1
00:00:00,500 --> 00:00:04,080
PROFESSOR: Today, we have to
discuss harmonic perturbations.

2
00:00:04,080 --> 00:00:08,700
So we've done Fermi's golden
rule for constant transitions.

3
00:00:08,700 --> 00:00:13,950
We saw transitions from a
discrete state to a continuum.

4
00:00:13,950 --> 00:00:16,470
And by integrating
over the continuum,

5
00:00:16,470 --> 00:00:20,460
we found a nice rule,
Fermi's golden rule,

6
00:00:20,460 --> 00:00:25,510
that govern the transition
rate for this process.

7
00:00:25,510 --> 00:00:28,020
So the only thing we
have to do different

8
00:00:28,020 --> 00:00:31,620
now is consider the case
that the perturbation is not

9
00:00:31,620 --> 00:00:35,230
just a step that gets
up and stays there,

10
00:00:35,230 --> 00:00:37,830
but it has a
frequency dependence.

11
00:00:37,830 --> 00:00:42,100
So that will bring a
couple of novel features.

12
00:00:42,100 --> 00:00:44,520
But at the end of the
day, as we will see,

13
00:00:44,520 --> 00:00:48,560
our Fermi's golden rule is
going to look pretty similar

14
00:00:48,560 --> 00:00:51,930
to the original
Fermi's golden rule.

15
00:00:51,930 --> 00:00:56,580
A nice application of
Fermi's golden rule

16
00:00:56,580 --> 00:01:01,550
is the calculation of the
ionization rate for hydrogen,

17
00:01:01,550 --> 00:01:04,010
in which you take
a hydrogen atom,

18
00:01:04,010 --> 00:01:09,820
you put it in an electric
field or send a light wave,

19
00:01:09,820 --> 00:01:13,350
and then suddenly the
electron and the hydrogen atom

20
00:01:13,350 --> 00:01:15,870
from the ground state ionizes.

21
00:01:15,870 --> 00:01:19,530
And we can compute already--
we have the technology

22
00:01:19,530 --> 00:01:23,460
to compute the ionization rate.

23
00:01:23,460 --> 00:01:26,740
That's a pretty
physical quantity.

24
00:01:26,740 --> 00:01:30,990
And that will be an example
we'll develop today.

25
00:01:30,990 --> 00:01:34,050
Those rates have
the funny situation

26
00:01:34,050 --> 00:01:37,380
that the calculation
can be somewhat

27
00:01:37,380 --> 00:01:39,930
involved and interesting.

28
00:01:39,930 --> 00:01:43,230
And the answers, generally,
by the time you simplify them,

29
00:01:43,230 --> 00:01:45,650
are pretty simple
and pretty nice.

30
00:01:45,650 --> 00:01:47,730
So it's a good idea.

31
00:01:47,730 --> 00:01:50,550
You have to have patience
with those calculations

32
00:01:50,550 --> 00:01:55,150
to simplify it till the end,
and that's pretty instructive.

33
00:01:55,150 --> 00:02:00,270
So we begin with
harmonic perturbations.

34
00:02:00,270 --> 00:02:05,160
So we did constant
perturbations already.

35
00:02:05,160 --> 00:02:09,389
So now harmonic perturbations.

36
00:02:13,660 --> 00:02:18,160
So our situation is
that in which age H of t

37
00:02:18,160 --> 00:02:23,230
is equal to a known
Hamiltonian plus delta H of t.

38
00:02:23,230 --> 00:02:29,620
And this time, delta H
of t is conventionally

39
00:02:29,620 --> 00:02:34,900
written as 2 H
prime cosine omega t

40
00:02:34,900 --> 00:02:40,930
for some t between t0
and 0, and 0 otherwise.

41
00:02:49,230 --> 00:02:53,220
All of us wonder why the 2 here.

42
00:02:53,220 --> 00:02:56,940
One reason for it-- it's
all convention, of course.

43
00:02:56,940 --> 00:03:01,410
You have your perturbation,
and what you call E H prime

44
00:03:01,410 --> 00:03:05,010
or what you call 2 H
prime is your choice.

45
00:03:05,010 --> 00:03:09,120
But this 2 has the
advantage that when

46
00:03:09,120 --> 00:03:13,560
you describe the cosine
in terms of exponentials--

47
00:03:13,560 --> 00:03:17,480
e to the i omega t plus e to the
minus i omega t-- the over 2,

48
00:03:17,480 --> 00:03:19,620
it cancels this one.

49
00:03:19,620 --> 00:03:26,100
And that makes Fermi's golden
rule, that will follow also

50
00:03:26,100 --> 00:03:29,820
and will be valid for
these perturbations,

51
00:03:29,820 --> 00:03:33,840
take exactly the
same form as it did

52
00:03:33,840 --> 00:03:36,030
for the case of
constant perturbation.

53
00:03:36,030 --> 00:03:42,450
So it's fairly convenient to
put that 2, and we'll put it in.

54
00:03:42,450 --> 00:03:48,830
Some books don't, and then they
have different looking formulas

55
00:03:48,830 --> 00:03:52,190
for Fermi golden rule
depending to which case you're

56
00:03:52,190 --> 00:03:53,300
talking about.

57
00:03:53,300 --> 00:03:58,050
Of course, when we mean that
this is the time dependence,

58
00:03:58,050 --> 00:04:02,490
we are implying that H
prime is time independent.

59
00:04:05,160 --> 00:04:07,190
Because the time
dependence is this one.

60
00:04:07,190 --> 00:04:11,000
That's what we're
interested in considering.

61
00:04:11,000 --> 00:04:14,720
Of course-- this has
been asked sometimes--

62
00:04:14,720 --> 00:04:18,500
H prime can depend on all
kinds of other thing-- position

63
00:04:18,500 --> 00:04:21,170
coordinates, some
other quantities.

64
00:04:21,170 --> 00:04:25,960
But we're focusing on time
here, so we'll leave it there.

65
00:04:25,960 --> 00:04:29,030
Moreover, for reasons
of convention,

66
00:04:29,030 --> 00:04:32,480
just let's always thing
of omega as positive.

67
00:04:32,480 --> 00:04:35,690
It wouldn't make a difference
if it would be negative here

68
00:04:35,690 --> 00:04:41,210
with the cosine function, but
let's just set by convention

69
00:04:41,210 --> 00:04:43,590
that omega is positive.

70
00:04:43,590 --> 00:04:46,340
Finally, we're going
to do transitions again

71
00:04:46,340 --> 00:04:49,220
from an initial
to a final state.

72
00:04:49,220 --> 00:04:51,080
So we will consider
the case when

73
00:04:51,080 --> 00:04:58,440
we go from an initial
state to a final state.

74
00:04:58,440 --> 00:05:03,650
And therefore, we will
work in this language

75
00:05:03,650 --> 00:05:07,610
with this constant
coefficients Cn's, these

76
00:05:07,610 --> 00:05:12,190
coefficients that multiply
the states in psi tilde.

77
00:05:12,190 --> 00:05:22,180
Psi tilde is equal to
Cn n of t, sum over n.

78
00:05:22,180 --> 00:05:29,930
And these Cn's,
at time equals 0,

79
00:05:29,930 --> 00:05:38,090
will be equal to delta ni,
which means that they are all 0

80
00:05:38,090 --> 00:05:42,500
except when we're talking
about Ci at 0 is equal to 1

81
00:05:42,500 --> 00:05:47,040
because we start with
an initial state.

82
00:05:47,040 --> 00:05:52,770
We had a general formula for
the transition coefficient.

83
00:05:52,770 --> 00:06:00,702
And Cm of 1 at time equal t--

84
00:06:00,702 --> 00:06:03,130
or I'll put t0--

85
00:06:03,130 --> 00:06:09,270
is equal to sum over n,
integral from 0 to t0 e

86
00:06:09,270 --> 00:06:22,550
to the i omega mn t prime, delta
H mn of t prime over i h bar,

87
00:06:22,550 --> 00:06:28,110
Cn at time equals 0, dt prime.

88
00:06:30,980 --> 00:06:35,880
This was our general formula
for transition coefficients.

89
00:06:35,880 --> 00:06:42,590
The Cm's is the coefficient
or the amplitude for the state

90
00:06:42,590 --> 00:06:47,190
to be found in the m
eigenstate at time t0

91
00:06:47,190 --> 00:06:50,230
to first order in
perturbation theory.

92
00:06:50,230 --> 00:06:53,090
And it depends on
where you started on.

93
00:06:53,090 --> 00:06:57,830
That's why the sum over n
here with initial states.

94
00:06:57,830 --> 00:07:01,010
But this sum is going
to collapse because we

95
00:07:01,010 --> 00:07:03,950
know we start with the state i.

96
00:07:03,950 --> 00:07:08,430
So when we substitute
Cn equal to this,

97
00:07:08,430 --> 00:07:12,410
the sum only works
when n is equal to i.

98
00:07:12,410 --> 00:07:15,740
So we'll put for ni's.

99
00:07:15,740 --> 00:07:19,640
And, of course, we're going to
also take for the final state

100
00:07:19,640 --> 00:07:21,140
to be f.

101
00:07:21,140 --> 00:07:29,830
So the formula now
reads Cf 1 at t0

102
00:07:29,830 --> 00:07:37,220
is equal to integral from 0
to t0 e to the I omega f--

103
00:07:37,220 --> 00:07:43,730
m was f-- i t prime.

104
00:07:43,730 --> 00:07:49,250
And now the delta H. The delta
H is this whole quantity,

105
00:07:49,250 --> 00:07:51,610
so we have to substitute it.

106
00:07:51,610 --> 00:07:58,830
So 2 H prime is the only part
that has matrix elements.

107
00:07:58,830 --> 00:08:01,060
The cosine omega t
is just a function.

108
00:08:01,060 --> 00:08:12,825
So it's H prime fi cosine
omega t prime, and dt prime.

109
00:08:20,060 --> 00:08:21,455
There's the i h bar.

110
00:08:24,600 --> 00:08:27,150
So I think I got
everything right.

111
00:08:27,150 --> 00:08:30,110
The sum collapsed.

112
00:08:30,110 --> 00:08:35,240
mn is being replaced
by the right labels.

113
00:08:35,240 --> 00:08:40,580
mn here, this is the expectation
value between m and n.

114
00:08:40,580 --> 00:08:43,130
And that becomes
between f and i.

115
00:08:43,130 --> 00:08:45,620
And it affects this
whole thing, but it just

116
00:08:45,620 --> 00:08:50,570
ends up affecting the
Hamiltonian H prime here.

117
00:08:50,570 --> 00:08:52,240
So I think we're OK.

118
00:08:52,240 --> 00:08:54,530
We have everything there.

119
00:08:54,530 --> 00:09:01,560
And Hfi, of course, doesn't
have time dependence.

120
00:09:01,560 --> 00:09:06,900
So we said H prime has
no time dependence.

121
00:09:06,900 --> 00:09:10,440
So that thing can go
out of the integral.

122
00:09:10,440 --> 00:09:14,410
So this will go out.

123
00:09:14,410 --> 00:09:17,890
And the integral is simple
because you have now

124
00:09:17,890 --> 00:09:24,200
Hfi prime over i h bar.

125
00:09:24,200 --> 00:09:26,470
And the 2, we leave
it for the cosine.

126
00:09:26,470 --> 00:09:28,370
So we get two integrals.

127
00:09:28,370 --> 00:09:42,100
t0 e to the i omega fi
plus omega t prime--

128
00:09:42,100 --> 00:09:45,520
from the first exponential
in the cosine--

129
00:09:45,520 --> 00:09:52,930
plus an e to the i omega
fi minus omega t prime--

130
00:09:52,930 --> 00:09:55,570
from the second
exponential in cosine--

131
00:09:55,570 --> 00:09:58,400
dt prime.

132
00:09:58,400 --> 00:09:59,830
Well, that's very nice.

133
00:09:59,830 --> 00:10:01,630
This is all doable.

134
00:10:01,630 --> 00:10:04,480
The Hfi doesn't
give us any trouble.

135
00:10:04,480 --> 00:10:05,500
It's a constant.

136
00:10:05,500 --> 00:10:07,660
It's out of the integral.

137
00:10:07,660 --> 00:10:10,300
It's all pretty nice and simple.

138
00:10:10,300 --> 00:10:14,840
So we can do these
two integrals.

139
00:10:14,840 --> 00:10:16,790
They're integrals
of exponentials,

140
00:10:16,790 --> 00:10:20,900
so it's just an exponential
divided by those coefficients.

141
00:10:20,900 --> 00:10:27,860
So I'll just do it and
evaluated it between t0 and 0.

142
00:10:27,860 --> 00:10:29,120
So what do we get?

143
00:10:29,120 --> 00:10:37,230
Minus i Hfi prime over
h bar, e to the i omega

144
00:10:37,230 --> 00:10:48,530
fi plus omega t0, minus 1,
over omega fi plus omega.

145
00:10:48,530 --> 00:10:51,440
You can imagine that
e to the i omega

146
00:10:51,440 --> 00:10:55,710
t integrates to e to the
i omega t over omega.

147
00:10:55,710 --> 00:10:58,430
So that's why that works.

148
00:10:58,430 --> 00:11:03,620
And the two limits are t0 and 0.

149
00:11:03,620 --> 00:11:13,540
Plus e to the i omega fi
minus omega t0, minus 1

150
00:11:13,540 --> 00:11:20,080
again, over omega
fi minus omega.

151
00:11:20,080 --> 00:11:22,170
Great.

152
00:11:22,170 --> 00:11:23,470
Our integral is there.

153
00:11:23,470 --> 00:11:24,070
It's done.

154
00:11:24,070 --> 00:11:27,730
And now it's time to
appreciate what it tells us,

155
00:11:27,730 --> 00:11:30,745
because it tells us something
very important, this formula.

156
00:11:41,180 --> 00:11:44,270
So you look at this
and you say, well, OK.

157
00:11:44,270 --> 00:11:50,290
This is the transition
amplitude to state omega--

158
00:11:50,290 --> 00:11:53,810
I'm sorry-- state
f, final state.

159
00:11:53,810 --> 00:11:57,670
And it depends on omega fi.

160
00:11:57,670 --> 00:12:09,150
And omega fi just Ef
minus Ei over h bar.

161
00:12:09,150 --> 00:12:12,830
So if I know my final
discrete state Ef,

162
00:12:12,830 --> 00:12:18,440
I can figure out what is
the transition probability.

163
00:12:18,440 --> 00:12:23,750
Now, these denominators are
intriguing because maybe

164
00:12:23,750 --> 00:12:26,580
if you adjust the
frequency omega--

165
00:12:26,580 --> 00:12:30,770
suppose you have an initial
state and a final state here.

166
00:12:30,770 --> 00:12:35,660
You may adjust the frequency
omega to match them,

167
00:12:35,660 --> 00:12:43,650
and in that case maybe make
the denominators equal to 0.

168
00:12:43,650 --> 00:12:49,960
And that's exactly what
kind of happens here.

169
00:12:49,960 --> 00:12:53,110
So, first of all, if you
look at this expression,

170
00:12:53,110 --> 00:12:59,850
it's a sum of two terms that are
added together and multiplied

171
00:12:59,850 --> 00:13:01,560
by a constant.

172
00:13:01,560 --> 00:13:05,850
As t0 really goes
to 0, completely

173
00:13:05,850 --> 00:13:12,100
goes to 0, t0, each
factor, actually,

174
00:13:12,100 --> 00:13:16,090
if you see the Taylor
expansion of the exponential,

175
00:13:16,090 --> 00:13:19,990
you cancel the 1 and you
then cancel the linear term

176
00:13:19,990 --> 00:13:21,410
with the denominator.

177
00:13:21,410 --> 00:13:24,420
This is just i t0.

178
00:13:24,420 --> 00:13:26,440
And this is also i t0.

179
00:13:26,440 --> 00:13:32,980
So they're comparable as
time is really going to 0.

180
00:13:32,980 --> 00:13:36,670
But time going to 0 is
never of interest for us.

181
00:13:36,670 --> 00:13:40,120
For us, we need to be
the time a little big

182
00:13:40,120 --> 00:13:43,180
already so that
our calculations,

183
00:13:43,180 --> 00:13:50,160
as we did in the constant
transitions that had lobes that

184
00:13:50,160 --> 00:13:54,730
decreases constants over t0--

185
00:13:54,730 --> 00:13:57,640
we needed the time to
be sufficiently large

186
00:13:57,640 --> 00:14:00,400
so that the lobes are narrow.

187
00:14:00,400 --> 00:14:01,870
And that, we could guarantee.

188
00:14:01,870 --> 00:14:05,180
So t0 going to 0 is
not very interesting.

189
00:14:05,180 --> 00:14:08,200
We need t0 a little bigger.

190
00:14:08,200 --> 00:14:12,550
Not too big that the
rate of a process

191
00:14:12,550 --> 00:14:15,640
overwhelms the probability.

192
00:14:15,640 --> 00:14:17,800
But we need a little big.

193
00:14:17,800 --> 00:14:20,610
So, in that case,
the numerators are

194
00:14:20,610 --> 00:14:22,450
going to be bounded numbers.

195
00:14:22,450 --> 00:14:25,580
You see, you have an
exponential minus 1.

196
00:14:25,580 --> 00:14:28,630
So that varies from--

197
00:14:28,630 --> 00:14:34,570
the magnitude of this thing
varies from 2 to 0, basically.

198
00:14:34,570 --> 00:14:36,400
In fact, in these
numerators, you

199
00:14:36,400 --> 00:14:42,680
can see, if the phase is 0
for some particular value--

200
00:14:42,680 --> 00:14:47,020
if the exponential has a phase
that's proportional to 2 pi,

201
00:14:47,020 --> 00:14:48,090
this is 0.

202
00:14:48,090 --> 00:14:50,500
And then sometimes this
exponential is minus 1,

203
00:14:50,500 --> 00:14:52,820
so it gets to minus 2.

204
00:14:52,820 --> 00:14:54,610
So it's finite.

205
00:14:54,610 --> 00:14:56,890
And the same is here.

206
00:14:56,890 --> 00:14:59,710
So these are bounded numerators.

207
00:14:59,710 --> 00:15:03,040
On the other hand, you
may have the possibility

208
00:15:03,040 --> 00:15:06,400
that these things become 0.

209
00:15:06,400 --> 00:15:09,460
And those are the cases
that are of interest to us,

210
00:15:09,460 --> 00:15:13,500
the cases when those
terms are going to be 0.