1
00:00:00,201 --> 00:00:00,700
Good.

2
00:00:00,700 --> 00:00:03,880
So let's do, then,
our transitions.

3
00:00:03,880 --> 00:00:06,160
So we do the constant
perturbation.

4
00:00:11,930 --> 00:00:15,200
Constant perturbation.

5
00:00:20,740 --> 00:00:25,300
So as we said, delta
H is equal to V,

6
00:00:25,300 --> 00:00:27,130
and it's time independent.

7
00:00:27,130 --> 00:00:28,900
It just begins at time 0.

8
00:00:31,780 --> 00:00:37,360
And we'll examine what's
going on by time t0.

9
00:00:37,360 --> 00:00:40,680
So what are we going to do?

10
00:00:40,680 --> 00:00:46,720
We're going to
examine a transition

11
00:00:46,720 --> 00:00:52,210
to go from some initial
state i, initial state,

12
00:00:52,210 --> 00:00:56,110
to a final state f.

13
00:00:56,110 --> 00:00:59,980
So we don't have to say much
about what the Hamiltonian is

14
00:00:59,980 --> 00:01:01,330
or anything.

15
00:01:01,330 --> 00:01:03,910
For us, V is going
to have a constant.

16
00:01:03,910 --> 00:01:06,280
It's going to have
some matrix elements

17
00:01:06,280 --> 00:01:09,570
that once we do an
example you can calculate,

18
00:01:09,570 --> 00:01:14,530
but for the time being we
need not know too much.

19
00:01:14,530 --> 00:01:18,940
So I'm going to
use the key formula

20
00:01:18,940 --> 00:01:24,040
that was derived already
about perturbation theory

21
00:01:24,040 --> 00:01:27,800
and how you get the
transition amplitude.

22
00:01:27,800 --> 00:01:35,060
So we know that the coefficient
c associated to the m state

23
00:01:35,060 --> 00:01:37,810
to first order in
perturbation theory

24
00:01:37,810 --> 00:01:47,410
at that time t0 can be computed
as a sum over all n integral

25
00:01:47,410 --> 00:02:00,580
from 0 to t0 e to the i omega
mn t prime delta Hmn t prime

26
00:02:00,580 --> 00:02:07,900
over ih bar Cn 0 dt prime.

27
00:02:07,900 --> 00:02:10,940
So let me just say a few words.

28
00:02:10,940 --> 00:02:16,800
You may have seen this
yesterday in recitation.

29
00:02:16,800 --> 00:02:18,030
Here it is.

30
00:02:18,030 --> 00:02:23,970
The state has described
in this usual language

31
00:02:23,970 --> 00:02:35,020
of psi tilde of t was described
as a sum of c ms of ts ms.

32
00:02:35,020 --> 00:02:38,980
And we calculate those
in perturbation theory.

33
00:02:38,980 --> 00:02:43,950
So Cn's, we know them
at time equals 0,

34
00:02:43,950 --> 00:02:45,960
we imagine we know
the initial state

35
00:02:45,960 --> 00:02:47,490
and we want to
calculate them later.

36
00:02:47,490 --> 00:02:50,700
Well, the first order
in perturbation theory,

37
00:02:50,700 --> 00:02:54,000
the Cn's are called Cn's 1's.

38
00:02:54,000 --> 00:02:55,740
And they're given
by this formula.

39
00:02:55,740 --> 00:02:57,570
First order in
perturbation theory,

40
00:02:57,570 --> 00:03:00,210
because we have
a single delta H,

41
00:03:00,210 --> 00:03:04,230
this omega mn is the energy
of m minus the energy

42
00:03:04,230 --> 00:03:07,037
of n over h bar.

43
00:03:07,037 --> 00:03:07,620
And that's it.

44
00:03:07,620 --> 00:03:09,105
Those are all our symbols.

45
00:03:12,260 --> 00:03:14,460
So what do we have to do now?

46
00:03:14,460 --> 00:03:20,490
We have a transition from some
initial state to a final state.

47
00:03:20,490 --> 00:03:22,980
So what does that mean?

48
00:03:22,980 --> 00:03:31,910
It means that Cn 0,
it only exists when

49
00:03:31,910 --> 00:03:34,320
n represents the initial state.

50
00:03:34,320 --> 00:03:38,700
So I'll just write delta ni.

51
00:03:38,700 --> 00:03:50,790
At time equals zero,
system is in the state i.

52
00:03:50,790 --> 00:03:56,670
And at time t0, we're
asking for the probability

53
00:03:56,670 --> 00:03:58,510
to go into final state.

54
00:03:58,510 --> 00:04:03,600
So instead of using n and
m, we're just using f and i.

55
00:04:03,600 --> 00:04:09,540
And therefore, we'll
have m equal f.

56
00:04:09,540 --> 00:04:12,900
With these two
facts, the formula

57
00:04:12,900 --> 00:04:18,959
becomes cf the amplitude to
first order in perturbation

58
00:04:18,959 --> 00:04:27,060
theory to be in the state f
at time t0 is the sum is gone,

59
00:04:27,060 --> 00:04:31,050
sum over n just
applies for n equals i.

60
00:04:31,050 --> 00:04:34,240
So i will go here.

61
00:04:34,240 --> 00:04:41,670
We'll have one over
ih bar 0 to t0 e

62
00:04:41,670 --> 00:04:45,490
to the i omega final
state to initial state.

63
00:04:45,490 --> 00:04:54,900
That's m and n t prime Vfi,
because the delta H is V,

64
00:04:54,900 --> 00:04:58,350
and we're going from
initial to final.

65
00:04:58,350 --> 00:05:02,430
So fi here.

66
00:05:02,430 --> 00:05:06,150
And then the t prime.

67
00:05:06,150 --> 00:05:07,240
It's all gone.

68
00:05:07,240 --> 00:05:10,710
It's all become
very simple, though.

69
00:05:10,710 --> 00:05:14,370
You would say too simple.

70
00:05:14,370 --> 00:05:17,270
We have this is
time independent,

71
00:05:17,270 --> 00:05:19,050
so it goes out of the interval.

72
00:05:19,050 --> 00:05:22,770
So we just have the interval
of an exponential here.

73
00:05:22,770 --> 00:05:26,390
That's very, very easy.

74
00:05:26,390 --> 00:05:31,910
What is Vfi case vi?

75
00:05:35,030 --> 00:05:35,530
So

76
00:05:35,530 --> 00:05:40,930
This goes out, and we just have
to integrate this function.

77
00:05:43,960 --> 00:05:49,600
I'll write it in a
way that is simpler.

78
00:05:49,600 --> 00:05:51,070
Maybe you skip a line.

79
00:05:51,070 --> 00:05:54,580
I don't want to just
count the factors of is

80
00:05:54,580 --> 00:05:55,870
and doing the integral.

81
00:05:55,870 --> 00:05:59,260
When you integrate this,
you get another exponential.

82
00:05:59,260 --> 00:06:01,480
You're going to get
the exponential at t0

83
00:06:01,480 --> 00:06:04,700
minus the exponential at zero.

84
00:06:04,700 --> 00:06:08,020
All right, so we don't
want to do our integral.

85
00:06:08,020 --> 00:06:10,080
So I'll just write the answer.

86
00:06:10,080 --> 00:06:12,790
I was saying we get
an exponential here

87
00:06:12,790 --> 00:06:16,630
at t0 minus the
value at zero, then

88
00:06:16,630 --> 00:06:21,490
you take half of the
exponential out the form a sign.

89
00:06:21,490 --> 00:06:26,560
This are simple matters, so I
will not do the integral here.

90
00:06:26,560 --> 00:06:34,690
You get Vfi over Ef
minus Ei e to the i omega

91
00:06:34,690 --> 00:06:44,560
fi t0 two minus two i sine
of omega fi t0 over two.

92
00:06:49,390 --> 00:06:50,530
You can believe that.

93
00:06:50,530 --> 00:06:53,560
I think you can
believe the sine,

94
00:06:53,560 --> 00:07:01,230
and I have everything here.

95
00:07:01,230 --> 00:07:08,550
The h bar helped turn the
omega fi into Ef minus Ei.

96
00:07:08,550 --> 00:07:20,460
So we can now compute the
transition probability

97
00:07:20,460 --> 00:07:22,950
to go from the initial
to the final stage.

98
00:07:22,950 --> 00:07:26,480
So we'll write it like this.

99
00:07:26,480 --> 00:07:31,100
I to f-- it's a little funny.

100
00:07:31,100 --> 00:07:31,980
I don't know.

101
00:07:31,980 --> 00:07:33,930
You can write it
whichever way you want.

102
00:07:33,930 --> 00:07:35,520
Some people like it like that.

103
00:07:35,520 --> 00:07:39,150
I'm going to do it in the sense
that the initial state always

104
00:07:39,150 --> 00:07:42,680
appears as a cat, the
final state as a bra.

105
00:07:42,680 --> 00:07:45,480
So you draw the arrow
like that, more or less

106
00:07:45,480 --> 00:07:48,340
to keep the sense of
order in your brain.

107
00:07:48,340 --> 00:07:50,190
But if it doesn't
help you, write it

108
00:07:50,190 --> 00:07:52,320
whichever way you want.

109
00:07:52,320 --> 00:08:03,300
Pfi at t0 one is
the norm squared

110
00:08:03,300 --> 00:08:06,570
of this coefficient, cf one.

111
00:08:06,570 --> 00:08:09,870
The probability to be
found in the final state

112
00:08:09,870 --> 00:08:12,970
has to do with the norm
squared of this thing.

113
00:08:12,970 --> 00:08:23,070
So it's this that's part of what
was reviewed yesterday squared.

114
00:08:26,890 --> 00:08:28,020
So what do we get?

115
00:08:28,020 --> 00:08:30,000
That simplifies quite a bit.

116
00:08:30,000 --> 00:08:38,970
We get Vfi squared times
four sine squared omega fi

117
00:08:38,970 --> 00:08:45,220
t0 over two over ef
minus ei squared.

118
00:08:47,940 --> 00:08:54,490
And this is unit free.

119
00:08:54,490 --> 00:08:56,890
This has units of energy.

120
00:08:56,890 --> 00:08:59,470
V is a variation
of the Hamiltonian.

121
00:08:59,470 --> 00:09:00,700
It has units of energy.

122
00:09:00,700 --> 00:09:04,420
When you put states,
states are normalized

123
00:09:04,420 --> 00:09:07,240
so it doesn't change the units.

124
00:09:07,240 --> 00:09:11,020
And this has units of energy
squared, this has no units,

125
00:09:11,020 --> 00:09:12,220
and this is the answer.

126
00:09:12,220 --> 00:09:14,620
A little strange.

127
00:09:14,620 --> 00:09:22,140
There's a periodic variation
on the transition probability,

128
00:09:22,140 --> 00:09:26,980
and what does it mean to
have a weak perturbation we

129
00:09:26,980 --> 00:09:28,210
can ask already?

130
00:09:28,210 --> 00:09:31,780
And the answer in
general is quite simple.

131
00:09:31,780 --> 00:09:34,020
It's a pragmatic answer.

132
00:09:34,020 --> 00:09:37,000
A perturbation is
weak if this answer

133
00:09:37,000 --> 00:09:39,400
is very little, very small.

134
00:09:39,400 --> 00:09:43,330
Suppose this probability
comes out to be three,

135
00:09:43,330 --> 00:09:45,490
you know it's already too big.

136
00:09:45,490 --> 00:09:49,060
But if this is 10 to the minus
sixth times this function,

137
00:09:49,060 --> 00:09:50,320
that's reasonable.

138
00:09:50,320 --> 00:09:53,020
You're shining atoms
and one in a million

139
00:09:53,020 --> 00:09:55,090
goes and gets ionized.

140
00:09:55,090 --> 00:09:56,710
That's a reasonable thing.

141
00:09:56,710 --> 00:10:01,750
So the perturbation theory
is valid for whatever time

142
00:10:01,750 --> 00:10:06,310
you use this formula as long
as this number is small,

143
00:10:06,310 --> 00:10:11,410
and this could be arranged by
having Vfi sufficiently small.

144
00:10:11,410 --> 00:10:14,470
I want to understand
this function better,

145
00:10:14,470 --> 00:10:18,170
because this is a transition
from initial state

146
00:10:18,170 --> 00:10:19,840
to final state that
looks like that.

147
00:10:19,840 --> 00:10:22,250
So let's understand it better.

148
00:10:22,250 --> 00:10:31,330
Suppose one, Ef is different
from Ei, then how does it look?

149
00:10:31,330 --> 00:10:34,630
Well, it looks like this.

150
00:10:34,630 --> 00:10:37,840
I brought some other chalk
not that it helps too much.

151
00:10:37,840 --> 00:10:45,600
But it looks like this
as a function of time.

152
00:10:49,050 --> 00:10:56,970
The height here
is height four Vfi

153
00:10:56,970 --> 00:11:04,350
squared over Ef minus Ei
squared, and it's oscillatory.

154
00:11:04,350 --> 00:11:09,900
It goes to zero again at
time 2 pi over omega fi.

155
00:11:13,010 --> 00:11:16,250
OK, so this is the oscillation.

156
00:11:16,250 --> 00:11:23,680
Actually for a small time,
this grows quadratically,

157
00:11:23,680 --> 00:11:26,400
and then it starts blowing up.

158
00:11:29,060 --> 00:11:32,200
So here while the
initial behavior

159
00:11:32,200 --> 00:11:35,470
would be quadratic
for small time,

160
00:11:35,470 --> 00:11:39,470
this actually is quadratic
as we will see in a second,

161
00:11:39,470 --> 00:11:43,480
but you can more or less see
by the expansion of the sine,

162
00:11:43,480 --> 00:11:48,880
then the initial quadratic
growth gets tamed and becomes

163
00:11:48,880 --> 00:11:52,630
an oscillation here.

164
00:11:52,630 --> 00:11:58,120
This is valid for all times if
this number is relatively small

165
00:11:58,120 --> 00:12:02,830
so that we believe
perturbation theory,

166
00:12:02,830 --> 00:12:06,910
and that's that for that case.

167
00:12:06,910 --> 00:12:10,900
It's also interesting
that this gets suppressed

168
00:12:10,900 --> 00:12:15,010
as the energy of the final
state is different, more

169
00:12:15,010 --> 00:12:17,810
and more different, from the
energy of the initial state.

170
00:12:17,810 --> 00:12:22,370
So it always oscillates, but
if the state your transition

171
00:12:22,370 --> 00:12:26,380
is very far away, it is going
to be extremely suppressed

172
00:12:26,380 --> 00:12:28,040
by the quadratic factors.

173
00:12:28,040 --> 00:12:30,250
So this is an
important suppression.

174
00:12:30,250 --> 00:12:36,920
This is saying that transitions
that change the energy

175
00:12:36,920 --> 00:12:40,540
are not that favored.

176
00:12:40,540 --> 00:12:45,640
A constant perturbation
doesn't supply really

177
00:12:45,640 --> 00:12:50,590
energy to produce transitions
that change the energy much,

178
00:12:50,590 --> 00:12:52,390
and they are suppressed.

179
00:12:52,390 --> 00:12:55,450
So they produce them,
but they are suppressed.

180
00:12:55,450 --> 00:12:57,550
The other case
that is of interest

181
00:12:57,550 --> 00:13:02,420
is the case when
Ef is equal to Ei.

182
00:13:02,420 --> 00:13:07,580
I'm not saying that the state
f is the same as the state i.

183
00:13:07,580 --> 00:13:08,470
Not at all.

184
00:13:08,470 --> 00:13:10,630
It's a different
state but happens

185
00:13:10,630 --> 00:13:16,010
to have the same energy,
and in that case,

186
00:13:16,010 --> 00:13:19,590
we must take the
limit as Ef goes

187
00:13:19,590 --> 00:13:26,660
to Ei, remember omega fi is
Ef minus Ei so over h bar.

188
00:13:26,660 --> 00:13:38,780
So the limit as ef goes to
Ei of this Pif is how much.

189
00:13:41,530 --> 00:13:44,390
I'll kind of do it
in my head here.

190
00:13:44,390 --> 00:13:50,750
We have an h bar here that
is going to be left over.

191
00:13:50,750 --> 00:13:55,010
One over h bar,
so h bar squared.

192
00:13:55,010 --> 00:13:56,300
This is going to cancel.

193
00:13:56,300 --> 00:13:59,480
The four is going
to cancel with this,

194
00:13:59,480 --> 00:14:04,640
and we're going
to get vfi squared

195
00:14:04,640 --> 00:14:11,000
over h squared t0 squared.

196
00:14:11,000 --> 00:14:14,750
OK, so here it is,
the quadratic behavior

197
00:14:14,750 --> 00:14:17,480
when the energy
of the final state

198
00:14:17,480 --> 00:14:21,390
is the same as the energy
of your original state.

199
00:14:21,390 --> 00:14:25,910
Now, the transition probability
starts to grow quadratically.

200
00:14:25,910 --> 00:14:30,590
That cannot be valid
for too long time,

201
00:14:30,590 --> 00:14:34,880
because eventually that
number grows without bound,

202
00:14:34,880 --> 00:14:37,530
and that number could
become as big as 1,

203
00:14:37,530 --> 00:14:46,110
and that transition
is not reasonable.

204
00:14:46,110 --> 00:14:59,660
So this is valid
up to some max t0.

205
00:14:59,660 --> 00:15:03,500
And it's up to you,
depends on what Vfi is,

206
00:15:03,500 --> 00:15:05,150
how long you can trust this.

207
00:15:05,150 --> 00:15:06,900
So this is a growth.

208
00:15:06,900 --> 00:15:10,800
This is the same
growth we observe here.

209
00:15:10,800 --> 00:15:15,580
The limit as Ef goes
to Ei go to zero

210
00:15:15,580 --> 00:15:18,950
is actually the same as the
limit as t goes to zero,

211
00:15:18,950 --> 00:15:22,940
so it's the same
quadratic behavior.

212
00:15:22,940 --> 00:15:27,150
So finally, what's
going to happen?

213
00:15:27,150 --> 00:15:30,950
What are we aiming here?

214
00:15:30,950 --> 00:15:34,850
Well, we're aiming
to the case where

215
00:15:34,850 --> 00:15:41,870
we have in the energy line,
we have the initial energy Ei

216
00:15:41,870 --> 00:15:49,070
and then we're going to have
a continuum of final states

217
00:15:49,070 --> 00:15:50,880
that overlap with Ei.

218
00:15:54,480 --> 00:15:57,040
They're all over there.

219
00:15:57,040 --> 00:16:02,530
That's Ef all over there.

220
00:16:02,530 --> 00:16:06,160
Of course with our box, if
you come with your microscope,

221
00:16:06,160 --> 00:16:07,400
you see lines here.

222
00:16:10,080 --> 00:16:11,850
But they're all there.

223
00:16:11,850 --> 00:16:16,170
There is a continuum
overlapping with this,

224
00:16:16,170 --> 00:16:21,210
and now we're going to attempt
to sum over the continuum.

225
00:16:21,210 --> 00:16:22,755
And what should we observe?

226
00:16:25,590 --> 00:16:29,640
We should observe that
when we add the continuum

227
00:16:29,640 --> 00:16:33,310
physically, what do we need?

228
00:16:33,310 --> 00:16:41,430
We need to find what is called
a transition rate, in which you

229
00:16:41,430 --> 00:16:47,160
have the probability of
transition per unit time

230
00:16:47,160 --> 00:16:48,402
is a constant.

231
00:16:48,402 --> 00:16:49,860
You see, you have
a phenom-- you're

232
00:16:49,860 --> 00:16:52,800
shining light on an atom.

233
00:16:52,800 --> 00:16:55,080
OK, you shine
light and you wait.

234
00:16:55,080 --> 00:16:57,640
Eventually the atom ionizes--

235
00:16:57,640 --> 00:17:00,220
photoelectric effect.

236
00:17:00,220 --> 00:17:03,320
But if you have a billion
atoms, then you can shine light

237
00:17:03,320 --> 00:17:06,520
and you're going to have a
transition rate, basically how

238
00:17:06,520 --> 00:17:10,300
many atoms are going
to happen to ionize.

239
00:17:10,300 --> 00:17:13,480
So in order to have
a transition rate,

240
00:17:13,480 --> 00:17:18,579
the probability
that you transition

241
00:17:18,579 --> 00:17:22,089
has to be proportional
to the time

242
00:17:22,089 --> 00:17:25,550
that the perturbation
has been acting.

243
00:17:25,550 --> 00:17:33,110
So the probability of transition
must grow linear in t.

244
00:17:33,110 --> 00:17:35,660
Therefore, you have
a transition rate

245
00:17:35,660 --> 00:17:39,320
which is the probability of
transition per unit time,

246
00:17:39,320 --> 00:17:42,770
so you can grow linear
in time, so per unit time

247
00:17:42,770 --> 00:17:44,720
you have a transition rate.

248
00:17:44,720 --> 00:17:48,050
So somehow look
what's happening here.

249
00:17:48,050 --> 00:17:53,630
When Ef is different than Ei,
the transition probability

250
00:17:53,630 --> 00:17:55,460
is not linear in time.

251
00:17:55,460 --> 00:17:57,110
It does this.

252
00:17:57,110 --> 00:18:00,800
When E approaches--

253
00:18:00,800 --> 00:18:04,430
Ef approaches Ei, the
probability of transition

254
00:18:04,430 --> 00:18:07,700
goes quadratic in
time, and what we want

255
00:18:07,700 --> 00:18:12,080
is a probability of transition
that grows linear in time.

256
00:18:12,080 --> 00:18:15,560
That would define
a transition rate.

257
00:18:15,560 --> 00:18:19,530
So how is that going to happen?

258
00:18:19,530 --> 00:18:22,940
Well, we'll see it happen
in front of our eyes.

259
00:18:22,940 --> 00:18:26,700
The magic of integration
is going to do it.

260
00:18:26,700 --> 00:18:28,400
And moreover, we're
going to see that

261
00:18:28,400 --> 00:18:32,480
consistent with this intuition,
most of the transitions

262
00:18:32,480 --> 00:18:37,100
that are relevant are happening
within Heisenberg's uncertainty

263
00:18:37,100 --> 00:18:43,610
principle of a little energy
interval here around Ei.

264
00:18:43,610 --> 00:18:47,720
So this will be considered to
be at the end of the day energy

265
00:18:47,720 --> 00:18:50,030
conserving transitions.

266
00:18:50,030 --> 00:18:55,280
The Hamiltonian, the delta V
helps the transition happen

267
00:18:55,280 --> 00:18:59,070
but doesn't supply energy
at the end of the day.

268
00:18:59,070 --> 00:19:02,290
So this is what
we're getting to.