WEBVTT
00:00:00.087 --> 00:00:00.670
PROFESSOR: OK.
00:00:00.670 --> 00:00:03.750
So what I'm going
to try to do now
00:00:03.750 --> 00:00:11.310
is set up again
this equation and do
00:00:11.310 --> 00:00:17.630
the analog of what
we're doing there
00:00:17.630 --> 00:00:25.940
and try to determine this
function fk in some nice way.
00:00:29.180 --> 00:00:29.695
All right.
00:00:33.690 --> 00:00:37.210
So let's think of this equation.
00:00:37.210 --> 00:00:38.970
I want to do it in pieces.
00:00:38.970 --> 00:00:50.410
So psi of r is going to
be equal to some formula.
00:00:50.410 --> 00:01:00.070
And then it has to be equal
to this right hand side.
00:01:00.070 --> 00:01:06.820
Let's write the e to the ikz
the way we've done it before.
00:01:06.820 --> 00:01:08.080
It's up there.
00:01:08.080 --> 00:01:09.910
So I'll write it here.
00:01:09.910 --> 00:01:15.520
Square root of 4 pi
over k square, sum of l
00:01:15.520 --> 00:01:22.090
equals 0 to infinity,
square root of 2l plus 1, i
00:01:22.090 --> 00:01:34.690
to the l yl 0 1 over 2 i e
to the ikr minus l pi over 2
00:01:34.690 --> 00:01:42.711
over r minus e the minus
ikr minus l pi over 2.
00:01:42.711 --> 00:01:43.210
Wow.
00:01:43.210 --> 00:01:47.320
It's tiring this r.
00:01:47.320 --> 00:01:55.990
Plus f of k of theta
e to the ikr over r.
00:02:07.883 --> 00:02:08.382
OK.
00:02:11.546 --> 00:02:13.360
So what do we have here?
00:02:13.360 --> 00:02:17.360
We've written the right
hand side of this equation.
00:02:17.360 --> 00:02:18.420
I copied it.
00:02:18.420 --> 00:02:24.765
I have not done anything except
taking r much greater than a.
00:02:28.150 --> 00:02:32.680
Because otherwise in the
plane wave into the ikz,
00:02:32.680 --> 00:02:35.770
I could not have expanded
the Bessel functions
00:02:35.770 --> 00:02:38.620
unless I took r greater than a.
00:02:38.620 --> 00:02:43.750
But that's good, because
we now have our waves.
00:02:43.750 --> 00:02:45.020
OK.
00:02:45.020 --> 00:02:47.600
We have our waves there.
00:02:47.600 --> 00:02:52.780
Now, look at this
right hand side.
00:02:52.780 --> 00:02:58.780
Where is the incoming wave
in this right hand side?
00:02:58.780 --> 00:03:00.720
The incoming wave is here.
00:03:05.020 --> 00:03:07.980
That's the only term
that is incoming,
00:03:07.980 --> 00:03:10.700
because this is
an outgoing wave,
00:03:10.700 --> 00:03:14.510
and this is an outgoing wave.
00:03:14.510 --> 00:03:19.960
So if I want to write
the left hand side,
00:03:19.960 --> 00:03:24.790
the incoming wave of
the left hand side
00:03:24.790 --> 00:03:27.000
has to be equal to this wave.
00:03:29.990 --> 00:03:33.650
And of course, the outgoing
wave of the left hand side
00:03:33.650 --> 00:03:37.280
will also have to be equal
to whatever is outgoing here,
00:03:37.280 --> 00:03:41.180
but the incoming must be this.
00:03:41.180 --> 00:03:43.640
So I'm going to write
this left hand side
00:03:43.640 --> 00:03:52.220
and already use this and put
4 pi over k square sum of l
00:03:52.220 --> 00:03:58.600
equals 0 to infinity 2l
plus 1 i to the l y l
00:03:58.600 --> 00:04:09.730
0 1 over 2 pi, big parentheses,
and one outgoing wave
00:04:09.730 --> 00:04:21.550
and one incoming wave minus
ikr minus l pi over 2 over r.
00:04:21.550 --> 00:04:26.380
And here, I don't
know what to put,
00:04:26.380 --> 00:04:29.710
but I've put already
there on the left hand
00:04:29.710 --> 00:04:34.540
side of this
equation for psi of r
00:04:34.540 --> 00:04:40.510
for the full solution, a wave
that matches the right hand
00:04:40.510 --> 00:04:46.930
side, because it has
the same incoming wave.
00:04:59.170 --> 00:05:03.910
And now, I'm going to use
some physical intuition
00:05:03.910 --> 00:05:08.240
to guess what we'll have
to put on this part.
00:05:08.240 --> 00:05:13.610
This is the step that requires
a little imagination, not too
00:05:13.610 --> 00:05:18.510
much, because we already
did something similar here.
00:05:18.510 --> 00:05:30.300
So what's happening
here and here is
00:05:30.300 --> 00:05:36.040
intuition I think you should
keep after weeks of this course
00:05:36.040 --> 00:05:38.390
when it's all forgotten,
there's some intuition
00:05:38.390 --> 00:05:40.880
that you should keep.
00:05:40.880 --> 00:05:46.520
And it's about this scattering
happening for each partial wave
00:05:46.520 --> 00:05:47.440
independent.
00:05:47.440 --> 00:05:49.325
Yes.
00:05:49.325 --> 00:05:50.780
AUDIENCE: 2i.
00:05:50.780 --> 00:05:53.080
PROFESSOR: 2i, yes.
00:05:53.080 --> 00:05:54.574
No, 2 pi there.
00:05:58.711 --> 00:05:59.210
Yes.
00:06:03.800 --> 00:06:05.180
Thank you.
00:06:05.180 --> 00:06:09.080
So here it is.
00:06:09.080 --> 00:06:15.320
This is a solution, and we've
got the intuition already.
00:06:15.320 --> 00:06:19.770
I will justify this later,
of course, very precisely.
00:06:19.770 --> 00:06:21.830
But I think that
this one you need
00:06:21.830 --> 00:06:24.920
to have a little bit
of an intuition of what
00:06:24.920 --> 00:06:26.540
you should do.
00:06:26.540 --> 00:06:30.650
And first, we said
each l works separately
00:06:30.650 --> 00:06:36.060
to create a solution of
the Schrodinger equation.
00:06:36.060 --> 00:06:40.050
That's superposition,
and it's [INAUDIBLE]..
00:06:40.050 --> 00:06:43.200
Each l is working separately.
00:06:43.200 --> 00:06:47.250
Each l is like a
scattering problem.
00:06:47.250 --> 00:06:54.140
Each l has a wave that comes
in and a wave that comes out,
00:06:54.140 --> 00:07:02.240
because these things, j and n,
have waves, and they have an
00:07:02.240 --> 00:07:03.320
in and out.
00:07:03.320 --> 00:07:08.080
So these have some in
wave and some out wave.
00:07:08.080 --> 00:07:12.210
And if each wave
works separately,
00:07:12.210 --> 00:07:18.470
it has an in wave and then out
wave, in a scattering problem,
00:07:18.470 --> 00:07:22.070
these waves must have
the same amplitude,
00:07:22.070 --> 00:07:25.940
because otherwise they wouldn't
have the same probability
00:07:25.940 --> 00:07:29.960
current, and probability
would get stuck.
00:07:29.960 --> 00:07:37.070
So this must be an outgoing
wave having this same amplitude
00:07:37.070 --> 00:07:37.965
as this wave.
00:07:40.790 --> 00:07:43.490
And by the argument
we have here,
00:07:43.490 --> 00:07:46.910
it just differs by a phase.
00:07:46.910 --> 00:08:00.650
So we'll put here e to the ikr
minus l pi over 2 plus 2i delta
00:08:00.650 --> 00:08:05.180
l over r.
00:08:05.180 --> 00:08:10.250
So that this wave, spherical
wave, that it's outgoing,
00:08:10.250 --> 00:08:13.460
it has the same
amplitude as this one,
00:08:13.460 --> 00:08:15.600
and cannot be the same.
00:08:15.600 --> 00:08:19.160
The only difference
can be a phase shift,
00:08:19.160 --> 00:08:20.670
and that's the phase shift.
00:08:23.750 --> 00:08:30.490
So your picture is scattering
in three dimensions.
00:08:30.490 --> 00:08:34.400
Looks like, OK, you
threw in a plane wave
00:08:34.400 --> 00:08:39.669
and out came a
spherical wave out.
00:08:39.669 --> 00:08:43.030
The other picture that is more
consistent with the way you
00:08:43.030 --> 00:08:48.220
solve it is that you have an
infinite set of partial waves
00:08:48.220 --> 00:08:52.735
for different l's, each
one scattering, the l
00:08:52.735 --> 00:08:57.460
equals 0, the equal 1,
the l equal 2, all of them
00:08:57.460 --> 00:09:00.320
scattering.
00:09:00.320 --> 00:09:02.770
So this corresponds
to an [? ansatz ?]
00:09:02.770 --> 00:09:07.960
in terms of phase
shifts, and now you
00:09:07.960 --> 00:09:11.230
can say you've
parameterized your ignorance
00:09:11.230 --> 00:09:13.000
in a physical way.
00:09:13.000 --> 00:09:17.290
You've discovered that all that
characterizes the scattering
00:09:17.290 --> 00:09:20.740
is, as it was in one
dimension, a phase shift.
00:09:20.740 --> 00:09:24.130
In one dimension, there
was a single phase shift,
00:09:24.130 --> 00:09:29.170
because you didn't have
all these general solutions
00:09:29.170 --> 00:09:31.060
that you had in
three dimensions.
00:09:31.060 --> 00:09:34.410
Your energy eigenstates
were momentum eigenstates
00:09:34.410 --> 00:09:37.510
there were
non-degenerate really.
00:09:37.510 --> 00:09:41.560
There was just a couple of
momentum eigenstates, wave in
00:09:41.560 --> 00:09:42.640
and waves out.
00:09:42.640 --> 00:09:44.980
Here is infinitely degenerate.
00:09:44.980 --> 00:09:46.940
There's spherical stuff.
00:09:46.940 --> 00:09:51.890
So there's a phase shift
for each value of l.
00:09:51.890 --> 00:09:55.550
So we've parameterized the
physics of the scattering
00:09:55.550 --> 00:09:59.360
problem in terms of phase
shift, and now, it's
00:09:59.360 --> 00:10:03.320
interesting to try to figure
out what is this quantity
00:10:03.320 --> 00:10:06.320
after all in terms
of the phase shifts.
00:10:06.320 --> 00:10:08.540
It's already here.
00:10:08.540 --> 00:10:10.680
We just have to solve it.
00:10:10.680 --> 00:10:16.010
So from this equation, I now
can say that this term cancels
00:10:16.010 --> 00:10:21.530
with this term, and now,
I can solve for this term
00:10:21.530 --> 00:10:26.870
f of theta e to the ikr by
collecting these other two
00:10:26.870 --> 00:10:27.665
terms together.
00:10:32.880 --> 00:10:41.240
And therefore, fk of theta e
to the ikr over r is equal,
00:10:41.240 --> 00:10:45.870
and it's exactly the same thing
I did here, cancel here, pass,
00:10:45.870 --> 00:10:49.380
and the mathematics is going
to be completely analogous,
00:10:49.380 --> 00:10:52.835
except that they have to
carry all that sum there.
00:10:57.570 --> 00:10:58.780
No big deal.
00:10:58.780 --> 00:11:00.210
So what is it?
00:11:00.210 --> 00:11:05.040
Square root of 4 pi
over k, sum from l
00:11:05.040 --> 00:11:12.820
equals 0 to infinity, 2l
plus 1, i to the l y l0.
00:11:16.340 --> 00:11:17.765
1 over 2i.
00:11:23.390 --> 00:11:27.230
OK, 1 over 2i.
00:11:27.230 --> 00:11:29.390
OK, I'll do it this way.
00:11:29.390 --> 00:11:39.290
e to the 2i delta l minus 1 e
to the ikr e to the minus il pi
00:11:39.290 --> 00:11:40.370
over 2.
00:11:40.370 --> 00:11:42.620
I think I got everything there.
00:11:42.620 --> 00:11:46.580
I put the first term
on the right hand side,
00:11:46.580 --> 00:11:48.410
I moved it to the
left hand side.
00:11:48.410 --> 00:11:51.960
The coefficient
was all the same.
00:11:51.960 --> 00:11:55.520
This was the coefficient they
both had the same coefficient.
00:11:55.520 --> 00:11:59.990
Then I just have to subtract
these two exponentials over r.
00:11:59.990 --> 00:12:04.040
So I did forget the r.
00:12:04.040 --> 00:12:06.500
So I just subtract
the two exponentials.
00:12:06.500 --> 00:12:13.380
Both exponentials have the e
to the ikr minus l pi over 2.
00:12:13.380 --> 00:12:17.390
And the difference is that the
first exponential on the top
00:12:17.390 --> 00:12:21.527
has the extra e to the
2i delta l minus 1,
00:12:21.527 --> 00:12:22.610
and the other one doesn't.
00:12:28.200 --> 00:12:29.635
So what do we get here?
00:12:33.740 --> 00:12:44.190
This part is e to the
i delta sine delta,
00:12:44.190 --> 00:12:55.030
and this part is e to
the minus i pi over 2
00:12:55.030 --> 00:13:03.150
to the power l, which
is minus i to the l,
00:13:03.150 --> 00:13:12.570
and i to the l times minus i
to the l is happily just 1.
00:13:12.570 --> 00:13:16.620
i times minus i is 1,
and 1 to the l is 1.
00:13:16.620 --> 00:13:25.740
So this term and
this term cancel.
00:13:25.740 --> 00:13:29.190
So finally, and I can
cancel happily they
00:13:29.190 --> 00:13:31.170
r dependence is all the same.
00:13:31.170 --> 00:13:36.510
I can cancel this r
dependence, this r dependence.
00:13:36.510 --> 00:13:41.820
And finally, we've
got fk of theta
00:13:41.820 --> 00:13:48.210
equals square root of
4 pi over k sum from l
00:13:48.210 --> 00:13:57.150
equals 0 to infinity square
root 2l plus 1 y L0 of theta e
00:13:57.150 --> 00:14:03.120
to the i delta l sine delta l.
00:14:10.370 --> 00:14:15.590
So that said, that's
our formula for fk
00:14:15.590 --> 00:14:17.240
in terms of the phase shift.
00:14:25.790 --> 00:14:26.980
So what have we achieved?
00:14:26.980 --> 00:14:30.720
We want f of k, because that
gives us the cross section.
00:14:30.720 --> 00:14:35.110
What we have figured out is
that the calculation of fk
00:14:35.110 --> 00:14:37.970
really requires knowing
the phase shift.
00:14:37.970 --> 00:14:43.720
And the phase shifts are defined
by that formula over there,
00:14:43.720 --> 00:14:49.240
where we have estimated
how one wave is connected
00:14:49.240 --> 00:14:52.450
to the other one, the
incoming and the outgoing
00:14:52.450 --> 00:14:57.220
for a given fixed f, for a
given partial wave, how they're
00:14:57.220 --> 00:15:00.780
offset by this phase shift.