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ROBERT FIELD: This
lecture is not

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00:00:24,550 --> 00:00:28,060
relevant to this
exam or any exam.

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00:00:28,060 --> 00:00:33,670
It's time-dependent quantum
mechanics, which you probably

11
00:00:33,670 --> 00:00:37,060
want to know about,
but it's a lot

12
00:00:37,060 --> 00:00:39,700
to digest at the
level of this course.

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00:00:39,700 --> 00:00:47,530
So I'm going to introduce a lot
of the tricks and terminology,

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00:00:47,530 --> 00:00:50,410
and I hope that some of
you will care about that

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00:00:50,410 --> 00:00:52,870
and will go on to use this.

16
00:00:52,870 --> 00:00:57,020
But mostly, this is
a first exposure,

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00:00:57,020 --> 00:00:59,520
and there's a lot of derivation.

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00:00:59,520 --> 00:01:02,510
And it's hard to see the
forest for the trees.

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00:01:02,510 --> 00:01:05,379
OK, so these are
the important things

20
00:01:05,379 --> 00:01:07,390
that I'm going to
cover in the lecture.

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00:01:07,390 --> 00:01:10,540
First, the dipole
approximation--

22
00:01:10,540 --> 00:01:14,950
how can we simplify the
interaction between molecules

23
00:01:14,950 --> 00:01:17,090
and electromagnetic radiation?

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00:01:17,090 --> 00:01:20,050
This is the main
simplification, and I'll

25
00:01:20,050 --> 00:01:21,970
explain where it comes from.

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00:01:21,970 --> 00:01:24,490
Then we have
transitions that occur.

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00:01:24,490 --> 00:01:28,000
And they're caused by a
time-dependent perturbation

28
00:01:28,000 --> 00:01:36,190
where the zero-order
Hamiltonian is time-independent,

29
00:01:36,190 --> 00:01:39,169
but the perturbation
term is time-dependent.

30
00:01:39,169 --> 00:01:40,210
And what does that cause?

31
00:01:40,210 --> 00:01:43,120
It causes transitions.

32
00:01:43,120 --> 00:01:45,700
We're going to express
the problem in terms

33
00:01:45,700 --> 00:01:49,510
of the eigenstates of the
time-independent Hamiltonian,

34
00:01:49,510 --> 00:01:52,750
the zero-order Hamiltonian,
and we know that these always

35
00:01:52,750 --> 00:01:55,600
have this time-dependent
factor if we're

36
00:01:55,600 --> 00:01:59,100
doing time-dependent
quantum mechanics.

37
00:01:59,100 --> 00:02:00,810
The two crucial
approximations are

38
00:02:00,810 --> 00:02:06,030
going to be the electromagnetic
field is weak and continuous.

39
00:02:06,030 --> 00:02:09,389
Now many experiments involve
short pulses and very intense

40
00:02:09,389 --> 00:02:13,260
pulses, and the time-dependent
quantum mechanics

41
00:02:13,260 --> 00:02:16,170
for those problems is
completely different,

42
00:02:16,170 --> 00:02:18,480
but you need to
understand this in order

43
00:02:18,480 --> 00:02:22,440
to understand what's
different about it.

44
00:02:22,440 --> 00:02:24,270
We also assume
that we're starting

45
00:02:24,270 --> 00:02:26,250
the system in a
single eigenstate,

46
00:02:26,250 --> 00:02:28,680
and that's pretty normal.

47
00:02:28,680 --> 00:02:33,030
But often, you're starting
the system in many eigenstates

48
00:02:33,030 --> 00:02:34,080
that are uncorrelated.

49
00:02:34,080 --> 00:02:35,350
We don't talk about that.

50
00:02:35,350 --> 00:02:38,100
That's something that has to do
with the density matrix, which

51
00:02:38,100 --> 00:02:41,560
is beyond the level
of this course.

52
00:02:41,560 --> 00:02:43,350
And one of the
things that happens

53
00:02:43,350 --> 00:02:47,700
is we get this thing
called linear response.

54
00:02:47,700 --> 00:02:51,540
Now I went for years
hearing the reverence

55
00:02:51,540 --> 00:02:55,150
that people apply
to linear response,

56
00:02:55,150 --> 00:02:58,020
but I hadn't a clue what it was.

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00:02:58,020 --> 00:03:00,270
So you can start out
knowing something

58
00:03:00,270 --> 00:03:02,220
about linear response.

59
00:03:02,220 --> 00:03:05,770
Now this all leads up
to Fermi's golden rule,

60
00:03:05,770 --> 00:03:10,050
which explains the rate
at which transitions

61
00:03:10,050 --> 00:03:13,690
occur between some initial
state and some final state.

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00:03:13,690 --> 00:03:15,420
And there is a lot
more complexity

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00:03:15,420 --> 00:03:18,130
in Fermi's golden rule than
what I'm going to present,

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00:03:18,130 --> 00:03:23,670
but this is the first
step in understanding it.

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00:03:23,670 --> 00:03:26,920
Then I'm going to
talk about where

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00:03:26,920 --> 00:03:30,460
do pure rotation transitions
come from and vibrational

67
00:03:30,460 --> 00:03:31,190
transitions.

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00:03:31,190 --> 00:03:33,460
Then at the end,
I'll show a movie

69
00:03:33,460 --> 00:03:37,420
which gives you a
sense of what goes on

70
00:03:37,420 --> 00:03:42,220
in making a transition
be strong and sharp.

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00:03:45,020 --> 00:03:49,520
OK, I'm a spectroscopist,
and I use spectroscopy

72
00:03:49,520 --> 00:03:53,840
to learn all sorts of
secrets that molecules keep.

73
00:03:53,840 --> 00:03:56,690
And in order to do
that, I need to record

74
00:03:56,690 --> 00:04:01,190
a spectrum, which basically is
you have some radiation source.

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00:04:01,190 --> 00:04:05,450
And you tune its frequency,
and things happen.

76
00:04:05,450 --> 00:04:08,610
And why do the things happen?

77
00:04:08,610 --> 00:04:12,860
How do we understand
the interaction

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00:04:12,860 --> 00:04:16,990
of electromagnetic
radiation and a molecule?

79
00:04:16,990 --> 00:04:19,010
And there's really two
ways to understand it.

80
00:04:26,150 --> 00:04:42,020
We have one-way molecules as
targets, photons as bullets,

81
00:04:42,020 --> 00:04:44,580
and it's a simple
geometric picture.

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00:04:44,580 --> 00:04:47,600
And the size of the target
is related to the transition

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00:04:47,600 --> 00:04:50,820
moments, and it works.

84
00:04:50,820 --> 00:04:52,170
It's very, very simple.

85
00:04:52,170 --> 00:04:54,410
There's no time-dependent
quantum mechanics.

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00:04:54,410 --> 00:04:56,270
It's probabilistic.

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00:04:56,270 --> 00:05:01,200
And for the first 45
years of my career,

88
00:05:01,200 --> 00:05:07,130
this is the way I handled an
understanding of transitions

89
00:05:07,130 --> 00:05:09,597
caused by electromagnetic
radiation.

90
00:05:09,597 --> 00:05:10,180
This is wrong.

91
00:05:12,910 --> 00:05:15,170
It has a wide applicability.

92
00:05:15,170 --> 00:05:18,170
But if you try to
take it too seriously,

93
00:05:18,170 --> 00:05:20,320
you will miss a
lot of good stuff.

94
00:05:24,030 --> 00:05:28,970
The other way is to
use the time-dependent

95
00:05:28,970 --> 00:05:34,530
in your equation, and
it looks complicated

96
00:05:34,530 --> 00:05:37,196
because we're going
to be combining

97
00:05:37,196 --> 00:05:38,820
the time-dependent
Schrodinger equation

98
00:05:38,820 --> 00:05:41,217
and the time-independent
Schrodinger equation.

99
00:05:41,217 --> 00:05:43,800
We're going to be thinking about
the electromagnetic radiation

100
00:05:43,800 --> 00:05:47,730
as waves rather than
photons, and that

101
00:05:47,730 --> 00:05:50,250
means there is constructive
and destructive interference.

102
00:05:50,250 --> 00:05:52,680
There's phase
information, which is not

103
00:05:52,680 --> 00:05:55,140
present in the
molecules-as-targets,

104
00:05:55,140 --> 00:05:57,610
photons-as-bullets picture.

105
00:05:57,610 --> 00:05:59,960
Now I don't want you
to say, well, I'm

106
00:05:59,960 --> 00:06:01,710
never going to think
this way because it's

107
00:06:01,710 --> 00:06:04,430
so easy to think about trends.

108
00:06:04,430 --> 00:06:07,080
And, you know, the Beer-Lambert
law, all these things that you

109
00:06:07,080 --> 00:06:12,930
use to describe the probability
of an absorption or emission

110
00:06:12,930 --> 00:06:18,190
transition, this sort of
thing is really useful.

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00:06:18,190 --> 00:06:31,540
OK, so this is the right
way, and the crucial step

112
00:06:31,540 --> 00:06:33,430
is the dipole approximation.

113
00:06:36,370 --> 00:06:47,140
So we have
electromagnetic radiation

114
00:06:47,140 --> 00:06:51,920
being a combination of electric
field and magnetic field,

115
00:06:51,920 --> 00:06:57,250
and we can describe the
electric field in terms of--

116
00:07:11,571 --> 00:07:12,070
OK.

117
00:07:14,710 --> 00:07:21,230
So this is a vector, and it's
a function of a vector in time.

118
00:07:21,230 --> 00:07:24,760
And there is some magnitude,
which is a vector.

119
00:07:24,760 --> 00:07:27,430
And its cosine of
this thing, this

120
00:07:27,430 --> 00:07:34,900
is the wave vector, which
is 2 pi over the wavelength,

121
00:07:34,900 --> 00:07:37,960
but it also has a direction.

122
00:07:37,960 --> 00:07:40,030
And it points in the
direction that the radiation

123
00:07:40,030 --> 00:07:42,360
is propagating.

124
00:07:42,360 --> 00:07:46,390
And this is the
position coordinate,

125
00:07:46,390 --> 00:07:47,720
and this is the frequency.

126
00:07:47,720 --> 00:07:54,150
So there's a similar expression
for the magnetic part--

127
00:07:59,130 --> 00:08:00,126
same thing.

128
00:08:07,120 --> 00:08:11,090
I should leave this exposed.

129
00:08:11,090 --> 00:08:13,910
So we know several things.

130
00:08:13,910 --> 00:08:17,060
We know for
electromagnetic radiation

131
00:08:17,060 --> 00:08:21,500
that the electric
field is always

132
00:08:21,500 --> 00:08:25,010
perpendicular to
the magnetic field.

133
00:08:25,010 --> 00:08:33,580
We know a relationship
between the constant term,

134
00:08:33,580 --> 00:08:38,090
and we have this k vector,
which points in the propagation

135
00:08:38,090 --> 00:08:40,659
direction.

136
00:08:40,659 --> 00:08:41,799
Now the question is--

137
00:08:50,710 --> 00:08:54,250
because we have--
we have a molecule

138
00:08:54,250 --> 00:08:56,530
and we have the
electromagnetic radiation.

139
00:08:59,910 --> 00:09:04,700
And so the question is, what's
a typical size for a molecule

140
00:09:04,700 --> 00:09:05,510
in the gas field?

141
00:09:05,510 --> 00:09:06,820
Well, anywhere?

142
00:09:06,820 --> 00:09:08,040
Pick a number.

143
00:09:08,040 --> 00:09:11,250
How big is a molecule?

144
00:09:11,250 --> 00:09:13,664
STUDENT: A couple angstroms?

145
00:09:13,664 --> 00:09:15,330
ROBERT FIELD: I like
a couple angstroms.

146
00:09:15,330 --> 00:09:17,010
That's a diatomic molecule.

147
00:09:17,010 --> 00:09:19,080
They're going to be
people who like proteins,

148
00:09:19,080 --> 00:09:22,560
and they're going to talk
about 10 or 100 nanometers.

149
00:09:22,560 --> 00:09:28,140
But typically, you can say
1 nanometer or 2 angstroms

150
00:09:28,140 --> 00:09:30,960
or something like that.

151
00:09:30,960 --> 00:09:33,940
Now we're going to shine
light at a molecule.

152
00:09:33,940 --> 00:09:37,550
What's the typical
wavelength of light that we

153
00:09:37,550 --> 00:09:41,740
use to record a spectrum?

154
00:09:41,740 --> 00:09:45,020
Visible wavelength.

155
00:09:45,020 --> 00:09:46,950
What is that?

156
00:09:46,950 --> 00:09:48,950
STUDENT: 400 to 700 nanometers?

157
00:09:48,950 --> 00:09:52,340
ROBERT FIELD: Yeah, so
the wavelength of light

158
00:09:52,340 --> 00:09:55,850
is on the order of,
say, 500 nanometers.

159
00:09:55,850 --> 00:10:04,130
And if it's in the infrared,
it might be 10,000 nanometers.

160
00:10:04,130 --> 00:10:07,100
If it were in the
visible, it might be 100--

161
00:10:07,100 --> 00:10:10,550
in the ultraviolet, it might
be as short as 100 nanometers.

162
00:10:10,550 --> 00:10:15,170
But the point is that this
wavelength is much, much larger

163
00:10:15,170 --> 00:10:18,470
than the size of a molecule,
so this picture here

164
00:10:18,470 --> 00:10:19,400
is complete garbage.

165
00:10:22,290 --> 00:10:27,550
The picture for the ratio--

166
00:10:27,550 --> 00:10:30,330
so even this is garbage.

167
00:10:30,330 --> 00:10:32,250
The electric field
or the magnetic field

168
00:10:32,250 --> 00:10:36,090
that the molecule sees is
constant over the length

169
00:10:36,090 --> 00:10:43,750
of the molecule to a
very good approximation.

170
00:10:43,750 --> 00:10:52,580
So now we have this
expression for the field,

171
00:10:52,580 --> 00:10:57,640
and this is a number which
is a very, very small number

172
00:10:57,640 --> 00:11:01,480
times a still
pretty small number.

173
00:11:01,480 --> 00:11:05,500
This k dot r is very small.

174
00:11:05,500 --> 00:11:11,750
It says we can expand
this in a power series

175
00:11:11,750 --> 00:11:16,520
and throw away everything
except the omega t.

176
00:11:16,520 --> 00:11:19,840
That's the dipole approximation.

177
00:11:19,840 --> 00:11:28,000
So all of a sudden, we
have as our electric field

178
00:11:28,000 --> 00:11:32,290
just E0 cosine omega t.

179
00:11:34,970 --> 00:11:37,430
That's fantastic.

180
00:11:37,430 --> 00:11:41,630
So we've gotten rid of the
spatial degree of freedom,

181
00:11:41,630 --> 00:11:44,430
and that enables us to
do all sorts of things

182
00:11:44,430 --> 00:11:47,700
that would have required
a lot more justification.

183
00:11:47,700 --> 00:11:54,320
Now sometimes we need to
keep higher order terms

184
00:11:54,320 --> 00:11:55,070
in this expansion.

185
00:11:55,070 --> 00:11:57,920
We've kept none of them,
just the zero order term.

186
00:11:57,920 --> 00:12:03,350
And so if we do, that's called
quadrupole or octopole or

187
00:12:03,350 --> 00:12:06,380
hexadecapole, and
there are transitions

188
00:12:06,380 --> 00:12:10,520
that are not dipole allowed
but are quadrupole allowed.

189
00:12:10,520 --> 00:12:14,120
And they're incredibly
weak because k

190
00:12:14,120 --> 00:12:16,100
dot r is really, really small.

191
00:12:19,520 --> 00:12:25,140
Now the intensity of
quadrupole-allowed transitions

192
00:12:25,140 --> 00:12:30,230
is on the order of a million
times smaller than dipole.

193
00:12:30,230 --> 00:12:31,570
So why go there?

194
00:12:31,570 --> 00:12:35,890
Well, sometimes the dipole
transitions are forbidden.

195
00:12:35,890 --> 00:12:38,550
And so if you're going to get
the molecule to talk to you,

196
00:12:38,550 --> 00:12:41,120
you're going to have
to somehow make use

197
00:12:41,120 --> 00:12:43,010
of the quadrupole transitions.

198
00:12:43,010 --> 00:12:45,770
But it's a completely
different kind of experiment

199
00:12:45,770 --> 00:12:48,290
because you have to have
an incredibly long path

200
00:12:48,290 --> 00:12:51,720
length and a relatively
high number density.

201
00:12:51,720 --> 00:12:54,170
And so you don't
want to go there,

202
00:12:54,170 --> 00:12:56,270
and that's something
that's beside--

203
00:12:56,270 --> 00:12:58,800
aside from what we care about.

204
00:12:58,800 --> 00:13:05,420
So now many of you are going to
be doing experiments involving

205
00:13:05,420 --> 00:13:13,280
light, and that will
involve the electric field.

206
00:13:13,280 --> 00:13:16,830
Some of you will be
doing magnetic resonance,

207
00:13:16,830 --> 00:13:19,680
and they will be
thinking entirely

208
00:13:19,680 --> 00:13:21,450
about the magnetic field.

209
00:13:24,690 --> 00:13:26,560
The theory is the same.

210
00:13:26,560 --> 00:13:32,330
It's just the main actor
is a little bit different.

211
00:13:32,330 --> 00:13:36,740
Now if we're dealing
with an electric field,

212
00:13:36,740 --> 00:13:40,670
we are interested
in the symmetry

213
00:13:40,670 --> 00:13:45,410
of this operator, which is
the electric field dotted

214
00:13:45,410 --> 00:13:51,390
into the molecular
dipole moment,

215
00:13:51,390 --> 00:13:56,610
and that operator
has odd parity.

216
00:13:56,610 --> 00:14:00,000
And so now I'm not going
to tell you what parity is.

217
00:14:00,000 --> 00:14:03,150
But because this has
odd parity, there

218
00:14:03,150 --> 00:14:06,840
are only transitions between
states of opposite parity,

219
00:14:06,840 --> 00:14:12,180
whereas this, the magnetic
operator, has even parity.

220
00:14:12,180 --> 00:14:15,000
And so they only have
transitions between states

221
00:14:15,000 --> 00:14:16,410
of the same parity.

222
00:14:16,410 --> 00:14:18,710
Now you want to be curious
about what parity is,

223
00:14:18,710 --> 00:14:20,250
and I'm not going to tell you.

224
00:14:20,250 --> 00:14:26,160
OK, so the problem
is tremendously

225
00:14:26,160 --> 00:14:29,280
simplified by the
fact that now we just

226
00:14:29,280 --> 00:14:33,900
have a time-dependent
field, which

227
00:14:33,900 --> 00:14:36,490
is constant over the molecule.

228
00:14:36,490 --> 00:14:40,110
So the molecule is seeing
an oscillatory field,

229
00:14:40,110 --> 00:14:43,140
but the whole molecule is
feeling that same field.

230
00:14:46,870 --> 00:14:50,010
OK, now we're ready to start
doing quantum mechanics.

231
00:14:57,540 --> 00:15:02,040
So the interaction term,
the thing that causes

232
00:15:02,040 --> 00:15:03,330
transitions to occur--

233
00:15:05,970 --> 00:15:11,570
the electric interaction term,
which we're going to call

234
00:15:11,570 --> 00:15:15,720
H1 because it's a perturbation.

235
00:15:15,720 --> 00:15:18,480
We're going to be doing
something in perturbation

236
00:15:18,480 --> 00:15:21,550
theory, but it's time-dependent
perturbation theory,

237
00:15:21,550 --> 00:15:24,300
which is a whole lot
more complicated and rich

238
00:15:24,300 --> 00:15:26,040
than ordinary time-independent.

239
00:15:26,040 --> 00:15:31,500
Now many of you have found
time-independent perturbation

240
00:15:31,500 --> 00:15:36,270
theory tedious and
algebraically complicated.

241
00:15:36,270 --> 00:15:37,770
Time-dependent
perturbation theory

242
00:15:37,770 --> 00:15:41,460
for these kinds of
operators is not tedious.

243
00:15:41,460 --> 00:15:43,050
It's really beautiful.

244
00:15:43,050 --> 00:15:45,510
And there are many, many cases.

245
00:15:45,510 --> 00:15:47,580
It's not just having
another variable.

246
00:15:47,580 --> 00:15:50,550
There's a lot of
really neat stuff.

247
00:15:50,550 --> 00:15:55,290
And what I'm going to present
today or I am presenting today

248
00:15:55,290 --> 00:16:00,180
is the theory for CW radiation--
that's continuous radiation--

249
00:16:00,180 --> 00:16:05,430
really weak, interacting
with a molecule or a system

250
00:16:05,430 --> 00:16:09,190
in a single quantum
state initially.

251
00:16:09,190 --> 00:16:11,560
And it's important.

252
00:16:11,560 --> 00:16:16,850
The really weak and the CW are
two really important features.

253
00:16:16,850 --> 00:16:20,030
And the single quantum
state is just a convenience.

254
00:16:20,030 --> 00:16:20,990
We can deal with that.

255
00:16:20,990 --> 00:16:22,930
That's not a big
deal, but it does

256
00:16:22,930 --> 00:16:29,050
involve using a different, more
physical, or a more correct

257
00:16:29,050 --> 00:16:32,050
definition of what we mean
by an average measurement

258
00:16:32,050 --> 00:16:35,240
on a system of many particles.

259
00:16:35,240 --> 00:16:38,170
And you'll hear the word
"density matrix" if you go on

260
00:16:38,170 --> 00:16:39,421
in physical chemistry.

261
00:16:39,421 --> 00:16:41,170
But I'm not going to
do anything about it,

262
00:16:41,170 --> 00:16:43,400
but that's how we deal with it.

263
00:16:43,400 --> 00:16:53,230
OK, so this is going
to be minus mu--

264
00:16:53,230 --> 00:16:59,720
it's a vector-- dot E of
t, which is also a vector.

265
00:16:59,720 --> 00:17:02,720
Now a dot product,
that looks really neat.

266
00:17:02,720 --> 00:17:06,500
However, this is a vector
in the molecular frame,

267
00:17:06,500 --> 00:17:09,150
and this is a vector in
the laboratory frame.

268
00:17:09,150 --> 00:17:12,530
So this dot product is a
whole bunch more complicated

269
00:17:12,530 --> 00:17:15,720
than you would think.

270
00:17:15,720 --> 00:17:17,910
Now I do want to
mention that when

271
00:17:17,910 --> 00:17:22,680
we talk about the rigid rotor,
the rigid rotor is telling

272
00:17:22,680 --> 00:17:26,550
us what is the probability
amplitude of the orientation

273
00:17:26,550 --> 00:17:30,420
of the molecular frame relative
to the laboratory frame.

274
00:17:30,420 --> 00:17:34,710
So that is where all this
information about these two

275
00:17:34,710 --> 00:17:38,980
different coordinate
systems reside,

276
00:17:38,980 --> 00:17:42,330
and we'll see a
little bit of that.

277
00:17:42,330 --> 00:17:49,012
OK, there's a similar expression
for the magnetic term.

278
00:17:49,012 --> 00:17:50,470
I'm just not going
to write it down

279
00:17:50,470 --> 00:17:53,290
because it's just too
much stuff to write down.

280
00:17:53,290 --> 00:17:57,580
So the Hamiltonian, the
time-independent Hamiltonian,

281
00:17:57,580 --> 00:18:04,200
can be expressed
as H0 plus H1 of t.

282
00:18:07,190 --> 00:18:10,460
This looks exactly like
time-independent perturbation

283
00:18:10,460 --> 00:18:14,420
theory, except this guy, which
makes all the complications is

284
00:18:14,420 --> 00:18:17,990
time dependent.

285
00:18:17,990 --> 00:18:21,690
But this says, OK, we
can find a whole set,

286
00:18:21,690 --> 00:18:26,280
a complete set of eignenenergies
and eigenfunctions.

287
00:18:26,280 --> 00:18:29,892
And we know how to write the
time-dependent Schrodinger--

288
00:18:29,892 --> 00:18:31,850
the solutions of the
time-dependent Schrodinger

289
00:18:31,850 --> 00:18:34,790
equation if this
is the whole game.

290
00:18:34,790 --> 00:18:37,250
So we're going to use
these as basis functions

291
00:18:37,250 --> 00:18:40,340
just as we did in ordinary
perturbation theory.

292
00:18:45,697 --> 00:18:55,220
So H0 times some
eigenfunction, which now I'm

293
00:18:55,220 --> 00:19:02,493
writing as explicitly
time-dependent is En phi n t

294
00:19:02,493 --> 00:19:08,180
equals 0 e to the minus
i En t over h-bar.

295
00:19:08,180 --> 00:19:09,890
So this is a solution.

296
00:19:09,890 --> 00:19:13,201
This thing is a solution to
the time-dependent Schrodinger

297
00:19:13,201 --> 00:19:13,700
equation.

298
00:19:18,870 --> 00:19:23,930
And so when the
external field is off,

299
00:19:23,930 --> 00:19:28,450
then the only states
that we consider

300
00:19:28,450 --> 00:19:31,840
are eigenstates of the
zero-order Hamiltonian,

301
00:19:31,840 --> 00:19:35,260
and they can be time dependent.

302
00:19:35,260 --> 00:19:49,570
But if we write psi n star
of t times psi n of t,

303
00:19:49,570 --> 00:19:54,380
well, that's not time dependent
if this is an eigenstate.

304
00:19:54,380 --> 00:19:58,220
So the only way we
get time dependence

305
00:19:58,220 --> 00:20:01,760
is by having this time-dependent
perturbation term.

306
00:20:06,050 --> 00:20:13,820
OK, so let's take
some initial state.

307
00:20:13,820 --> 00:20:20,870
And let us call that initial
state some arbitrary state.

308
00:20:20,870 --> 00:20:24,830
And we can always write
this as a superposition

309
00:20:24,830 --> 00:20:30,455
of zero-order states.

310
00:20:37,270 --> 00:20:45,260
OK, and now, unfortunately,
both the coefficients

311
00:20:45,260 --> 00:20:48,020
in this linear combination
and the functions

312
00:20:48,020 --> 00:20:49,261
are time dependent.

313
00:20:51,910 --> 00:20:54,480
So this means when we're
going to be applying

314
00:20:54,480 --> 00:20:56,400
the time-dependent
Schrodinger equation,

315
00:20:56,400 --> 00:20:59,500
we take a partial derivative
with respect to t,

316
00:20:59,500 --> 00:21:03,310
we get derivatives
with this and this.

317
00:21:03,310 --> 00:21:05,260
So it's an extra
level of complexity,

318
00:21:05,260 --> 00:21:08,130
but we can deal
with it, because one

319
00:21:08,130 --> 00:21:10,560
of the things that we
keep coming back to

320
00:21:10,560 --> 00:21:14,920
is that everything we
talk about is expressed

321
00:21:14,920 --> 00:21:19,290
as a linear combination of
t equals zero eigenstates

322
00:21:19,290 --> 00:21:21,489
of the zero-order Hamiltonian.

323
00:21:26,540 --> 00:21:28,590
OK, so the time-dependent
Schrodinger equation--

324
00:21:28,590 --> 00:21:34,120
i h-bar partial with
respect to t of the--

325
00:21:38,800 --> 00:21:42,120
yeah, of the wave
function is equal to--

326
00:21:56,600 --> 00:22:00,570
OK, that's our friend
or our new friend

327
00:22:00,570 --> 00:22:04,600
because the old
friend was too simple.

328
00:22:04,600 --> 00:22:10,130
And so, well, we can represent
this partial derivative

329
00:22:10,130 --> 00:22:15,210
just using dots because
the equations I'm

330
00:22:15,210 --> 00:22:18,000
going to be putting on
the board are hideous,

331
00:22:18,000 --> 00:22:23,940
and so we want to use
every abbreviation we can.

332
00:22:23,940 --> 00:22:27,960
This is written as a product
of time-dependent coefficients

333
00:22:27,960 --> 00:22:29,670
and time-dependent functions.

334
00:22:29,670 --> 00:22:32,580
When we apply the
derivative to it,

335
00:22:32,580 --> 00:22:36,580
we're going to get
derivatives of each.

336
00:22:36,580 --> 00:23:03,205
And so that's the
left-hand side.

337
00:23:07,630 --> 00:23:10,600
OK, and let's look at this
left-hand side for a minute.

338
00:23:15,760 --> 00:23:18,230
OK, so we've got something
that we don't really

339
00:23:18,230 --> 00:23:22,671
know what to do with, but this
guy, we know that this is--

340
00:23:22,671 --> 00:23:26,750
this time-dependent
wave function

341
00:23:26,750 --> 00:23:30,860
is something that we can use
the time-dependent Schrodinger

342
00:23:30,860 --> 00:23:33,860
equation on and get
a simplification.

343
00:23:33,860 --> 00:23:34,819
So the left-hand side--

344
00:23:34,819 --> 00:23:36,401
I haven't written
the right-hand side.

345
00:23:36,401 --> 00:23:38,090
I'm just working on
the left-hand side

346
00:23:38,090 --> 00:23:40,890
of what we get when we start
to write this equation.

347
00:23:40,890 --> 00:23:53,230
And what we get is we know that
the time dependence of this

348
00:23:53,230 --> 00:24:01,570
is equal to 1 over i h-bar
times the Hamiltonian operating

349
00:24:01,570 --> 00:24:04,790
on phi n.

350
00:24:07,736 --> 00:24:10,620
Is that what I want?

351
00:24:10,620 --> 00:24:12,310
I can't read my
notes so I have to--

352
00:24:12,310 --> 00:24:14,350
I have to be--

353
00:24:14,350 --> 00:24:16,930
yeah, so we've just taken
that 1 over i h-bar.

354
00:24:20,740 --> 00:24:25,150
This is going to be the
time-independent Hamiltonian,

355
00:24:25,150 --> 00:24:26,370
the zero-order Hamiltonian.

356
00:24:26,370 --> 00:24:27,970
And we know what we get here.

357
00:24:33,000 --> 00:24:33,500
Yes?

358
00:24:33,500 --> 00:24:35,060
STUDENT: So all
your phi n's, those

359
00:24:35,060 --> 00:24:36,642
are the zero-order solutions?

360
00:24:36,642 --> 00:24:37,850
ROBERT FIELD: That's correct.

361
00:24:37,850 --> 00:24:39,475
STUDENT: So they're
unperturbed states?

362
00:24:39,475 --> 00:24:43,370
ROBERT FIELD: They're
unperturbed eigenstates of H0.

363
00:24:43,370 --> 00:24:50,210
And if it's psi n of t, it
has the e to the i En of t--

364
00:24:52,940 --> 00:24:56,510
En t over h-bar factor
implicit, and we're

365
00:24:56,510 --> 00:24:58,700
going to be using that.

366
00:24:58,700 --> 00:25:06,910
All right, so what
we get when we

367
00:25:06,910 --> 00:25:09,400
take that partial derivative,
we get a simplification.

368
00:25:19,510 --> 00:25:22,120
OK, let me just write
the right-hand side

369
00:25:22,120 --> 00:25:23,140
of this equation too.

370
00:25:23,140 --> 00:25:30,140
So we have the simplified
left-hand side,

371
00:25:30,140 --> 00:25:34,210
which is psi n c--

372
00:25:34,210 --> 00:25:37,270
I've never lectured on
time-dependent perturbation

373
00:25:37,270 --> 00:25:38,650
theory before.

374
00:25:38,650 --> 00:25:41,740
And so although I
think I understand it,

375
00:25:41,740 --> 00:25:47,140
it's not as available in
core as it ought to be.

376
00:25:47,140 --> 00:25:52,030
OK, so we have this minus--

377
00:25:56,829 --> 00:25:58,120
where did the wave function go?

378
00:26:04,630 --> 00:26:07,930
Well, there's got
to be a phi in here

379
00:26:07,930 --> 00:26:21,380
and then minus i over h-bar En
cn t over the times phi n of t.

380
00:26:21,380 --> 00:26:30,010
That's the left-hand
side in the bracket here.

381
00:26:30,010 --> 00:26:33,050
OK, and the right-hand side
of the original equation,

382
00:26:33,050 --> 00:26:56,940
that is just some n cn t
En plus H1 of t phi n t.

383
00:26:56,940 --> 00:27:01,420
OK, it takes a
little imagination,

384
00:27:01,420 --> 00:27:05,260
but this and the
terms associated

385
00:27:05,260 --> 00:27:07,360
with that are the same.

386
00:27:07,360 --> 00:27:10,330
This happened when we did
non-degenerate perturbation

387
00:27:10,330 --> 00:27:11,110
theory.

388
00:27:11,110 --> 00:27:14,320
We looked at the
lambdas of one equation.

389
00:27:14,320 --> 00:27:18,910
There was a cancellation
of two ugly terms.

390
00:27:18,910 --> 00:27:21,820
And so what ends
up happening is we

391
00:27:21,820 --> 00:27:26,720
get a tremendous
simplification of the problem.

392
00:27:26,720 --> 00:27:49,530
And so the left-hand side of
the equation has the form,

393
00:27:49,530 --> 00:27:53,640
and the right-hand side has
the form over here without

394
00:27:53,640 --> 00:27:54,680
the extra term--

395
00:27:54,680 --> 00:28:09,820
sum over n, cn of t
H1 of t psi n of t.

396
00:28:18,640 --> 00:28:23,337
OK, and now we
have this equation.

397
00:28:23,337 --> 00:28:24,920
We have this simple
thing here, and we

398
00:28:24,920 --> 00:28:27,110
have this ugly thing here.

399
00:28:27,110 --> 00:28:40,160
And we want to simplify this
by multiplying on the left

400
00:28:40,160 --> 00:28:43,230
by psi F of t so--

401
00:28:46,520 --> 00:28:48,206
and integrating
with respect to tau.

402
00:28:51,860 --> 00:28:54,750
F is for final.

403
00:28:54,750 --> 00:28:56,820
So we're interested
in the transition

404
00:28:56,820 --> 00:28:59,560
from some initial state
to some final state.

405
00:28:59,560 --> 00:29:02,270
So we're going to massage this.

406
00:29:02,270 --> 00:29:05,340
And when we do that, we get--

407
00:29:09,690 --> 00:29:12,510
I've clearly skipped a step,
but it doesn't matter--

408
00:29:12,510 --> 00:29:26,790
i h-bar cf dot of t is equal
to this integral sum c n of t

409
00:29:26,790 --> 00:29:30,060
integral cf of--

410
00:29:30,060 --> 00:29:42,820
phi f of t H1 f of
t phi n of t, e tau.

411
00:29:46,170 --> 00:29:50,940
This is a very
important equation

412
00:29:50,940 --> 00:29:54,240
because we have a simple
derivative of the coefficient

413
00:29:54,240 --> 00:29:57,220
that we want, and it's
expressed as an integral.

414
00:29:57,220 --> 00:30:00,940
And we have an integral
between an eigenstate

415
00:30:00,940 --> 00:30:04,440
of the zero-order Hamiltonian
and another eigenstate.

416
00:30:04,440 --> 00:30:17,430
And this is just H1 f n of t.

417
00:30:17,430 --> 00:30:21,715
OK, so we have these guys.

418
00:30:27,810 --> 00:30:29,830
So what we want to
know is, all right,

419
00:30:29,830 --> 00:30:33,590
this is the thing that's
making stuff happen.

420
00:30:33,590 --> 00:30:38,010
This is a matrix
element of this term.

421
00:30:38,010 --> 00:30:48,600
Well, H1 of t, which is
equal to v cosine omega t

422
00:30:48,600 --> 00:30:57,810
can be written as v times
1/2 e to the i omega t plus e

423
00:30:57,810 --> 00:31:00,930
to the minus i omega t.

424
00:31:00,930 --> 00:31:04,530
This is really neat
because you notice

425
00:31:04,530 --> 00:31:08,520
we have these complex
oscillating field terms,

426
00:31:08,520 --> 00:31:11,850
and we have on each of
these wave functions

427
00:31:11,850 --> 00:31:14,970
a complex oscillating term.

428
00:31:14,970 --> 00:31:20,700
And what ends up happening
is that we get this equation.

429
00:31:20,700 --> 00:31:27,340
i h-bar cf dot of
t is equal to--

430
00:31:27,340 --> 00:31:29,390
and this is-- you
know, it's ugly.

431
00:31:29,390 --> 00:31:30,060
It gets big.

432
00:31:30,060 --> 00:31:32,040
A lot of stuff
has to be written,

433
00:31:32,040 --> 00:31:34,750
and I have to transfer
from my notes to here.

434
00:31:34,750 --> 00:31:36,720
And then you have to
transfer to your paper.

435
00:31:36,720 --> 00:31:38,670
And there is going to be--

436
00:31:38,670 --> 00:31:40,290
there will be printed
lecture notes.

437
00:31:40,290 --> 00:31:42,840
And in fact, there may actually
be printed lecture notes

438
00:31:42,840 --> 00:31:44,470
for this lecture.

439
00:31:44,470 --> 00:31:47,430
But if they're not,
they will be soon.

440
00:31:47,430 --> 00:31:51,600
OK, and so we get this
differential equation, which

441
00:31:51,600 --> 00:32:03,540
is the sum over n c n of t
integral psi f star of t times

442
00:32:03,540 --> 00:32:10,710
1/2 v, v to the i omega t
plus e to the minus i omega

443
00:32:10,710 --> 00:32:18,530
t times psi n of t, e tau.

444
00:32:21,510 --> 00:32:24,510
Well, these guys
have time dependence,

445
00:32:24,510 --> 00:32:26,780
and so we can put that in.

446
00:32:26,780 --> 00:32:39,210
And now this integral has
the form psi f star 0 1/2 v,

447
00:32:39,210 --> 00:32:54,520
and we have e to the minus
I omega and f minus omega t.

448
00:32:54,520 --> 00:32:58,070
Omega nf, the difference in--

449
00:32:58,070 --> 00:33:05,210
so omega nf is En
minus Ef over h-bar.

450
00:33:08,970 --> 00:33:14,670
And so we have minus this
oscillating term, minus omega

451
00:33:14,670 --> 00:33:24,930
t, and then we have e to the
minus i nft plus omega t.

452
00:33:30,080 --> 00:33:34,620
So here this isn't well, it's
so ugly because of my stupidity

453
00:33:34,620 --> 00:33:35,120
here.

454
00:33:35,120 --> 00:33:38,360
But what we have here
is a resonance integral.

455
00:33:38,360 --> 00:33:42,740
We have something
that's oscillating fast

456
00:33:42,740 --> 00:33:45,880
minus something that's
oscillating fast.

457
00:33:45,880 --> 00:33:48,140
And we have the same
thing plus something

458
00:33:48,140 --> 00:33:50,930
that's oscillating fast.

459
00:33:50,930 --> 00:33:55,550
So those terms are zero because
we have an integral that

460
00:33:55,550 --> 00:33:57,740
is oscillating.

461
00:33:57,740 --> 00:33:58,392
I'm sorry.

462
00:33:58,392 --> 00:34:00,350
It's oscillating between
positive and negative,

463
00:34:00,350 --> 00:34:01,670
positive, negative.

464
00:34:04,880 --> 00:34:10,489
And as long as omega is
different from omega nf,

465
00:34:10,489 --> 00:34:16,040
those integrals are zero because
this integrand, as we integrate

466
00:34:16,040 --> 00:34:20,420
to t equals infinity
or to any time,

467
00:34:20,420 --> 00:34:24,300
is oscillating about
zero, and it's small.

468
00:34:24,300 --> 00:34:30,650
However, if omega is the
same as minus omega nf

469
00:34:30,650 --> 00:34:35,270
or plus omega nf, well, then
this thing is 1 times t.

470
00:34:38,120 --> 00:34:40,400
It gets really big.

471
00:34:40,400 --> 00:34:42,920
Now we're talking about
coefficients, which

472
00:34:42,920 --> 00:34:45,139
are related to probabilities.

473
00:34:45,139 --> 00:34:47,060
And so these coefficients
had better not

474
00:34:47,060 --> 00:34:51,989
go get really big because
probability is always

475
00:34:51,989 --> 00:34:54,980
going to be less than 1.

476
00:34:54,980 --> 00:34:57,200
OK, so what we're
going to do now

477
00:34:57,200 --> 00:35:00,410
is collect the rubble in
a form that it turns out

478
00:35:00,410 --> 00:35:01,380
to be really useful.

479
00:35:13,510 --> 00:35:17,310
So we have an
equation for the time

480
00:35:17,310 --> 00:35:23,580
dependence of a final state, and
it's expressed as a sum over n.

481
00:35:23,580 --> 00:35:27,570
But if we say, oh, let's
make our initial state

482
00:35:27,570 --> 00:35:30,850
just one of those.

483
00:35:30,850 --> 00:35:33,690
So our initial state is--

484
00:35:33,690 --> 00:35:37,990
let's call it ci.

485
00:35:37,990 --> 00:35:41,380
And we say, well,
the system is not

486
00:35:41,380 --> 00:35:47,540
in any other state other than
the i state, and this is weak.

487
00:35:47,540 --> 00:35:51,040
So we can neglect all
of the other states

488
00:35:51,040 --> 00:35:54,370
where n is not equal to i.

489
00:35:54,370 --> 00:36:01,870
And if they're not
there, cn has to be 1,

490
00:36:01,870 --> 00:36:04,240
so we can forget about it.

491
00:36:04,240 --> 00:36:07,150
So we end up with this
incredibly wonderful

492
00:36:07,150 --> 00:36:09,170
simple equation.

493
00:36:09,170 --> 00:36:10,630
So we make the two
approximations.

494
00:36:10,630 --> 00:36:13,690
Single state, the
perturbation is really weak,

495
00:36:13,690 --> 00:36:20,720
and we get cf of t
is equal to the vfi--

496
00:36:20,720 --> 00:36:22,510
the off-diagonal
matrix element--

497
00:36:22,510 --> 00:36:30,578
over 2i h-bar times the
integral from 0 to t e

498
00:36:30,578 --> 00:36:44,720
to the minus i omega i
f t minus omega times

499
00:36:44,720 --> 00:36:52,930
e to the i omega
i f plus omega dt.

500
00:36:57,350 --> 00:37:00,260
Well, all complexity is gone.

501
00:37:00,260 --> 00:37:03,690
We have the amount
of the final state,

502
00:37:03,690 --> 00:37:07,650
and it's expressed by a
matrix element and some time

503
00:37:07,650 --> 00:37:09,500
dependence.

504
00:37:09,500 --> 00:37:18,580
And this is a resonant situation
where if omega t, omega t--

505
00:37:18,580 --> 00:37:24,570
if omega is equal
to omega i f, fine.

506
00:37:24,570 --> 00:37:29,490
Then this is zero,
the exponent is zero,

507
00:37:29,490 --> 00:37:32,770
we get t here from that.

508
00:37:32,770 --> 00:37:35,150
And we get zero from that one
because that's oscillating

509
00:37:35,150 --> 00:37:36,441
so fast it doesn't do anything.

510
00:37:39,330 --> 00:37:45,430
But that's a problem because
this c is a probability.

511
00:37:45,430 --> 00:37:50,580
And so the square of c had
better not be larger than one,

512
00:37:50,580 --> 00:37:54,240
and this is cruising
to be larger than 1.

513
00:37:54,240 --> 00:37:56,940
But we don't care about cw.

514
00:37:56,940 --> 00:37:58,530
What we really
care about-- well,

515
00:37:58,530 --> 00:38:03,000
what is the rate as
opposed to the probability?

516
00:38:03,000 --> 00:38:11,130
OK, because the rate
of increase of state f

517
00:38:11,130 --> 00:38:16,080
is something that we can
calculate from this integral

518
00:38:16,080 --> 00:38:19,880
simply by taking the--

519
00:38:19,880 --> 00:38:23,695
we multiply the
integral by 1 over T

520
00:38:23,695 --> 00:38:29,850
if the limit T goes to infinity.

521
00:38:32,820 --> 00:38:34,550
And now we get a
new equation, which

522
00:38:34,550 --> 00:38:40,210
is called Fermi's golden rule.

523
00:38:40,210 --> 00:38:43,100
OK, so I'm skipping
some steps, and I'm

524
00:38:43,100 --> 00:38:45,260
doing things in the wrong order.

525
00:38:45,260 --> 00:38:48,920
But so first of
all, the probability

526
00:38:48,920 --> 00:38:52,250
of the transition from
the i state to the f state

527
00:38:52,250 --> 00:38:53,820
as a function of time.

528
00:38:53,820 --> 00:38:56,330
So the probability is
going to keep growing.

529
00:38:56,330 --> 00:39:00,230
That's why we want to do this
trick with dividing by t.

530
00:39:00,230 --> 00:39:01,801
What time is it?

531
00:39:01,801 --> 00:39:02,300
OK.

532
00:39:09,740 --> 00:39:17,150
That's just cf of t
squared, and that's just

533
00:39:17,150 --> 00:39:28,760
the fi over 4 h-bar squared
times this integral 0 to t e

534
00:39:28,760 --> 00:39:39,200
to the plus and e to the
minus term dt squared.

535
00:39:39,200 --> 00:39:45,340
OK, the integrals survive only
if omega is equal to omega i f.

536
00:39:49,630 --> 00:39:52,580
And if we convert to a
rate so that the rate is

537
00:39:52,580 --> 00:40:04,890
going to be Wfi, which is
going to be Vfi over 4 h-bar

538
00:40:04,890 --> 00:40:14,331
squared times the sum of
two delta functions Vi

539
00:40:14,331 --> 00:40:27,440
minus Ef minus omega plus sum
of Ei minus Ef plus omega.

540
00:40:30,420 --> 00:40:34,530
So the rate is just
this simple thing--

541
00:40:34,530 --> 00:40:36,990
the square matrix
element and a delta

542
00:40:36,990 --> 00:40:39,570
function-- saying either it's
an absorption or emission

543
00:40:39,570 --> 00:40:42,250
transition on resonance,
and we're cooked.

544
00:40:42,250 --> 00:40:46,700
OK, so now I want to
show some pictures

545
00:40:46,700 --> 00:40:54,540
of a movie, which will make this
whole thing make more sense.

546
00:40:54,540 --> 00:40:57,240
This is for a
vibrational transition.

547
00:40:57,240 --> 00:41:00,815
So we have the electric field--

548
00:41:03,770 --> 00:41:06,530
the dipole interacting
with the electric field.

549
00:41:06,530 --> 00:41:10,410
And now let's just turn
on the time dependence.

550
00:41:10,410 --> 00:41:14,100
OK, so this is the
interaction term.

551
00:41:14,100 --> 00:41:18,050
We add that interaction term
to the zero-order Hamiltonian,

552
00:41:18,050 --> 00:41:24,650
and so we end up getting a
big effect of the potential.

553
00:41:24,650 --> 00:41:27,450
The potential's going
like this, like that.

554
00:41:27,450 --> 00:41:30,660
And so the eigenfunctions
of that potential

555
00:41:30,660 --> 00:41:35,260
are going to be profoundly
affected, and so let's do that.

556
00:41:35,260 --> 00:41:38,760
Let's go to the next.

557
00:41:38,760 --> 00:41:42,980
All right, so here now we
have a realistic small field,

558
00:41:42,980 --> 00:41:44,870
and now this is small.

559
00:41:44,870 --> 00:41:46,640
And you can hardly
see this thing moving.

560
00:41:51,520 --> 00:42:00,830
OK, now what we have is
the wave function of this.

561
00:42:00,830 --> 00:42:12,610
And what we see is if omega
is 1/4 the energy, if omega

562
00:42:12,610 --> 00:42:16,120
is much smaller than the
vibrational frequency

563
00:42:16,120 --> 00:42:21,730
or much larger, we get very
little effect of the time

564
00:42:21,730 --> 00:42:24,750
dependent.

565
00:42:24,750 --> 00:42:27,330
You can see that the
wave function is just

566
00:42:27,330 --> 00:42:28,560
moving a little bit.

567
00:42:28,560 --> 00:42:30,630
The potential is
jiggling around,

568
00:42:30,630 --> 00:42:33,990
whether the perturbation
is strong or weak.

569
00:42:33,990 --> 00:42:35,590
It's not on resonance.

570
00:42:35,590 --> 00:42:41,810
And now let's go
to the resonance.

571
00:42:41,810 --> 00:42:43,940
Now what's happening
is the potential

572
00:42:43,940 --> 00:42:46,640
is moving not too much,
but the wave function

573
00:42:46,640 --> 00:42:50,050
is diving all over the place.

574
00:42:50,050 --> 00:42:51,810
And if we ask, well,
what does that really

575
00:42:51,810 --> 00:42:59,230
look like as a sum
of terms, the thing

576
00:42:59,230 --> 00:43:01,810
that's different from the
zero-order wave function

577
00:43:01,810 --> 00:43:03,730
is this.

578
00:43:03,730 --> 00:43:08,070
So zero-order wave function
is one nodeless thing.

579
00:43:08,070 --> 00:43:14,850
This is the time-dependent term,
and it looks like V equals 1 So

580
00:43:14,850 --> 00:43:19,060
what this shows is,
yes, there are--

581
00:43:19,060 --> 00:43:23,890
if we have a time-dependent
field, and it's resonant,

582
00:43:23,890 --> 00:43:25,720
then we get a very
strong interaction

583
00:43:25,720 --> 00:43:27,700
even though the field is weak.

584
00:43:27,700 --> 00:43:31,570
And it causes the appearance
of the other level

585
00:43:31,570 --> 00:43:32,841
but oscillating.

586
00:43:35,610 --> 00:43:39,030
And so resonance is
really important,

587
00:43:39,030 --> 00:43:41,450
and selection rule
is really important.

588
00:43:41,450 --> 00:43:46,650
The selection rule for the
vibrational transitions

589
00:43:46,650 --> 00:43:49,620
has to do with the form.

590
00:43:49,620 --> 00:43:54,790
Oh, I shouldn't
be rushing at all.

591
00:43:54,790 --> 00:43:59,915
OK, so let's draw a picture.

592
00:44:02,840 --> 00:44:06,210
And this is the part that has
puzzled me for a long time,

593
00:44:06,210 --> 00:44:09,620
but I've got it now.

594
00:44:09,620 --> 00:44:20,170
So here we have a
picture of the molecule.

595
00:44:20,170 --> 00:44:22,800
And this end is positive,
and this end is negative.

596
00:44:22,800 --> 00:44:25,280
And we have a
positive electrode,

597
00:44:25,280 --> 00:44:27,570
and we have a
negative electrode.

598
00:44:27,570 --> 00:44:30,050
So that's an electric field.

599
00:44:30,050 --> 00:44:35,800
And so now the positive
electrode is saying,

600
00:44:35,800 --> 00:44:38,690
you better go away, and
you better come here.

601
00:44:38,690 --> 00:44:42,370
So it's trying to
use compressed bond.

602
00:44:42,370 --> 00:44:44,470
And now this field
oscillates, and so it's

603
00:44:44,470 --> 00:44:48,120
compressing and expanding.

604
00:44:48,120 --> 00:44:50,960
Now that's what's going on.

605
00:44:50,960 --> 00:44:53,260
But how does quantum
mechanics account for it?

606
00:44:53,260 --> 00:44:57,100
Well, quantum mechanics says
in order for the bond length

607
00:44:57,100 --> 00:45:00,710
to change, we have to
mix in some other state.

608
00:45:00,710 --> 00:45:06,460
So we have the ground state,
and we have an excited state

609
00:45:06,460 --> 00:45:08,630
that looks like that.

610
00:45:08,630 --> 00:45:11,560
And so the field is
mixing these two.

611
00:45:11,560 --> 00:45:19,600
Now that means that the operator
is mu 0 plus derivative of mu

612
00:45:19,600 --> 00:45:25,600
with respect to the
electric field times q.

613
00:45:25,600 --> 00:45:31,750
So this is the thing that
allows some mixing of an excited

614
00:45:31,750 --> 00:45:33,680
state into the ground state.

615
00:45:33,680 --> 00:45:37,210
This is our friend the
harmonic oscillator--

616
00:45:37,210 --> 00:45:39,310
operator, displacement operator.

617
00:45:39,310 --> 00:45:42,440
It has selection rules delta
v equals plus or minus 1,

618
00:45:42,440 --> 00:45:45,740
and that's all.

619
00:45:45,740 --> 00:45:50,630
So a vibrational
transition is caused

620
00:45:50,630 --> 00:45:55,640
by the derivative of the--

621
00:45:58,839 --> 00:46:05,150
yeah, that's-- no, derivative of
the dipole moment with respect

622
00:46:05,150 --> 00:46:07,650
to Q.

623
00:46:07,650 --> 00:46:12,750
So did I have it right?

624
00:46:12,750 --> 00:46:14,100
Yes.

625
00:46:14,100 --> 00:46:16,170
So this is something--

626
00:46:16,170 --> 00:46:20,040
we can calculate how the
dipole moment depends

627
00:46:20,040 --> 00:46:21,810
on the displacement
from equilibrium,

628
00:46:21,810 --> 00:46:26,060
but this is the operator that
causes the mixing of states.

629
00:46:26,060 --> 00:46:30,690
So one of the things I've
loved to do over the years

630
00:46:30,690 --> 00:46:34,670
is to write a cumulative
exam in which I ask, well,

631
00:46:34,670 --> 00:46:38,110
what is it that causes a
vibrational transition?

632
00:46:38,110 --> 00:46:40,290
What does a molecule
have to have in order

633
00:46:40,290 --> 00:46:42,480
to have a vibrational
transition?

634
00:46:42,480 --> 00:46:44,250
And also what does
a molecule have

635
00:46:44,250 --> 00:46:47,610
to have to have a
rotational transition?

636
00:46:47,610 --> 00:46:50,100
Well, this is what causes
the rotational transition

637
00:46:50,100 --> 00:46:55,210
because we can think of the
dipole moment interacting

638
00:46:55,210 --> 00:46:59,950
with a field, which is going
like that or like that.

639
00:46:59,950 --> 00:47:04,720
And so what that does is it
causes a torque on the system.

640
00:47:04,720 --> 00:47:07,330
It doesn't change
the dipole moment,

641
00:47:07,330 --> 00:47:09,640
doesn't stretch the molecule.

642
00:47:09,640 --> 00:47:17,650
It causes a
transition, and this is

643
00:47:17,650 --> 00:47:23,920
expressed in terms
of the interaction

644
00:47:23,920 --> 00:47:32,230
mu dot E. This dot
product, this cosine theta,

645
00:47:32,230 --> 00:47:34,060
is the operator
that causes this.

646
00:47:34,060 --> 00:47:44,830
We call the relationship between
the laboratory and the body

647
00:47:44,830 --> 00:47:46,690
fixed coordinate
system is determined

648
00:47:46,690 --> 00:47:50,720
by the cosine of some angle,
and the cosine of the angle

649
00:47:50,720 --> 00:47:55,970
is what's responsible for a
pure rotational transition.

650
00:47:55,970 --> 00:47:57,720
And we have
vibrational transitions

651
00:47:57,720 --> 00:48:00,240
where they are derivative
of the dipole with respect

652
00:48:00,240 --> 00:48:02,160
to the coordinate.

653
00:48:02,160 --> 00:48:06,980
Now let's say we have nitrogen--

654
00:48:06,980 --> 00:48:11,830
no dipole moment, no derivative
of the dipole moment.

655
00:48:11,830 --> 00:48:13,380
Suppose we have CO.

656
00:48:13,380 --> 00:48:15,850
CO has a very
small dipole moment

657
00:48:15,850 --> 00:48:19,570
and a huge derivative of the
dipole moment with respect

658
00:48:19,570 --> 00:48:21,100
to displacement.

659
00:48:21,100 --> 00:48:24,670
And so CO has really strong
vibrational transitions

660
00:48:24,670 --> 00:48:27,430
and rather weak
rotational transitions.

661
00:48:27,430 --> 00:48:34,180
So if it happened that CO had
zero permanent dipole moment,

662
00:48:34,180 --> 00:48:36,490
it would have no
rotational transition.

663
00:48:36,490 --> 00:48:39,100
But as you go up to higher
V's, then it would not be zero.

664
00:48:39,100 --> 00:48:41,840
And you would see
rotational transitions.

665
00:48:41,840 --> 00:48:44,800
And so there's all sorts of
insights that come from this.

666
00:48:44,800 --> 00:48:48,460
And so now we know what
causes transitions.

667
00:48:48,460 --> 00:48:51,140
There is some
operator, which causes

668
00:48:51,140 --> 00:48:53,980
mixing of some wave functions.

669
00:48:53,980 --> 00:48:55,810
And the time-dependent
perturbation theory

670
00:48:55,810 --> 00:49:00,910
when it's resonant
mixes only one state.

671
00:49:00,910 --> 00:49:03,130
We have selection rules
which we understand just

672
00:49:03,130 --> 00:49:04,510
by looking at the wave fun--

673
00:49:04,510 --> 00:49:08,290
looking at the matrix
elements, and now we

674
00:49:08,290 --> 00:49:11,770
have a big
understanding of what is

675
00:49:11,770 --> 00:49:13,330
going to appear in a spectrum.

676
00:49:13,330 --> 00:49:15,170
What are the intensities
in the spectrum?

677
00:49:15,170 --> 00:49:17,140
What are the transitions?

678
00:49:17,140 --> 00:49:18,980
Which transitions are
going to be allowed?

679
00:49:18,980 --> 00:49:22,600
Which are going to be forbidden?

680
00:49:22,600 --> 00:49:24,170
And that's kind of useful.

681
00:49:24,170 --> 00:49:28,790
So there is this
tremendously tedious algebra,

682
00:49:28,790 --> 00:49:31,540
which I didn't do a very
good job displaying,

683
00:49:31,540 --> 00:49:33,760
but you don't need it
because, at the end,

684
00:49:33,760 --> 00:49:36,640
you get Fermi's golden
rule, which says

685
00:49:36,640 --> 00:49:40,310
transitions occur on resonance.

686
00:49:40,310 --> 00:49:42,400
Now if you're a little
bit off resonance,

687
00:49:42,400 --> 00:49:46,780
well, then the stationary phase
in the oscillating exponential

688
00:49:46,780 --> 00:49:50,590
persists for a while,
and then it goes away.

689
00:49:50,590 --> 00:49:54,100
And so you get a little bit
of slightly off-resonance

690
00:49:54,100 --> 00:49:58,180
transition probability, and
you get other things too.

691
00:49:58,180 --> 00:50:01,780
But you already now have
enough to understand basically

692
00:50:01,780 --> 00:50:06,940
everything you need
to begin to make sense

693
00:50:06,940 --> 00:50:13,190
of the interaction of radiation
with molecules correctly,

694
00:50:13,190 --> 00:50:15,020
and this isn't
bullets and targets.

695
00:50:15,020 --> 00:50:19,370
This is waves with
phases, and so there

696
00:50:19,370 --> 00:50:21,980
are all sorts of things you have
to do to be honest about it.

697
00:50:21,980 --> 00:50:24,530
But you know what
the actors are,

698
00:50:24,530 --> 00:50:26,810
and that's really
a useful thing.

699
00:50:26,810 --> 00:50:31,150
And you're never going to
be tested on this from me.

700
00:50:31,150 --> 00:50:34,390
OK, good luck on the
exam tomorrow night.