1
00:00:00,500 --> 00:00:02,230
PROFESSOR: OK, so
we're good to consider,

2
00:00:02,230 --> 00:00:03,960
therefore, the two cases.

3
00:00:03,960 --> 00:00:08,109
So let's consider
the first case,

4
00:00:08,109 --> 00:00:10,420
that this can be
pretty important,

5
00:00:10,420 --> 00:00:19,380
this case one, when omega
fi plus omega is nearly 0.

6
00:00:19,380 --> 00:00:22,560
In this case, what's happening?

7
00:00:22,560 --> 00:00:32,340
You have Ef minus Ei plus
h bar omega is nearly 0,

8
00:00:32,340 --> 00:00:41,700
or Ef roughly equal to
Ei minus h bar omega.

9
00:00:41,700 --> 00:00:46,900
So what has happened if you have
a number-- so the question is,

10
00:00:46,900 --> 00:00:51,540
when this happens, which
means that your omega--

11
00:00:51,540 --> 00:00:53,550
that is, your perturbation--

12
00:00:53,550 --> 00:01:00,000
is tailored to produce this,
then Ef is Ei minus h omega.

13
00:01:00,000 --> 00:01:03,450
So you can think of
the energy scale here.

14
00:01:03,450 --> 00:01:06,200
And Ef is lower than Ei.

15
00:01:09,270 --> 00:01:12,340
And the difference
is h bar omega.

16
00:01:12,340 --> 00:01:13,780
So what is this process?

17
00:01:16,790 --> 00:01:21,760
This process is called
stimulated emission.

18
00:01:21,760 --> 00:01:25,000
And why is that called
stimulated emission?

19
00:01:25,000 --> 00:01:29,830
Because you're going
from a state of energy Ei

20
00:01:29,830 --> 00:01:33,790
to a state of energy Ef
that has lower energy.

21
00:01:33,790 --> 00:01:40,270
In that process, you're
releasing energy h bar omega

22
00:01:40,270 --> 00:01:43,780
to the perturbation.

23
00:01:43,780 --> 00:01:49,350
So it's almost--
you would say it's

24
00:01:49,350 --> 00:01:52,660
stimulated because the
perturbation did it.

25
00:01:52,660 --> 00:01:55,750
Still, it almost seems
like something the system

26
00:01:55,750 --> 00:01:58,480
would do by itself.

27
00:01:58,480 --> 00:01:59,770
It has higher energy.

28
00:01:59,770 --> 00:02:04,330
It can go to lower energy and
emit something with frequency

29
00:02:04,330 --> 00:02:06,430
with energy h omega.

30
00:02:06,430 --> 00:02:10,180
But it does that only because
there is a perturbation.

31
00:02:10,180 --> 00:02:15,050
If there was no perturbation,
this would not happen.

32
00:02:15,050 --> 00:02:19,130
That's why, say if you consider
a hydrogen atom or a system,

33
00:02:19,130 --> 00:02:22,720
it's not that the electron
that is in a higher state just

34
00:02:22,720 --> 00:02:26,020
jumps by itself down.

35
00:02:26,020 --> 00:02:29,230
It jumps by itself out
because it can couple

36
00:02:29,230 --> 00:02:30,910
to other degrees of freedom.

37
00:02:30,910 --> 00:02:34,090
And in particular, it can couple
to an electromagnetic field

38
00:02:34,090 --> 00:02:35,450
and send out the photon.

39
00:02:35,450 --> 00:02:41,650
So here, you have stimulated
emission of energy h bar omega.

40
00:02:41,650 --> 00:02:47,680
So stimulated
emission-- emission.

41
00:02:54,720 --> 00:03:00,630
And then the other term
corresponds to the case

42
00:03:00,630 --> 00:03:08,450
when omega fi minus omega
is equal to 0, in which case

43
00:03:08,450 --> 00:03:14,610
Ef is equal Ei plus h bar omega.

44
00:03:14,610 --> 00:03:23,460
This time, Ef is higher
than Ei h bar omega.

45
00:03:23,460 --> 00:03:25,890
And this process is
called absorption--

46
00:03:28,750 --> 00:03:29,750
absorption.

47
00:03:34,160 --> 00:03:39,900
You absorb energy h bar
omega from the perturbation.

48
00:03:39,900 --> 00:03:46,790
The perturbation is capable of
giving the system energy h bar

49
00:03:46,790 --> 00:03:53,720
omega to enable a transition
between Ei and Ef.

50
00:03:53,720 --> 00:04:00,600
So this harmonic perturbation
has two tricks up its sleeve.

51
00:04:00,600 --> 00:04:05,900
It can push you up
by giving you energy,

52
00:04:05,900 --> 00:04:10,520
or it can stimulate you
to go to a lower state

53
00:04:10,520 --> 00:04:14,060
and give energy to
the system represented

54
00:04:14,060 --> 00:04:16,670
by the perturbation.

55
00:04:16,670 --> 00:04:21,230
So two good things it
can do, and both cases

56
00:04:21,230 --> 00:04:22,440
are pretty important.

57
00:04:22,440 --> 00:04:26,190
So we're going to develop
one of these cases.

58
00:04:26,190 --> 00:04:28,670
The other one has exactly
the same formulas.

59
00:04:28,670 --> 00:04:32,790
And we need the Fermi golden
rule for this situation.

60
00:04:32,790 --> 00:04:35,510
So that's what
I'll do right now.

61
00:04:35,510 --> 00:04:39,680
To fix our notation, let's
just do the case of absorption.

62
00:04:42,200 --> 00:04:46,190
But both are going to be
taken care simultaneously.

63
00:04:46,190 --> 00:04:54,980
So absorption-- so what is C?

64
00:04:54,980 --> 00:04:59,210
So when you're doing
absorption, you're saying, OK.

65
00:04:59,210 --> 00:05:03,510
I'm basically having
this process in which Ef

66
00:05:03,510 --> 00:05:07,730
is related to Ei in this way.

67
00:05:07,730 --> 00:05:12,040
This term is becoming 0.

68
00:05:12,040 --> 00:05:14,310
And this term is negligible.

69
00:05:14,310 --> 00:05:18,822
So you can completely
ignore one term

70
00:05:18,822 --> 00:05:20,030
when you're doing absorption.

71
00:05:20,030 --> 00:05:22,010
And you can ignore
the other term

72
00:05:22,010 --> 00:05:24,770
when you're doing a
stimulated emission.

73
00:05:24,770 --> 00:05:27,080
You would say, oh,
but it's not exact.

74
00:05:27,080 --> 00:05:28,540
What about if I keep it?

75
00:05:28,540 --> 00:05:33,140
Well, there are many things
that we don't do exact.

76
00:05:33,140 --> 00:05:37,520
This term is much
bigger than the other--

77
00:05:37,520 --> 00:05:39,680
in principle,
infinitely bigger--

78
00:05:39,680 --> 00:05:41,430
because we're going
to be integrated

79
00:05:41,430 --> 00:05:43,360
over a narrow set of states.

80
00:05:43,360 --> 00:05:47,270
So it would make no sense
to keep those other things.

81
00:05:47,270 --> 00:05:50,720
Those other terms would
be much smaller, probably,

82
00:05:50,720 --> 00:05:53,970
at even the next order
of perturbation theory.

83
00:05:53,970 --> 00:05:59,040
So we keep, therefore,
the second term.

84
00:05:59,040 --> 00:06:01,110
And so what do we have?

85
00:06:01,110 --> 00:06:13,130
Cf of t naught 1 minus
H fi prime over h bar, e

86
00:06:13,130 --> 00:06:21,800
to the i omega fi
minus omega t over 2.

87
00:06:21,800 --> 00:06:24,240
You don't see that
term, of course.

88
00:06:24,240 --> 00:06:29,030
But I'm going to take
out half of this phase

89
00:06:29,030 --> 00:06:33,880
so that I get a sine
function out here.

90
00:06:33,880 --> 00:06:35,700
So I took out half of the phase.

91
00:06:35,700 --> 00:06:44,910
And then I get 2i sine omega
fi minus omega t over 2.

92
00:06:48,660 --> 00:06:52,140
OK, that's this thing.

93
00:06:52,140 --> 00:06:56,870
So the probability to go
from initial to final state

94
00:06:56,870 --> 00:07:07,580
at time t naught would be
the C f1 of t naught squared.

95
00:07:07,580 --> 00:07:18,590
And that's 4H fi prime,
because of the two here.

96
00:07:18,590 --> 00:07:24,660
It's 4H fi prime
squared over h squared

97
00:07:24,660 --> 00:07:34,130
sine squared of omega fi
minus omega over 2 times time

98
00:07:34,130 --> 00:07:40,260
over omega fi minus
omega squared.

99
00:07:40,260 --> 00:07:50,320
And this is all for Ef roughly
equal to Ei plus h bar omega.

100
00:07:50,320 --> 00:07:51,760
So I didn't do much.

101
00:07:51,760 --> 00:07:55,270
I really didn't do
all that much so far.

102
00:07:55,270 --> 00:08:00,370
I just took the second term,
rewrote it with a sine,

103
00:08:00,370 --> 00:08:02,410
and calculated its norm squared.

104
00:08:02,410 --> 00:08:06,340
That's the probability to go
from initial to final states.

105
00:08:14,510 --> 00:08:20,160
So the last step is
integrating over a final state.

106
00:08:20,160 --> 00:08:24,960
So you have this sum over final
states of this probability

107
00:08:24,960 --> 00:08:27,960
to go to the final state.

108
00:08:27,960 --> 00:08:32,730
And we'll write it as the
integral over the set of states

109
00:08:32,730 --> 00:08:39,840
rho of e final dE
final probability

110
00:08:39,840 --> 00:08:44,505
final to initial of t naught.

111
00:08:47,930 --> 00:08:50,940
And this is the same calculation
we were doing before.

112
00:08:54,040 --> 00:09:02,130
And for this calculation, we
just substitute what P fi is.

113
00:09:02,130 --> 00:09:06,480
We assume that as
we integrate, we're

114
00:09:06,480 --> 00:09:11,280
going to have a narrow range
so that this function, rho

115
00:09:11,280 --> 00:09:17,630
of Ef and h prime fi can
go out of the integral.

116
00:09:17,630 --> 00:09:19,465
So what are we going to have?

117
00:09:22,280 --> 00:09:24,050
I'll write it here.

118
00:09:24,050 --> 00:09:30,790
4-- from the matrix
element, from this thing--

119
00:09:30,790 --> 00:09:35,240
H fi prime squared
over h squared.

120
00:09:39,960 --> 00:09:42,430
That goes out.

121
00:09:42,430 --> 00:09:45,020
Then the rho of Ef goes out.

122
00:09:45,020 --> 00:09:48,840
So I'll write it rho of Ef.

123
00:09:48,840 --> 00:09:51,070
And rather than
leaving it like Ef,

124
00:09:51,070 --> 00:09:56,040
like that, because it seems
like a variable, still,

125
00:09:56,040 --> 00:09:58,050
of integration--
if I take it out,

126
00:09:58,050 --> 00:10:00,900
I should be explicit
what this Ef is.

127
00:10:00,900 --> 00:10:03,015
And in this case, Ef--

128
00:10:03,015 --> 00:10:04,720
we're doing absorption.

129
00:10:04,720 --> 00:10:10,590
So this is equal to
Ei plus h bar omega.

130
00:10:10,590 --> 00:10:13,290
That's this central
contribution.

131
00:10:13,290 --> 00:10:16,440
And that's where
you're taking rho out.

132
00:10:16,440 --> 00:10:20,790
So this is pretty important.

133
00:10:20,790 --> 00:10:24,840
This rho is evaluated
at the final energy,

134
00:10:24,840 --> 00:10:31,560
which in this process is
h omega in addition to Ei.

135
00:10:31,560 --> 00:10:37,185
And then the rest of
the integral dE f--

136
00:10:37,185 --> 00:10:40,500
you still have dE f--
and the sine function.

137
00:10:40,500 --> 00:10:49,500
So sine squared of omega
fi minus omega t over--

138
00:10:49,500 --> 00:10:50,780
t naught, I'm sorry.

139
00:10:50,780 --> 00:10:54,150
I'm missing t naught everywhere.

140
00:10:54,150 --> 00:10:56,770
But I did copy it.

141
00:10:56,770 --> 00:10:58,890
t naught and t
naught's up there.

142
00:11:02,440 --> 00:11:10,150
Over 2 over omega fi
minus omega squared.

143
00:11:10,150 --> 00:11:14,270
And this is the
story of the lobes.

144
00:11:14,270 --> 00:11:20,410
This is the function that has
the lobes as a function of Ef.

145
00:11:20,410 --> 00:11:23,020
As a function of
Ef, this quantity

146
00:11:23,020 --> 00:11:26,440
varies and starts
developing the lobes.

147
00:11:26,440 --> 00:11:33,620
And the lobes happen
for values of Ef

148
00:11:33,620 --> 00:11:38,310
that are separated by some
h bar divided by t naught.

149
00:11:38,310 --> 00:11:41,970
So this is the
exact same integral

150
00:11:41,970 --> 00:11:45,480
we analyzed for the
constant transitions.

151
00:11:45,480 --> 00:11:47,400
And it is the same
integral that we

152
00:11:47,400 --> 00:11:53,370
argued that could be done as a
sine squared x over x squared.

153
00:11:53,370 --> 00:11:56,770
So I will not do it again.

154
00:11:56,770 --> 00:12:02,440
This integral gives h bar
linear in t naught over 2 pi.

155
00:12:05,670 --> 00:12:08,190
When you study this,
you will have the notes

156
00:12:08,190 --> 00:12:09,570
and you can review it.

157
00:12:09,570 --> 00:12:11,850
If you've taken
notes, you will see

158
00:12:11,850 --> 00:12:14,460
that is exactly
the same integral

159
00:12:14,460 --> 00:12:20,850
we had last time, except that
we didn't have the omega, which

160
00:12:20,850 --> 00:12:22,650
really doesn't change things.

161
00:12:22,650 --> 00:12:25,680
It just shifts the zero.

162
00:12:25,680 --> 00:12:28,920
So the peak of this
contribution is when

163
00:12:28,920 --> 00:12:31,620
omega fi is equal to omega.

164
00:12:31,620 --> 00:12:36,930
Before when we did it was
when omega fi was equal to 0.

165
00:12:36,930 --> 00:12:39,690
That's why, in the
constant transitions

166
00:12:39,690 --> 00:12:42,780
you conserved energy.

167
00:12:42,780 --> 00:12:49,950
So with this result in
there, the whole answer

168
00:12:49,950 --> 00:12:53,235
here is 2 pi over h bar--

169
00:12:59,820 --> 00:13:04,050
h bar only because one
h bar cancels, indeed--

170
00:13:04,050 --> 00:13:16,690
rho at Ef equal Ei
plus h omega H fi prime

171
00:13:16,690 --> 00:13:19,840
squared times t naught.

172
00:13:19,840 --> 00:13:26,300
That's all that is left,
which is great because that

173
00:13:26,300 --> 00:13:29,270
is our Fermi's golden rule.

174
00:13:29,270 --> 00:13:32,510
Remember, that's a
transition probability.

175
00:13:32,510 --> 00:13:36,350
And the rate is obtained
by dividing this

176
00:13:36,350 --> 00:13:41,150
by the time that has elapsed to
find the probability per unit

177
00:13:41,150 --> 00:13:41,930
time.

178
00:13:41,930 --> 00:13:44,550
That's the rate.

179
00:13:44,550 --> 00:13:47,330
It's probability for
transition per unit,

180
00:13:47,330 --> 00:13:58,970
times 2 pi over h bar rho
at Ef equal Ei plus h bar

181
00:13:58,970 --> 00:14:10,670
omega times the matrix element
f H prime initial squared.

182
00:14:10,670 --> 00:14:16,490
That's Fermi's golden rule
for harmonic transitions,

183
00:14:16,490 --> 00:14:19,700
and in particular,
for absorption.

184
00:14:19,700 --> 00:14:24,650
But stimulated emission,
the calculation

185
00:14:24,650 --> 00:14:26,720
is completely analogous.

186
00:14:26,720 --> 00:14:32,300
So the end result for stimulated
emission is just minus H omega.

187
00:14:32,300 --> 00:14:35,850
The final energy
now is this one.

188
00:14:35,850 --> 00:14:38,150
So the top sine--

189
00:14:38,150 --> 00:14:46,630
plus is for absorption, minus
for stimulated emission.

190
00:14:56,490 --> 00:14:59,910
And if you want a
reminder here, it

191
00:14:59,910 --> 00:15:10,350
was that delta H was 2H
prime cosine omega t.

192
00:15:10,350 --> 00:15:16,710
So this reminds you that
this H prime you have here

193
00:15:16,710 --> 00:15:19,620
is the matrix element
of the perturbation

194
00:15:19,620 --> 00:15:20,820
with this convention.

195
00:15:20,820 --> 00:15:23,670
This omega you have
here is the frequency

196
00:15:23,670 --> 00:15:26,040
of the perturbing Hamiltonian.

197
00:15:26,040 --> 00:15:27,510
And that's what has happened.

198
00:15:30,810 --> 00:15:34,920
So this is Fermi's golden rule.

199
00:15:34,920 --> 00:15:35,610
It's over.

200
00:15:35,610 --> 00:15:40,050
We've done it, done it
basically for two cases--

201
00:15:40,050 --> 00:15:44,010
the constant perturbation and
the harmonic perturbation.

202
00:15:44,010 --> 00:15:45,640
And there's a lot
of physics here

203
00:15:45,640 --> 00:15:51,750
that we will be exploring
starting now, but continuing

204
00:15:51,750 --> 00:15:58,260
with atoms and radiation in
general, atomic transitions.

205
00:15:58,260 --> 00:16:00,870
W is the rate.

206
00:16:00,870 --> 00:16:05,484
So this is called the
transition rate per unit time--

207
00:16:05,484 --> 00:16:14,290
transition rate per unit time.

208
00:16:14,290 --> 00:16:18,940
So the probability of
transition or probability

209
00:16:18,940 --> 00:16:23,060
of transition per unit time--

210
00:16:23,060 --> 00:16:30,130
probability of transition
per unit time--

211
00:16:33,510 --> 00:16:36,060
that's probably more
understandable than the word

212
00:16:36,060 --> 00:16:37,260
rate.

213
00:16:37,260 --> 00:16:43,020
So this P fi gives you the
probability of transition

214
00:16:43,020 --> 00:16:46,200
after a time t naught.

215
00:16:46,200 --> 00:16:48,570
Happily, it's
proportional to t naught,

216
00:16:48,570 --> 00:16:52,020
so you divide by the
time that has elapsed,

217
00:16:52,020 --> 00:16:55,710
and it's a probability of
transition per unit time.

218
00:16:55,710 --> 00:16:58,290
So once you compute
this number, you

219
00:16:58,290 --> 00:17:01,770
get the probability of
transition per unit time.

220
00:17:01,770 --> 00:17:06,150
So if this W is something
and you have a billion atoms,

221
00:17:06,150 --> 00:17:09,500
you multiply that probability
by the number of atoms you have,

222
00:17:09,500 --> 00:17:12,590
and you know how many
have decayed already.