1
00:00:00,500 --> 00:00:05,280
PROFESSOR: How do we think
of the Born approximation?

2
00:00:11,420 --> 00:00:16,059
Well, we can imagine a
diagrammatic expression

3
00:00:16,059 --> 00:00:17,940
when we use propagators.

4
00:00:17,940 --> 00:00:22,540
Propagators are things
that take a signal

5
00:00:22,540 --> 00:00:25,960
and propagate it
in some direction.

6
00:00:25,960 --> 00:00:36,190
So think of your scattering
center here, and, of course,

7
00:00:36,190 --> 00:00:38,320
we were supposed
to look far away

8
00:00:38,320 --> 00:00:41,650
but I don't have enough
room on the blackboard.

9
00:00:41,650 --> 00:00:46,090
I'll take this point
to be far away.

10
00:00:46,090 --> 00:00:52,310
And now what does it
say, this equation?

11
00:00:52,310 --> 00:00:59,080
It says that the first Born
approximation, you get that r,

12
00:00:59,080 --> 00:01:00,490
the incident wave.

13
00:01:00,490 --> 00:01:03,040
The incident wave has reached r.

14
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This is the incident wave of r.

15
00:01:05,230 --> 00:01:13,980
So I will represent this as a
wave that reaches the point r.

16
00:01:13,980 --> 00:01:17,390
That's the first term
on the right hand

17
00:01:17,390 --> 00:01:22,700
side, a line reaching r.

18
00:01:22,700 --> 00:01:28,280
Because a wave has come in
that direction and reached r.

19
00:01:28,280 --> 00:01:34,890
On the other hand, this
is a little different.

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00:01:34,890 --> 00:01:40,010
What is the physical
interpretation of this term?

21
00:01:40,010 --> 00:01:43,160
This is a wave that
reached r prime.

22
00:01:46,350 --> 00:01:53,100
And this wave was
shaken by the potential

23
00:01:53,100 --> 00:01:58,830
and created the source, as
if the potential of r prime

24
00:01:58,830 --> 00:02:01,740
was a medium.

25
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The wave reached r prime, shook
the potential, and then a wave

26
00:02:06,960 --> 00:02:11,880
appeared that was propagated
by the propagator, the Green's

27
00:02:11,880 --> 00:02:12,380
function.

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It took it from r
prime, it took it to r.

29
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So our interpretation
of this diagrammatically

30
00:02:20,730 --> 00:02:24,690
is that another part
of the incoming wave

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00:02:24,690 --> 00:02:30,950
came here and got to
the point r prime.

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00:02:30,950 --> 00:02:36,930
At the point r
prime, it interacted

33
00:02:36,930 --> 00:02:42,440
with the potential,
which is a medium,

34
00:02:42,440 --> 00:02:53,230
and out came a wave
in the direction of r

35
00:02:53,230 --> 00:02:55,860
and reached there.

36
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And r prime is supposed
to be integrated.

37
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But that's kind
of in the figure.

38
00:03:06,130 --> 00:03:09,960
So this is the
zeroth order term,

39
00:03:09,960 --> 00:03:16,080
this is the first order term,
wave that reaches r prime,

40
00:03:16,080 --> 00:03:17,970
wave that reaches [INAUDIBLE].

41
00:03:17,970 --> 00:03:20,250
Now I can include
the second term.

42
00:03:20,250 --> 00:03:22,870
I could include
it in this diagram

43
00:03:22,870 --> 00:03:24,580
but I would confuse things.

44
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So let's draw another one.

45
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[INAUDIBLE] thing
doesn't look the same.

46
00:03:37,940 --> 00:03:41,370
I won't get them
to look the same.

47
00:03:41,370 --> 00:03:44,460
Doesn't matter.

48
00:03:44,460 --> 00:03:51,030
Here is r still, and what
is our second wave doing?

49
00:03:51,030 --> 00:03:54,390
Our second wave, we
could put it here

50
00:03:54,390 --> 00:04:00,750
and imagine that you have
here the e to the ikr.

51
00:04:00,750 --> 00:04:09,580
So here you would have e to
the ik dot r double prime.

52
00:04:09,580 --> 00:04:16,130
So the wave reaches the point
r double prime, which is here.

53
00:04:22,840 --> 00:04:26,320
And from the point
r double prime,

54
00:04:26,320 --> 00:04:28,450
it interacts with the medium--

55
00:04:28,450 --> 00:04:30,340
that's that heavy dot--

56
00:04:30,340 --> 00:04:34,090
and gets propagated
into r prime, which

57
00:04:34,090 --> 00:04:36,980
is also inside the material.

58
00:04:36,980 --> 00:04:42,345
So another wave,
and here is r prime.

59
00:04:46,490 --> 00:04:50,990
And once it's
propagated into r prime,

60
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it interacts with
the material u--

61
00:04:54,050 --> 00:04:55,570
that's the heavy dot--

62
00:04:55,570 --> 00:04:57,815
and gets propagated into r.

63
00:05:04,060 --> 00:05:08,040
So that's the second
order Feynman diagram.

64
00:05:08,040 --> 00:05:12,940
The zeroth order first, Feynman
diagram, the wave just reaches.

65
00:05:12,940 --> 00:05:16,180
First order, the wave
comes, interacts once

66
00:05:16,180 --> 00:05:20,170
with the potential, and
re-emits a wave over here.

67
00:05:20,170 --> 00:05:24,130
Second order comes here,
interacts with the potential,

68
00:05:24,130 --> 00:05:27,970
emits a wave, interacts
at another point,

69
00:05:27,970 --> 00:05:31,180
emits another wave,
and gets to you.

70
00:05:31,180 --> 00:05:35,530
That's a pictorial description
of a Born approximation, which

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makes it intuitive,
makes it clear.

72
00:05:38,470 --> 00:05:44,230
It is like Feynman diagrams of
field theory, in which you have

73
00:05:44,230 --> 00:05:47,350
some set of initial state,
some set of final state,

74
00:05:47,350 --> 00:05:54,820
and you sort of find the
ways the particles arrange

75
00:05:54,820 --> 00:05:56,090
themselves.

76
00:05:56,090 --> 00:05:59,080
If you have an electron
and a photon scattering,

77
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you have an electron here and
a photon scattering, and then

78
00:06:07,090 --> 00:06:10,860
photon and electron
out scattering.

79
00:06:10,860 --> 00:06:12,130
What do they do?

80
00:06:12,130 --> 00:06:15,370
And then Feynman method,
you think, OK, they

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00:06:15,370 --> 00:06:17,620
do all what they
can possibly do.

82
00:06:17,620 --> 00:06:22,120
They get the photon, interacts
with electron, electron

83
00:06:22,120 --> 00:06:24,940
propagates and
re-emits a photon,

84
00:06:24,940 --> 00:06:31,420
or it may be that the
photon gets emitted first

85
00:06:31,420 --> 00:06:35,020
and then the incoming
photon hits the other place

86
00:06:35,020 --> 00:06:37,240
and it goes that way.

87
00:06:37,240 --> 00:06:39,170
All these things can happen.

88
00:06:39,170 --> 00:06:41,980
And here you have
all the ways in which

89
00:06:41,980 --> 00:06:46,060
the waves can scatter
and re-scatter back

90
00:06:46,060 --> 00:06:48,040
to reach the observer.

91
00:06:48,040 --> 00:06:50,890
Basically, you're
finding all possible ways

92
00:06:50,890 --> 00:06:54,550
the waves interact with
the medium and reach you.

93
00:06:54,550 --> 00:06:57,100
And that's a very intuitive way.

94
00:06:57,100 --> 00:06:59,440
You think of it,
there the material,

95
00:06:59,440 --> 00:07:02,540
you're there, what do I get?

96
00:07:02,540 --> 00:07:05,500
Well, you get the waves
that just hit you directly.

97
00:07:05,500 --> 00:07:08,890
And then you get the waves that
hit the material and scatter

98
00:07:08,890 --> 00:07:09,532
to you.

99
00:07:09,532 --> 00:07:10,990
And then you get
the waves that hit

100
00:07:10,990 --> 00:07:14,830
the material, scattered, hit
the material again, and then got

101
00:07:14,830 --> 00:07:15,400
to you.

102
00:07:15,400 --> 00:07:18,430
And then you get the waves
that did that several times.

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00:07:18,430 --> 00:07:20,680
A very nice, simple picture.

104
00:07:23,500 --> 00:07:28,990
So that is the
Born approximation.

105
00:07:28,990 --> 00:07:30,860
We can do one example.

106
00:07:30,860 --> 00:07:33,580
Let's do one example.

107
00:07:33,580 --> 00:07:39,480
And assume, simplify
this formula.

108
00:07:39,480 --> 00:07:41,010
So we did this formula.

109
00:07:41,010 --> 00:07:45,480
And if the potential is
not spherically symmetric,

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you can't do more with it.

111
00:07:48,460 --> 00:07:52,080
But if the potential is
spherically symmetric,

112
00:07:52,080 --> 00:07:58,890
that first Born formula,
which is a very famous result,

113
00:07:58,890 --> 00:08:03,370
admits simplification and you
can do some of the integral.

114
00:08:03,370 --> 00:08:06,030
You can do the angular
part of the integral

115
00:08:06,030 --> 00:08:10,110
because the potential is
spherically symmetric,

116
00:08:10,110 --> 00:08:13,410
and you know how
things depend on angle.

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00:08:13,410 --> 00:08:19,590
So it's a nice thing to have a
boxed formula that already does

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00:08:19,590 --> 00:08:24,030
for you this calculation when
the potential is spherically

119
00:08:24,030 --> 00:08:24,760
symmetric.

120
00:08:24,760 --> 00:08:33,900
So if v of r is equal to v of r,
which is spherically symmetric,

121
00:08:33,900 --> 00:08:41,220
we go back to that formula and
we would have f of k of theta,

122
00:08:41,220 --> 00:08:49,350
we claim, we'll see if that
is true, minus 1 over 4 pi.

123
00:08:49,350 --> 00:08:55,020
Now this u there, remember,
this u was just the scaling of v

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00:08:55,020 --> 00:08:59,670
in order to make the Schrodinger
equation look simple.

125
00:08:59,670 --> 00:09:06,690
So that involves a factor
of 2 m over h bar squared.

126
00:09:06,690 --> 00:09:15,360
You still have the cube r prime,
e to the minus i k dot r prime,

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00:09:15,360 --> 00:09:16,590
v of r prime.

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00:09:24,200 --> 00:09:31,770
So what is the intuition
of such formulas?

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00:09:31,770 --> 00:09:36,150
I think it's important
for you, maybe

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00:09:36,150 --> 00:09:39,590
you haven't seen
these formulas before,

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00:09:39,590 --> 00:09:41,900
if you look at the
formula like this.

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00:09:44,850 --> 00:09:47,910
What do you expect the
answer to depend on?

133
00:09:54,470 --> 00:09:57,900
What is that integral
going to depend on?

134
00:10:03,120 --> 00:10:03,650
Any opinion?

135
00:10:14,390 --> 00:10:16,710
Well, it certainly will
depend on the potential,

136
00:10:16,710 --> 00:10:24,030
but now if it's very symmetric,
I might as well delete that.

137
00:10:24,030 --> 00:10:24,830
Is that right?

138
00:10:24,830 --> 00:10:28,710
We said it's
spherically symmetric.

139
00:10:28,710 --> 00:10:30,980
So it will depend
on the potential.

140
00:10:30,980 --> 00:10:33,220
That's, of course, true.

141
00:10:33,220 --> 00:10:37,750
But will it depend
on the vector k?

142
00:10:37,750 --> 00:10:40,420
Yes, it better depend
on the vector k.

143
00:10:40,420 --> 00:10:44,980
But all of the vector k
or some of the vector k?

144
00:10:54,130 --> 00:10:57,325
It's good to think about that
before doing the integral.

145
00:11:05,650 --> 00:11:07,920
Well, the interesting
thing is when

146
00:11:07,920 --> 00:11:11,340
you have an integral in this
form in which this doesn't

147
00:11:11,340 --> 00:11:17,960
depend on the direction
of r, this is our prime,

148
00:11:17,960 --> 00:11:21,140
but in fact, you
see, there's nowhere

149
00:11:21,140 --> 00:11:25,310
in this formula anything
to do with r and r prime.

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00:11:25,310 --> 00:11:27,720
This is an integral
all over space.

151
00:11:27,720 --> 00:11:32,910
So in fact, I can also
delete now the primes,

152
00:11:32,910 --> 00:11:38,690
integrate them
over all of space.

153
00:11:38,690 --> 00:11:41,860
There's no r on the left
hand side, there's no r left,

154
00:11:41,860 --> 00:11:44,400
so what for carrying primes?

155
00:11:44,400 --> 00:11:47,380
No need for that.

156
00:11:47,380 --> 00:11:53,290
Now what I want to explain here
is that this integral can only

157
00:11:53,290 --> 00:11:57,640
depend on the magnitude
of the vector k.

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00:11:57,640 --> 00:12:01,000
Cannot depend on the
direction of the vector k.

159
00:12:01,000 --> 00:12:06,100
And because it depends on the
magnitude of the vector k,

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00:12:06,100 --> 00:12:10,910
it only depends on theta,
which is what we wanted to say.

161
00:12:10,910 --> 00:12:12,590
So if before, you write

162
00:12:12,590 --> 00:12:15,230
For theta, you could have
asked, well, how do you know?

163
00:12:15,230 --> 00:12:19,100
This interval can depend
on phi, k dependent on phi,

164
00:12:19,100 --> 00:12:26,090
but no, this integral doesn't
depend on the direction of k.

165
00:12:26,090 --> 00:12:30,380
And it doesn't because
you can choose your axis

166
00:12:30,380 --> 00:12:33,840
to do this integral
whichever way you want.

167
00:12:33,840 --> 00:12:36,480
This is a dummy
variable of integration.

168
00:12:36,480 --> 00:12:40,790
So if vector K points
in this direction,

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00:12:40,790 --> 00:12:45,140
you could choose the z-axis
to go there, and that's it.

170
00:12:45,140 --> 00:12:49,640
It's only going to depend on
the magnitude of that vector.

171
00:12:49,640 --> 00:12:55,080
So that's an important
concept in here.

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00:12:55,080 --> 00:12:57,380
Oh, and you can
think it another way.

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00:12:57,380 --> 00:13:02,780
This answer, if I would have
put another vector k prime here

174
00:13:02,780 --> 00:13:10,220
that is obtained by
rotation, think of k prime,

175
00:13:10,220 --> 00:13:18,370
I put here k prime, which
is obtained by a rotation,

176
00:13:18,370 --> 00:13:23,980
acted by a rotation matrix on
the vector, say, from the left.

177
00:13:23,980 --> 00:13:27,610
Well, you would say, OK, if
it's acted by a rotation,

178
00:13:27,610 --> 00:13:32,620
I could instead make the
rotation act on the vector r.

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00:13:32,620 --> 00:13:35,890
This potential doesn't
change on the rotations

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00:13:35,890 --> 00:13:39,560
and this measure is
invariant on the rotation.

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00:13:39,560 --> 00:13:42,490
So at the end of the
day, this integral

182
00:13:42,490 --> 00:13:45,010
does not depend on
the rotation you did.

183
00:13:45,010 --> 00:13:48,550
So bottom line,
this integral just

184
00:13:48,550 --> 00:13:52,780
doesn't depend on
the direction of k,

185
00:13:52,780 --> 00:13:57,310
and we can do it by putting k--

186
00:13:57,310 --> 00:14:00,700
or saying k is
here and now making

187
00:14:00,700 --> 00:14:06,651
it explicit by putting r prime
here and calling this angle

188
00:14:06,651 --> 00:14:07,150
theta.

189
00:14:10,670 --> 00:14:12,990
And we just do it.

190
00:14:12,990 --> 00:14:17,760
So if we call it that
way, this integral

191
00:14:17,760 --> 00:14:28,370
is, you have minus m
over 2 pi h squared,

192
00:14:28,370 --> 00:14:30,600
and you have the
volume integral, which

193
00:14:30,600 --> 00:14:35,030
is 2 pi for the phi integral.

194
00:14:35,030 --> 00:14:43,400
So we'll have a theta and a phi
that you integrate with respect

195
00:14:43,400 --> 00:14:47,690
to k, and the phi integral
will always give you 2 pi

196
00:14:47,690 --> 00:14:50,400
and will leave the
theta integral here.

197
00:14:50,400 --> 00:14:51,510
So 2 pi.

198
00:14:51,510 --> 00:14:56,510
And the r integral
is r squared dr.

199
00:14:56,510 --> 00:15:02,315
The theta integral in spherical
coordinates is v cos theta.

200
00:15:06,560 --> 00:15:11,820
And you have here e to the
minus i length of this,

201
00:15:11,820 --> 00:15:17,340
length of that, cosine
theta times v of r.

202
00:15:23,060 --> 00:15:26,690
So things simplify even more.

203
00:15:26,690 --> 00:15:35,280
So the 2 pis can [INAUDIBLE]
minus m over h squared.

204
00:15:35,280 --> 00:15:46,380
The radial integral, it's 0 to
infinity, r squared vr, v of r,

205
00:15:46,380 --> 00:15:51,380
and you have the
angular integral,

206
00:15:51,380 --> 00:15:54,870
and that angular integral
is easily doable.

207
00:15:54,870 --> 00:15:56,285
It's that and this.

208
00:15:59,130 --> 00:16:05,140
You can [INAUDIBLE] cosine theta
u and just do the integral.

209
00:16:05,140 --> 00:16:14,030
That integral gives you
2 sin of Kr over kr.

210
00:16:21,230 --> 00:16:32,320
And our result, therefore,
is, you shouldn't confuse

211
00:16:32,320 --> 00:16:37,750
the capital K with
the lowercase k, which

212
00:16:37,750 --> 00:16:41,920
is the magnitude of the momentum
or the magnitude of the energy.

213
00:16:47,740 --> 00:16:49,540
So we're done, basically.

214
00:16:52,990 --> 00:16:57,780
And what do we have
for fk of theta?

215
00:16:57,780 --> 00:17:06,970
We have minus 2m
over K h squared,

216
00:17:06,970 --> 00:17:14,740
integral from 0 to infinity
of r, dr, one r got cancelled,

217
00:17:14,740 --> 00:17:23,750
the k went out, r dr,
v of r, sine of Kr.

218
00:17:30,770 --> 00:17:36,710
So the Fourier transform
got partially done.

219
00:17:36,710 --> 00:17:44,240
And here, K, as you remember,
is 2 k sine theta over 2.

220
00:17:50,970 --> 00:17:54,340
So there we get the Born
approximation simplified

221
00:17:54,340 --> 00:17:56,470
as much as we could.

222
00:17:56,470 --> 00:17:58,770
We couldn't do
much better anymore

223
00:17:58,770 --> 00:18:02,800
without knowing what
the potential is.

224
00:18:02,800 --> 00:18:04,710
But once you know
the potential, it

225
00:18:04,710 --> 00:18:07,830
has become just the
simple integral.

226
00:18:07,830 --> 00:18:15,270
So while the concepts that led
to this result are interesting,

227
00:18:15,270 --> 00:18:17,820
and they involve
nice approximations,

228
00:18:17,820 --> 00:18:21,430
by the time you have this
result, this is very useful.

229
00:18:21,430 --> 00:18:24,870
You can put an arbitrary
potential v of r,

230
00:18:24,870 --> 00:18:28,110
calculate the
integral, and out you

231
00:18:28,110 --> 00:18:32,600
get the scattering amplitude.

232
00:18:32,600 --> 00:18:37,320
So this is done, for example,
for Yukawa potential,

233
00:18:37,320 --> 00:18:44,640
v of r minus 1 over r--

234
00:18:44,640 --> 00:18:46,800
well, we will put the beta--

235
00:18:50,190 --> 00:18:56,370
e to the minus mu r.

236
00:18:56,370 --> 00:18:58,530
That's called a
Yukawa potential.

237
00:19:05,050 --> 00:19:11,810
It's the kind of modification
of the electromagnetic potential

238
00:19:11,810 --> 00:19:16,460
in the limit in which the
photon would acquire a mass.

239
00:19:16,460 --> 00:19:19,190
That was sort of the
intuition of Yukawa.

240
00:19:19,190 --> 00:19:26,250
If you see-- this has
units of [INAUDIBLE]

241
00:19:26,250 --> 00:19:29,560
physically mass and h
bars and things like that.

242
00:19:29,560 --> 00:19:36,420
But if you let this thing
go to 0, if mu goes to zero,

243
00:19:36,420 --> 00:19:42,200
you recover coulomb potential.

244
00:19:42,200 --> 00:19:46,870
So that's the relevance
of the Yukawa potential.

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00:19:46,870 --> 00:19:50,060
Yukawa invented it
while thinking of pions

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and thinking of the forces
transmitted by pions, who

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are light hadrons and strongly
interacting particles,

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but with low mass,
relatively low mass

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compared to a proton, one
fifth of it or even less,

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maybe, one seventh, one eighth.

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00:20:11,030 --> 00:20:13,850
And then you would have
a potential of this for

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00:20:13,850 --> 00:20:16,100
and you can study
scattering when

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00:20:16,100 --> 00:20:19,900
you have this kind of formula.

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00:20:19,900 --> 00:20:24,430
So this is solved in many books.

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00:20:24,430 --> 00:20:29,150
It's also in the textbook,
and in this case, f of k,

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the integral can
be done exactly.

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You can plug v of r there.

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And the answer is
that fk of theta

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00:20:38,660 --> 00:20:53,171
is equal to minus 2 m beta over
h squared mu squared plus K

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squared.

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00:20:58,870 --> 00:21:02,540
And K squared would
be this quantity.

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00:21:02,540 --> 00:21:05,410
So there is a theta
dependence, and you

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00:21:05,410 --> 00:21:09,830
can look at this formula,
compare with the Coulomb k,

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00:21:09,830 --> 00:21:16,350
[? take ?] limits, plot it,
it's kind of interesting.

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So it's easily done, and you
can play with many potentials.

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All right, so we're
done with scattering.

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Done like three lectures
and a half on this subject,

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and I'm going to now turn into
a new subject, our last subject,

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which is we're going
to try to explain

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00:21:37,720 --> 00:21:41,760
some of the basics of
identical particles.