1
00:00:00,570 --> 00:00:05,640
PROFESSOR: Einstein's argument,
we consider again our atoms.

2
00:00:05,640 --> 00:00:08,900
And this time we're going to
use the thermal equilibrium.

3
00:00:08,900 --> 00:00:14,460
We're going to really
make use of that fact.

4
00:00:14,460 --> 00:00:20,870
So, again, we have
level b and level a.

5
00:00:20,870 --> 00:00:23,280
And this is Einstein's argument.

6
00:00:33,570 --> 00:00:37,310
And we're going to
have no populations.

7
00:00:37,310 --> 00:00:41,450
If we discuss equilibrium, we
can consider a system in a box.

8
00:00:41,450 --> 00:00:44,300
And there's a few billion atoms.

9
00:00:44,300 --> 00:00:47,150
And there's some
atoms whose electron

10
00:00:47,150 --> 00:00:49,160
is going to be in state b.

11
00:00:49,160 --> 00:00:52,880
Some atoms whose electrons
are going to be in state a.

12
00:00:52,880 --> 00:00:59,270
Let's call these
numbers Nb and Na.

13
00:00:59,270 --> 00:01:11,090
And we also have
photons at temperature T

14
00:01:11,090 --> 00:01:20,440
and have a beta parameter 1 over
the Boltzmann constant times T.

15
00:01:20,440 --> 00:01:23,450
So this is the process
we're going to consider.

16
00:01:23,450 --> 00:01:26,335
So what's going to happen?

17
00:01:28,910 --> 00:01:34,460
If we came up from the edition
that we've built from 806,

18
00:01:34,460 --> 00:01:36,320
we would think,
OK, there's going

19
00:01:36,320 --> 00:01:42,800
to be absorption process and
stimulated emission process.

20
00:01:42,800 --> 00:01:44,770
And between the
two, they're going

21
00:01:44,770 --> 00:01:47,060
to be able to reach equilibrium.

22
00:01:47,060 --> 00:01:52,800
We don't believe this is not a
process that can equilibrate.

23
00:01:52,800 --> 00:01:54,760
So this would be our intuition.

24
00:01:54,760 --> 00:01:58,640
The intuition for
people that lived

25
00:01:58,640 --> 00:02:04,250
at the beginning of last
century was rather different.

26
00:02:04,250 --> 00:02:08,810
They felt that there would
be absorption process

27
00:02:08,810 --> 00:02:12,590
and there was
spontaneous emission.

28
00:02:12,590 --> 00:02:16,010
The thing that you probably
intuitively would think,

29
00:02:16,010 --> 00:02:20,000
if you are at a high level,
you spontaneously decay.

30
00:02:20,000 --> 00:02:23,330
So the thing that
was not known to them

31
00:02:23,330 --> 00:02:26,660
was this stimulated emission.

32
00:02:26,660 --> 00:02:31,700
That is what Einstein is
credited for discovering here.

33
00:02:31,700 --> 00:02:35,090
People felt, and I think the
paper finds that makes clear

34
00:02:35,090 --> 00:02:37,460
that the intuition,
is that you have

35
00:02:37,460 --> 00:02:41,990
spontaneous emission,
in which spontaneously

36
00:02:41,990 --> 00:02:47,570
by some kind of instability,
the higher state goes

37
00:02:47,570 --> 00:02:48,890
to the bottom.

38
00:02:48,890 --> 00:02:51,860
And you have absorption.

39
00:02:51,860 --> 00:02:54,410
But then Einstein figured
out that you couldn't achieve

40
00:02:54,410 --> 00:02:56,222
equilibrium in that way.

41
00:02:56,222 --> 00:02:57,680
Well, the way we're
going to do it,

42
00:02:57,680 --> 00:03:00,650
we're going to put the
two things we know--

43
00:03:00,650 --> 00:03:03,440
the absorption
and the stimulated

44
00:03:03,440 --> 00:03:06,380
emission-- and see
that we don't get it

45
00:03:06,380 --> 00:03:08,450
to work, the equilibrium.

46
00:03:08,450 --> 00:03:14,210
But then when we add the
spontaneous emission, we will.

47
00:03:14,210 --> 00:03:18,260
And as it turns out,
the spontaneous emission

48
00:03:18,260 --> 00:03:21,080
is a little harder to calculate.

49
00:03:21,080 --> 00:03:24,500
If we were to do it
in 806, it probably

50
00:03:24,500 --> 00:03:28,120
would be a matter of
two lectures involving

51
00:03:28,120 --> 00:03:29,470
an electromagnetic field.

52
00:03:29,470 --> 00:03:31,370
So we will not do it.

53
00:03:31,370 --> 00:03:36,140
But the good thing is that
Einstein's argument tells you

54
00:03:36,140 --> 00:03:40,160
the speed of spontaneous
emission, the rate

55
00:03:40,160 --> 00:03:43,310
of spontaneous
emission, in terms

56
00:03:43,310 --> 00:03:47,660
of this rate of stimulated
emission or absorption.

57
00:03:47,660 --> 00:03:50,300
So it does the
calculation for you

58
00:03:50,300 --> 00:03:54,750
by some other
thermodynamical means.

59
00:03:54,750 --> 00:03:59,285
So we're going to
use here three facts.

60
00:04:02,230 --> 00:04:03,580
One is that the--

61
00:04:03,580 --> 00:04:10,960
three facts-- one is the
populations are in equilibrium.

62
00:04:19,730 --> 00:04:26,430
So Na dot is they stop
changing is equal to 0.

63
00:04:26,430 --> 00:04:29,690
And Nb dot is equal to 0.

64
00:04:29,690 --> 00:04:34,040
They don't stop changing
because nothing happens.

65
00:04:34,040 --> 00:04:37,010
All the time there
will be emission,

66
00:04:37,010 --> 00:04:38,450
and there will be absorption.

67
00:04:38,450 --> 00:04:42,020
But if you reach equilibrium,
the number of atoms

68
00:04:42,020 --> 00:04:45,810
remain the same on every state.

69
00:04:45,810 --> 00:04:49,370
So that's our statement
that the populations

70
00:04:49,370 --> 00:04:52,610
achieve equilibrium.

71
00:04:52,610 --> 00:04:58,040
The second statement is that the
equilibrium is thermodynamical.

72
00:04:58,040 --> 00:05:02,510
So it's thermal equilibrium.

73
00:05:06,800 --> 00:05:14,540
That is Nb over Na, for example,
is the Boltzmann factor e

74
00:05:14,540 --> 00:05:20,990
to the minus beta, Eb over
e to the minus beta Ea.

75
00:05:20,990 --> 00:05:35,430
And this is equal to e to the
minus beta h bar omega ba.

76
00:05:35,430 --> 00:05:39,600
You get e to the minus
beta, eb minus ea.

77
00:05:39,600 --> 00:05:44,700
But eb minus ea
is h bar omega ba.

78
00:05:44,700 --> 00:05:48,690
So that's thermal equilibrium.

79
00:05:48,690 --> 00:05:53,370
And the last thing
that we need is

80
00:05:53,370 --> 00:05:55,590
a statement about the photons.

81
00:05:55,590 --> 00:05:58,860
What do they do when
they have equilibrium?

82
00:05:58,860 --> 00:06:03,340
And that was known already
due the work of Planck

83
00:06:03,340 --> 00:06:06,070
and others, black
body equilibrium.

84
00:06:06,070 --> 00:06:09,510
So we need to know something
about the thermal radiation.

85
00:06:17,500 --> 00:06:22,620
And the way one describes
this is in terms of a function

86
00:06:22,620 --> 00:06:25,420
U of omega d omega.

87
00:06:25,420 --> 00:06:28,240
In the black body
radiation, there

88
00:06:28,240 --> 00:06:30,725
are at a given
temperature, there

89
00:06:30,725 --> 00:06:32,920
are photons with
very little energy.

90
00:06:32,920 --> 00:06:36,280
There are some largest
number of photons

91
00:06:36,280 --> 00:06:38,740
with some energy associated
with temperature,

92
00:06:38,740 --> 00:06:39,790
and then it decays.

93
00:06:39,790 --> 00:06:43,040
So you have photons
of all energy.

94
00:06:43,040 --> 00:06:47,290
So if you want a description of
what's going on in black body

95
00:06:47,290 --> 00:06:51,730
radiation, you can
consider the energy

96
00:06:51,730 --> 00:06:55,270
in the photons in
the frequency range.

97
00:06:55,270 --> 00:06:57,490
But you even must
be more precise.

98
00:06:57,490 --> 00:07:03,550
It is the energy per unit
volume in a frequency range.

99
00:07:03,550 --> 00:07:07,740
Because if it's different,
the energy of the black body

100
00:07:07,740 --> 00:07:10,600
cavities, this room
or it's a little box.

101
00:07:10,600 --> 00:07:16,340
So it's an energy per
volume per frequency range.

102
00:07:16,340 --> 00:07:18,085
And that's what
this quantity is.

103
00:07:18,085 --> 00:07:20,220
So let me write it.

104
00:07:20,220 --> 00:07:41,450
Energy per unit volume in
the frequency range dw.

105
00:07:41,450 --> 00:07:46,610
In other words, it's kind of a
proxy for the number of photons

106
00:07:46,610 --> 00:07:47,270
available.

107
00:07:47,270 --> 00:07:51,950
All the photos have at some
value of the frequency,

108
00:07:51,950 --> 00:07:53,030
of energy e omega.

109
00:07:53,030 --> 00:07:55,850
So if you know the
energy, you basically

110
00:07:55,850 --> 00:08:01,550
are getting here the number of
photons with frequency omega

111
00:08:01,550 --> 00:08:04,805
in that range per unit
volume, all that stuff.

112
00:08:07,590 --> 00:08:11,750
So this has a formula.

113
00:08:11,750 --> 00:08:16,290
And the formula that
was known to people

114
00:08:16,290 --> 00:08:25,680
was this quantities and then
omega cube, d omega, over e

115
00:08:25,680 --> 00:08:30,230
to the beta h bar omega minus 1.

116
00:08:34,649 --> 00:08:40,390
So this is the basis
of the calculation.

117
00:08:43,200 --> 00:08:44,070
What do we do?

118
00:08:44,070 --> 00:08:46,670
We have to consider
the possible processes.

119
00:08:49,630 --> 00:08:54,120
OK, so our processes
are absorption.

120
00:08:58,050 --> 00:09:02,700
And in this case,
we go from a to b.

121
00:09:02,700 --> 00:09:10,320
And let's try to
write a rate for them.

122
00:09:13,380 --> 00:09:17,700
So what would the
rate depend on?

123
00:09:17,700 --> 00:09:22,060
Well, here's some
little assumptions.

124
00:09:22,060 --> 00:09:26,890
Certainly, if you don't have
particles in the a state,

125
00:09:26,890 --> 00:09:28,470
you cannot have this process.

126
00:09:28,470 --> 00:09:33,780
So this process, the total
rate that we observe in the box

127
00:09:33,780 --> 00:09:35,820
will depend on an Na.

128
00:09:35,820 --> 00:09:38,760
The more particles
you have in this state

129
00:09:38,760 --> 00:09:45,240
a, the larger the probability
that you get the transitions,

130
00:09:45,240 --> 00:09:46,435
and the larger the rate.

131
00:09:49,900 --> 00:09:55,240
It will also be affected
by the number of photons

132
00:09:55,240 --> 00:10:01,090
available at that
frequency that can

133
00:10:01,090 --> 00:10:05,360
produce a transition in
proportional to that.

134
00:10:05,360 --> 00:10:12,200
And finally, the quantity
that our study of perturbation

135
00:10:12,200 --> 00:10:15,910
theory will tell us
about, but at that time,

136
00:10:15,910 --> 00:10:23,380
finds that was not known, it's
a transition coefficient, Bab,

137
00:10:23,380 --> 00:10:25,940
he called it.

138
00:10:25,940 --> 00:10:27,780
And this is the unknown one.

139
00:10:31,280 --> 00:10:32,890
And this is what we don't know.

140
00:10:32,890 --> 00:10:35,430
We know U. We assume we know Na.

141
00:10:35,430 --> 00:10:41,640
This is the transition
rate per atom

142
00:10:41,640 --> 00:10:44,560
and then multiplied by
the number of atoms.

143
00:10:44,560 --> 00:10:50,340
So this is transition
rate per atom.

144
00:10:58,330 --> 00:11:05,750
Then we have the process
of spontaneous emission.

145
00:11:05,750 --> 00:11:11,480
And then we will be
another coefficient, Bba.

146
00:11:11,480 --> 00:11:15,230
And it would depend on
the number of particles

147
00:11:15,230 --> 00:11:19,580
that are in the state b,
because spontaneous emissions

148
00:11:19,580 --> 00:11:22,040
that transition from b to a.

149
00:11:24,710 --> 00:11:27,940
We call it-- oh, not
spontaneous stimulated.

150
00:11:27,940 --> 00:11:29,870
I'm sorry-- stimulated emission.

151
00:11:33,450 --> 00:11:35,150
This is the one
we're considering.

152
00:11:35,150 --> 00:11:37,790
It's stimulated
by the radiation.

153
00:11:37,790 --> 00:11:40,670
So it's also proportional
to the number

154
00:11:40,670 --> 00:11:44,960
of photons present
and proportional

155
00:11:44,960 --> 00:11:47,210
to the number of
atoms that can be

156
00:11:47,210 --> 00:11:54,470
convinced to do the transition
times another coefficient, Bba.

157
00:11:54,470 --> 00:12:02,660
So this is, I think,
what we in 806 would do.

158
00:12:02,660 --> 00:12:04,570
We would consider
this two processes

159
00:12:04,570 --> 00:12:08,190
and attempt to make it work.

160
00:12:08,190 --> 00:12:10,116
And let's see what we get then.

161
00:12:14,680 --> 00:12:16,800
We're trying to get equilibrium.

162
00:12:16,800 --> 00:12:24,160
So we want the transitions to
equilibrate and, therefore,

163
00:12:24,160 --> 00:12:28,490
the populations not to change.

164
00:12:28,490 --> 00:12:30,350
So let's look-- for example--

165
00:12:30,350 --> 00:12:36,460
you could look at either one,
but you can look at Nb dot.

166
00:12:36,460 --> 00:12:40,300
It should be 0.

167
00:12:40,300 --> 00:12:49,590
But it's equal to the
rate of absorption

168
00:12:49,590 --> 00:12:56,800
minus the rate of
stimulated emission.

169
00:13:02,430 --> 00:13:06,350
You see because the
number of particles in b

170
00:13:06,350 --> 00:13:09,800
change because you get
some new particles in state

171
00:13:09,800 --> 00:13:12,500
b due to the absorption process.

172
00:13:12,500 --> 00:13:14,840
And it happens with this rate.

173
00:13:14,840 --> 00:13:19,670
And you lose some particle
because some atoms

174
00:13:19,670 --> 00:13:22,660
do the transition from the
higher level to the lower

175
00:13:22,660 --> 00:13:23,910
level.

176
00:13:23,910 --> 00:13:26,060
So what is the
rate of absorption?

177
00:13:26,060 --> 00:13:28,250
We have it here.

178
00:13:28,250 --> 00:13:37,946
It's this one, Bab
U of omega ba Na.

179
00:13:37,946 --> 00:13:44,900
And this one is Bba, the
same U of omega ba Nb.

180
00:13:47,550 --> 00:13:50,700
And we can factor the U out.

181
00:13:50,700 --> 00:13:53,900
And this is the wrong
calculation, I must say,

182
00:13:53,900 --> 00:13:57,560
because we're missing
that extra process that

183
00:13:57,560 --> 00:14:02,600
was intuitive to
Einstein, but to us it's

184
00:14:02,600 --> 00:14:06,050
a little less clear--

185
00:14:06,050 --> 00:14:10,700
Nb times U of omega ba.

186
00:14:16,000 --> 00:14:20,500
OK, I can do a one
more little thing.

187
00:14:20,500 --> 00:14:21,970
I can factor an Na.

188
00:14:21,970 --> 00:14:35,330
And this becomes Bab minus Bba
to the minus beta h bar omega

189
00:14:35,330 --> 00:14:41,020
Ba U of omega ba.

190
00:14:41,020 --> 00:14:50,140
OK, I use the ratio of Na
over Nb being thermodynamical.

191
00:14:50,140 --> 00:14:56,080
So Nb over Na was used
from point two to get this.

192
00:14:56,080 --> 00:14:57,930
And this should be equal to 0.

193
00:15:00,910 --> 00:15:04,270
But this equation
can't be satisfied.

194
00:15:07,170 --> 00:15:08,840
What do you have here?

195
00:15:08,840 --> 00:15:12,730
You should be able to
equilibrate at any temperature.

196
00:15:12,730 --> 00:15:15,820
On the other hand,
what is our intuition

197
00:15:15,820 --> 00:15:20,320
about these quantities,
Bab and Bba?

198
00:15:20,320 --> 00:15:22,690
They should be
temperature independent.

199
00:15:22,690 --> 00:15:29,200
These are properties of the
geometry of those states

200
00:15:29,200 --> 00:15:32,510
and the overlaps of
the wave functions.

201
00:15:32,510 --> 00:15:35,350
These are atomic
physics properties

202
00:15:35,350 --> 00:15:38,030
of the levels of the particles.

203
00:15:38,030 --> 00:15:40,590
We will calculate them.

204
00:15:40,590 --> 00:15:44,440
And here is the input of
how many photons there

205
00:15:44,440 --> 00:15:46,480
are, how many atoms there are.

206
00:15:46,480 --> 00:15:49,300
And the number of
photons certainly

207
00:15:49,300 --> 00:15:50,560
depend on the temperature.

208
00:15:50,560 --> 00:15:52,780
The number of atoms
for equilibrium

209
00:15:52,780 --> 00:15:54,260
depend on the temperature.

210
00:15:54,260 --> 00:15:57,220
But this is a factor
that says, well,

211
00:15:57,220 --> 00:16:01,210
how likely is the transition
once you have a photon

212
00:16:01,210 --> 00:16:02,890
and once you have an atom?

213
00:16:02,890 --> 00:16:07,990
And that depends like we did
for the ionization, calculated

214
00:16:07,990 --> 00:16:10,450
some matrix elements
that are totally

215
00:16:10,450 --> 00:16:12,800
independent of temperature.

216
00:16:12,800 --> 00:16:17,830
So these numbers are totally
independent of temperature.

217
00:16:17,830 --> 00:16:20,740
And we're asking
this to be 0, which

218
00:16:20,740 --> 00:16:23,655
requires this factor to be 0.

219
00:16:23,655 --> 00:16:26,300
And this depends on temperature.

220
00:16:26,300 --> 00:16:31,510
So you cannot attain
equilibrium with this way.

221
00:16:31,510 --> 00:16:43,600
So it's impossible to satisfy
for all temperatures given

222
00:16:43,600 --> 00:16:48,990
that Bab and Bba are constants.

223
00:17:04,369 --> 00:17:06,589
So we're missing a process.

224
00:17:06,589 --> 00:17:12,700
This is the process that
Einstein thought was intuitive,

225
00:17:12,700 --> 00:17:16,214
the process of
spontaneous emission.

226
00:17:19,569 --> 00:17:24,490
So we add one more process.

227
00:17:24,490 --> 00:17:29,745
It's called
spontaneous emission.

228
00:17:36,820 --> 00:17:40,795
And it's a process
also from b to a.

229
00:17:45,373 --> 00:17:48,670
And it's going to have a rate.

230
00:17:48,670 --> 00:17:52,120
But it's not going
to depend, that rate,

231
00:17:52,120 --> 00:17:55,570
on the number of photons,
because it's happening

232
00:17:55,570 --> 00:17:57,940
independently of the photons.

233
00:17:57,940 --> 00:18:01,420
So we don't have this U factor.

234
00:18:01,420 --> 00:18:06,790
We do have the Nb because
each of the b atoms

235
00:18:06,790 --> 00:18:09,160
can spontaneously decay.

236
00:18:09,160 --> 00:18:11,330
But we don't have the U factor.

237
00:18:14,880 --> 00:18:16,710
So what do we have?

238
00:18:16,710 --> 00:18:23,320
A rate, which is the
term by a coefficient

239
00:18:23,320 --> 00:18:27,940
that Einstein called it a,
that's why the name a and b

240
00:18:27,940 --> 00:18:32,440
coefficients of
Einstein, a times and b.

241
00:18:35,490 --> 00:18:38,310
So that's the
spontaneous emission rate

242
00:18:38,310 --> 00:18:42,510
per atom multiplied by
the number of atoms.

243
00:18:42,510 --> 00:18:47,630
So we go back to our equation--

244
00:18:47,630 --> 00:18:52,130
rate of absorption minus
rate of stimulated emission

245
00:18:52,130 --> 00:18:57,470
minus the rate of
spontaneous emission.

246
00:18:57,470 --> 00:19:00,020
So I'll write it here.

247
00:19:00,020 --> 00:19:07,310
0 is equal to Nb dot
equal minus A Nb--

248
00:19:07,310 --> 00:19:10,240
that's the spontaneous emission.

249
00:19:10,240 --> 00:19:11,900
We write it first.

250
00:19:11,900 --> 00:19:13,990
And then we'll write
the other two--

251
00:19:13,990 --> 00:19:30,520
Bba and Nb U omega ba
plus Bab U of omega ba Na.