1
00:00:05,150 --> 00:00:07,640
MARKUS KLUTE: Now back to 8.701.

2
00:00:07,640 --> 00:00:12,520
So in this section, we'll talk
about relativistic kinematics.

3
00:00:12,520 --> 00:00:15,575
Let me start by saying that
one of my favorite classes here

4
00:00:15,575 --> 00:00:19,730
at MIT is a class called
8.20, special relativity,

5
00:00:19,730 --> 00:00:25,100
where we teach students about
special relativity, of course,

6
00:00:25,100 --> 00:00:28,280
but Einstein and paradoxes.

7
00:00:28,280 --> 00:00:30,080
And it's one of my
favorite classes.

8
00:00:30,080 --> 00:00:33,290
And in their class,
there's a component

9
00:00:33,290 --> 00:00:36,650
on particle physics, which
has to do with just using

10
00:00:36,650 --> 00:00:39,890
relativistic kinematics
in order to understand how

11
00:00:39,890 --> 00:00:44,240
to create antimatter,
how to collide beams,

12
00:00:44,240 --> 00:00:46,190
how we can analyze decays.

13
00:00:46,190 --> 00:00:48,360
And in this
introductory section,

14
00:00:48,360 --> 00:00:51,530
we're going to do a
very similar thing.

15
00:00:51,530 --> 00:00:55,820
I trust that you all had some
sort of class introduction

16
00:00:55,820 --> 00:01:01,160
of special relativity, some of
you maybe general relativity.

17
00:01:01,160 --> 00:01:03,770
What we want to do here
is review this content

18
00:01:03,770 --> 00:01:09,930
very briefly, but then more
use it in a number of examples.

19
00:01:09,930 --> 00:01:12,860
So in particle physics,
nuclear physics,

20
00:01:12,860 --> 00:01:15,530
we often deal particles
who will travel

21
00:01:15,530 --> 00:01:17,060
close to the speed of light.

22
00:01:17,060 --> 00:01:20,300
The photon travels at
the speed of light.

23
00:01:20,300 --> 00:01:26,390
We typically define the velocity
as v/c in natural units.

24
00:01:26,390 --> 00:01:30,980
Beta is the velocity,
gamma is defined by 1

25
00:01:30,980 --> 00:01:34,820
over square root 1 minus
the velocity squared.

26
00:01:34,820 --> 00:01:37,580
Beta is always
smaller or equal to 1,

27
00:01:37,580 --> 00:01:40,550
smaller for massive particle.

28
00:01:40,550 --> 00:01:44,340
And gamma is always
equal or greater than 1.

29
00:01:44,340 --> 00:01:47,660
The total energy of a
particle with 0 mass--

30
00:01:47,660 --> 00:01:52,750
sorry, with non-zero mass, is
then given by gamma times m

31
00:01:52,750 --> 00:01:55,250
c squared.

32
00:01:55,250 --> 00:01:58,850
And the momentum is given by
gamma times mv or gamma times

33
00:01:58,850 --> 00:02:01,490
m beta.

34
00:02:01,490 --> 00:02:05,270
The total energy
squared of a particle,

35
00:02:05,270 --> 00:02:08,449
considering one massive
particle, or one particle,

36
00:02:08,449 --> 00:02:11,810
is given by energy squared
equal momentum squared

37
00:02:11,810 --> 00:02:13,710
plus mass squared.

38
00:02:13,710 --> 00:02:17,210
And if you now consider
a particle with 0 mass,

39
00:02:17,210 --> 00:02:19,980
you see that the energy
and the momentum are equal.

40
00:02:19,980 --> 00:02:21,800
If you consider a
particle at rest,

41
00:02:21,800 --> 00:02:24,770
meaning the momentum is
0, you see that the energy

42
00:02:24,770 --> 00:02:25,880
is equal to the mass.

43
00:02:25,880 --> 00:02:29,420
You get Einstein's famous
formula, the energy is equal--

44
00:02:29,420 --> 00:02:31,040
E equal m c squared.

45
00:02:31,040 --> 00:02:33,650
Energy is equal to the
mass or the equivalence

46
00:02:33,650 --> 00:02:35,920
between those two.

47
00:02:35,920 --> 00:02:39,800
But we want to fully
understand and control

48
00:02:39,800 --> 00:02:41,950
our Lorentz transformations.

49
00:02:41,950 --> 00:02:46,640
Here shown for example, we have
a boost or a transformation

50
00:02:46,640 --> 00:02:48,092
in x-direction.

51
00:02:48,092 --> 00:02:49,550
So you see that
energy and momentum

52
00:02:49,550 --> 00:02:53,360
transform like time and space.

53
00:02:53,360 --> 00:02:55,820
And you just-- I really
encourage you to just review

54
00:02:55,820 --> 00:02:59,330
this in more general
cases, but you can always,

55
00:02:59,330 --> 00:03:01,310
when you have a boost
in one direction,

56
00:03:01,310 --> 00:03:05,632
just do rotation and get to
this more simplified case.

57
00:03:05,632 --> 00:03:07,340
So here is the first
example I would like

58
00:03:07,340 --> 00:03:09,590
you to actually go through.

59
00:03:09,590 --> 00:03:11,090
The Lorentz
transformation here, I

60
00:03:11,090 --> 00:03:14,420
decided to use the z-direction,
just to change things up

61
00:03:14,420 --> 00:03:15,350
a little bit.

62
00:03:15,350 --> 00:03:19,130
And the velocity of the
boosted frame is vb.

63
00:03:19,130 --> 00:03:25,460
So we want to calculate the
quantity m squared s squared

64
00:03:25,460 --> 00:03:27,920
in the transformed frame.

65
00:03:27,920 --> 00:03:30,620
And what you will find, if you
actually do the calculation--

66
00:03:30,620 --> 00:03:34,250
and the solutions are
in the backup slides--

67
00:03:34,250 --> 00:03:37,220
it's that z quantity
doesn't change

68
00:03:37,220 --> 00:03:38,540
in the Lorentz transformation.

69
00:03:38,540 --> 00:03:40,130
It is invariant.

70
00:03:40,130 --> 00:03:43,820
And we'll talk about an
invariant mass in this context.

71
00:03:46,820 --> 00:03:49,030
So now in particle
physics, we often

72
00:03:49,030 --> 00:03:52,300
have the case that we
are not considering just

73
00:03:52,300 --> 00:03:54,685
one particle and want to
describe just one particular

74
00:03:54,685 --> 00:03:58,690
and measure it, but often
the case of particles,

75
00:03:58,690 --> 00:04:02,450
or multiple particles, which
are involved in the reaction.

76
00:04:02,450 --> 00:04:04,600
So we can look at the
total energy, just the sum

77
00:04:04,600 --> 00:04:07,465
of the energy of all
particles, and total momentum,

78
00:04:07,465 --> 00:04:10,210
the sum of the momentum
of all particles.

79
00:04:10,210 --> 00:04:13,250
And those two quantities
are always conserved.

80
00:04:13,250 --> 00:04:14,650
They are not invariant.

81
00:04:14,650 --> 00:04:18,940
So be aware of the distinction
between conserved properties

82
00:04:18,940 --> 00:04:20,829
and invariant properties.

83
00:04:20,829 --> 00:04:24,640
Invariant here means
perform a transformation

84
00:04:24,640 --> 00:04:27,070
like the Lorentz
transformation, and the property

85
00:04:27,070 --> 00:04:28,480
doesn't change.

86
00:04:28,480 --> 00:04:32,680
Conserved here means we have a
reaction, and in that reaction

87
00:04:32,680 --> 00:04:34,420
the property is not changing.

88
00:04:34,420 --> 00:04:37,520
Those are two different,
distinct things.

89
00:04:37,520 --> 00:04:40,900
So now you can look at the
invariant property, or the one

90
00:04:40,900 --> 00:04:45,430
which is conserved in
this collision, which

91
00:04:45,430 --> 00:04:49,410
is this mass term or
mass-squared term.

92
00:04:49,410 --> 00:04:53,620
The total mass, we'll
define this total mass as

93
00:04:53,620 --> 00:04:57,440
equal to the energy squared
minus momentum squared.

94
00:04:57,440 --> 00:04:59,300
And then you can
consider the two cases

95
00:04:59,300 --> 00:05:03,680
of a laboratory frame and the
so-called center-of-mass frame.

96
00:05:03,680 --> 00:05:07,220
So in the laboratory
frame, you have a particle.

97
00:05:07,220 --> 00:05:10,080
It's moving when we
observe this particle,

98
00:05:10,080 --> 00:05:12,650
and then it decays into, in
this example, three daughter

99
00:05:12,650 --> 00:05:14,020
particles.

100
00:05:14,020 --> 00:05:16,940
In the center-of-mass
frame in this example,

101
00:05:16,940 --> 00:05:20,050
we put ourselves into the
rest frame of the particle

102
00:05:20,050 --> 00:05:21,740
we are interested in.

103
00:05:21,740 --> 00:05:25,943
And then that frame then,
three particles emerge.

104
00:05:25,943 --> 00:05:27,610
And we can describe
the three particles.

105
00:05:27,610 --> 00:05:30,520
So the momenta between the
three daughter particles

106
00:05:30,520 --> 00:05:32,440
are not going to be [INAUDIBLE].

107
00:05:32,440 --> 00:05:38,050
But because this total mass
is an invariant property,

108
00:05:38,050 --> 00:05:40,000
it's the same in both frames.

109
00:05:40,000 --> 00:05:43,600
And it's equal to the mass
of the parent particle

110
00:05:43,600 --> 00:05:47,110
which we [INAUDIBLE] So when you
measure the energy and momentum

111
00:05:47,110 --> 00:05:51,820
of the daughter particles, you
can infer in any frame the mass

112
00:05:51,820 --> 00:05:56,110
of the parent particle by
calculating the total mass.

113
00:05:56,110 --> 00:05:59,440
And so you can infer
from those measurements

114
00:05:59,440 --> 00:06:03,040
the identity of the
mother particle.

115
00:06:03,040 --> 00:06:05,635
And that's, for example, how
we discover the Higgs boson.

116
00:06:05,635 --> 00:06:09,580
We measure the Higgs boson
decay into a pair of photons,

117
00:06:09,580 --> 00:06:13,030
and then we calculate the
mass of those two photons

118
00:06:13,030 --> 00:06:14,650
in our laboratory frame.

119
00:06:14,650 --> 00:06:17,650
And that mass, then, is
equal to the Higgs mass.

120
00:06:20,810 --> 00:06:24,410
So now here we want to compare
or look into those two cases

121
00:06:24,410 --> 00:06:25,770
a little bit more.

122
00:06:25,770 --> 00:06:29,270
The first case is a case
where we have a particle 1

123
00:06:29,270 --> 00:06:31,490
colliding with a
particle 2, where

124
00:06:31,490 --> 00:06:33,260
the particle 2 is at rest.

125
00:06:33,260 --> 00:06:36,200
Particle 1 has a
certain energy E1.

126
00:06:36,200 --> 00:06:37,480
And the second example--

127
00:06:37,480 --> 00:06:39,520
this is called a
fixed-target experiment.

128
00:06:39,520 --> 00:06:44,920
So the second particle is fixed,
the first one is colliding.

129
00:06:44,920 --> 00:06:48,010
The second example is the one
where you have two particles,

130
00:06:48,010 --> 00:06:51,670
and both have energies, and
we bring them to collision.

131
00:06:51,670 --> 00:06:53,950
Often, the two
particles are in nature,

132
00:06:53,950 --> 00:06:57,010
like two protons, an
electron and positron,

133
00:06:57,010 --> 00:06:59,200
and the energies of
the beams are the same.

134
00:06:59,200 --> 00:07:01,030
But this doesn't
have to be the case.

135
00:07:01,030 --> 00:07:04,360
Later in the class, we'll
look at heavy iron collisions.

136
00:07:04,360 --> 00:07:06,490
It's the collision of
heavy ions like lead.

137
00:07:06,490 --> 00:07:08,230
It's a proton.

138
00:07:08,230 --> 00:07:09,880
And here the masses
are different,

139
00:07:09,880 --> 00:07:12,507
and the energy of the
particles can be different.

140
00:07:15,260 --> 00:07:15,760
All right.

141
00:07:15,760 --> 00:07:17,440
And here's another exercise now.

142
00:07:17,440 --> 00:07:20,530
So we want to actually
create a Z boson, which

143
00:07:20,530 --> 00:07:23,110
has a mass of about 91 GeV.

144
00:07:23,110 --> 00:07:26,582
Note I dropped the c squared,
1 over c squared here.

145
00:07:26,582 --> 00:07:28,540
And you want to produce
this particle colliding

146
00:07:28,540 --> 00:07:31,440
a positron with an election.

147
00:07:31,440 --> 00:07:37,390
This has happened at LEP at
CERN in the late '80s and '90s.

148
00:07:37,390 --> 00:07:42,970
The center-of-mass energy, often
what's called square root of s,

149
00:07:42,970 --> 00:07:44,878
is equal to 91 GeV.

150
00:07:44,878 --> 00:07:46,420
So that's the energy
we need in order

151
00:07:46,420 --> 00:07:48,310
to produce this new particle.

152
00:07:48,310 --> 00:07:50,230
The mass of the electron
and the positron

153
00:07:50,230 --> 00:07:55,540
are 511 KeV or 0.511 MeV.

154
00:07:55,540 --> 00:08:01,570
So the energy needed
is 45 GeV, 45.5 GeV.

155
00:08:01,570 --> 00:08:04,990
However, that was
the setup at LEP,

156
00:08:04,990 --> 00:08:06,800
where you have two
beams colliding.

157
00:08:06,800 --> 00:08:09,280
So we have this
center-of-mass energy

158
00:08:09,280 --> 00:08:12,790
being given by the energy
directly given approximately

159
00:08:12,790 --> 00:08:14,860
by the energy of the two beams.

160
00:08:14,860 --> 00:08:17,830
So now, imagine somebody
would have proposed

161
00:08:17,830 --> 00:08:21,790
a fixed-target experiment, where
you have stationary electrons,

162
00:08:21,790 --> 00:08:27,860
for example electrons in
atoms, just a gas of some sort,

163
00:08:27,860 --> 00:08:31,120
and then you have produced
positrons in a beam,

164
00:08:31,120 --> 00:08:34,000
you accelerate them and
bring them to collision.

165
00:08:34,000 --> 00:08:38,740
So the question now
is, how large does it--

166
00:08:38,740 --> 00:08:41,559
do you need an energy
of this positron beam?

167
00:08:41,559 --> 00:08:44,890
How large does it have to be
in order to produce a Z boson?

168
00:08:44,890 --> 00:08:46,750
So again, this is
something I would

169
00:08:46,750 --> 00:08:49,690
like you to actually
explore and just write down.

170
00:08:49,690 --> 00:08:54,790
Solutions for this example
are also in the backup.

171
00:08:54,790 --> 00:08:56,950
So now, you know,
there is a number

172
00:08:56,950 --> 00:08:59,080
of interesting
examples just coming

173
00:08:59,080 --> 00:09:02,780
from E equal m c squared,
and from being able to use--

174
00:09:02,780 --> 00:09:04,660
and that can be
answered by being

175
00:09:04,660 --> 00:09:06,950
able to use Lorentz
transformation.

176
00:09:06,950 --> 00:09:09,610
And so now here I give you
just a set of examples.

177
00:09:09,610 --> 00:09:13,210
And you should work on
them on your own time.

178
00:09:13,210 --> 00:09:15,910
Maybe we'll touch on
them in recitation.

179
00:09:15,910 --> 00:09:18,830
The first one is
rather straightforward.

180
00:09:18,830 --> 00:09:22,150
Again, we are talking
about LEP at CERN.

181
00:09:22,150 --> 00:09:24,190
After the Z bosons
were produced,

182
00:09:24,190 --> 00:09:26,770
when it was trying
to go to high energy

183
00:09:26,770 --> 00:09:30,280
to find some new physics,
some new particle, for example

184
00:09:30,280 --> 00:09:31,150
the Higgs boson.

185
00:09:31,150 --> 00:09:33,730
The Higgs boson
might be produced

186
00:09:33,730 --> 00:09:38,180
by a process which is called
Higgs-Strahlung process.

187
00:09:38,180 --> 00:09:39,800
We will look at this later.

188
00:09:39,800 --> 00:09:41,680
So you have an
electron and a positron

189
00:09:41,680 --> 00:09:45,170
colliding to virtual Z boson.

190
00:09:45,170 --> 00:09:48,700
So that is a Z boson which
is heavier than 91 GeV.

191
00:09:48,700 --> 00:09:50,470
We'll see later how
that's possible.

192
00:09:50,470 --> 00:09:55,290
But then the virtual Z boson
can radiate a Higgs boson.

193
00:09:55,290 --> 00:09:56,950
So that's why it's
called Strahlung,

194
00:09:56,950 --> 00:10:00,890
like the German word for
radiation, Strahlung process.

195
00:10:00,890 --> 00:10:04,210
And so electrons
and positrons were

196
00:10:04,210 --> 00:10:09,880
accelerated to 100 GeV each,
center-of-mass energy 200 GeV.

197
00:10:09,880 --> 00:10:13,960
What was the gamma factor
for those electrons?

198
00:10:13,960 --> 00:10:16,000
Another question which
is quite exciting

199
00:10:16,000 --> 00:10:17,800
is, how much energy
do you need in order

200
00:10:17,800 --> 00:10:21,610
to split a proton and a
neutron, which is a bound state?

201
00:10:21,610 --> 00:10:23,830
It's called a deuteron.

202
00:10:23,830 --> 00:10:27,070
And it's a fundamental--
the important particle

203
00:10:27,070 --> 00:10:29,500
in the evolution
of our universe,

204
00:10:29,500 --> 00:10:35,140
in the sense that in order to
generate higher mass or higher

205
00:10:35,140 --> 00:10:37,630
proton number
elements, a neutron is

206
00:10:37,630 --> 00:10:39,550
rather important in this.

207
00:10:39,550 --> 00:10:42,130
And so just by knowing
the mass of the proton,

208
00:10:42,130 --> 00:10:44,890
the mass of the neutron, and
the mass of the deuteron,

209
00:10:44,890 --> 00:10:50,690
you can now calculate how
much energy is in the--

210
00:10:50,690 --> 00:10:54,890
binded-- what is the binding
energy between those particles.

211
00:10:54,890 --> 00:10:58,400
We'll talk a lot about models
to calculate binding energy when

212
00:10:58,400 --> 00:11:01,430
we talk about nuclear physics.

213
00:11:01,430 --> 00:11:03,050
But here, just
from the kinematics

214
00:11:03,050 --> 00:11:04,910
you can-- and from
E equal m c squared,

215
00:11:04,910 --> 00:11:06,830
you can calculate
how much energy

216
00:11:06,830 --> 00:11:12,390
needs to be in this
binded or compound state.

217
00:11:12,390 --> 00:11:15,690
From atomic physics, you
might remember or know

218
00:11:15,690 --> 00:11:17,770
that particles can--

219
00:11:17,770 --> 00:11:20,380
excited particles
can emit photons.

220
00:11:20,380 --> 00:11:22,890
And so now you have a particle.

221
00:11:22,890 --> 00:11:25,830
It goes-- it [INAUDIBLE]
excites, radiates a photon.

222
00:11:25,830 --> 00:11:27,200
What happens now to the photon?

223
00:11:27,200 --> 00:11:31,650
I mentioned this happening in a
big gas or in some solid state.

224
00:11:31,650 --> 00:11:34,950
Can the photon be reabsorbed
by the same medium,

225
00:11:34,950 --> 00:11:38,620
or even by the same particle?

226
00:11:38,620 --> 00:11:41,260
It's not a trivial
question, but what

227
00:11:41,260 --> 00:11:43,330
is the conditions under
which-- so for example,

228
00:11:43,330 --> 00:11:46,390
imagine you have a gas
as an excited particle.

229
00:11:46,390 --> 00:11:47,830
And it emits a photon.

230
00:11:47,830 --> 00:11:50,770
And so now the photon
sees the rest of the gas.

231
00:11:50,770 --> 00:11:54,650
Can that rest of the
gas absorb the photon?

232
00:11:54,650 --> 00:11:55,700
Interesting question.

233
00:11:55,700 --> 00:11:58,460
It's not trivial.

234
00:11:58,460 --> 00:12:00,470
Another interesting
question, I think,

235
00:12:00,470 --> 00:12:05,300
is you're trying to produce
new forms of matter.

236
00:12:05,300 --> 00:12:07,060
Like you just
produced a Z boson,

237
00:12:07,060 --> 00:12:09,890
but you can also
produce antiprotons.

238
00:12:09,890 --> 00:12:14,150
So what is the minimal energy
in a proton on a fixed-target

239
00:12:14,150 --> 00:12:15,240
experiment--

240
00:12:15,240 --> 00:12:19,820
so again, you have a target
of protons in some form,

241
00:12:19,820 --> 00:12:22,070
you shoot a proton
against this target,

242
00:12:22,070 --> 00:12:25,460
and you try to
produce an antiproton.

243
00:12:25,460 --> 00:12:27,830
So that means that you
have to put produce--

244
00:12:27,830 --> 00:12:31,250
in this collision, you have two
protons in the initial state,

245
00:12:31,250 --> 00:12:34,640
you have to have a proton,
a proton, another proton,

246
00:12:34,640 --> 00:12:38,030
and an antiproton
in your final state.

247
00:12:38,030 --> 00:12:40,820
But what-- how much energy is
needed for the [INAUDIBLE] beam

248
00:12:40,820 --> 00:12:45,192
in order to succeed
with this collision?

249
00:12:45,192 --> 00:12:47,340
My counting is incorrect here.

250
00:12:47,340 --> 00:12:50,780
So this should be
5, but OK, fine.

251
00:12:50,780 --> 00:12:51,550
Decays.

252
00:12:51,550 --> 00:12:53,440
So assume a pion decays at rest.

253
00:12:53,440 --> 00:12:55,040
So a pion is at rest.

254
00:12:55,040 --> 00:12:58,310
You look at a pion, that's
the compound state of meson

255
00:12:58,310 --> 00:13:00,470
out of an up quark
and a down quark.

256
00:13:00,470 --> 00:13:03,560
And it might decay in an
electron and a positron.

257
00:13:03,560 --> 00:13:07,350
Whatever the dynamics
is in these decays,

258
00:13:07,350 --> 00:13:09,530
if you just look at
the kinematics of this,

259
00:13:09,530 --> 00:13:11,750
how fast are the decay products?

260
00:13:11,750 --> 00:13:13,520
In order to calculate
that, you need

261
00:13:13,520 --> 00:13:15,980
to look at the pion
mass, electron mass

262
00:13:15,980 --> 00:13:18,870
we just discussed, and the
positron has the same mass.

263
00:13:18,870 --> 00:13:20,930
So how fast are
electron and positron

264
00:13:20,930 --> 00:13:22,520
coming out of a pion decay?

265
00:13:22,520 --> 00:13:25,280
Assume that the pion is at rest.

266
00:13:25,280 --> 00:13:27,950
And you can use
momentum conservation

267
00:13:27,950 --> 00:13:33,810
and calculate the speed of
the electrons and positrons.

268
00:13:33,810 --> 00:13:35,910
Again, one of those
minimal-energy proton

269
00:13:35,910 --> 00:13:38,490
colliding experiments,
very similar setup.

270
00:13:38,490 --> 00:13:42,840
But here we try to produce a
proton, a neutron, and a pion

271
00:13:42,840 --> 00:13:45,030
out of proton-proton collision.

272
00:13:45,030 --> 00:13:47,970
And then the last one is the
so-called Compton effect,

273
00:13:47,970 --> 00:13:51,600
where you have a photon which
scatters of an electron target.

274
00:13:51,600 --> 00:13:53,400
And so you have an
incoming photon,

275
00:13:53,400 --> 00:13:54,900
and the electron is at rest.

276
00:13:54,900 --> 00:13:57,690
And then you look at
the scattered photon

277
00:13:57,690 --> 00:14:01,680
angle, scattered electron
angle, and in that collision

278
00:14:01,680 --> 00:14:04,270
the energy of the photon
is going to change.

279
00:14:04,270 --> 00:14:08,250
So the energy of the
photon is h times mu or h

280
00:14:08,250 --> 00:14:10,630
over lambda, the wavelength.

281
00:14:10,630 --> 00:14:13,590
And so the question is, how does
the wavelength of the photon

282
00:14:13,590 --> 00:14:16,720
change in this
kind of condition.

283
00:14:16,720 --> 00:14:19,140
So those are just
examples in how

284
00:14:19,140 --> 00:14:22,380
you can use relativistic
kinematics in order

285
00:14:22,380 --> 00:14:26,340
to calculate very important
aspects of collisions

286
00:14:26,340 --> 00:14:29,910
in particle physics without any
understanding, at this point,

287
00:14:29,910 --> 00:14:33,480
of the underlying dynamics,
the underlying forces,

288
00:14:33,480 --> 00:14:39,940
the underlying conservation
laws, and so on.

289
00:14:39,940 --> 00:14:42,900
So later in this
class, we'll discuss

290
00:14:42,900 --> 00:14:46,170
what is the likelihood of a
pion to decay into an electron

291
00:14:46,170 --> 00:14:50,820
positron, and why that is
actually not that likely.

292
00:14:50,820 --> 00:14:54,480
And also, the collision
rates, lifetimes of particles.

293
00:14:54,480 --> 00:14:56,640
But here we are just
looking at the kinematic

294
00:14:56,640 --> 00:15:01,320
of those processes and calculate
how much energy is involved

295
00:15:01,320 --> 00:15:06,140
and what is the momentum
of resulting particles.

296
00:15:06,140 --> 00:15:08,530
So I'll stop here.

297
00:15:08,530 --> 00:15:10,990
If you scroll down
on the slides,

298
00:15:10,990 --> 00:15:14,090
you'll find solutions
to two of the problems.

299
00:15:14,090 --> 00:15:17,490
And we'll discuss
them in recitation.