.
00:20:57.870 --> 00:21:03.390
If we have that, we can plug
into the Schrodinger equation,
00:21:03.390 --> 00:21:04.790
and see what we get.
00:21:04.790 --> 00:21:10.010
Well, this operator
will act on this f.
00:21:10.010 --> 00:21:15.520
This will have the
operator L squared,
00:21:15.520 --> 00:21:18.900
but L squared over Ylm,
you know what it is.
00:21:18.900 --> 00:21:23.040
And v of r is multiplicative,
so it's no big problem.
00:21:23.040 --> 00:21:24.230
So what do we have?
00:21:24.230 --> 00:21:30.070
We have minus h squared
over 2m 1 over r.
00:21:30.070 --> 00:21:32.720
Now I can talk
normal derivatives.
00:21:32.720 --> 00:21:46.900
d r squared r times fElm
plus 1 over 2mr squared.
00:21:46.900 --> 00:21:49.690
And now have L squared
acting on this,
00:21:49.690 --> 00:21:54.680
but L squared acting on the
Ylm is just this factor.
00:21:54.680 --> 00:22:03.640
So we have h squared l times
l plus 1 times the fElm.
00:22:08.850 --> 00:22:12.910
Now I didn't put the
Ylm in the first term
00:22:12.910 --> 00:22:15.200
because I'm going to
cancel it throughout.
00:22:15.200 --> 00:22:28.700
So we have this term here plus
v of r fElm equals E fElm.
00:22:34.140 --> 00:22:39.670
That is substituting into the
equation h psi equal E psi.
00:22:39.670 --> 00:22:42.490
So first term here.
00:22:42.490 --> 00:22:46.300
Second term, it acted on
the spherical harmonic.
00:22:46.300 --> 00:22:47.840
v of r is multiplicative.
00:22:47.840 --> 00:22:50.030
E on that.
00:22:50.030 --> 00:22:52.500
But then what you
see immediately
00:22:52.500 --> 00:22:55.960
is that this differential
equation doesn't depend on m.
00:22:59.550 --> 00:23:04.200
It was L squared, but no
Lz in the Hamiltonian.
00:23:04.200 --> 00:23:06.610
So no m dependent.
00:23:06.610 --> 00:23:12.290
So actually we
were overly proven
00:23:12.290 --> 00:23:17.370
in thinking that f
was a function of m.
00:23:17.370 --> 00:23:21.390
What we really have
is that psi Elm
00:23:21.390 --> 00:23:28.530
is equal to a function of E
and l or r Ylm of theta phi.
00:23:31.350 --> 00:23:33.760
And then the
differential equation
00:23:33.760 --> 00:23:37.230
is minus h squared over 2m.
00:23:37.230 --> 00:23:40.220
Let's multiply all by r.
00:23:40.220 --> 00:23:44.942
d second dr squared of r fEl.
00:23:50.410 --> 00:23:51.540
Plus look here.
00:23:55.760 --> 00:24:00.070
The r that I'm multiplying
is going to go into the f.
00:24:00.070 --> 00:24:01.700
Here it's going
to go into the f.
00:24:01.700 --> 00:24:03.260
Here it's going
to go into the f.
00:24:03.260 --> 00:24:04.680
It's an overall thing.
00:24:04.680 --> 00:24:09.100
But here we keep h
squared l times l
00:24:09.100 --> 00:24:26.090
plus 1 over 2mr squared rfEl
plus v of r fEl rfEl equal
00:24:26.090 --> 00:24:27.555
e times rfEl.
00:24:36.470 --> 00:24:42.825
So what you see here is that
this function is quite natural.
00:24:46.390 --> 00:24:53.432
So it suggests the
definition of uEl to be rfEl.
00:24:59.530 --> 00:25:03.060
So that the differential
equation now finally becomes
00:25:03.060 --> 00:25:13.930
minus h squared over 2m d
second dr squared of uEl plus
00:25:13.930 --> 00:25:18.035
there's the u here, the u
here, and this potential
00:25:18.035 --> 00:25:20.310
that has two terms.
00:25:20.310 --> 00:25:26.920
So this will be v of
r plus h squared l
00:25:26.920 --> 00:25:36.036
times l plus 1 over 2mr
squared uEl equals E times eEl.
00:25:40.540 --> 00:25:42.920
And this is the famous
radial equation.
00:25:47.100 --> 00:25:50.870
It's an equation for you.
00:25:50.870 --> 00:25:57.250
And here, this whole
thing is sometimes
00:25:57.250 --> 00:25:58.940
called the effective potential.
00:26:04.670 --> 00:26:07.280
So look what we've got.
00:26:07.280 --> 00:26:16.056
This f, if you wish here,
is now of the form uEl
00:26:16.056 --> 00:26:24.040
of r Ylm over r theta phi.
00:26:24.040 --> 00:26:25.960
f is u over r.
00:26:25.960 --> 00:26:30.520
So this is the way we've
written the solution,
00:26:30.520 --> 00:26:35.010
and u satisfies this equation,
which is a one dimensional
00:26:35.010 --> 00:26:38.856
Schrodinger equation
for the radius r.
00:26:38.856 --> 00:26:42.990
One dimensional equation
with an effective potential
00:26:42.990 --> 00:26:44.990
that depends on L.
00:26:44.990 --> 00:26:49.580
So actually the first
thing you have to notice
00:26:49.580 --> 00:26:52.890
is that the central
potential problem
00:26:52.890 --> 00:27:00.080
has turned into an
infinite collection of one
00:27:00.080 --> 00:27:01.510
dimensional problems.
00:27:01.510 --> 00:27:04.956
One for each value of l.
00:27:04.956 --> 00:27:09.940
For different values of l, you
have a different potential.
00:27:09.940 --> 00:27:12.310
Now they're not
all that different.
00:27:12.310 --> 00:27:15.530
They have different
intensity of this term.
00:27:15.530 --> 00:27:20.200
For l equals 0, well
you have some solutions.
00:27:20.200 --> 00:27:23.780
And for l equal 1, the answer
could be quite different.
00:27:23.780 --> 00:27:26.410
For l equal 2, still different.
00:27:26.410 --> 00:27:30.210
And you have to solve
an infinite number
00:27:30.210 --> 00:27:35.330
of one dimensional problems.
00:27:35.330 --> 00:27:39.380
That's what the Schrodinger
equation has turned into.
00:27:39.380 --> 00:27:45.210
So we filled all
these blackboards.
00:27:45.210 --> 00:27:48.420
Let's see, are there questions?
00:27:48.420 --> 00:27:50.470
Anything so far?
00:28:01.068 --> 00:28:01.567
Yes?
00:28:07.536 --> 00:28:09.326
AUDIENCE: You might
get to this later,
00:28:09.326 --> 00:28:13.303
but what does it mean
in our wave equations,
00:28:13.303 --> 00:28:18.780
in our wave function there,
psi of Elm is equal to fEl,
00:28:18.780 --> 00:28:22.516
and the spherical harmonic
of that one mean that one has
00:28:22.516 --> 00:28:24.320
an independence and
the other doesn't.
00:28:24.320 --> 00:28:27.550
Can they be separated
on the basis of m?
00:28:27.550 --> 00:28:33.520
PROFESSOR: So it is just a fact
that the radial solution is
00:28:33.520 --> 00:28:37.010
independent of n, so it's
an important property.
00:28:37.010 --> 00:28:39.900
n is fairly simple.
00:28:39.900 --> 00:28:43.500
The various state, the
states with angular momentum
00:28:43.500 --> 00:28:48.180
l, but different m's just differ
in their angular dependence,
00:28:48.180 --> 00:28:52.010
not in the radial dependence.
00:28:52.010 --> 00:28:54.230
And practically,
it means that you
00:28:54.230 --> 00:28:59.960
have an infinite set of one
dimensional problems labeled
00:28:59.960 --> 00:29:04.790
by l, and not labeled by m,
which conceivably could have
00:29:04.790 --> 00:29:07.240
happened, but it doesn't happen.
00:29:07.240 --> 00:29:10.380
So just a major simplicity.
00:29:10.380 --> 00:29:11.614
Yes?
00:29:11.614 --> 00:29:13.030
AUDIENCE: Does the
radial equation
00:29:13.030 --> 00:29:15.784
have all the same properties as
a one dimensional Schrodinger
00:29:15.784 --> 00:29:16.284
equation?
00:29:16.284 --> 00:29:18.720
Or does the divergence in
the effect [INAUDIBLE] 0
00:29:18.720 --> 00:29:19.220
change that?
00:29:19.220 --> 00:29:21.470
PROFESSOR: Well,
it changes things,
00:29:21.470 --> 00:29:24.640
but the most serious
change is the fact
00:29:24.640 --> 00:29:28.650
that, in one
dimensional problems,
00:29:28.650 --> 00:29:32.640
x goes from minus
infinity to infinity.
00:29:32.640 --> 00:29:35.100
And here it goes
from 0 to infinity,
00:29:35.100 --> 00:29:38.510
so we need to worry
about what happens at 0.
00:29:38.510 --> 00:29:41.590
Basically that's the
main complication.
00:29:41.590 --> 00:29:46.670
One dimensional potential,
but it really just
00:29:46.670 --> 00:29:48.370
can't go below 0.
00:29:48.370 --> 00:29:53.470
r is a radial variable,
and we can't forget that.
00:29:53.470 --> 00:29:54.290
Yes?
00:29:54.290 --> 00:29:58.224
AUDIENCE: The potential v of r
will depend on whatever problem
00:29:58.224 --> 00:29:59.140
you're solving, right?
00:29:59.140 --> 00:30:00.139
PROFESSOR: That's right.
00:30:00.139 --> 00:30:04.670
AUDIENCE: Could you find
the v of r [INAUDIBLE]?
00:30:04.670 --> 00:30:07.090
PROFESSOR: Well
that doesn't quite
00:30:07.090 --> 00:30:09.620
make sense as a Hamiltonian.
00:30:09.620 --> 00:30:13.270
You see, if you
have a v of r, it's
00:30:13.270 --> 00:30:17.180
something that is supposed to
be v of r for any wave function.
00:30:17.180 --> 00:30:18.380
That's the definition.
00:30:18.380 --> 00:30:20.900
So it can depend
on some parameter,
00:30:20.900 --> 00:30:25.230
but that parameter cannot be
the l of the particular wave
00:30:25.230 --> 00:30:28.092
function.
00:30:28.092 --> 00:30:32.370
AUDIENCE: [INAUDIBLE] or
something that would interact
00:30:32.370 --> 00:30:32.870
with the--
00:30:32.870 --> 00:30:34.790
PROFESSOR: If you
have magnetic fields,
00:30:34.790 --> 00:30:38.860
things change, because
then you can split levels
00:30:38.860 --> 00:30:40.900
with respect to m.
00:30:40.900 --> 00:30:43.640
Break degeneracies and
things change indeed.
00:30:46.780 --> 00:30:49.960
We'll take care of those by
using perturbation theory
00:30:49.960 --> 00:30:50.760
mostly.
00:30:50.760 --> 00:30:53.840
Use this solution and
then perturbation theory.
00:30:53.840 --> 00:31:01.420
OK, so let's proceed
a little more on this.
00:31:01.420 --> 00:31:07.710
So the first thing that we
want to talk a little about
00:31:07.710 --> 00:31:11.440
is the normalization and
some boundary conditions,
00:31:11.440 --> 00:31:14.950
because otherwise
we can't really
00:31:14.950 --> 00:31:17.610
understand what's going on.
00:31:17.610 --> 00:31:22.630
And happily the discussion
is not that complicated.
00:31:22.630 --> 00:31:24.450
So we want to normalize.
00:31:24.450 --> 00:31:25.600
So what do we want?
00:31:25.600 --> 00:31:39.230
Integral d cube x psi Elm
of x squared equals 1.
00:31:39.230 --> 00:31:44.200
So clearly we want to go
into angular variables.
00:31:44.200 --> 00:31:53.290
So again, this is r squared
dr integral d solid angle,
00:31:53.290 --> 00:31:54.530
r squared Er.
00:31:54.530 --> 00:32:10.800
And this thing is now uEl
squared absolute value
00:32:10.800 --> 00:32:12.250
over r squared.
00:32:12.250 --> 00:32:14.230
Look at the right
most blackboard.
00:32:14.230 --> 00:32:19.570
uEl of r, I must square it
because the wave function
00:32:19.570 --> 00:32:20.770
is squared.
00:32:20.770 --> 00:32:22.720
Over r squared.
00:32:22.720 --> 00:32:31.565
And then I have Ylm star of
theta phi Ylm of theta phi.
00:32:34.560 --> 00:32:37.330
And if this is supposed
to be normalized,
00:32:37.330 --> 00:32:39.245
this is supposed
to be the number 1.
00:32:42.280 --> 00:32:46.220
Well happily, this
part, this is why
00:32:46.220 --> 00:32:50.570
we needed to talk a little
about spherical harmonics.
00:32:50.570 --> 00:32:55.630
This integral is 1, because
it corresponds precisely
00:32:55.630 --> 00:32:59.650
to l equal l prime
m equal m prime.
00:32:59.650 --> 00:33:04.610
And look how lucky
or nice this is.
00:33:04.610 --> 00:33:07.020
r squared cancels
with r squared,
00:33:07.020 --> 00:33:10.120
so the final condition
is the integral from 0
00:33:10.120 --> 00:33:23.000
to infinity dr uEl of r
squared is equal to 1,
00:33:23.000 --> 00:33:26.770
which shows that
kind of the u really
00:33:26.770 --> 00:33:30.120
plays a role for wave
function and a line.
00:33:30.120 --> 00:33:32.760
And even though it was
a little complicated,
00:33:32.760 --> 00:33:35.520
there was the r here,
and angular dependence,
00:33:35.520 --> 00:33:38.530
and everything, a
good wave function
00:33:38.530 --> 00:33:43.240
is one that is just
think of psi as being u.
00:33:43.240 --> 00:33:46.630
A one dimensional wave
function psi being u,
00:33:46.630 --> 00:33:49.784
and if you can integrate
it square, you've got it.
00:33:49.784 --> 00:33:50.700
AUDIENCE: [INAUDIBLE].
00:33:53.890 --> 00:33:57.900
PROFESSOR: Because I
had to square this,
00:33:57.900 --> 00:33:59.662
so there was u over r.
00:33:59.662 --> 00:34:03.280
AUDIENCE: But
that's [INAUDIBLE].
00:34:03.280 --> 00:34:05.000
PROFESSOR: Oh, I'm sorry.
00:34:05.000 --> 00:34:09.449
That parenthesis is a remnant.
00:34:09.449 --> 00:34:11.284
I tried to erase it a little.
00:34:14.600 --> 00:34:16.830
It's not squared anymore.
00:34:16.830 --> 00:34:20.500
The square is on the
absolute value is r squared.
00:34:24.980 --> 00:34:27.276
So this is good news
for our interpretation.
00:34:34.230 --> 00:34:39.050
So now before I discuss the
peculiarities of the boundary
00:34:39.050 --> 00:34:45.352
conditions, I want to
introduce really the main point
00:34:45.352 --> 00:34:47.310
that we're going to
illustrate in this lecture.
00:34:47.310 --> 00:34:51.029
This is the thing that
should remain in your heads.
00:34:54.900 --> 00:35:00.390
It's a picture, but
it's an important one.
00:35:03.470 --> 00:35:06.330
When you want to
organize the spectrum,
00:35:06.330 --> 00:35:08.465
you'll draw the
following diagram.
00:35:12.740 --> 00:35:17.820
Energy is here and l here.
00:35:17.820 --> 00:35:19.590
And it's a funny
kind of diagram.
00:35:19.590 --> 00:35:21.660
It's not like a curve or a plot.
00:35:21.660 --> 00:35:26.760
It's like a histogram or
kind of thing like that.
00:35:26.760 --> 00:35:33.656
So what will happen is that you
have a one dimensional problem.
00:35:37.480 --> 00:35:44.200
If these potentials are normal,
there will be bound states.
00:35:44.200 --> 00:35:46.940
And let's consider the
case of bound states
00:35:46.940 --> 00:35:50.840
for the purposes of this
graph, just bound states.
00:35:50.840 --> 00:35:53.670
Now you look at
this, and you say OK,
00:35:53.670 --> 00:35:55.110
what am I supposed to do?
00:35:55.110 --> 00:36:00.335
I'm going to have states
for all values of l,
00:36:00.335 --> 00:36:03.560
and m, and probably
some energies.
00:36:03.560 --> 00:36:06.780
So m doesn't affect
the radial equation.
00:36:06.780 --> 00:36:08.160
That's very important.
00:36:08.160 --> 00:36:11.310
But l does, so I have
a different problem
00:36:11.310 --> 00:36:12.890
to solve for different l.
00:36:12.890 --> 00:36:18.790
So I will make my histogram
here and put here l
00:36:18.790 --> 00:36:22.570
equals 0 at this region.
00:36:22.570 --> 00:36:27.890
l equals 1, l equals 2,
l equals 3, and go on.
00:36:30.490 --> 00:36:35.200
Now suppose I fix an l.
00:36:35.200 --> 00:36:36.520
l is fixed.
00:36:36.520 --> 00:36:41.010
Now it's a Schrodinger equation
for a one dimensional problem.
00:36:41.010 --> 00:36:47.390
You would expect that if the
potential suitably grows, which
00:36:47.390 --> 00:36:51.790
is a typical case,
E will be quantized.
00:36:51.790 --> 00:36:54.830
And there will not
be degeneracies,
00:36:54.830 --> 00:36:57.740
because the bound state
spectrum in one dimension
00:36:57.740 --> 00:36:59.530
is not degenerate.
00:36:59.530 --> 00:37:03.730
So I should expect
that for each l there
00:37:03.730 --> 00:37:08.500
are going to be energy values
that are going to appear.
00:37:08.500 --> 00:37:14.780
So for l equals 0, I expect that
there will be some energy here
00:37:14.780 --> 00:37:16.600
for which I've got a state.
00:37:16.600 --> 00:37:19.490
And that line means
I got a state.
00:37:19.490 --> 00:37:23.385
And there's some energy here
that could be called E1,
00:37:23.385 --> 00:37:35.720
0 is the first energy that
is allowed with l equals 0.
00:37:35.720 --> 00:37:39.570
Then there will be
another one here maybe.
00:37:39.570 --> 00:37:44.480
E-- I'll write it down-- 2,0.
00:37:44.480 --> 00:37:53.620
So basically I'm labeling the
energies with En,l which means
00:37:53.620 --> 00:37:56.970
the first solution
with l equals 0,
00:37:56.970 --> 00:37:58.980
the second solution
with l equals 0,
00:37:58.980 --> 00:38:04.430
the third solution E 3,0.
00:38:04.430 --> 00:38:08.460
Then you come to l
equals 1, and you
00:38:08.460 --> 00:38:11.640
must solve the equation again.
00:38:11.640 --> 00:38:15.530
And then for l equal 1, there
will be the lowest energy,
00:38:15.530 --> 00:38:19.290
the ground state energy of
the l equal 1 potential,
00:38:19.290 --> 00:38:21.240
and then higher and higher.
00:38:21.240 --> 00:38:25.130
Since the l equal 1
potential is higher
00:38:25.130 --> 00:38:30.270
than the l equals 0
potential, it's higher up.
00:38:30.270 --> 00:38:33.010
The energies should
be higher up,
00:38:33.010 --> 00:38:35.600
at least the first
one should be.
00:38:35.600 --> 00:38:38.980
And therefore the
first one could
00:38:38.980 --> 00:38:44.190
be a little higher than this,
or maybe by some accident
00:38:44.190 --> 00:38:49.180
it just fits here, or
maybe it should fit here.
00:38:49.180 --> 00:38:51.540
Well, we don't know
but know, but there's
00:38:51.540 --> 00:38:55.760
no obvious reason why it
should, so I'll put it here.
00:38:55.760 --> 00:38:56.750
l equals 1.
00:38:56.750 --> 00:39:04.150
And this would be E1,1.
00:39:04.150 --> 00:39:07.460
The first state with l equals 1.
00:39:07.460 --> 00:39:11.150
Then here it could be E2,1.
00:39:11.150 --> 00:39:16.040
The second state with l
equal 1 and higher up.
00:39:16.040 --> 00:39:21.570
And then for l equal-- my
diagram is a little too big.
00:39:21.570 --> 00:39:25.200
E1,1.
00:39:25.200 --> 00:39:25.930
E2,1.
00:39:25.930 --> 00:39:30.230
And then you have states here,
so maybe this one, l equals 2,
00:39:30.230 --> 00:39:31.590
I don't know where it goes.
00:39:31.590 --> 00:39:36.350
It just has to be higher than
this one, so I'll put it here.
00:39:36.350 --> 00:39:39.760
And this will be E1,2.
00:39:39.760 --> 00:39:43.230
Maybe there's an E2,2.
00:39:43.230 --> 00:39:49.980
And here an E1,3.
00:39:49.980 --> 00:39:54.210
But this is the answer
to your problem.
00:39:54.210 --> 00:39:59.520
That's the energy levels
of a central potential.
00:39:59.520 --> 00:40:02.510
So it's a good,
nice little diagram
00:40:02.510 --> 00:40:07.210
in which you put the states,
you put the little line wherever
00:40:07.210 --> 00:40:08.590
you find the state.
00:40:08.590 --> 00:40:12.140
And for l equals 0,
you have those states.
00:40:12.140 --> 00:40:17.760
Now because there's no
degeneracies in the bound
00:40:17.760 --> 00:40:21.800
states of a one
dimensional potential,
00:40:21.800 --> 00:40:27.000
I don't have two lines here
that coincide, because there's
00:40:27.000 --> 00:40:30.180
no two states with
the same energy here.
00:40:30.180 --> 00:40:34.030
It's just one state.
00:40:34.030 --> 00:40:35.200
And this one here.
00:40:35.200 --> 00:40:38.460
I cannot have two things there.
00:40:38.460 --> 00:40:41.500
That's pretty important to.
00:40:41.500 --> 00:40:46.230
So you have a list
of states here.
00:40:46.230 --> 00:40:50.260
And just one state here, one
state, but as you can see,
00:40:50.260 --> 00:40:54.500
you're probably are catching
me in a little wrong play
00:40:54.500 --> 00:40:58.420
of words, because I say
there's one state here.
00:40:58.420 --> 00:41:01.130
Yes, it's one state,
because it's l equals 0.
00:41:01.130 --> 00:41:02.570
One state, one state.
00:41:02.570 --> 00:41:05.890
But this state,
which is one single--
00:41:05.890 --> 00:41:11.410
this should be called one
single l equal 1 multiplet.
00:41:11.410 --> 00:41:15.660
So this is not really one
state at the end of the day.
00:41:15.660 --> 00:41:19.940
It's one state of the one
dimensional radial equation,
00:41:19.940 --> 00:41:24.600
but you know that l
equals 1 comes accompanied
00:41:24.600 --> 00:41:28.250
with three values of m.
00:41:28.250 --> 00:41:31.750
So there's three states
that are degenerate,
00:41:31.750 --> 00:41:33.560
because they have
the same energy.
00:41:33.560 --> 00:41:37.430
The energy doesn't depend on l.
00:41:37.430 --> 00:41:43.440
So this thing is an
l equal 1 multiplet,
00:41:43.440 --> 00:41:47.730
which means really three states.
00:41:47.730 --> 00:41:50.090
And this is three states.
00:41:50.090 --> 00:41:52.420
And this is three states.
00:41:52.420 --> 00:41:57.860
And this is 1 l equal
2 multiplet, which
00:41:57.860 --> 00:42:03.160
has possibility of m equals
2, 1, 0 minus 1 and minus 2.
00:42:03.160 --> 00:42:06.960
So in this state is just
one l equal 2 multiplet,
00:42:06.960 --> 00:42:12.180
but it really means five states
of the central potential.
00:42:12.180 --> 00:42:15.860
Five degenerate
states, because the m
00:42:15.860 --> 00:42:17.810
doesn't change the energy.
00:42:17.810 --> 00:42:19.480
And this is five states.
00:42:19.480 --> 00:42:21.850
And this is seven states.
00:42:21.850 --> 00:42:28.250
One l equal 3 multiplet,
which contains seven states.
00:42:28.250 --> 00:42:30.870
OK, so questions?
00:42:30.870 --> 00:42:35.240
This is the most
important graph.
00:42:35.240 --> 00:42:37.330
If you have that
picture in your head,
00:42:37.330 --> 00:42:39.900
then you can
understand really where
00:42:39.900 --> 00:42:41.850
you're going with any potential.
00:42:41.850 --> 00:42:44.334
Any confusion here
above the notation?
00:42:44.334 --> 00:42:44.834
Yes?
00:42:44.834 --> 00:42:46.929
AUDIENCE: So normally
when we think about a one
00:42:46.929 --> 00:42:49.220
dimensional problem, we say
that there's no degeneracy.
00:42:49.220 --> 00:42:51.770
Not really.
00:42:51.770 --> 00:42:53.287
No multiple degeneracy,
so should we
00:42:53.287 --> 00:42:56.816
think of the radial equation as
having copies for each m value
00:42:56.816 --> 00:42:59.760
and each having the
same eigenvalue?
00:42:59.760 --> 00:43:02.450
PROFESSOR: I don't
think it's necessary.
00:43:02.450 --> 00:43:05.930
You see, you've got your uEl.
00:43:05.930 --> 00:43:07.660
And you have here you solutions.
00:43:07.660 --> 00:43:10.700
Once the uEl is
good, you're supposed
00:43:10.700 --> 00:43:13.340
to be able to put any Ylm.
00:43:13.340 --> 00:43:18.060
So put l, and now the m's that
are allowed are solutions.
00:43:18.060 --> 00:43:19.560
You're solving the problem.
00:43:19.560 --> 00:43:26.100
So think of a master radial
function as good for a fixed l,
00:43:26.100 --> 00:43:30.030
and therefore it works
for all values of m.
00:43:30.030 --> 00:43:33.310
But don't try to think of
many copies of this equation.
00:43:33.310 --> 00:43:37.010
I don't think it would help you.
00:43:37.010 --> 00:43:38.060
Any other questions?
00:43:42.956 --> 00:43:44.932
Yes?
00:43:44.932 --> 00:43:47.155
AUDIENCE: Sorry to ask,
but if you could just
00:43:47.155 --> 00:43:49.790
review how is degeneracy
built one more time?
00:43:49.790 --> 00:43:50.670
PROFESSOR: Yeah.
00:43:50.670 --> 00:43:53.390
Remember last time we
were talking about,
00:43:53.390 --> 00:44:02.020
for example, what is a
j equal to multiplet.
00:44:02.020 --> 00:44:05.860
Well, these were a collection
of states jm with j
00:44:05.860 --> 00:44:09.310
equals 2 an m sum value.
00:44:09.310 --> 00:44:14.320
And they are all obtained by
acting with angular momentum
00:44:14.320 --> 00:44:15.990
operators in each other.
00:44:15.990 --> 00:44:17.260
And there are five states.
00:44:17.260 --> 00:44:28.310
The 2,2, the 2,1, the 2,0, the
2, minus 1, and the 2, minus 2.
00:44:28.310 --> 00:44:30.240
And all these
states are obtained
00:44:30.240 --> 00:44:35.630
by acting with, say, lowering
operators l minus and this.
00:44:35.630 --> 00:44:38.930
Now all these angular
momentum operators,
00:44:38.930 --> 00:44:43.560
all of the Li's commute
with the Hamiltonian.
00:44:43.560 --> 00:44:46.200
Therefore all of
these states are
00:44:46.200 --> 00:44:49.720
obtained by acting with Li
must have the same energy.
00:44:49.720 --> 00:44:52.390
That's why we say that
this comes in a multiplet.
00:44:52.390 --> 00:44:59.100
So when you get j-- in this case
we'll call it l-- l equals 2.
00:44:59.100 --> 00:45:01.880
You get five states.
00:45:01.880 --> 00:45:04.980
They correspond to the
various values of m.
00:45:04.980 --> 00:45:07.320
So when you did that
radial equation that
00:45:07.320 --> 00:45:10.820
has a solution for
l equals 2, you're
00:45:10.820 --> 00:45:12.450
getting the full multiplet.
00:45:12.450 --> 00:45:14.460
You're getting five states.
00:45:14.460 --> 00:45:16.970
1 l equal 2 multiplet.
00:45:16.970 --> 00:45:19.280
That's why one line here.
00:45:19.280 --> 00:45:21.555
That is equivalent
to five states.
00:45:24.740 --> 00:45:31.770
OK, so that diagram, of course,
is really quite important.
00:45:31.770 --> 00:45:41.430
So now we want to understand
the boundary conditions.
00:45:41.430 --> 00:45:43.050
So we have here this.
00:45:43.050 --> 00:45:46.890
So this probably
shouldn't erase yet.
00:45:46.890 --> 00:45:48.480
Let's do the
boundary conditions.
00:45:58.390 --> 00:46:03.315
So behavior here
at r equals to 0.
00:46:08.720 --> 00:46:11.300
At r going to 0.
00:46:17.360 --> 00:46:21.780
The first claim is that
surprisingly, you would think,
00:46:21.780 --> 00:46:24.560
well, normalization is king.
00:46:24.560 --> 00:46:26.760
If it's normalized, it's good.
00:46:26.760 --> 00:46:29.470
So just any number.
00:46:29.470 --> 00:46:33.910
Just don't let it diverge
near 0, and that will be OK.
00:46:33.910 --> 00:46:36.940
But it turns out
that that's not true.
00:46:36.940 --> 00:46:38.290
It's not right.
00:46:38.290 --> 00:46:56.100
And you need the limit as r goes
to 0 of uEl of r be equal to 0.
00:46:56.100 --> 00:47:03.610
And we'll take this and
explore the simplest case.
00:47:03.610 --> 00:47:08.860
That is corresponds to
saying what if the limit of r
00:47:08.860 --> 00:47:15.270
goes to 0 or uEl of
r was a constant?
00:47:15.270 --> 00:47:17.870
What goes wrong?
00:47:17.870 --> 00:47:21.250
Certainly normalization
doesn't go wrong.
00:47:21.250 --> 00:47:23.880
It can be a constant.
00:47:23.880 --> 00:47:26.830
u could be like that, and
it would be normalized,
00:47:26.830 --> 00:47:29.270
and that doesn't go wrong.
00:47:29.270 --> 00:47:31.670
So let's look at
the wave function.
00:47:31.670 --> 00:47:35.440
What happens with this?
00:47:35.440 --> 00:47:38.160
I actually will
take for simplicity,
00:47:38.160 --> 00:47:45.840
because we'll analyze it later,
the example of l equals 0.
00:47:45.840 --> 00:47:48.720
So let's put even 0.
00:47:48.720 --> 00:47:49.750
l equals 0.
00:47:53.830 --> 00:48:04.006
Well, suppose you look
at the wave function now,
00:48:04.006 --> 00:48:06.700
and how does it look?
00:48:06.700 --> 00:48:16.270
Psi of E0-- if l is equal
to 0, m must be equal to 0--
00:48:16.270 --> 00:48:21.990
would be this u over
r times a constant.
00:48:21.990 --> 00:48:25.520
So a constant, because
y 0, 0 is a constant.
00:48:25.520 --> 00:48:30.640
And then you uE0 of r over r.
00:48:30.640 --> 00:48:41.050
So when r approaches 0, psi
goes like c prime over r,
00:48:41.050 --> 00:48:44.140
some other constant over r.
00:48:44.140 --> 00:48:46.240
So I'm doing
something very simple.
00:48:46.240 --> 00:48:53.360
I'm saying if uE0 is approaching
the constant at the origin,
00:48:53.360 --> 00:48:57.930
if it's uE0, well, this is
a constant because it's 0,0.
00:48:57.930 --> 00:48:59.800
So this is going to constant.
00:48:59.800 --> 00:49:01.890
So at the end of the
day, the wave function
00:49:01.890 --> 00:49:03.360
looks like 1 over r.
00:49:08.280 --> 00:49:16.820
But this is impossible, because
the Schrodinger equation H
00:49:16.820 --> 00:49:26.070
psi has minus h squared over 2m
Laplacian on psi plus dot dot
00:49:26.070 --> 00:49:26.570
dot.
00:49:30.150 --> 00:49:39.330
And the up Laplacian of 1 over
r is minus 4 pi times a delta
00:49:39.330 --> 00:49:43.355
function at x equals 0.
00:49:43.355 --> 00:49:49.220
So this means that the
Schrodinger equation,
00:49:49.220 --> 00:49:52.680
you think oh I put
psi equals c over r.
00:49:52.680 --> 00:49:56.300
Well, if you calculate the
Laplacian, it seems to be 0.
00:49:56.300 --> 00:49:59.680
But if you're more careful,
as you know for [? emm ?]
00:49:59.680 --> 00:50:03.750
the Laplacian of 1 over r is
minus 4 pi times the delta
00:50:03.750 --> 00:50:05.240
function.
00:50:05.240 --> 00:50:09.990
So in the Schrodinger
equation, the kinetic term
00:50:09.990 --> 00:50:12.330
produces a delta function.
00:50:12.330 --> 00:50:14.720
There's no reason
to believe there's
00:50:14.720 --> 00:50:17.600
a delta function
in the potential.
00:50:17.600 --> 00:50:21.310
We'll not try such
crazy potentials.
00:50:21.310 --> 00:50:25.460
A delta function in a one
dimensional potential,
00:50:25.460 --> 00:50:27.800
you've got the solution.
00:50:27.800 --> 00:50:31.480
A delta function in a
three dimensional potential
00:50:31.480 --> 00:50:35.235
is absolutely crazy.
00:50:35.235 --> 00:50:38.760
It has infinite number
of bound states,
00:50:38.760 --> 00:50:40.840
and they just go
all the way down
00:50:40.840 --> 00:50:43.030
to energies of minus infinity.
00:50:43.030 --> 00:50:46.190
It's a very horrendous
thing, a delta function
00:50:46.190 --> 00:50:49.660
in three dimensions,
for quantum mechanics.
00:50:49.660 --> 00:50:54.070
So this thing, there's no delta
function in the potential.
00:50:54.070 --> 00:50:56.630
And you've got a delta
function from the kinetic term.
00:50:56.630 --> 00:50:58.800
You're not going to
be able to cancel it.
00:50:58.800 --> 00:51:01.840
This is not a solution.
00:51:07.570 --> 00:51:14.060
So you really cannot
approach a constant there.
00:51:14.060 --> 00:51:16.190
It's quite bad.
00:51:16.190 --> 00:51:20.130
So the wave functions
will have to vanish,
00:51:20.130 --> 00:51:25.990
and we can prove that, or at
least under some circumstances
00:51:25.990 --> 00:51:26.920
prove it.
00:51:26.920 --> 00:51:30.310
And as all these
things are, they all
00:51:30.310 --> 00:51:33.500
depend on how crazy
potentials you want to accept.
00:51:33.500 --> 00:51:37.110
So we should say something.
00:51:37.110 --> 00:51:41.990
So I'll say something
about these potentials,
00:51:41.990 --> 00:51:46.606
and we'll prove a result.
00:51:50.830 --> 00:52:09.460
So my statement will be the
centrifugal barrier, which
00:52:09.460 --> 00:52:14.860
is a name for this
part of the potential,
00:52:14.860 --> 00:52:23.370
dominates as r goes to 0.
00:52:23.370 --> 00:52:27.050
If this doesn't happen,
all bets are off.
00:52:27.050 --> 00:52:33.720
So let's assume that v of
r, maybe it's 1 over r,
00:52:33.720 --> 00:52:36.210
but it's not worse
than 1 over r squared.
00:52:36.210 --> 00:52:39.370
It's 1 over r
cubed, for example,
00:52:39.370 --> 00:52:41.430
or something like that.
00:52:41.430 --> 00:52:43.790
You would have to
analyze it from scratch
00:52:43.790 --> 00:52:45.080
if it would be that bad.
00:52:45.080 --> 00:52:50.260
But I will assume that the
centrifugal barrier dominates.
00:52:50.260 --> 00:52:54.270
And then look at the
differential equation.
00:52:54.270 --> 00:52:56.430
Well, what differential
equation do I have?
00:52:56.430 --> 00:53:04.475
Well, I have this and this.
00:53:04.475 --> 00:53:07.210
This thing is less
important than that,
00:53:07.210 --> 00:53:10.970
and this is also less
important, because this is u
00:53:10.970 --> 00:53:12.530
divided by r squared.
00:53:12.530 --> 00:53:14.250
And here is just u.
00:53:14.250 --> 00:53:17.380
So this is certainly
less important than that,
00:53:17.380 --> 00:53:19.610
and this is less
important than that,
00:53:19.610 --> 00:53:22.740
and if I want to have
some variation of u,
00:53:22.740 --> 00:53:26.170
or understand how it
varies, I must keep this.
00:53:26.170 --> 00:53:31.880
So at this order, I should
keep just the kinetic term
00:53:31.880 --> 00:53:38.490
h squared over 2m d
second dr squared u of El.
00:53:42.400 --> 00:53:49.840
And h squared l times l
plus 1 over 2 mr squared.
00:53:49.840 --> 00:53:55.770
And I will try to cancel these
two to explore how the wave
00:53:55.770 --> 00:53:59.360
function looks near or equal 0.
00:53:59.360 --> 00:54:02.300
These are the two most important
terms of the differential
00:54:02.300 --> 00:54:06.050
equation, so I have the
right to keep those, and try
00:54:06.050 --> 00:54:12.450
to balance them out to leading
order, and see what I get.
00:54:12.450 --> 00:54:16.960
So all the h squared
over 2m's go away.
00:54:16.960 --> 00:54:26.300
So this is equivalent to
d second uEl dr squared is
00:54:26.300 --> 00:54:32.165
equal to l times l plus
1 uEl over r squared.
00:54:35.810 --> 00:54:39.400
And this is solved
by a power uEl.
00:54:43.080 --> 00:54:50.900
You can try r to the
s, some number s.
00:54:50.900 --> 00:55:01.390
And then this thing gives
you s times s minus 1.
00:55:01.390 --> 00:55:06.195
Taking two derivatives is
equal to l times l plus 1.
00:55:11.580 --> 00:55:14.830
As you take two derivatives,
you lose two powers of r,
00:55:14.830 --> 00:55:17.960
so it will work out.
00:55:17.960 --> 00:55:21.190
And from here, you see
that the possible solutions
00:55:21.190 --> 00:55:25.725
are s equals l plus 1.
00:55:25.725 --> 00:55:29.100
And s equals 2 minus l.
00:55:33.660 --> 00:55:39.310
So this corresponds
to a uEl that
00:55:39.310 --> 00:55:44.890
goes like r to the
l plus 1, or a uEl
00:55:44.890 --> 00:55:48.958
that goes like 1
over r to the l.
00:55:52.782 --> 00:55:56.660
This Is far too singular.
00:55:56.660 --> 00:56:01.020
For l equals 0, we argued
that the wave function
00:56:01.020 --> 00:56:02.586
should go like a constant.
00:56:06.900 --> 00:56:09.920
I'm sorry, cannot
go like a constant.
00:56:09.920 --> 00:56:11.860
Must vanish.
00:56:11.860 --> 00:56:13.480
This is not possible.
00:56:13.480 --> 00:56:14.510
It's not a solution.
00:56:14.510 --> 00:56:16.020
It must vanish.
00:56:16.020 --> 00:56:22.200
For l equals 0, uE0 goes
like r and vanishes.
00:56:22.200 --> 00:56:25.650
So that's consistent,
and this is good.
00:56:25.650 --> 00:56:29.650
For l equals 0, this would
be like a constant as well
00:56:29.650 --> 00:56:30.400
and would be fine.
00:56:30.400 --> 00:56:34.790
But for l equals 1
already, this is 1 over r,
00:56:34.790 --> 00:56:36.035
and this is not normalizable.
00:56:38.900 --> 00:56:50.210
So this time this is not
normalizable for l greater
00:56:50.210 --> 00:56:52.320
or equal than one.
00:56:52.320 --> 00:56:59.170
So this is the answer
[INAUDIBLE] this assumption,
00:56:59.170 --> 00:57:01.880
which is a very
reasonable assumption.
00:57:01.880 --> 00:57:05.480
But if you don't have
that you have to beware.
00:57:08.010 --> 00:57:16.490
OK, this is our
condition for u there.
00:57:16.490 --> 00:57:24.090
And so uEl goes like
this as r goes to 0.
00:57:24.090 --> 00:57:27.840
It would be the whole answer.
00:57:27.840 --> 00:57:37.910
So f, if you care about f still,
which is what appears here,
00:57:37.910 --> 00:57:40.970
goes like u divided by r.
00:57:40.970 --> 00:57:50.562
So fEl goes like cr to the l.
00:57:54.258 --> 00:58:02.460
And when l is equal to 0,
f behaves like a constant.
00:58:02.460 --> 00:58:06.400
u vanishes for l
equal to 0, but f
00:58:06.400 --> 00:58:08.760
goes like a
constant, which means
00:58:08.760 --> 00:58:13.540
that if you take 0
orbital angular momentum,
00:58:13.540 --> 00:58:16.670
you may have some
probability of finding
00:58:16.670 --> 00:58:21.660
the particle at the origin,
because this f behaves
00:58:21.660 --> 00:58:25.520
like a constant for l equals 0.
00:58:25.520 --> 00:58:28.720
On the other hand,
for any higher l,
00:58:28.720 --> 00:58:31.920
f will also vanish
at the origin.
00:58:31.920 --> 00:58:36.840
And that is intuitively said
that the centrifugal barrier
00:58:36.840 --> 00:58:39.990
prevents the particle
from reaching the origin.
00:58:39.990 --> 00:58:42.840
There's a barrier,
a potential barrier.
00:58:42.840 --> 00:58:47.020
This potential is
1 over r squared.
00:58:47.020 --> 00:58:49.420
Doesn't let you go to
close to the origin.
00:58:49.420 --> 00:58:53.480
But that potential
disappears for l equals 0,
00:58:53.480 --> 00:58:56.840
and therefore the particle
can reach the origin.
00:58:56.840 --> 00:59:00.500
But only for l equals 0
it can reach the origin.
00:59:03.120 --> 00:59:08.710
OK, one more thing.
00:59:08.710 --> 00:59:13.320
Behavior near infinity
is of interest as well.
00:59:20.130 --> 00:59:25.260
So what happens for
r goes to infinity?
00:59:32.440 --> 00:59:34.850
Well, for r goes to
infinity, you also
00:59:34.850 --> 00:59:39.770
have to be a little
careful what you assume.
00:59:39.770 --> 00:59:44.000
I wish I could tell you it's
always like this, but it's not.
00:59:44.000 --> 00:59:46.455
It's rich in all
kinds of problems.
00:59:49.040 --> 00:59:51.050
So there's two
cases where there's
00:59:51.050 --> 00:59:53.500
an analysis that is simple.
00:59:53.500 --> 00:59:59.890
Suppose v of r is equal to 0
for r greater than some r0.
01:00:02.520 --> 01:00:12.896
Or r times v of f goes to
0 as r goes to infinity.
01:00:12.896 --> 01:00:13.645
Two possibilities.
01:00:16.660 --> 01:00:22.440
The potential is plane
0 after some distance.
01:00:22.440 --> 01:00:30.140
Or the potential
multiplied by r goes to 0
01:00:30.140 --> 01:00:31.310
as r goes to infinity.
01:00:31.310 --> 01:00:36.660
And you would say, look, you've
missed the most important case.
01:00:36.660 --> 01:00:40.390
The hydrogen atom, the
potential is 1 over r.
01:00:40.390 --> 01:00:43.640
r times v of r doesn't go to 0.
01:00:43.640 --> 01:00:45.850
And indeed, what I'm
going to write here
01:00:45.850 --> 01:00:49.380
doesn't quite apply to the wave
functions of the hydrogen atom.
01:00:49.380 --> 01:00:51.440
They're a little unusual.
01:00:51.440 --> 01:00:56.805
The potential of the hydrogen
atom is felt quite far away.
01:00:59.480 --> 01:01:04.520
So never the less, if you
have those conditions,
01:01:04.520 --> 01:01:13.000
we can ignore the potential
as we go far away.
01:01:13.000 --> 01:01:17.595
And we'll consider the
following situation.
01:01:33.100 --> 01:01:37.930
Look that the centrifugal
barrier satisfies this as well.
01:01:37.930 --> 01:01:41.330
So the full effective
potential satisfies.
01:01:41.330 --> 01:01:44.870
If v of r satisfies
that, r times 1
01:01:44.870 --> 01:01:48.780
over r squared of effective
potential also satisfies that.
01:01:48.780 --> 01:01:52.250
So we can ignore
all the potential,
01:01:52.250 --> 01:01:58.170
and we're left
ignore the effective.
01:01:58.170 --> 01:02:01.440
And therefore we're left
with minus h squared
01:02:01.440 --> 01:02:09.793
over 2m d second uEl dr
squared is equal to EuEl.
01:02:13.900 --> 01:02:17.920
And that's a very
trivial equation.
01:02:17.920 --> 01:02:19.481
Yes, Matt?
01:02:19.481 --> 01:02:20.954
AUDIENCE: When you
say v of r goes
01:02:20.954 --> 01:02:23.409
to 0 for r greater
than [INAUDIBLE] 0.
01:02:23.409 --> 01:02:25.870
Are you effectively
[INAUDIBLE] the potential?
01:02:25.870 --> 01:02:30.870
PROFESSOR: Right, there may
be some potentials like this.
01:02:30.870 --> 01:02:34.340
A potential that is like that.
01:02:34.340 --> 01:02:38.180
An attractive potential, and it
vanishes after some distance.
01:02:38.180 --> 01:02:41.940
Or a repulsive potential that
vanishes after some distance.
01:02:41.940 --> 01:02:44.440
AUDIENCE: But say the potential
was a [INAUDIBLE] potential.
01:02:44.440 --> 01:02:47.072
Are you just approximating it
to 0 after it's [INAUDIBLE]?
01:02:47.072 --> 01:02:49.280
PROFESSOR: Well, if I'm in
the [INAUDIBLE] potential,
01:02:49.280 --> 01:02:53.460
unfortunately I'm
neither here nor here,
01:02:53.460 --> 01:02:55.830
so this doesn't apply.
01:02:55.830 --> 01:02:58.390
So the [INAUDIBLE]
potential is an exception.
01:02:58.390 --> 01:02:59.994
The solutions are
a little more--
01:02:59.994 --> 01:03:02.160
AUDIENCE: The conditions
you're saying. [INAUDIBLE].
01:03:02.160 --> 01:03:05.230
PROFESSOR: So these
are conditions
01:03:05.230 --> 01:03:08.490
that allow me to say something.
01:03:08.490 --> 01:03:10.900
If they're not
satisfied, I sort of
01:03:10.900 --> 01:03:14.810
have to analyze
them case by case.
01:03:14.810 --> 01:03:18.110
That's the price we have to pay.
01:03:18.110 --> 01:03:22.280
It's a little more complicated
than you would think naively.
01:03:22.280 --> 01:03:27.000
Now here, it's interesting to
consider two possibilities.
01:03:27.000 --> 01:03:30.800
The case when E is less
than 0, or the case when
01:03:30.800 --> 01:03:33.660
E is greater than 0.
01:03:33.660 --> 01:03:37.910
So scattering solutions
or bound state solutions.
01:03:37.910 --> 01:03:42.800
For these ones, if the
energy is less than 0
01:03:42.800 --> 01:03:47.480
and there's no potential, you're
in the forbidden zone far away,
01:03:47.480 --> 01:03:51.080
so you must have a
decaying exponential.
01:03:51.080 --> 01:03:57.200
El goes like exponential
of minus square root
01:03:57.200 --> 01:04:03.280
of 2m E over h squared r.
01:04:03.280 --> 01:04:04.845
That solves that equation.
01:04:07.962 --> 01:04:10.040
You see, the solution
of these things
01:04:10.040 --> 01:04:15.810
are either exponential
decays or exponential growths
01:04:15.810 --> 01:04:19.040
and oscillatory solutions,
sines and cosines,
01:04:19.040 --> 01:04:21.790
or E to the i things.
01:04:21.790 --> 01:04:27.840
So here we have a decay,
because with energy less than 0,
01:04:27.840 --> 01:04:29.080
the potential is 0.
01:04:29.080 --> 01:04:32.850
So you're in a forbidden region,
so you must decay like that.
01:04:32.850 --> 01:04:35.520
In this hydrogen
atom what happens
01:04:35.520 --> 01:04:39.090
is that there's a power
of r multiplying here.
01:04:39.090 --> 01:04:43.710
Like r to the n, or r to the
k or something like that.
01:04:43.710 --> 01:04:51.216
If E is less than 0, you
have uE equal exponential
01:04:51.216 --> 01:05:04.561
of plus minus ikr, where k
is square root of 2m E over h
01:05:04.561 --> 01:05:05.060
squared.
01:05:08.420 --> 01:05:11.670
And those, again,
solve that equation.
01:05:11.670 --> 01:05:16.280
And they are sort of
wave solutions far away.
01:05:22.350 --> 01:05:25.150
Now with this
information, the behavior
01:05:25.150 --> 01:05:31.460
of the u's near the origin, the
behavior of the u's far away,
01:05:31.460 --> 01:05:35.040
you can then make
qualitative plots
01:05:35.040 --> 01:05:37.820
of how solutions would
look at the origin.
01:05:37.820 --> 01:05:39.610
They grow up like r to the l.
01:05:39.610 --> 01:05:42.150
Then it's a one
dimensional potential,
01:05:42.150 --> 01:05:46.310
so they oscillate maybe, but
then decay exponentially.
01:05:46.310 --> 01:05:47.940
And the kind of
thing you used to do
01:05:47.940 --> 01:05:51.170
in 804 of plotting
how things look,
01:05:51.170 --> 01:05:55.550
it's feasible at this stage.
01:05:55.550 --> 01:05:59.580
So it's about time
to do examples.
01:05:59.580 --> 01:06:01.260
I have three examples.
01:06:01.260 --> 01:06:04.900
Given time, maybe
I'll get to two.
01:06:04.900 --> 01:06:06.030
That's OK.
01:06:06.030 --> 01:06:08.320
The last example is
kind of the cutest,
01:06:08.320 --> 01:06:13.550
but maybe it's OK to
leave it for Monday.
01:06:13.550 --> 01:06:17.100
So are there
questions about this
01:06:17.100 --> 01:06:19.015
before we begin our examples?
01:06:24.840 --> 01:06:26.262
Andrew?
01:06:26.262 --> 01:06:30.472
AUDIENCE: What is consumption
of [INAUDIBLE] [? barrier ?]
01:06:30.472 --> 01:06:30.972
dominates.
01:06:30.972 --> 01:06:33.340
But why is that a
reasonable assumptions?
01:06:33.340 --> 01:06:35.720
PROFESSOR: Well,
potentials that are just
01:06:35.720 --> 01:06:41.450
too singular at the
origin are not common.
01:06:41.450 --> 01:06:45.820
Just doesn't happen.
01:06:45.820 --> 01:06:50.770
So mathematically
you could try them,
01:06:50.770 --> 01:06:54.730
but I actually don't
know of useful examples
01:06:54.730 --> 01:06:57.315
if a potential is very
singular at the origin.
01:07:00.595 --> 01:07:03.976
AUDIENCE: [INAUDIBLE] in
the potential [INAUDIBLE]
01:07:03.976 --> 01:07:05.425
the centrifugal barrier.
01:07:05.425 --> 01:07:08.323
That [INAUDIBLE].
01:07:08.323 --> 01:07:09.310
PROFESSOR: Right.
01:07:09.310 --> 01:07:12.900
An effective potential, the
potential doesn't blow up--
01:07:12.900 --> 01:07:17.880
your potential doesn't blow
up more than 1 over r squared
01:07:17.880 --> 01:07:19.090
or something like that.
01:07:19.090 --> 01:07:24.640
So we'll just take it like that.
01:07:24.640 --> 01:07:31.720
OK, our first example
is the free particle.
01:07:31.720 --> 01:07:33.260
You would say come on.
01:07:33.260 --> 01:07:34.890
That's ridiculous.
01:07:34.890 --> 01:07:35.525
Too simple.
01:07:38.510 --> 01:07:42.980
But it's fairly non-trivial
in spherical coordinates.
01:07:42.980 --> 01:07:45.720
And you say, well, so what.
01:07:45.720 --> 01:07:49.550
Free particles, you say
what the momentum is.
01:07:49.550 --> 01:07:51.070
You know the energy.
01:07:51.070 --> 01:07:52.780
How do you label the states?
01:07:52.780 --> 01:07:55.250
You label them by three momenta.
01:07:55.250 --> 01:07:58.540
Or energy and direction.
01:07:58.540 --> 01:08:01.830
So momentum
eigenstates, for example
01:08:01.830 --> 01:08:04.000
But in spherical
coordinates, these will not
01:08:04.000 --> 01:08:06.080
be momentum
eigenstates, and these
01:08:06.080 --> 01:08:08.220
are interesting
because they allow
01:08:08.220 --> 01:08:12.450
us to solve for more
complicated problems, in fact.
01:08:12.450 --> 01:08:15.390
And they allow you to
understand scattering out
01:08:15.390 --> 01:08:16.810
of central potential.
01:08:16.810 --> 01:08:18.725
So these are actually
pretty important.
01:08:22.410 --> 01:08:25.010
You can label these
things by three numbers.
01:08:25.010 --> 01:08:27.729
p1, p2, p3.
01:08:27.729 --> 01:08:33.840
Or energy and theta and phi,
the directions of the momenta.
01:08:33.840 --> 01:08:39.700
What we're going to label
them are by energy l and m.
01:08:39.700 --> 01:08:45.399
So you might say how do we
compare all these infinities,
01:08:45.399 --> 01:08:48.779
but it somehow works out.
01:08:48.779 --> 01:08:53.600
There's the same number of
states really in either way.
01:08:53.600 --> 01:08:57.279
So what do we have?
01:08:57.279 --> 01:09:01.300
It's a potential
that v is equal to 0.
01:09:01.300 --> 01:09:03.790
So let's write the
differential equation.
01:09:03.790 --> 01:09:06.390
v is equal to 0.
01:09:06.390 --> 01:09:07.920
But not v effective.
01:09:07.920 --> 01:09:16.760
So you have minus h squared
over 2m d second uEl
01:09:16.760 --> 01:09:23.950
dr squared plus h
squared over 2m l times
01:09:23.950 --> 01:09:32.279
l plus 1 over r
squared uEl equal EuEl.
01:09:32.279 --> 01:09:35.494
This is actually
quite interesting.
01:09:35.494 --> 01:09:39.590
As you will see, it's a bit
puzzling the first time.
01:09:39.590 --> 01:09:43.649
Well, let's cancel
this h squared over 2m,
01:09:43.649 --> 01:09:47.270
because they're
kind of annoying.
01:09:47.270 --> 01:09:54.090
So we'll put d second uEl over
dr squared with a minus-- I'll
01:09:54.090 --> 01:10:03.350
keep that minus-- plus l times
l plus 1 over r squared uEl.
01:10:03.350 --> 01:10:08.610
And here I'll put k
squared times uEl.
01:10:08.610 --> 01:10:13.070
And k squared is the
same k as before.
01:10:13.070 --> 01:10:16.880
And E is positive because
you have a free particle.
01:10:16.880 --> 01:10:19.330
E is positive.
01:10:19.330 --> 01:10:25.940
And k squared is given by
this, 2m E over h squared.
01:10:25.940 --> 01:10:28.145
So this is the equation
we have to solve.
01:10:33.330 --> 01:10:40.990
And it's kind of interesting,
because on the one hand,
01:10:40.990 --> 01:10:45.590
there is an energy on
the right hand side.
01:10:45.590 --> 01:10:48.700
And then you would say, look,
it looks like this just typical
01:10:48.700 --> 01:10:51.290
one dimensional
Schrodinger equation.
01:10:51.290 --> 01:10:53.610
Therefore that
energy probably is
01:10:53.610 --> 01:10:57.250
quantized because it shows
in the right hand side.
01:10:57.250 --> 01:11:03.340
Why wouldn't it be quantized
if it just shows this way?
01:11:03.340 --> 01:11:06.990
On the other hand, it
shouldn't be quantized.
01:11:06.990 --> 01:11:12.550
So what is it about this
differential equation that
01:11:12.550 --> 01:11:16.780
shows that the energy
never gets quantized?
01:11:16.780 --> 01:11:20.330
Well, the fact is that
the energy in some sense
01:11:20.330 --> 01:11:22.850
doesn't show up in this
differential equation.
01:11:22.850 --> 01:11:27.630
You think it's here, but
it's not really there.
01:11:27.630 --> 01:11:29.710
What does that mean?
01:11:29.710 --> 01:11:31.760
It actually means
that you can define
01:11:31.760 --> 01:11:41.240
a new variable rho
equal kr, scale r.
01:11:41.240 --> 01:11:47.870
And basically chain rule
or your intuition, this k
01:11:47.870 --> 01:11:49.500
goes down here.
01:11:49.500 --> 01:11:53.810
k squared r squared k squared
r squared, it's all rho.
01:11:53.810 --> 01:11:56.730
So chain rule or
changing variables
01:11:56.730 --> 01:12:03.650
will turn this equation into
a minus d second uEl d rho
01:12:03.650 --> 01:12:08.850
squared plus l times
l plus 1 rho squared
01:12:08.850 --> 01:12:14.900
is equal to-- times uEl--
is equal to uEl here.
01:12:20.450 --> 01:12:23.270
And the energy has
disappeared from the equation
01:12:23.270 --> 01:12:27.400
by rescaling, a trivial
rescaling of coordinates.
01:12:27.400 --> 01:12:30.860
That doesn't mean that
the energy is not there.
01:12:30.860 --> 01:12:35.510
It is there, because
you will find solutions
01:12:35.510 --> 01:12:37.720
that depend on rho,
and then you will
01:12:37.720 --> 01:12:40.830
put rho equal kr and
the energies there.
01:12:40.830 --> 01:12:44.070
But there's no
quantization of energy,
01:12:44.070 --> 01:12:49.490
because the energy doesn't
show in this equation anymore.
01:12:49.490 --> 01:12:55.310
It's kind of a neat thing, or
rather conceptually interesting
01:12:55.310 --> 01:12:59.350
thing that energy is
not there anymore.
01:12:59.350 --> 01:13:04.560
And then you look at this
differential equation,
01:13:04.560 --> 01:13:09.720
and you realize that
it's a nasty one.
01:13:09.720 --> 01:13:15.280
So this equation is
quite easy without this.
01:13:15.280 --> 01:13:18.130
It's a power solution.
01:13:18.130 --> 01:13:23.700
It's quite easy without this,
it's exponentials are this.
01:13:23.700 --> 01:13:26.580
But whenever you have a
differential equation that
01:13:26.580 --> 01:13:31.930
has two derivatives,
a term with 1
01:13:31.930 --> 01:13:35.440
over x squared
times the function,
01:13:35.440 --> 01:13:40.600
and a term with 1
times the function,
01:13:40.600 --> 01:13:42.255
you're in Bessel territory.
01:13:44.960 --> 01:13:47.300
All these functions
have Bessel things.
01:13:49.820 --> 01:13:53.940
And then you have another
term like 1 over x d dx.
01:13:53.940 --> 01:13:57.540
That is not a problem, but the
presence of these two things,
01:13:57.540 --> 01:14:01.170
one with 1 over x squared
and one with this,
01:14:01.170 --> 01:14:02.640
complicates this equation.
01:14:02.640 --> 01:14:05.270
So Bessel, without
this, would be
01:14:05.270 --> 01:14:08.880
exponential solution without
this would be powers.
01:14:08.880 --> 01:14:12.610
In the end, the fact is that
this is spherical Bessel,
01:14:12.610 --> 01:14:15.200
and it's a little complicated.
01:14:15.200 --> 01:14:16.860
Not terribly complicated.
01:14:16.860 --> 01:14:20.280
The solutions are
spherical Bessel functions,
01:14:20.280 --> 01:14:22.970
which are not all that bad.
01:14:22.970 --> 01:14:25.190
And let me say what they are.
01:14:38.550 --> 01:14:41.960
So what are the
solutions to this thing?
01:14:41.960 --> 01:14:45.390
In fact, the solutions
that are easier to find
01:14:45.390 --> 01:14:54.960
is that the uEl's are r
times the Bessel function
01:14:54.960 --> 01:14:58.020
jl is called spherical
Bessel functions.
01:14:58.020 --> 01:15:01.860
So it's not capital
j that people
01:15:01.860 --> 01:15:06.320
use for the normal
Bessel, but lower case l.
01:15:06.320 --> 01:15:08.640
Of kr.
01:15:08.640 --> 01:15:13.430
As you know, you solve this,
and the solutions for this
01:15:13.430 --> 01:15:18.050
would be of the
form rho jl for rho.
01:15:18.050 --> 01:15:22.530
But rho is kr, so we don't
care about the constant,
01:15:22.530 --> 01:15:25.590
because this is a
homogeneous linear equation.
01:15:25.590 --> 01:15:26.890
So some number here.
01:15:26.890 --> 01:15:29.320
You could put a
constant if you wish.
01:15:29.320 --> 01:15:32.130
But that's the solution.
01:15:32.130 --> 01:15:34.820
Therefore your
complete solutions
01:15:34.820 --> 01:15:42.590
is like the psi's of
Elm would be u divided
01:15:42.590 --> 01:15:50.406
by r, which is jl of kr
times Ylm's of theta phi.
01:15:50.406 --> 01:15:51.780
These are the
complete solutions.
01:15:56.280 --> 01:15:59.400
This is a second order
differential equation.
01:15:59.400 --> 01:16:03.870
Therefore it has to
have two solutions.
01:16:03.870 --> 01:16:07.960
And this is what is called a
regular solution at the origin.
01:16:07.960 --> 01:16:12.250
The Bessel functions
come in j and n type.
01:16:12.250 --> 01:16:15.670
And the n type is
singular at the origins,
01:16:15.670 --> 01:16:17.340
so we won't care about it.
01:16:20.810 --> 01:16:23.870
So what do we get from here?
01:16:23.870 --> 01:16:27.780
Well, some behavior
that is well known.
01:16:27.780 --> 01:16:36.110
Rho jl of rho behaves like
rho to the l plus 2 over 2l
01:16:36.110 --> 01:16:42.690
plus 1 double factorial
as rho goes to 0.
01:16:42.690 --> 01:16:46.540
So that's a fact about
these Bessel functions.
01:16:46.540 --> 01:16:50.570
They behave that
way, which is good,
01:16:50.570 --> 01:16:56.640
because rho jl
behaves like that,
01:16:56.640 --> 01:17:00.770
so u behaves like r
to the l plus 1, which
01:17:00.770 --> 01:17:03.700
is what we derived
a little time ago.
01:17:03.700 --> 01:17:05.950
So this behavior of
the Bessel function
01:17:05.950 --> 01:17:09.370
is indeed consistent
with our solution.
01:17:09.370 --> 01:17:13.890
Moreover, there's another
behavior that is interesting.
01:17:13.890 --> 01:17:15.910
This Bessel
function, by the time
01:17:15.910 --> 01:17:19.580
it's written like
that, when you go
01:17:19.580 --> 01:17:25.390
far off to infinity
jl of rho, it
01:17:25.390 --> 01:17:31.950
behaves like sine of
rho minus l pi over 2.
01:17:35.900 --> 01:17:40.120
This is as rho goes to infinity.
01:17:40.120 --> 01:17:45.080
So as rho goes to
infinity, this is
01:17:45.080 --> 01:17:47.550
behaving like a
trigonometric function.
01:17:47.550 --> 01:17:54.680
It's consistent with this,
because rho-- this is rho jl
01:17:54.680 --> 01:17:58.120
is what we call u essentially.
01:17:58.120 --> 01:18:02.470
So u behaves like this
with rho equal kr.
01:18:02.470 --> 01:18:03.730
And that's consistent.
01:18:03.730 --> 01:18:07.350
This superposition of
a sine and a cosine.
01:18:07.350 --> 01:18:10.570
But it's kind of interesting
though that this l pi over 2
01:18:10.570 --> 01:18:12.940
shows up here.
01:18:12.940 --> 01:18:15.890
You see the fact
that this function
01:18:15.890 --> 01:18:17.940
has to vanish at the origin.
01:18:17.940 --> 01:18:20.800
It vanishes at the origin
and begins to vary.
01:18:20.800 --> 01:18:23.990
And by the time you go
far away, you contract.
01:18:23.990 --> 01:18:28.970
And the way it
behaves is this way.
01:18:28.970 --> 01:18:32.580
The face is determined.
01:18:32.580 --> 01:18:36.060
So that actually gives
a lot of opportunity
01:18:36.060 --> 01:18:43.190
to physicists because
the free particle--
01:18:43.190 --> 01:18:51.090
so for the free
particle, uEl behaves
01:18:51.090 --> 01:18:59.285
like sine of kr minus l pi
over 2 as r goes to infinity.
01:19:05.560 --> 01:19:20.590
So from that people have
asked the following question.
01:19:20.590 --> 01:19:25.720
What if you have a
potential that, for example
01:19:25.720 --> 01:19:30.140
for simplicity, a potential
that is localized.
01:19:30.140 --> 01:19:32.650
Well, if this
potential is localized,
01:19:32.650 --> 01:19:35.850
the solution far
away is supposed
01:19:35.850 --> 01:19:39.280
to be a superposition
of sines and cosines.
01:19:39.280 --> 01:19:42.750
So if there is no
potential, the solution
01:19:42.750 --> 01:19:44.780
is supposed to be this.
01:19:44.780 --> 01:19:48.270
Now another superposition
of sines and cosines,
01:19:48.270 --> 01:19:50.330
at the end of the
day, can always
01:19:50.330 --> 01:19:55.040
be written as some sine of
this thing plus a change
01:19:55.040 --> 01:20:01.230
in this phase So in
general, uEl will
01:20:01.230 --> 01:20:11.100
go like sine of kr minus l pi
over 2 plus a shift, a phase
01:20:11.100 --> 01:20:17.140
shift, delta l that can
depend on the energy.
01:20:21.490 --> 01:20:26.170
So if you haven't tried to
find the radial solutions
01:20:26.170 --> 01:20:28.600
of a problem with
some potential,
01:20:28.600 --> 01:20:32.400
if the potential is 0,
there's no such term.
01:20:32.400 --> 01:20:37.780
But if the potential is
here, it will have an effect
01:20:37.780 --> 01:20:41.330
and will give you a phase shift.
01:20:41.330 --> 01:20:44.550
So if you're doing particle
scattering experiments,
01:20:44.550 --> 01:20:47.300
you're sending
waves from far away
01:20:47.300 --> 01:20:50.560
and you just see how the
wave behaves far away,
01:20:50.560 --> 01:20:54.190
you do have measurement
information on this phase
01:20:54.190 --> 01:20:55.140
shift.
01:20:55.140 --> 01:20:58.310
And from this phase shift,
you can learn something
01:20:58.310 --> 01:21:00.900
about the potential.
01:21:00.900 --> 01:21:05.850
So this is how this
problem of free particle
01:21:05.850 --> 01:21:10.130
suddenly becomes very
important and very interesting.
01:21:10.130 --> 01:21:13.370
For example, as a way
through the behavior at
01:21:13.370 --> 01:21:17.410
infinity learning something
about the potential.
01:21:17.410 --> 01:21:20.800
For example, if the
potential is attractive,
01:21:20.800 --> 01:21:26.020
it pulls the wave function
in and produces some sign
01:21:26.020 --> 01:21:30.520
of delta that the corresponds
to a positive delta.
01:21:30.520 --> 01:21:33.120
If the potential
is repulsive, it
01:21:33.120 --> 01:21:36.580
pushes the wave
function out, repels it
01:21:36.580 --> 01:21:39.120
and produces a delta
that is negative.
01:21:39.120 --> 01:21:42.740
You can track those
signs thinking carefully.
01:21:42.740 --> 01:21:48.100
But the potentials will teach
you something about delta.
01:21:48.100 --> 01:21:53.460
The other case that this is
interesting-- I will just
01:21:53.460 --> 01:21:58.400
introduce it and stop, because
we might as well stop--
01:21:58.400 --> 01:22:03.880
is a very important case.
01:22:03.880 --> 01:22:05.890
The square well.
01:22:05.890 --> 01:22:09.250
Well, we've studied
in one dimension
01:22:09.250 --> 01:22:11.640
the infinite square well.
01:22:11.640 --> 01:22:14.580
That's one potential that
you now how to solve,
01:22:14.580 --> 01:22:18.440
and sines and
cosines is very easy.
01:22:18.440 --> 01:22:21.740
Now imagine a
spherical square well,
01:22:21.740 --> 01:22:25.770
which is some sort of cavity
in which a particle is
01:22:25.770 --> 01:22:28.500
free to move here, but
the potential becomes
01:22:28.500 --> 01:22:30.900
infinite at the boundary.
01:22:33.740 --> 01:22:38.940
It's a hollow sphere,
so the potential v of r
01:22:38.940 --> 01:22:42.180
is equal to 0 for r less than a.
01:22:42.180 --> 01:22:46.090
And it's infinity
for r greater than a.
01:22:46.090 --> 01:22:51.590
So it's like a bag, a
balloon with solid walls
01:22:51.590 --> 01:22:53.460
impossible to penetrate.
01:22:53.460 --> 01:22:58.590
So this is the most
symmetric simple potential
01:22:58.590 --> 01:23:00.200
you could imagine in the world.
01:23:02.990 --> 01:23:05.520
And we're going to solve it.
01:23:05.520 --> 01:23:07.190
How can we solve this?
01:23:07.190 --> 01:23:12.900
Well, we did 2/3 of the
work already in solving it.
01:23:12.900 --> 01:23:14.050
Why?
01:23:14.050 --> 01:23:18.890
Because inside here
the potential is 0,
01:23:18.890 --> 01:23:22.060
so the particle is free.
01:23:22.060 --> 01:23:29.480
So inside here the solutions
are of the form uEl
01:23:29.480 --> 01:23:33.180
go like rjl of kr.
01:23:36.130 --> 01:23:40.750
And the only thing you will need
is that they vanish at the end.
01:23:40.750 --> 01:23:44.410
So you will fix
this by demanding
01:23:44.410 --> 01:23:57.050
that ka is a number z such--
well, the jl of ka will be 0.
01:23:57.050 --> 01:23:59.460
So that the wave
function vanishes
01:23:59.460 --> 01:24:03.290
at this point where the
potential becomes infinite.
01:24:03.290 --> 01:24:05.530
So you've solved
most of the problem.
01:24:05.530 --> 01:24:10.700
And we'll discuss it in detail,
because it's an important one.
01:24:10.700 --> 01:24:16.940
But this is the most symmetric
potential, you may think.
01:24:16.940 --> 01:24:23.380
This potential is very
symmetric, very pretty,
01:24:23.380 --> 01:24:27.520
but nothing to write home about.
01:24:27.520 --> 01:24:30.250
If you tried to
look-- and we're going
01:24:30.250 --> 01:24:34.060
to calculate this diagram.
01:24:34.060 --> 01:24:35.820
You would say well
it's so symmetric
01:24:35.820 --> 01:24:39.950
that something pretty
is going to happen here.
01:24:39.950 --> 01:24:41.720
Nothing happens.
01:24:41.720 --> 01:24:44.290
These states will show up.
01:24:44.290 --> 01:24:49.040
And these ones will show
up, and no state ever
01:24:49.040 --> 01:24:50.460
will match another one.
01:24:50.460 --> 01:24:55.130
There's no pattern, or
rhyme, or reason for it.
01:24:55.130 --> 01:24:57.630
On the other hand,
if you would have
01:24:57.630 --> 01:25:07.550
taken a potential v of r
of the form beta r squared,
01:25:07.550 --> 01:25:10.960
that potential will
exhibit enormous amounts
01:25:10.960 --> 01:25:14.130
of degeneracies all over.
01:25:14.130 --> 01:25:17.480
And we will have to
understand why that happens.
01:25:17.480 --> 01:25:19.500
So we'll see you next Monday.
01:25:19.500 --> 01:25:21.390
Enjoy your break.
01:25:21.390 --> 01:25:26.540
Homework will only happen
late after Thanksgiving.
01:25:26.540 --> 01:25:28.940
And just have a great time.
01:25:28.940 --> 01:25:32.190
Thank you for coming today,
and will see you soon.
01:25:32.190 --> 01:25:33.740
[APPLAUSE]