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00:00:00,630 --> 00:00:05,790
PROFESSOR: We have to set up
a little better the geometry

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00:00:05,790 --> 00:00:08,820
of the calculation.

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00:00:08,820 --> 00:00:13,890
And for that we have to
think of various angles.

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00:00:13,890 --> 00:00:18,940
We oriented there the electric
field along the z direction.

5
00:00:18,940 --> 00:00:24,940
But that's not going to be too
convenient for our calculation.

6
00:00:24,940 --> 00:00:30,850
So these calculations are
a bit of an art to do them.

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00:00:30,850 --> 00:00:33,840
They're not that trivial.

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00:00:33,840 --> 00:00:38,100
Not terribly difficult. But
I think if you appreciate it,

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00:00:38,100 --> 00:00:41,580
next time you ever have
to do one of these things

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00:00:41,580 --> 00:00:43,990
it will become clear.

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00:00:43,990 --> 00:00:48,300
So first your have
to think physically.

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Is there an angle
in this problem?

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Is there any important
angle happening here?

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It's a question to you.

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I think if you figure out that
you have a chance of doing

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00:01:15,020 --> 00:01:18,410
a diagram that reflects this.

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Is there a physical angle,
you think, in this process?

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A relevant angle?

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00:01:25,920 --> 00:01:26,610
Yes, Lou?

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00:01:26,610 --> 00:01:31,179
AUDIENCE: [INAUDIBLE]

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PROFESSOR: Perfect.

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Yeah.

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The relevant angle is
you have a directional

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ready for the electric field.

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So if the electron
comes off there's

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going to be an angle with
respect to that electric field.

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So that's our physical
angle in this question.

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00:01:47,270 --> 00:01:52,670
So let's try to
draw this in a way.

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So I will draw it this way.

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And I brought colored
chalk for this.

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I'll put the z-axis here, and
I'll put the electron momentum

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in this direction.

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k is the lateral momentum.

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Now the electric
field is going to come

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00:02:18,630 --> 00:02:21,840
at some angle with
respect to the--

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00:02:21,840 --> 00:02:25,140
or the electron momentum is
at some angle with respect

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to the electric field.

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So that angle will
presumably stay there

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for the rest of the calculation.

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So let's do it with green.

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00:02:36,100 --> 00:02:40,910
So here is my electric field.

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And it's going to have an
angle theta with respect

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to the direction of the electric
field or the photon incident

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00:02:52,830 --> 00:02:53,460
direction.

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00:02:56,760 --> 00:03:03,500
Now, this is actually more like
the polarization of the photon.

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The electric field is the
direction of the polarization

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of the photon.

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So one more vector, however,
the position that we

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have to integrate over--

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because we have the whole
hydrogen atom and the whole

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00:03:23,450 --> 00:03:25,010
of space to integrate.

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So x has a position.

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00:03:26,940 --> 00:03:29,940
So I'll use another color here.

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Unfortunately, the colors
are not that different.

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Here is r, the vector r.

56
00:03:38,680 --> 00:03:44,140
So now I have several angles.

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I have theta for
the electric field.

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But now that I've put all this
axis I not only have theta,

59
00:03:55,040 --> 00:04:00,080
but I also have phi
for the electric field.

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00:04:00,080 --> 00:04:14,010
And for r, I will have
theta prime and phi prime.

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00:04:16,750 --> 00:04:23,160
So r has a theta
prime and a phi prime.

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00:04:23,160 --> 00:04:25,300
We usually have theta and phi.

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But the answers, at
the end of the day,

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00:04:27,960 --> 00:04:29,465
are going to depend on theta.

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00:04:33,634 --> 00:04:38,550
And if you have theta here,
we want the theta to remain.

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00:04:38,550 --> 00:04:41,210
So theta prime and
phi prime are going

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00:04:41,210 --> 00:04:43,650
to be our variables
of integration

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00:04:43,650 --> 00:04:46,860
because you integrate over r.

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00:04:46,860 --> 00:04:50,370
Finally, we have one more
angle that we have to define.

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00:04:50,370 --> 00:04:55,380
So we have theta phi,
theta prime, phi prime,

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00:04:55,380 --> 00:04:57,480
and the angle that was here--

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00:05:01,950 --> 00:05:06,330
this was the angle between
r and the direction

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00:05:06,330 --> 00:05:07,600
of the electric field.

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00:05:07,600 --> 00:05:10,600
And we called it theta.

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00:05:10,600 --> 00:05:13,480
So we have to give
it a new name here.

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00:05:13,480 --> 00:05:19,720
And this angle, I'll
call it gamma here.

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00:05:24,770 --> 00:05:26,600
The angle between E and r.

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OK.

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00:05:29,656 --> 00:05:37,060
So We have everything
defined here.

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00:05:37,060 --> 00:05:39,280
Maybe I should list it.

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00:05:39,280 --> 00:05:47,070
E has angles theta and
phi, the direction of E. r

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00:05:47,070 --> 00:05:52,840
has angles theta
prime and phi prime.

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00:05:52,840 --> 00:06:00,460
And gamma is the
angle between E and r.

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00:06:00,460 --> 00:06:07,670
And k along z hat.

85
00:06:07,670 --> 00:06:10,780
So this is our situation.

86
00:06:19,660 --> 00:06:22,030
So what are we
trying to calculate?

87
00:06:22,030 --> 00:06:24,820
Well, we want to use
Fermi's golden rule.

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So we need to calculate
final H prime initial.

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That's the matrix element
of the Hamiltonian H

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prime between the final
and initial states.

91
00:06:47,300 --> 00:06:49,490
So what is this?

92
00:06:49,490 --> 00:06:51,560
Well, these are
all wave functions

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that depend all over space, and
this is a function of space.

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00:06:55,990 --> 00:06:56,900
So let's do it.

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00:06:56,900 --> 00:07:02,330
This is integral d cubed x.

96
00:07:02,330 --> 00:07:06,280
Let's put the final state first
because it shows up there.

97
00:07:06,280 --> 00:07:13,460
So it's 1 over L cubed e
to the minus i k dot r.

98
00:07:16,280 --> 00:07:21,970
So here is our final
state, u final.

99
00:07:21,970 --> 00:07:23,795
Then the Hamiltonian.

100
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That's simple.

101
00:07:25,010 --> 00:07:29,785
e E0 r cosine what?

102
00:07:33,190 --> 00:07:36,520
Cosine gamma.

103
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Is that right?

104
00:07:38,420 --> 00:07:40,670
This is what we had there.

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It's the angle between
the electric field and r.

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It was theta to begin
but now has become gamma.

107
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And then the final state.

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So this is our H prime.

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00:08:00,040 --> 00:08:07,030
And then the initial state
is 1 over pi a0 cubed

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00:08:07,030 --> 00:08:11,430
e to the minus r over a0.

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00:08:11,430 --> 00:08:12,370
OK.

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00:08:12,370 --> 00:08:15,220
This is our task.

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This is a matrix,
and here it is.

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An integral of a plane wave
against an electron wave

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00:08:22,810 --> 00:08:26,500
function and an extra
r dependence here.

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This could range from undoable
to difficult, basically.

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00:08:33,114 --> 00:08:37,210
And happily, it's just
a little difficult.

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00:08:37,210 --> 00:08:39,220
But this is an integral--

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00:08:39,220 --> 00:08:42,470
we'll see what are the
challenges on this integral.

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00:08:42,470 --> 00:08:46,060
So let's take a
few constants out.

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00:08:46,060 --> 00:08:54,050
So e, E0, pi, a0,
that all will go out.

122
00:08:54,050 --> 00:09:03,760
So this is e E0 over square
root of pi l cubed a0 cubed.

123
00:09:03,760 --> 00:09:10,390
So I took the l's out,
the e, E, this thing.

124
00:09:10,390 --> 00:09:11,040
All right.

125
00:09:14,480 --> 00:09:18,590
Let's write the integral
more explicitly.

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00:09:18,590 --> 00:09:26,240
This is r squared dr
sine the volume element.

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00:09:26,240 --> 00:09:31,250
Sine theta d theta d phi.

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00:09:31,250 --> 00:09:36,350
But I'm integrating over x,
which is integrating over r.

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00:09:36,350 --> 00:09:39,240
And r is theta
prime and phi prime.

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So these are all these ones.

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I'm integrating over all values
of theta prime and phi prime.

132
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Then what do I have?

133
00:09:52,750 --> 00:09:57,240
I have this exponent, k dot r.

134
00:09:57,240 --> 00:10:01,675
Well, my diagram shows
how k dot r is easy.

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00:10:01,675 --> 00:10:05,410
It involves a cosine
of theta prime.

136
00:10:05,410 --> 00:10:12,780
So it's e to the minus
ik, the magnitude of k,

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00:10:12,780 --> 00:10:16,590
the magnitude of r
cosine theta prime,

138
00:10:16,590 --> 00:10:19,860
because after all, k
was along the z-axis.

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00:10:19,860 --> 00:10:24,498
r is along the phi
theta prime direction.

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00:10:27,780 --> 00:10:28,320
OK.

141
00:10:28,320 --> 00:10:30,020
We're progressing.

142
00:10:30,020 --> 00:10:36,630
This, this, that term,
it's r cosine gamma.

143
00:10:39,190 --> 00:10:44,470
And the final term is not that
difficult. e to the minus r

144
00:10:44,470 --> 00:10:47,080
over a0.

145
00:10:47,080 --> 00:10:48,370
That's what we have to do.

146
00:10:48,370 --> 00:10:52,810
An integral over phi
prime, theta prime, and r.

147
00:10:56,070 --> 00:10:57,440
This is our challenge.

148
00:10:59,950 --> 00:11:06,740
And the reason it's a
challenge is the cosine gamma

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00:11:06,740 --> 00:11:09,650
because this gamma is the angle.

150
00:11:09,650 --> 00:11:13,880
It depends on theta, depends
on theta prime, depends on phi,

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00:11:13,880 --> 00:11:16,280
phi prime.

152
00:11:16,280 --> 00:11:18,350
That's the problem.

153
00:11:18,350 --> 00:11:21,950
If we can solve that
cosine gamma thing

154
00:11:21,950 --> 00:11:24,140
we can do this integral.

155
00:11:24,140 --> 00:11:27,200
And I think even if you
were doing it numerically,

156
00:11:27,200 --> 00:11:32,130
that cosine gamma there is
a little bit of a headache.

157
00:11:32,130 --> 00:11:34,460
You don't want to do an
integral that you can really

158
00:11:34,460 --> 00:11:36,590
do like this numerically.

159
00:11:36,590 --> 00:11:38,910
So you really want to do it.

160
00:11:38,910 --> 00:11:40,820
So what we need is to calculate.

161
00:11:40,820 --> 00:11:43,700
So we can begin by
saying, I'm going

162
00:11:43,700 --> 00:11:46,640
to calculate what
cosine gamma is.

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00:11:51,280 --> 00:11:55,420
And here is the way you
can calculate cosine gamma.

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00:11:55,420 --> 00:11:59,560
When you have two
unit vectors, cosine

165
00:11:59,560 --> 00:12:02,110
of the angle between
two unit vectors

166
00:12:02,110 --> 00:12:06,550
is just the dot product
of those two unit vectors.

167
00:12:06,550 --> 00:12:12,060
So for gamma, we can consider
a unit vector along e

168
00:12:12,060 --> 00:12:18,730
and a unit vector along r
and take the dot product.

169
00:12:18,730 --> 00:12:23,460
And remember, for an
arbitrary unit vector,

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00:12:23,460 --> 00:12:36,420
it's sine theta cos phi sine
theta sine phi cos theta.

171
00:12:36,420 --> 00:12:39,030
This is the theta
phi decomposition

172
00:12:39,030 --> 00:12:41,620
of an arbitrary unit vector.

173
00:12:41,620 --> 00:12:49,590
So cosine gamma
is the dot product

174
00:12:49,590 --> 00:12:59,680
of a vector n along the e
times a vector n along r.

175
00:12:59,680 --> 00:13:03,460
And vector n along
e and r are just

176
00:13:03,460 --> 00:13:08,810
with theta in one case and phi
and theta prime and phi prime.

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00:13:08,810 --> 00:13:11,710
So when I make
the dot product, I

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00:13:11,710 --> 00:13:23,980
get sine theta, cos phi, sine
theta prime, cos phi prime.

179
00:13:23,980 --> 00:13:28,150
That is the product
of the x components.

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00:13:28,150 --> 00:13:40,950
Plus sine theta, sine phi prime
times sine theta and sine phi.

181
00:13:40,950 --> 00:13:48,790
Sine theta prime sine phi
prime, plus cos theta,

182
00:13:48,790 --> 00:13:49,950
cos theta prime.

183
00:13:53,420 --> 00:13:53,920
OK.

184
00:13:53,920 --> 00:13:57,490
Doesn't look much easier,
but at least it's explicit.

185
00:13:57,490 --> 00:13:59,385
But it's actually much easier.

186
00:14:13,080 --> 00:14:15,720
And why is that?

187
00:14:15,720 --> 00:14:21,290
Because there's a lot of
factors in common here.

188
00:14:24,550 --> 00:14:32,535
In fact, sine theta and sine
theta prime are in both.

189
00:14:35,600 --> 00:14:41,880
So what you get
here is cosine gamma

190
00:14:41,880 --> 00:14:48,540
is equal to sine theta
sine theta prime.

191
00:14:48,540 --> 00:14:50,970
And then you have
cos phi, cos phi

192
00:14:50,970 --> 00:14:54,570
prime, plus sine
phi, sine phi prime.

193
00:14:54,570 --> 00:15:07,630
And that's cosine of phi minus
phi prime plus cos theta cos

194
00:15:07,630 --> 00:15:08,290
theta prime.

195
00:15:12,600 --> 00:15:13,100
OK.

196
00:15:13,100 --> 00:15:16,040
So that's cosine gamma.

197
00:15:16,040 --> 00:15:21,350
Now suppose you were to put
this whole thing in here.

198
00:15:21,350 --> 00:15:26,070
It's a big mess but all
quantities that we know.

199
00:15:26,070 --> 00:15:28,520
But there's one
nice thing, though--

200
00:15:28,520 --> 00:15:31,430
a very nice thing happening.

201
00:15:31,430 --> 00:15:34,145
Think of the interval
over d phi prime.

202
00:15:40,040 --> 00:15:42,560
This does not
depend on phi prime.

203
00:15:42,560 --> 00:15:44,410
This does not
depend on phi prime.

204
00:15:44,410 --> 00:15:47,950
But cosine gamma can
depend on phi prime.

205
00:15:47,950 --> 00:15:55,160
But here it has this thing,
cosine of phi minus phi prime.

206
00:15:55,160 --> 00:15:59,540
So when you try to do the
integral of this phi prime

207
00:15:59,540 --> 00:16:03,650
with this term,
this term will give,

208
00:16:03,650 --> 00:16:06,680
eventually, the integral
of d phi prime times

209
00:16:06,680 --> 00:16:10,370
cosine of phi minus phi prime.

210
00:16:10,370 --> 00:16:13,100
There is a lot of messy things.

211
00:16:13,100 --> 00:16:15,440
But this integral
is 0 because you

212
00:16:15,440 --> 00:16:19,040
are averaging over a full term.

213
00:16:19,040 --> 00:16:28,260
So happily, all this
term will not contribute.

214
00:16:28,260 --> 00:16:32,210
And that's what makes
the integral doable.

215
00:16:32,210 --> 00:16:34,700
So what do we have then?

216
00:16:34,700 --> 00:16:42,290
Our whole matrix
element, f H prime i,

217
00:16:42,290 --> 00:16:52,475
has become e E0 over square
root of pi l cube a0 cube.

218
00:16:55,070 --> 00:16:59,000
And now you just
have to integrate.

219
00:16:59,000 --> 00:17:01,430
This is the only term
that contributes.

220
00:17:10,630 --> 00:17:14,859
And when you put
this term, for sure

221
00:17:14,859 --> 00:17:19,210
you can now do the
d phi prime integral

222
00:17:19,210 --> 00:17:21,410
because that term
is phi dependent.

223
00:17:21,410 --> 00:17:23,890
So that integral gives
you just a factor

224
00:17:23,890 --> 00:17:32,210
of 2 pi from the
integral of d phi.

225
00:17:32,210 --> 00:17:36,290
And from this
thing, cosine theta

226
00:17:36,290 --> 00:17:39,830
is the angle between the
electric field and k.

227
00:17:39,830 --> 00:17:42,140
So it's a constant
for your integral.

228
00:17:42,140 --> 00:17:44,930
You're integrating over
theta prime and phi prime.

229
00:17:44,930 --> 00:17:47,330
So cosine theta also goes out.

230
00:17:50,100 --> 00:17:52,530
And here is the
integral that remains.

231
00:17:52,530 --> 00:17:58,020
r cube dr. It was r squared,
but there was an extra r

232
00:17:58,020 --> 00:17:59,740
from the perturbation.

233
00:17:59,740 --> 00:18:06,690
r cubed dr into the
minus a over r--

234
00:18:06,690 --> 00:18:09,690
r over a0.

235
00:18:09,690 --> 00:18:17,170
And this thing you can
pass two cosine variables.

236
00:18:17,170 --> 00:18:19,340
It's pretty useful.

237
00:18:19,340 --> 00:18:25,290
So this goes minus
1 to 1 d cos theta

238
00:18:25,290 --> 00:18:33,395
prime, cos theta prime e to
the minus ikr cos theta prime.

239
00:18:36,950 --> 00:18:39,420
This cos theta prime
came from here.

240
00:18:43,260 --> 00:18:47,010
And that's it.

241
00:18:47,010 --> 00:19:01,160
So this is a nice result.
It looks still difficult,

242
00:19:01,160 --> 00:19:03,180
but we've made great progress.

243
00:19:03,180 --> 00:19:06,800
And in fact, we've dealt
with a really difficult part

244
00:19:06,800 --> 00:19:09,830
of this problem, which
is orienting yourself

245
00:19:09,830 --> 00:19:13,110
of how you're going to
approach the matrix element.

246
00:19:13,110 --> 00:19:18,470
So to finish up I'll just
give you a little more

247
00:19:18,470 --> 00:19:19,100
of the answer.

248
00:19:19,100 --> 00:19:20,700
We'll complete the discussion.

249
00:19:20,700 --> 00:19:25,550
We need probably 15 more
minutes to finish it up.

250
00:19:25,550 --> 00:19:27,680
So the only thing
I'm going to say now

251
00:19:27,680 --> 00:19:31,700
is that if you
look at the notes,

252
00:19:31,700 --> 00:19:34,565
every integral here
is easily doable.

253
00:19:38,090 --> 00:19:42,260
So basically, there's
two integrals.

254
00:19:42,260 --> 00:19:48,350
And the way to do them is
first do the r integral.

255
00:19:48,350 --> 00:19:51,560
You will have to
have these terms.

256
00:19:51,560 --> 00:19:53,780
And then do the theta integral.

257
00:19:53,780 --> 00:19:59,550
And they are kind of
simple, both of them.

258
00:19:59,550 --> 00:20:03,200
You have to keep up
a lot of constants.

259
00:20:03,200 --> 00:20:06,930
But here is the answer
for the matrix element.

260
00:20:06,930 --> 00:20:11,590
So that part of the integral
I think you all can do.

261
00:20:11,590 --> 00:20:14,720
But you have to take your time.

262
00:20:14,720 --> 00:20:23,210
i 32 square root of pi e E0 a0.

263
00:20:23,210 --> 00:20:25,180
I did a lot of work
here, actually,

264
00:20:25,180 --> 00:20:29,720
in writing it in a
comprehensible way

265
00:20:29,720 --> 00:20:32,330
because it's pretty messy.

266
00:20:32,330 --> 00:20:40,850
l cube a0 cube, 1
over 1 plus k squared

267
00:20:40,850 --> 00:20:46,850
a0 squared cube cosine theta.

268
00:20:46,850 --> 00:20:49,115
By now, it starts to
simplify a little.

269
00:20:53,520 --> 00:20:54,160
OK.

270
00:20:54,160 --> 00:20:59,690
That was actually plenty of work
to get it to write it this way.

271
00:20:59,690 --> 00:21:03,220
I feel pretty happy
about that writing.

272
00:21:03,220 --> 00:21:04,600
Why?

273
00:21:04,600 --> 00:21:07,210
First, OK, there
is these numbers.

274
00:21:07,210 --> 00:21:08,350
Nothing I can do about it.

275
00:21:08,350 --> 00:21:10,510
But there's a
multitude of constants

276
00:21:10,510 --> 00:21:13,520
that I have simplified and
done all kinds of things.

277
00:21:13,520 --> 00:21:14,530
But it was not worth it.

278
00:21:14,530 --> 00:21:20,470
First, ka0, that is unit free.

279
00:21:20,470 --> 00:21:22,240
This is unit free.

280
00:21:22,240 --> 00:21:24,490
This is unit free.

281
00:21:24,490 --> 00:21:27,400
k is 1 over length.

282
00:21:27,400 --> 00:21:32,110
ka0 to the 4 is the length cube,
and here is the length cube.

283
00:21:32,110 --> 00:21:34,990
No units here either.

284
00:21:34,990 --> 00:21:38,840
Here, nice units.

285
00:21:38,840 --> 00:21:40,810
This is units of energy.

286
00:21:40,810 --> 00:21:41,840
Why?

287
00:21:41,840 --> 00:21:44,930
Electric field times
distance is potential,

288
00:21:44,930 --> 00:21:49,660
times electric charge is energy.

289
00:21:49,660 --> 00:21:51,260
Energy.

290
00:21:51,260 --> 00:21:53,630
So this is how this should be.

291
00:21:53,630 --> 00:21:59,180
The matrix element of an energy
between normalizable states

292
00:21:59,180 --> 00:22:00,380
should be an energy.

293
00:22:00,380 --> 00:22:03,270
And that has become clear here.

294
00:22:03,270 --> 00:22:06,620
The next steps that we have to
do, which we'll do next time,

295
00:22:06,620 --> 00:22:12,120
is to integrate over states, and
put in the density of states,

296
00:22:12,120 --> 00:22:13,550
and do a little simplification.

297
00:22:13,550 --> 00:22:15,350
But now it's all trivial.

298
00:22:15,350 --> 00:22:17,480
We don't really have
to integrate anymore

299
00:22:17,480 --> 00:22:21,640
because Fermi's golden
rule did the job for us.