1
00:00:00,792 --> 00:00:01,910
PROFESSOR: OK.

2
00:00:01,910 --> 00:00:06,650
So we have set up the problem.

3
00:00:09,410 --> 00:00:12,250
We now have to understand
what is the physics

4
00:00:12,250 --> 00:00:14,880
of this f of theta and phi.

5
00:00:14,880 --> 00:00:19,060
In a way, solving this
scattering problem

6
00:00:19,060 --> 00:00:25,060
means solving for
the thetas and phis.

7
00:00:25,060 --> 00:00:28,240
Now, you would say, OK,
that seems reasonable.

8
00:00:28,240 --> 00:00:31,222
You know how the
solution looks far away?

9
00:00:31,222 --> 00:00:33,430
And that's where you're
going to do the measurements.

10
00:00:33,430 --> 00:00:35,800
That's where you
have the detectors.

11
00:00:35,800 --> 00:00:40,840
So we have to find what
f of theta and phi is.

12
00:00:40,840 --> 00:00:44,260
But suppose you had it.

13
00:00:44,260 --> 00:00:48,130
What have you learned if you
have the f of theta and phi?

14
00:00:48,130 --> 00:00:52,120
So this is what we
need to compute,

15
00:00:52,120 --> 00:00:57,400
and we will build towards
the computation of this using

16
00:00:57,400 --> 00:01:01,690
what is called partial waves--

17
00:01:01,690 --> 00:01:04,205
waves and phase shifts.

18
00:01:08,360 --> 00:01:12,200
This will allow us to calculate
the f of theta and phi.

19
00:01:12,200 --> 00:01:17,000
But doesn't tell you yet
what f of theta and phi is.

20
00:01:17,000 --> 00:01:21,190
So the nice thing of
this f of theta and phi

21
00:01:21,190 --> 00:01:25,910
is that it gives you the
cross section of the process.

22
00:01:25,910 --> 00:01:30,260
So we need to understand
what is the cross section.

23
00:01:30,260 --> 00:01:33,500
So what is a cross section?

24
00:01:33,500 --> 00:01:39,050
It's a way to quantify the
effect of the scattering

25
00:01:39,050 --> 00:01:41,630
center on the particles.

26
00:01:41,630 --> 00:01:44,605
Suppose you have your
scattering center

27
00:01:44,605 --> 00:01:48,350
and you shoot particles in.

28
00:01:48,350 --> 00:01:50,900
Then you can say, OK, let me--

29
00:01:50,900 --> 00:01:55,340
let's see with my detectors
how many particles go off

30
00:01:55,340 --> 00:01:59,140
at an angle theta?

31
00:01:59,140 --> 00:02:02,570
When you say three
particles per second,

32
00:02:02,570 --> 00:02:05,140
I'm finding as you
should in particles,

33
00:02:05,140 --> 00:02:07,510
I'm getting three
particles per second

34
00:02:07,510 --> 00:02:11,260
in this detector at this angle.

35
00:02:11,260 --> 00:02:18,820
My detector at this angle
covers a solid angle of 100,

36
00:02:18,820 --> 00:02:22,810
and it's getting three
particles per second.

37
00:02:22,810 --> 00:02:29,050
Then you can say, OK, what
is the area of a target that

38
00:02:29,050 --> 00:02:33,340
out of the incoming
beam captures

39
00:02:33,340 --> 00:02:35,470
three particles per second?

40
00:02:35,470 --> 00:02:38,740
And maybe-- there are lots
of particles coming in,

41
00:02:38,740 --> 00:02:42,370
but if you produce
some area, that area

42
00:02:42,370 --> 00:02:45,970
will capture three
particles per second.

43
00:02:45,970 --> 00:02:50,470
And that would be what we call
the differential cross section.

44
00:02:50,470 --> 00:02:57,010
The value of the area
that captures or just

45
00:02:57,010 --> 00:03:00,490
whose flux of particles
is precisely what you're

46
00:03:00,490 --> 00:03:02,000
getting out there.

47
00:03:02,000 --> 00:03:06,100
So if you're looking at some
process in which there's

48
00:03:06,100 --> 00:03:11,410
this incoming particles and
you look at some angle--

49
00:03:11,410 --> 00:03:15,600
d omega-- where
there's a detector,

50
00:03:15,600 --> 00:03:18,670
a detector that
capture this angle,

51
00:03:18,670 --> 00:03:24,440
you can associate to
this object and d sigma--

52
00:03:24,440 --> 00:03:27,400
which is a differential
cross section--

53
00:03:27,400 --> 00:03:28,705
with units of area.

54
00:03:32,430 --> 00:03:36,930
And the physical interpretation
of the d sigma for this--

55
00:03:36,930 --> 00:03:42,600
the omega-- is that
sigma is the area that

56
00:03:42,600 --> 00:03:47,460
captures the amount
of flux that you

57
00:03:47,460 --> 00:03:52,410
see going in this direction.

58
00:03:52,410 --> 00:03:54,800
So it's a way this area--

59
00:03:54,800 --> 00:03:58,040
this differential cross
sections-- give you

60
00:03:58,040 --> 00:04:02,060
an iv, or a concrete
representation,

61
00:04:02,060 --> 00:04:07,210
of how big the target is as
seen by the particles that

62
00:04:07,210 --> 00:04:09,110
are coming in.

63
00:04:09,110 --> 00:04:13,490
I'll write it in a way
that makes it clear.

64
00:04:13,490 --> 00:04:19,444
So d sigma, which is called the
differential cross section--

65
00:04:19,444 --> 00:04:30,420
differential cross section--
is equal number of particles--

66
00:04:30,420 --> 00:04:33,890
there's lots of words here,
but it's good to write them--

67
00:04:33,890 --> 00:04:49,530
scattered per unit time
into the solid angle--

68
00:04:49,530 --> 00:05:06,925
d omega-- divided by the flux
of incident particles, which

69
00:05:06,925 --> 00:05:11,500
is equal to the
number of particles

70
00:05:11,500 --> 00:05:17,320
per unit area per unit time.

71
00:05:17,320 --> 00:05:20,350
So it's this ratio.

72
00:05:20,350 --> 00:05:23,260
Number of particles
scattered per unit time

73
00:05:23,260 --> 00:05:26,020
into solid angle, d omega--

74
00:05:26,020 --> 00:05:29,290
particles have no units.

75
00:05:29,290 --> 00:05:33,430
Integers scatter per
unit time is 1 over time

76
00:05:33,430 --> 00:05:37,420
into the solid angle-- d
omega has no units, either.

77
00:05:37,420 --> 00:05:43,030
The solid angle floods of
incident particles is--

78
00:05:43,030 --> 00:05:46,130
particles that has no
units over area over time.

79
00:05:46,130 --> 00:05:49,510
So here you had over time,
here over time, they cancel,

80
00:05:49,510 --> 00:05:51,760
this is 1 over area.

81
00:05:51,760 --> 00:05:56,330
That ratio, therefore,
has units of area.

82
00:05:56,330 --> 00:05:58,930
And this is what
I was telling you.

83
00:05:58,930 --> 00:06:01,010
The differential
cross section, which

84
00:06:01,010 --> 00:06:04,330
is an area, multiplied
by the flux,

85
00:06:04,330 --> 00:06:08,590
gives you the number of
particles per unit time

86
00:06:08,590 --> 00:06:11,410
that are crossing at
differential cross section

87
00:06:11,410 --> 00:06:12,340
area.

88
00:06:12,340 --> 00:06:14,770
And those are set
equal to the number

89
00:06:14,770 --> 00:06:20,590
of particles per unit time that
end up within this solid angle.

90
00:06:20,590 --> 00:06:27,050
So for a given
little solid angle,

91
00:06:27,050 --> 00:06:30,020
you get the d sigma,
which is small, as well.

92
00:06:30,020 --> 00:06:33,470
That's why it's called
differential cross section.

93
00:06:33,470 --> 00:06:40,880
So let's try to show that this
d sigma is really determined

94
00:06:40,880 --> 00:06:43,250
by f of theta and phi.

95
00:06:43,250 --> 00:06:49,050
This angle-- solid
angle, d omega--

96
00:06:49,050 --> 00:06:54,180
is happening at some
theta and some phi.

97
00:06:54,180 --> 00:06:58,430
That's a position, the center
of the little solid angle.

98
00:06:58,430 --> 00:07:03,090
A solid angle, d
omega, at theta phi.

99
00:07:06,230 --> 00:07:09,320
So this is our goal
now, to just calculate

100
00:07:09,320 --> 00:07:12,770
the differential cross section.

101
00:07:12,770 --> 00:07:13,610
OK.

102
00:07:13,610 --> 00:07:16,940
So we have to
compute this fluxes.

103
00:07:16,940 --> 00:07:19,700
We can do it here.

104
00:07:19,700 --> 00:07:23,240
Now, we're not working
with wave pockets.

105
00:07:23,240 --> 00:07:25,800
We're working with
energy eigenstates,

106
00:07:25,800 --> 00:07:28,700
and we're going to have a
little bit of a fun intuition.

107
00:07:28,700 --> 00:07:30,650
We're not-- we're
going to get it right,

108
00:07:30,650 --> 00:07:35,000
but you have to appreciate
it's a little funny.

109
00:07:35,000 --> 00:07:38,630
If I have a little
volume, and I say

110
00:07:38,630 --> 00:07:41,930
that I find the probability
to find the particle there

111
00:07:41,930 --> 00:07:46,040
to be 1/2, I would
say that there's

112
00:07:46,040 --> 00:07:48,980
a half of a particle
in that little volume.

113
00:07:48,980 --> 00:07:52,040
If I have a volume
that whose probability

114
00:07:52,040 --> 00:07:57,440
to find a particle is 1, I say,
OK, I have 1 particle in here.

115
00:07:57,440 --> 00:08:00,350
That's the rough intuition.

116
00:08:00,350 --> 00:08:03,860
So for wave functions
that are not normalizable,

117
00:08:03,860 --> 00:08:05,480
it's almost like--

118
00:08:05,480 --> 00:08:09,020
suppose size squared
is equal to 1.

119
00:08:09,020 --> 00:08:10,940
In a momentum, I
can say you would

120
00:08:10,940 --> 00:08:15,140
be saying like you have a
particle for every unit volume

121
00:08:15,140 --> 00:08:18,020
where size squared
integrates to 1.

122
00:08:18,020 --> 00:08:20,510
So that's roughly the intuition.

123
00:08:20,510 --> 00:08:24,830
And it's correct,
really, to use it there.

124
00:08:24,830 --> 00:08:27,950
If you think of the flux
of incident particles--

125
00:08:27,950 --> 00:08:29,180
so incident flux--

126
00:08:32,840 --> 00:08:40,360
flux-- we think of it as
the probability current,

127
00:08:40,360 --> 00:08:43,960
which is number of particles
per unit time per unit area.

128
00:08:43,960 --> 00:08:52,090
So it's h bar over m, imaginary
part of the incident flux.

129
00:08:52,090 --> 00:08:56,620
So I should use the incident
wave function, this phi

130
00:08:56,620 --> 00:09:00,760
of r, gradient phi of r.

131
00:09:06,840 --> 00:09:13,260
And phi was e to the ikz.

132
00:09:13,260 --> 00:09:17,310
That was the incident particle.

133
00:09:17,310 --> 00:09:20,460
So this Laplacian gives
you-- not Laplacian--

134
00:09:20,460 --> 00:09:25,815
this gradient takes a
gradient, it produces an ik,

135
00:09:25,815 --> 00:09:30,930
the rest gets canceled, and
the imaginary part gives you k.

136
00:09:30,930 --> 00:09:38,550
So this is h bar k over m
times the unit vector z.

137
00:09:38,550 --> 00:09:42,950
So that's the incident flux.

138
00:09:42,950 --> 00:09:47,870
This can be thought as the
number of particles per unit

139
00:09:47,870 --> 00:09:50,510
area and per unit time.

140
00:09:50,510 --> 00:09:53,270
You can also think
of it intuitively

141
00:09:53,270 --> 00:09:58,430
as rho times v.
The incident flux

142
00:09:58,430 --> 00:10:01,430
is like the incident
current, the current density,

143
00:10:01,430 --> 00:10:05,755
it's rho times v.
rho is phi squared--

144
00:10:08,310 --> 00:10:11,610
phi squared.

145
00:10:11,610 --> 00:10:12,930
But that's equal to 1.

146
00:10:16,340 --> 00:10:18,430
And the velocity
of the particles

147
00:10:18,430 --> 00:10:22,940
is the momentum over the mass.

148
00:10:22,940 --> 00:10:27,870
But that's hk over m.

149
00:10:27,870 --> 00:10:30,450
And if I put the direction,
I will have to put the z.

150
00:10:33,740 --> 00:10:37,550
So it's kind of the
same thing, rho v,

151
00:10:37,550 --> 00:10:39,710
the incident flux are all there.

152
00:10:39,710 --> 00:10:43,460
So this is the denominator
of that big formula

153
00:10:43,460 --> 00:10:45,380
we have there--

154
00:10:45,380 --> 00:10:46,610
the incident flux.

155
00:10:49,600 --> 00:10:54,070
Then we need the
number of particles

156
00:10:54,070 --> 00:10:59,920
scattered per unit time.

157
00:10:59,920 --> 00:11:03,040
We could do it with
a flux calculation.

158
00:11:03,040 --> 00:11:07,210
Also, in spherical
coordinates, but I can also

159
00:11:07,210 --> 00:11:09,730
do it intuitively.

160
00:11:09,730 --> 00:11:11,320
So I'll do it intuitively.

161
00:11:11,320 --> 00:11:16,615
You can try doing it using
a probability current.

162
00:11:19,150 --> 00:11:21,320
So intuitively what do we have?

163
00:11:21,320 --> 00:11:25,640
We have a d omega here.

164
00:11:25,640 --> 00:11:29,650
And let's consider a
little volume element here.

165
00:11:39,270 --> 00:11:41,150
So a little volume element.

166
00:11:41,150 --> 00:11:42,590
Here you have r--

167
00:11:42,590 --> 00:11:45,360
distance r-- a little vr.

168
00:11:45,360 --> 00:11:52,350
So a small volume element here.

169
00:11:52,350 --> 00:11:56,540
And let's calculate how
many particles there are

170
00:11:56,540 --> 00:11:58,430
in this small little element.

171
00:11:58,430 --> 00:12:05,450
So dn is number of particles
in the little volume--

172
00:12:08,045 --> 00:12:08,545
volume.

173
00:12:11,430 --> 00:12:15,660
So I must square
the wave function

174
00:12:15,660 --> 00:12:19,270
and multiply by the volume.

175
00:12:19,270 --> 00:12:20,950
That's what you would do.

176
00:12:20,950 --> 00:12:24,600
So what is the square
of the wave function?

177
00:12:24,600 --> 00:12:28,580
Now, we are talking about
the scattered wave function.

178
00:12:28,580 --> 00:12:36,170
So I must take this part
of the wave function

179
00:12:36,170 --> 00:12:40,680
and square it and
multiply by the volume.

180
00:12:40,680 --> 00:12:45,110
So we will have f of
theta and phi times

181
00:12:45,110 --> 00:12:50,490
e to the ikr over
r, all squared.

182
00:12:50,490 --> 00:12:53,690
So that's the wave
function squared.

183
00:12:53,690 --> 00:13:02,860
That's psi carat squared times
the air-- volume of this pill

184
00:13:02,860 --> 00:13:08,075
box is r squared
d omega times dr.

185
00:13:08,075 --> 00:13:12,740
r squared d omega is the
surface area of that little box.

186
00:13:12,740 --> 00:13:16,370
dr is the little length there.

187
00:13:16,370 --> 00:13:20,820
And, therefore, we have
that volume over there.

188
00:13:20,820 --> 00:13:24,510
So things simplify
the n is equal,

189
00:13:24,510 --> 00:13:26,910
therefore, the r
squared cancels,

190
00:13:26,910 --> 00:13:38,130
and it's just f of theta
phi squared, d omega, dr.

191
00:13:38,130 --> 00:13:50,450
So with all these
particles-- this

192
00:13:50,450 --> 00:13:52,820
is the number of
particles inside the box.

193
00:13:52,820 --> 00:13:55,670
But all these particles
will go through

194
00:13:55,670 --> 00:14:05,880
in a little time dt, which is
equal to dr over the velocity.

195
00:14:05,880 --> 00:14:08,030
This is the time to
go through the box--

196
00:14:11,900 --> 00:14:18,460
through the box.

197
00:14:18,460 --> 00:14:21,380
dt is dr over the velocity.

198
00:14:21,380 --> 00:14:25,670
And the velocity is hk over m.

199
00:14:25,670 --> 00:14:30,880
So dr times the momentum over
the mass, which is hk over m.

200
00:14:34,550 --> 00:14:35,050
OK.

201
00:14:35,050 --> 00:14:37,540
So we have dn, dt.

202
00:14:37,540 --> 00:14:38,870
This is number of particles.

203
00:14:38,870 --> 00:14:44,230
We divide them to form number
of particles per unit time.

204
00:14:44,230 --> 00:14:54,190
So the ratio dn dt is the
numerator of this quantity

205
00:14:54,190 --> 00:14:56,370
here--

206
00:14:56,370 --> 00:15:04,670
is numerator dn [? dt. ?]

207
00:15:04,670 --> 00:15:07,310
And what do we get?

208
00:15:07,310 --> 00:15:14,850
The dr cancels and
we're dividing the ndt--

209
00:15:14,850 --> 00:15:17,470
this factor goes
into the numerator,

210
00:15:17,470 --> 00:15:26,670
so we get h bar k over m f
of theta phi squared d omega.

211
00:15:34,000 --> 00:15:40,930
So this is the numerator of
that equation for the d sigma.

212
00:15:40,930 --> 00:15:43,630
So let's compute-- finish that.

213
00:15:43,630 --> 00:15:46,942
We finally-- I'm going to
get off our identification.

214
00:15:53,550 --> 00:15:58,140
And what do we get?

215
00:16:07,696 --> 00:16:15,220
We get d sigma is
the numerator there,

216
00:16:15,220 --> 00:16:24,880
which is this quantity, h
bar k over m f of theta phi

217
00:16:24,880 --> 00:16:34,790
squared times d omega over h
bar k over m, which is nice.

218
00:16:34,790 --> 00:16:37,340
That all going to cancel.

219
00:16:37,340 --> 00:16:42,990
And then we get the nice
formula that we were after.

220
00:16:42,990 --> 00:16:50,900
This differential
cross section is just

221
00:16:50,900 --> 00:16:54,350
determined by the
function f of theta phi

222
00:16:54,350 --> 00:16:56,300
that we need to calculate.

223
00:16:56,300 --> 00:16:58,790
f of theta phi is our goal.

224
00:16:58,790 --> 00:17:03,950
If we have it, we have the
differential cross section.

225
00:17:03,950 --> 00:17:08,089
Many people write this
formula this way--

226
00:17:08,089 --> 00:17:14,245
d sigma, d omega is
equal to f of theta phi.

227
00:17:23,924 --> 00:17:25,550
I think that's OK.

228
00:17:25,550 --> 00:17:28,580
I think in many ways
this is a little clearer.

229
00:17:28,580 --> 00:17:32,090
This small little
area associated

230
00:17:32,090 --> 00:17:35,030
to a small little
angle is given by that.

231
00:17:35,030 --> 00:17:40,045
This is a ratio of differentials
more than a derivative.

232
00:17:40,045 --> 00:17:44,010
A derivative with
respect to solid angle

233
00:17:44,010 --> 00:17:46,760
doesn't mean too much, I think.

234
00:17:46,760 --> 00:17:51,590
So-- but this is another
way people write it.

235
00:17:51,590 --> 00:17:55,725
And, finally, people
integrate this total cross

236
00:17:55,725 --> 00:17:59,930
section is the integral
of the differential

237
00:17:59,930 --> 00:18:02,930
cross section over--

238
00:18:02,930 --> 00:18:05,360
looking at all solid angles.

239
00:18:05,360 --> 00:18:15,720
So this means integrating f
of theta phi squared d omega.

240
00:18:15,720 --> 00:18:17,875
And that's the
total cross section.

241
00:18:31,100 --> 00:18:31,600
OK.

242
00:18:36,390 --> 00:18:39,520
So we've set up the problem.

243
00:18:39,520 --> 00:18:44,190
We-- what are the main
things that we've learned?

244
00:18:44,190 --> 00:18:47,040
We have the scattering center.

245
00:18:47,040 --> 00:18:51,990
We have our physical
condition of a wave coming in.

246
00:18:51,990 --> 00:18:55,800
Our physical condition
that the wave comes out.

247
00:18:55,800 --> 00:18:58,650
Those are represented here.

248
00:18:58,650 --> 00:19:01,020
And this situation
is saying that this

249
00:19:01,020 --> 00:19:05,190
is an approximate
solution for a wave--

250
00:19:05,190 --> 00:19:11,100
a wave coming out modulated
by a theta and phi dependence

251
00:19:11,100 --> 00:19:14,820
and your incoming wave.

252
00:19:14,820 --> 00:19:20,580
This modulation is the thing
that captures the effect

253
00:19:20,580 --> 00:19:26,460
of the potential and associates
to the strength of this

254
00:19:26,460 --> 00:19:29,640
potential and its ability
to scatter a differ--

255
00:19:29,640 --> 00:19:32,260
a cross section,
which is measurable--

256
00:19:32,260 --> 00:19:36,150
which means a probability
of interaction--

257
00:19:36,150 --> 00:19:39,330
capture the probability
of interaction.

258
00:19:39,330 --> 00:19:41,790
This cross sections
are very important.

259
00:19:41,790 --> 00:19:44,970
When you design an experiment
with an accelerator,

260
00:19:44,970 --> 00:19:48,990
you want to have a cross
section is sufficiently large,

261
00:19:48,990 --> 00:19:51,690
because this cross section
is going to tell you

262
00:19:51,690 --> 00:19:55,090
how many particles your
detector is going to get.

263
00:19:55,090 --> 00:19:58,440
Whether with the
flux that you have,

264
00:19:58,440 --> 00:20:00,780
with the beam intensity
that you have,

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00:20:00,780 --> 00:20:06,470
you're going to get one Higgs
a day, or 500 Higgs's a day.

266
00:20:06,470 --> 00:20:09,780
And it made a
difference for the LHC.

267
00:20:09,780 --> 00:20:12,450
It began with a
few Higgs's a day,

268
00:20:12,450 --> 00:20:16,190
and later, it's getting a
few hundred Higgs's a day.

269
00:20:16,190 --> 00:20:20,570
So-- so those quantities
are pretty important.