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ROBERT FIELD: Now if you
looked at today's notes,

9
00:00:27,310 --> 00:00:31,930
you'll notice that they're
very long and very complicated.

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00:00:31,930 --> 00:00:35,170
And that's not intended
to be intimidating.

11
00:00:35,170 --> 00:00:39,030
It's just to show you the
power of perturbation theory.

12
00:00:39,030 --> 00:00:44,360
So we have talked about
the two-level problem.

13
00:00:44,360 --> 00:00:48,130
The two-level problem is
one that's exactly solved.

14
00:00:48,130 --> 00:00:50,790
It's one of our favorite
exactly solved problems,

15
00:00:50,790 --> 00:00:53,920
although it doesn't seem to
have any physical relevance.

16
00:00:53,920 --> 00:00:57,520
It's just a nice
numerical exercise.

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00:00:57,520 --> 00:01:00,300
So we can take a two
by two Hamiltonian

18
00:01:00,300 --> 00:01:03,220
and exactly diagonalize it.

19
00:01:03,220 --> 00:01:05,489
And that's done
using a unitary--

20
00:01:05,489 --> 00:01:07,800
or actually, in the
case that we looked at,

21
00:01:07,800 --> 00:01:10,330
orthogonal-- transformation.

22
00:01:10,330 --> 00:01:16,340
And so it's a rotation
in state space.

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00:01:16,340 --> 00:01:19,820
And the rotation angle
is an explicit function

24
00:01:19,820 --> 00:01:23,160
of the parameters
in the Hamiltonian.

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00:01:23,160 --> 00:01:26,830
Now all good things
come to an end.

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00:01:26,830 --> 00:01:32,020
We cannot do this for anything
more than a two-level problem.

27
00:01:32,020 --> 00:01:35,020
But we can take the formalism
for the two-level problem

28
00:01:35,020 --> 00:01:38,020
and say, oh, well,
we can describe

29
00:01:38,020 --> 00:01:41,290
a transformation that
diagonalizes the end level

30
00:01:41,290 --> 00:01:43,440
problem.

31
00:01:43,440 --> 00:01:47,010
And it has the same
property of being unitary.

32
00:01:49,860 --> 00:01:54,090
But the good thing about it
is it's solved by a computer.

33
00:01:54,090 --> 00:01:58,030
You tell a computer
what the operator is,

34
00:01:58,030 --> 00:02:00,260
and it will diagonalize it.

35
00:02:00,260 --> 00:02:02,510
And not only will
it diagonalize it,

36
00:02:02,510 --> 00:02:05,330
but it will give
you the eigenvalues

37
00:02:05,330 --> 00:02:09,300
and the eigenvectors.

38
00:02:09,300 --> 00:02:14,640
It gets those eigenvectors by
applying the same two level

39
00:02:14,640 --> 00:02:19,960
unitary transformation
many, many times.

40
00:02:19,960 --> 00:02:23,760
So if you have a
six-level problem

41
00:02:23,760 --> 00:02:26,130
or a hundred-level
problem, the computer

42
00:02:26,130 --> 00:02:31,530
just cranks and cranks
on each two by two

43
00:02:31,530 --> 00:02:33,000
and does transformations.

44
00:02:33,000 --> 00:02:37,920
And it keeps doing that
until the off diagonal matrix

45
00:02:37,920 --> 00:02:40,660
element is small--

46
00:02:40,660 --> 00:02:43,000
remaining off diagonal
matrix element--

47
00:02:43,000 --> 00:02:45,580
is small compared
to the difference

48
00:02:45,580 --> 00:02:51,110
in energy between the
eigenvalues it connects.

49
00:02:51,110 --> 00:02:55,550
And so you tell it, I want
this convergence criterion

50
00:02:55,550 --> 00:02:58,070
to be a part in a hundred
or a part in a million.

51
00:02:58,070 --> 00:03:00,080
And it just does
this and does this.

52
00:03:00,080 --> 00:03:03,620
And eventually, it gives
you the transformation

53
00:03:03,620 --> 00:03:04,775
and the eigenvalues.

54
00:03:07,440 --> 00:03:12,290
So that means that it doesn't
matter how big the problem is.

55
00:03:12,290 --> 00:03:18,590
You just have to have a
computer that's patient or fast.

56
00:03:18,590 --> 00:03:22,640
And it will crank
out the results.

57
00:03:22,640 --> 00:03:28,640
Now you need to know how to use
the t matrices, or the t dagger

58
00:03:28,640 --> 00:03:33,770
matrix, because these things
enable you to solve basically

59
00:03:33,770 --> 00:03:37,130
any problem in time
independent quantum mechanics

60
00:03:37,130 --> 00:03:40,070
and a lot of problems in time
dependent quantum mechanics.

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00:03:43,410 --> 00:03:46,110
So there is another
way, and that's

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00:03:46,110 --> 00:03:49,950
going to solve an
n-level problem.

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00:03:49,950 --> 00:03:54,580
And that's the subject of this
lecture and the next lecture.

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00:03:54,580 --> 00:03:57,830
It's called non-degenerate
perturbation theory.

65
00:03:57,830 --> 00:04:04,320
Now our goal-- well, let's ask.

66
00:04:04,320 --> 00:04:06,710
I'm a spectroscopist.

67
00:04:06,710 --> 00:04:09,984
What do spectroscopists do?

68
00:04:09,984 --> 00:04:13,224
Now, there's a silly,
stupid answer to that.

69
00:04:13,224 --> 00:04:14,890
I want something a
little more profound.

70
00:04:19,260 --> 00:04:19,929
Yes?

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00:04:19,929 --> 00:04:21,160
AUDIENCE: Find energy levels?

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00:04:21,160 --> 00:04:24,620
ROBERT FIELD: Yeah,
we record spectra.

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00:04:24,620 --> 00:04:26,460
We get energy levels.

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00:04:26,460 --> 00:04:27,470
And we get intensities.

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00:04:27,470 --> 00:04:29,600
We get all sorts of things.

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00:04:29,600 --> 00:04:32,450
We could also be working in
the time domain, and we could

77
00:04:32,450 --> 00:04:35,420
we could be looking at some
kind of quantum beating

78
00:04:35,420 --> 00:04:38,300
system or some decaying system.

79
00:04:38,300 --> 00:04:41,330
We make these measurements.

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00:04:41,330 --> 00:04:45,620
But this is not why we do it.

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00:04:45,620 --> 00:04:47,335
Remember, yes--

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00:04:47,335 --> 00:04:49,210
AUDIENCE: All spectra
have buried information

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00:04:49,210 --> 00:04:52,030
about the physical
parameters of the system.

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00:04:52,030 --> 00:04:54,500
ROBERT FIELD: Exactly, exactly--

85
00:04:54,500 --> 00:04:58,810
that's what's kept me
going for my entire career.

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00:04:58,810 --> 00:05:00,820
I use the word encoded
rather than buried.

87
00:05:00,820 --> 00:05:03,010
But I like that.

88
00:05:03,010 --> 00:05:06,010
I might start using
buried as well.

89
00:05:06,010 --> 00:05:11,610
Yes, we're not allowed to
look inside of small things.

90
00:05:11,610 --> 00:05:13,260
We're not allowed to
determine the wave

91
00:05:13,260 --> 00:05:16,280
function by any experiment.

92
00:05:16,280 --> 00:05:21,080
But we are able to observe the
energy levels and properties

93
00:05:21,080 --> 00:05:23,120
of the Hamiltonian.

94
00:05:23,120 --> 00:05:26,180
And we can regard
the Hamiltonian

95
00:05:26,180 --> 00:05:30,410
as a fit model, a model where
there are adjustable parameters

96
00:05:30,410 --> 00:05:35,240
which are the structural
parameters, like the force

97
00:05:35,240 --> 00:05:39,120
constants and the reduced
masses and whatever

98
00:05:39,120 --> 00:05:44,210
we need to describe
everything in this problem.

99
00:05:44,210 --> 00:05:46,550
And today's lecture
is mostly going

100
00:05:46,550 --> 00:05:51,860
to be on the interactions
between normal modes

101
00:05:51,860 --> 00:05:54,550
of a polyatomic molecule.

102
00:05:54,550 --> 00:05:57,130
Now you might ask, why am I
doing this rather than talking

103
00:05:57,130 --> 00:06:07,351
about just an ordinary,
anharmonic oscillator.

104
00:06:07,351 --> 00:06:09,850
And the reason is, I'm going
to do that in the next lecture.

105
00:06:09,850 --> 00:06:12,460
And it's in your problem sets.

106
00:06:12,460 --> 00:06:14,380
You're going to do that.

107
00:06:14,380 --> 00:06:19,750
So I'm going to show a little
higher order of information.

108
00:06:19,750 --> 00:06:22,480
So we go from what
we can observe

109
00:06:22,480 --> 00:06:24,460
to a representation
of it, which we

110
00:06:24,460 --> 00:06:27,610
call the effective
Hamiltonian, where

111
00:06:27,610 --> 00:06:30,250
in this effective Hamiltonian
we have determined

112
00:06:30,250 --> 00:06:32,980
the values of all of
the important structural

113
00:06:32,980 --> 00:06:35,100
parameters.

114
00:06:35,100 --> 00:06:41,270
And from that, we can calculate
everything, everything

115
00:06:41,270 --> 00:06:45,390
that we could possibly
observe, including

116
00:06:45,390 --> 00:06:48,740
things we didn't observe.

117
00:06:48,740 --> 00:06:50,540
So it's really powerful.

118
00:06:50,540 --> 00:06:55,240
It's a way of taking the
totality of observations

119
00:06:55,240 --> 00:06:57,710
that you're going
to make and saying,

120
00:06:57,710 --> 00:07:01,090
yes, I have looked
inside this molecule.

121
00:07:01,090 --> 00:07:04,870
And I've determined everything
that I'm allowed to determine.

122
00:07:04,870 --> 00:07:08,100
And I can calculate
the wave function.

123
00:07:08,100 --> 00:07:10,590
The wave function is an
experimentally determined wave

124
00:07:10,590 --> 00:07:14,260
function, but only indirect.

125
00:07:14,260 --> 00:07:16,390
This is amazingly powerful.

126
00:07:16,390 --> 00:07:21,490
And basically, everybody
who deals with spectra

127
00:07:21,490 --> 00:07:25,860
is doing this whether
they know it or not.

128
00:07:25,860 --> 00:07:29,390
And so I want to be able
to give you the tools

129
00:07:29,390 --> 00:07:35,120
to be able to take any arbitrary
spectrum and extract from it

130
00:07:35,120 --> 00:07:36,860
the crucial information.

131
00:07:40,680 --> 00:07:43,590
In the last lecture,
we talked a little bit

132
00:07:43,590 --> 00:07:54,430
about matrix mechanics, and
that involved linear algebra.

133
00:08:02,180 --> 00:08:07,040
And you have a wonderful
handout on linear algebra, which

134
00:08:07,040 --> 00:08:10,400
will give you more
than enough to be

135
00:08:10,400 --> 00:08:13,100
able to do any quantum
mechanical problem we're going

136
00:08:13,100 --> 00:08:16,264
to be facing in this class.

137
00:08:16,264 --> 00:08:17,180
And there is notation.

138
00:08:21,650 --> 00:08:24,680
And the notation is
unfamiliar, and you

139
00:08:24,680 --> 00:08:26,630
have to learn how to use it.

140
00:08:26,630 --> 00:08:28,850
And so we have for example--

141
00:08:28,850 --> 00:08:34,232
the analogy to the Schrodinger
equation in matrix language--

142
00:08:38,860 --> 00:08:44,155
where we have the Hamiltonian,
an eigenvector, an eigenvalue,

143
00:08:44,155 --> 00:08:45,760
and this eigenvector.

144
00:08:45,760 --> 00:08:49,760
So it's an eigenvalue
or eigenvector equation.

145
00:08:49,760 --> 00:08:53,040
And this is the form of
the Schrodinger equation

146
00:08:53,040 --> 00:08:56,860
and we can pretend that it is
the real Schrodinger equation.

147
00:08:56,860 --> 00:08:59,420
And we can use the
standard approaches,

148
00:08:59,420 --> 00:09:04,810
but it's useful to work
in matrix notation.

149
00:09:04,810 --> 00:09:08,120
So we can solve for
the energy levels,

150
00:09:08,120 --> 00:09:14,440
and we can solve for basically
all of the eigenvectors.

151
00:09:14,440 --> 00:09:17,890
And our friend is this
unitary transformation where

152
00:09:17,890 --> 00:09:22,960
t dagger is equal to t inverse.

153
00:09:22,960 --> 00:09:30,730
And t times t inverse is
equal to this thing 1111100,

154
00:09:30,730 --> 00:09:32,455
or the unit matrix.

155
00:09:35,420 --> 00:09:36,560
I thought I saw a hand up.

156
00:09:36,560 --> 00:09:40,880
That was just-- that
was not serious.

157
00:09:43,790 --> 00:09:50,540
And so these unitary
transformations

158
00:09:50,540 --> 00:09:53,750
have this special
convenient property.

159
00:09:53,750 --> 00:09:59,150
And we use these
t daggers and ts

160
00:09:59,150 --> 00:10:02,930
to diagonalize the Hamiltonian.

161
00:10:02,930 --> 00:10:05,090
Now this is just a
little bit of review.

162
00:10:05,090 --> 00:10:10,100
So how did we derive the thing
that we are going to solve?

163
00:10:10,100 --> 00:10:15,950
Well, we took this
equation and we inserted 1

164
00:10:15,950 --> 00:10:19,490
between the Hamiltonian
and the vector.

165
00:10:24,140 --> 00:10:28,640
And then this is 1.

166
00:10:28,640 --> 00:10:31,380
So we don't need to
put it over here.

167
00:10:31,380 --> 00:10:35,610
And then we left
multiply by t dagger,

168
00:10:35,610 --> 00:10:37,530
and we put parentheses
around things.

169
00:10:49,180 --> 00:10:56,020
So this is now what we
were calling h tilde.

170
00:10:56,020 --> 00:10:59,160
And this is c tilde.

171
00:10:59,160 --> 00:11:02,640
And this is now an
equation that says, OK, we

172
00:11:02,640 --> 00:11:07,950
can transform the Hamiltonian
into diagonal form--

173
00:11:07,950 --> 00:11:12,060
E1, En, zeros.

174
00:11:15,630 --> 00:11:20,800
And when we have this
in diagonal form,

175
00:11:20,800 --> 00:11:33,540
we can say, well, this equation
is just E1, 100 et cetera.

176
00:11:33,540 --> 00:11:37,980
So for any eigenvalue,
we have an eigenvector.

177
00:11:43,200 --> 00:11:46,740
Now what we'd really
like to know is, well,

178
00:11:46,740 --> 00:11:51,240
how do we get these eigenvectors
from the unitary transformation

179
00:11:51,240 --> 00:11:56,475
that diagonalizes H. We don't
calculate this unitary-- yes?

180
00:11:56,475 --> 00:11:59,700
AUDIENCE: Is that i or--?

181
00:11:59,700 --> 00:12:02,160
ROBERT FIELD: This is a 1,
this is a 1, this is a 1.

182
00:12:02,160 --> 00:12:06,810
AUDIENCE: I mean in your c
tilde, third from the top.

183
00:12:09,114 --> 00:12:10,155
ROBERT FIELD: That's a j.

184
00:12:13,200 --> 00:12:16,260
This is a particular
eigenvalue, and this

185
00:12:16,260 --> 00:12:17,730
is the eigenvector in that.

186
00:12:21,740 --> 00:12:25,370
Now it's easy for me
to get screwed up,

187
00:12:25,370 --> 00:12:30,830
and it's easy for you to wonder,
what the hell am I doing,

188
00:12:30,830 --> 00:12:32,330
until you've done it.

189
00:12:32,330 --> 00:12:34,760
And then it's
completely transparent.

190
00:12:34,760 --> 00:12:36,530
It's really quite simple.

191
00:12:36,530 --> 00:12:38,750
But it's just this
extra notation.

192
00:12:42,380 --> 00:12:46,200
And so it's possible to show--
and I don't want to do it.

193
00:12:46,200 --> 00:12:48,500
I did it last time.

194
00:12:48,500 --> 00:12:51,980
You can show-- and
this is important--

195
00:12:51,980 --> 00:13:04,280
that this transformation
t dagger c equals c.

196
00:13:04,280 --> 00:13:07,930
Do I want?-- above, above.

197
00:13:07,930 --> 00:13:13,810
This is telling you that
the columns of t dagger

198
00:13:13,810 --> 00:13:15,610
are the eigenvectors.

199
00:13:15,610 --> 00:13:20,170
They are linear combination
of the basis vectors

200
00:13:20,170 --> 00:13:21,970
that correspond to
each eigenvector.

201
00:13:25,080 --> 00:13:27,610
And you can look at this in
more detail in the notes.

202
00:13:30,420 --> 00:13:33,090
Now this is great
because mostly you

203
00:13:33,090 --> 00:13:36,090
want to know what
the eigenvectors are

204
00:13:36,090 --> 00:13:39,030
for a particular eigenvalue.

205
00:13:39,030 --> 00:13:43,650
But sometimes, when
you're doing dynamics,

206
00:13:43,650 --> 00:13:47,030
you prepare, not an
eigenvector, but you

207
00:13:47,030 --> 00:13:52,400
prepare a t equals
0, a basis state.

208
00:13:52,400 --> 00:13:55,950
This is the most
common, doable problem,

209
00:13:55,950 --> 00:13:59,070
and it's also something that
one does in experiments.

210
00:13:59,070 --> 00:14:04,020
You can set up a problem so
that, with a short pulse,

211
00:14:04,020 --> 00:14:07,020
you prepare the system at
t equals 0 in something

212
00:14:07,020 --> 00:14:09,320
that's not an eigenstate.

213
00:14:09,320 --> 00:14:11,570
And usually it's a basis state--

214
00:14:11,570 --> 00:14:16,970
it's one of the eigenvalues
of an exactly solved problem.

215
00:14:16,970 --> 00:14:20,550
But your problems are
not exactly solved ones,

216
00:14:20,550 --> 00:14:23,460
but your experiment
selects that.

217
00:14:23,460 --> 00:14:25,860
And this, not being
an eigenstate,

218
00:14:25,860 --> 00:14:28,740
needs to be expressed
as a linear combination

219
00:14:28,740 --> 00:14:31,140
of eigenstates so
that you can actually

220
00:14:31,140 --> 00:14:37,450
calculate this,
which will describe

221
00:14:37,450 --> 00:14:40,720
how the system is behaving.

222
00:14:40,720 --> 00:14:45,050
And I like asking exam
problems like this

223
00:14:45,050 --> 00:14:49,790
because it's easy to
get hopelessly involved

224
00:14:49,790 --> 00:14:55,750
in ordinary algebra rather
than just using linear algebra.

225
00:14:55,750 --> 00:15:03,750
So if this is the transformation
to the eigenbasis,

226
00:15:03,750 --> 00:15:09,640
then the columns of t are
the transformation back

227
00:15:09,640 --> 00:15:13,180
to the 0 order bases.

228
00:15:13,180 --> 00:15:17,005
And the columns of t are
the rows of t dagger.

229
00:15:21,150 --> 00:15:22,350
Now why should you care?

230
00:15:22,350 --> 00:15:25,290
Because you don't know how to
calculate the elements of the t

231
00:15:25,290 --> 00:15:30,260
matrices yet, but that's what
perturbation theory is for.

232
00:15:30,260 --> 00:15:32,360
With perturbation
theory, it doesn't

233
00:15:32,360 --> 00:15:38,540
matter that the computer could
solve for the t and t dagger

234
00:15:38,540 --> 00:15:40,970
matrices.

235
00:15:40,970 --> 00:15:42,860
But there's no insight.

236
00:15:42,860 --> 00:15:44,000
You just get the numbers.

237
00:15:46,720 --> 00:15:49,270
I like to say that
spectroscopy is not

238
00:15:49,270 --> 00:15:53,770
about creating archival tables
of observed transitions and

239
00:15:53,770 --> 00:15:55,720
observed transition intensities.

240
00:15:55,720 --> 00:15:58,000
It's about understanding
how things work.

241
00:16:00,740 --> 00:16:05,770
And so depending on whether
you're doing a time domain

242
00:16:05,770 --> 00:16:09,280
experiment or frequency
domain experiment,

243
00:16:09,280 --> 00:16:13,060
you're going to want to use
either the columns of t dagger

244
00:16:13,060 --> 00:16:14,740
or the rows of t dagger.

245
00:16:18,290 --> 00:16:22,780
Now at some point
in your life, you

246
00:16:22,780 --> 00:16:25,990
have to be exposed to
non-degenerate perturbation

247
00:16:25,990 --> 00:16:29,440
theory because it
is really powerful.

248
00:16:29,440 --> 00:16:31,640
But it's also incredibly ugly.

249
00:16:34,210 --> 00:16:35,860
So we have a whole
bunch of problems--

250
00:16:35,860 --> 00:16:38,970
I'll go over here--

251
00:16:38,970 --> 00:16:42,070
that are related to
the particle in a box

252
00:16:42,070 --> 00:16:47,150
or the particle in an infinite
box or the harmonic oscillator.

253
00:16:47,150 --> 00:16:51,220
And so one of the
things that you might do

254
00:16:51,220 --> 00:16:56,340
is round off the corners
because physical systems don't

255
00:16:56,340 --> 00:17:00,110
have discontinuities.

256
00:17:00,110 --> 00:17:03,880
Well, that's going to be a very
modest change to the energy

257
00:17:03,880 --> 00:17:07,089
levels and wave functions.

258
00:17:07,089 --> 00:17:11,970
Another thing you might
do is have a barrier,

259
00:17:11,970 --> 00:17:16,030
and you could put
the barrier anywhere.

260
00:17:16,030 --> 00:17:19,900
What does the barrier
or an extra well do?

261
00:17:19,900 --> 00:17:22,285
Or you could do
something like this.

262
00:17:25,180 --> 00:17:28,030
So the particle-in-a-box is a
whole family of problems that

263
00:17:28,030 --> 00:17:31,780
you could solve using
perturbation theory by saying,

264
00:17:31,780 --> 00:17:35,310
OK, the extra stuff
is the perturbation.

265
00:17:35,310 --> 00:17:37,660
And we have to work
out the matrix elements

266
00:17:37,660 --> 00:17:39,850
of the Hamiltonian
that correspond

267
00:17:39,850 --> 00:17:43,360
to these extra things
and then figure out

268
00:17:43,360 --> 00:17:47,960
what to do to get the
eigenvalues and eigenfunctions.

269
00:17:47,960 --> 00:17:49,690
Now, for the
harmonic oscillator--

270
00:17:49,690 --> 00:17:52,780
again, you could put a
barrier in the middle

271
00:17:52,780 --> 00:17:56,020
or you could make it an
asymmetric like almost

272
00:17:56,020 --> 00:18:00,730
all molecular potentials are,
where this is dissociation

273
00:18:00,730 --> 00:18:04,420
and this is two closed cells
colliding with each other

274
00:18:04,420 --> 00:18:06,740
and doing very hard repulsion.

275
00:18:06,740 --> 00:18:08,930
And so that is harmonic
near the bottom,

276
00:18:08,930 --> 00:18:11,180
but it's not harmonic elsewhere.

277
00:18:11,180 --> 00:18:13,990
And so how do you
represent this?

278
00:18:13,990 --> 00:18:18,610
And the Morse oscillator
is a cheap way

279
00:18:18,610 --> 00:18:21,610
of generating something
with this shape

280
00:18:21,610 --> 00:18:24,220
and then doing the perturbation
theory to understand

281
00:18:24,220 --> 00:18:28,900
how the nonharmonic
aspect of the Morse

282
00:18:28,900 --> 00:18:31,550
can affect the energy levels.

283
00:18:31,550 --> 00:18:35,200
Or how did the energy levels
determine, say, the association

284
00:18:35,200 --> 00:18:38,670
energy of this molecule?

285
00:18:38,670 --> 00:18:41,840
How is that dissociation energy
encoded in the energy level

286
00:18:41,840 --> 00:18:42,340
pattern?

287
00:18:45,900 --> 00:18:55,050
Now for polyatomic molecules,
if you have n atoms,

288
00:18:55,050 --> 00:18:59,540
there are 3n minus 6
vibrational normal modes.

289
00:18:59,540 --> 00:19:00,750
Well, how do I know that?

290
00:19:00,750 --> 00:19:05,590
n atoms, there's 3n
degrees of freedom.

291
00:19:05,590 --> 00:19:10,420
There's 3 translations
and 3 rotations.

292
00:19:10,420 --> 00:19:12,570
And so that leaves 3n minus 6.

293
00:19:12,570 --> 00:19:15,250
And all of that is vibrations.

294
00:19:15,250 --> 00:19:20,770
So we have many normal modes,
and it's not too surprising

295
00:19:20,770 --> 00:19:23,470
that, if you stretch
one normal mode,

296
00:19:23,470 --> 00:19:26,770
it'll affect the
frequency of another.

297
00:19:26,770 --> 00:19:30,260
And we'd like to know that.

298
00:19:30,260 --> 00:19:32,480
And so perturbation
theory is really

299
00:19:32,480 --> 00:19:34,820
valuable for
polyatomic molecules.

300
00:19:34,820 --> 00:19:36,560
And that's the bulk
of the examples

301
00:19:36,560 --> 00:19:41,880
that I worked in the non-lecture
notes for this lecture.

302
00:19:41,880 --> 00:19:43,745
But there's also dynamics.

303
00:19:48,690 --> 00:19:54,540
So you prepare some
initial state of t equals 0

304
00:19:54,540 --> 00:19:58,080
and you want to know
how it's evolving.

305
00:19:58,080 --> 00:20:03,030
But often that initial state
is an eigenstate of one

306
00:20:03,030 --> 00:20:05,100
of the exactly solved problems.

307
00:20:05,100 --> 00:20:06,990
And you want to be
able to re-express

308
00:20:06,990 --> 00:20:12,360
that in terms of the
eigenstates of the real problem.

309
00:20:12,360 --> 00:20:17,880
And so you want to know C J--

310
00:20:17,880 --> 00:20:18,450
I'm sorry.

311
00:20:27,430 --> 00:20:32,280
You want to know
J, CJ eigenstates.

312
00:20:32,280 --> 00:20:35,110
So once you have
this, then you know

313
00:20:35,110 --> 00:20:38,110
how to write the
time dependent wave

314
00:20:38,110 --> 00:20:40,750
function because you are
dealing with eigenstates

315
00:20:40,750 --> 00:20:42,639
and they have eigenenergies.

316
00:20:42,639 --> 00:20:44,180
And so you just
write this thing out.

317
00:20:44,180 --> 00:20:45,580
It's just mechanical and boring.

318
00:20:49,660 --> 00:20:53,390
It's also true that
molecules rotate.

319
00:20:53,390 --> 00:20:56,110
And when they rotate,
there's centrifugal force

320
00:20:56,110 --> 00:21:01,220
and their internuclear
distances change.

321
00:21:01,220 --> 00:21:04,460
And we can then
calculate how that

322
00:21:04,460 --> 00:21:06,950
will affect the rotational
energy levels using

323
00:21:06,950 --> 00:21:07,870
perturbation theory.

324
00:21:11,220 --> 00:21:14,610
And then there's
the origin of life.

325
00:21:14,610 --> 00:21:17,700
Gases are not
supportive of life.

326
00:21:20,460 --> 00:21:23,580
You need two particles to
come together and start

327
00:21:23,580 --> 00:21:25,620
to condense into a liquid.

328
00:21:25,620 --> 00:21:27,150
That's the beginning.

329
00:21:27,150 --> 00:21:30,630
Perturbation theory explains
the long range interactions

330
00:21:30,630 --> 00:21:36,430
by which all gas phase particles
attract each other weakly.

331
00:21:36,430 --> 00:21:38,170
So that's important too.

332
00:21:38,170 --> 00:21:41,140
And so you'll be able
to do all of this stuff.

333
00:21:47,190 --> 00:21:50,310
So here we have non-degenerate
perturbation theory.

334
00:21:53,360 --> 00:21:59,490
And it is a mind numbing,
formal derivation.

335
00:21:59,490 --> 00:22:01,385
So we start out with
this rotary equation.

336
00:22:05,790 --> 00:22:09,330
And we say, well, let us
expand the Hamiltonian.

337
00:22:18,030 --> 00:22:19,590
And let's put a
little thing here.

338
00:22:26,450 --> 00:22:27,500
This is our friend.

339
00:22:27,500 --> 00:22:30,660
This is an exactly
solved problem.

340
00:22:30,660 --> 00:22:37,080
This is what's new, and this
is what's new and small.

341
00:22:37,080 --> 00:22:41,180
And we could
probably neglect it.

342
00:22:41,180 --> 00:22:44,060
And we do the same thing
to the energy levels.

343
00:22:57,460 --> 00:23:01,160
So these are the energy levels
for the exactly solved problem.

344
00:23:01,160 --> 00:23:02,620
And these are the
energy levels--

345
00:23:02,620 --> 00:23:05,120
the first order
corrected energy levels.

346
00:23:05,120 --> 00:23:07,710
And these are the second
order corrected energy levels.

347
00:23:12,002 --> 00:23:13,960
And we do the same thing
for the wave function.

348
00:23:36,962 --> 00:23:39,420
Now we write the Schrodinger
equation with these three term

349
00:23:39,420 --> 00:23:40,390
expressions.

350
00:23:40,390 --> 00:23:44,010
Now I'm also going
to say, we're never

351
00:23:44,010 --> 00:23:46,620
going to consider
this one either.

352
00:23:46,620 --> 00:23:51,030
So life is simpler without
them, but in the notes,

353
00:23:51,030 --> 00:23:52,170
I included them all.

354
00:23:55,110 --> 00:23:58,170
And so what we do is now
we write the full equation

355
00:23:58,170 --> 00:24:01,770
and we sort it into sub
equations corresponding

356
00:24:01,770 --> 00:24:05,250
to powers of lambda.

357
00:24:05,250 --> 00:24:09,850
So the lambda to the 0
equation is really easy.

358
00:24:09,850 --> 00:24:19,760
It's just H0 psi 0
is equal to E0 psi 0.

359
00:24:19,760 --> 00:24:21,980
We could put n's on this.

360
00:24:21,980 --> 00:24:24,620
And this is the
exactly solved problem.

361
00:24:24,620 --> 00:24:28,550
It says, yeah, you
build your foundation

362
00:24:28,550 --> 00:24:31,550
from the lambda to
0 equation, and it's

363
00:24:31,550 --> 00:24:34,990
just what you know already.

364
00:24:34,990 --> 00:24:44,240
And what you're going to do is
now use the psi n 0 and en 0

365
00:24:44,240 --> 00:24:45,470
to do everything else.

366
00:24:48,700 --> 00:24:51,180
The lambda to the one equa--

367
00:24:51,180 --> 00:24:53,960
well, you might ask,
well, what is lambda.

368
00:24:53,960 --> 00:24:56,030
It's just a mathematical trick.

369
00:24:56,030 --> 00:24:58,490
It has no significance
whatsoever.

370
00:24:58,490 --> 00:25:01,510
It's a smallness
parameter, but it's also

371
00:25:01,510 --> 00:25:04,360
something where you can
say, these equations

372
00:25:04,360 --> 00:25:07,800
will be true for
any value of lambda.

373
00:25:07,800 --> 00:25:11,250
And so it's just a way of
separating the equations

374
00:25:11,250 --> 00:25:13,440
into things that have
a structure that you

375
00:25:13,440 --> 00:25:15,690
can manipulate.

376
00:25:15,690 --> 00:25:18,960
People have waxed eloquent
about the meaning of lambda,

377
00:25:18,960 --> 00:25:20,790
and it really doesn't
have any meaning.

378
00:25:23,520 --> 00:25:31,110
So the lambda to the
1 equation-- well,

379
00:25:31,110 --> 00:25:36,360
we collect terms on the left
hand side that have 1 lambda

380
00:25:36,360 --> 00:25:37,690
and on the right hand side.

381
00:25:37,690 --> 00:25:40,020
And so the lambda
to the 1 equation

382
00:25:40,020 --> 00:25:55,840
is going to be, say, H0
psi 1 is equal to E0 psi 1.

383
00:26:02,270 --> 00:26:03,012
I'm sorry.

384
00:26:03,012 --> 00:26:04,220
There's more to it than that.

385
00:26:25,750 --> 00:26:26,620
There's four terms.

386
00:26:31,200 --> 00:26:32,910
We're just collecting
the terms that

387
00:26:32,910 --> 00:26:37,930
have 1 lambda on the left-hand
side and right-hand side.

388
00:26:37,930 --> 00:26:40,480
And we've got this equation.

389
00:26:40,480 --> 00:26:42,560
What are we going to do with it?

390
00:26:42,560 --> 00:26:46,730
Well, one thing we can do with
it is multiply on the left.

391
00:26:46,730 --> 00:26:49,260
Let's just put
some indices here.

392
00:26:49,260 --> 00:26:57,740
So we have n, n, n.

393
00:26:57,740 --> 00:27:05,360
We're going to multiply on the
left and integrate by psi n 0.

394
00:27:16,320 --> 00:27:18,970
So we're going to
get a bunch of terms.

395
00:27:18,970 --> 00:27:41,370
So we will have psi n 0 H 0 psi
n 1 plus psi n 0 H 1 psi n 0.

396
00:27:41,370 --> 00:27:43,990
We have integral, integral.

397
00:27:43,990 --> 00:27:51,610
And on the left hand side,
we have E n 0 integral psi n

398
00:27:51,610 --> 00:27:55,930
0 psi n 1.

399
00:27:55,930 --> 00:28:07,660
And we have E 1 psi n 0 psi n 0.

400
00:28:13,920 --> 00:28:15,600
Well, this is kind
of an ugly term.

401
00:28:15,600 --> 00:28:17,420
It's a psi 0 and a psi 1.

402
00:28:20,380 --> 00:28:27,610
But we know that H, when
operating on psi 0, gives E 0.

403
00:28:27,610 --> 00:28:44,230
So we're going to get E n 0
integral psi n 0 psi n 1--

404
00:28:44,230 --> 00:28:45,160
same two over here.

405
00:28:47,920 --> 00:28:49,890
Cancel them.

406
00:28:49,890 --> 00:28:57,970
And so we get a simple equation
that is just H1 n n is equal to

407
00:28:57,970 --> 00:29:01,310
E1 n.

408
00:29:01,310 --> 00:29:05,340
So we've gotten now
the diagonal matrix

409
00:29:05,340 --> 00:29:07,680
element of the
perturbation term is

410
00:29:07,680 --> 00:29:10,230
equal to the first-order
correction to the energy.

411
00:29:13,510 --> 00:29:16,330
And we can continue.

412
00:29:16,330 --> 00:29:19,930
The algebra isn't beautiful, but
we end up getting the following

413
00:29:19,930 --> 00:29:20,681
equations.

414
00:29:32,730 --> 00:29:42,430
We have E n 1 is H 1 nm.

415
00:29:42,430 --> 00:30:00,030
We have psi n 1 is m not
equal to n psi m 0 H nm.

416
00:30:11,110 --> 00:30:14,420
So what did I do here?

417
00:30:14,420 --> 00:30:16,060
I said, the wave function--

418
00:30:19,100 --> 00:30:20,400
we have completeness.

419
00:30:20,400 --> 00:30:25,790
So if we want the first order
corrections to the nth wave

420
00:30:25,790 --> 00:30:31,340
function, we can write this as
a linear combination of the zero

421
00:30:31,340 --> 00:30:33,350
order wave functions.

422
00:30:33,350 --> 00:30:36,900
And when we do that, we
end up with this formula.

423
00:30:36,900 --> 00:30:40,980
This is the mixing coefficient,
and these are the state-- now

424
00:30:40,980 --> 00:30:42,436
why do I exclude n?

425
00:30:42,436 --> 00:30:43,560
Because we already have it.

426
00:30:48,480 --> 00:30:53,110
And then we get the
second-order correction

427
00:30:53,110 --> 00:31:03,710
to the energy, which is m
not equal to n h n m 1 H m n

428
00:31:03,710 --> 00:31:06,665
1 over En 0.

429
00:31:10,630 --> 00:31:11,130
That's it.

430
00:31:13,689 --> 00:31:14,480
That's all we need.

431
00:31:18,880 --> 00:31:23,010
Now it does say non-degenerate
perturbation theory.

432
00:31:23,010 --> 00:31:25,300
And so it's subject
to the requirement

433
00:31:25,300 --> 00:31:34,250
that H1 n m over E n minus E m.

434
00:31:41,780 --> 00:31:46,730
So if the energy
denominator is near 0,

435
00:31:46,730 --> 00:31:48,110
we know we're in trouble.

436
00:31:50,930 --> 00:31:56,120
But for the vast majority
of energy levels,

437
00:31:56,120 --> 00:31:59,160
this term is much less than 1.

438
00:31:59,160 --> 00:32:03,620
And so that means we can deal
with all of the interactions

439
00:32:03,620 --> 00:32:07,700
among the non-degenerate
levels in one fell swoop.

440
00:32:10,220 --> 00:32:19,000
Now this is an infinite sum,
and this is an infinite sum.

441
00:32:19,000 --> 00:32:21,910
So we don't like infinities.

442
00:32:21,910 --> 00:32:25,810
But we can say, all right,
here's the Hamiltonian.

443
00:32:25,810 --> 00:32:27,980
It's an infinite Hamiltonian.

444
00:32:27,980 --> 00:32:30,280
And we're interested in
this little corner of it.

445
00:32:33,850 --> 00:32:38,940
And so all of the interactions
among these states with all

446
00:32:38,940 --> 00:32:42,610
of the infinite
others get subsumed

447
00:32:42,610 --> 00:32:44,800
into this infinite sum.

448
00:32:44,800 --> 00:32:45,730
It's a small number.

449
00:32:49,410 --> 00:32:52,790
And then we're just interested
in this little subspace

450
00:32:52,790 --> 00:32:57,070
of the energy levels that we're
sampling in our experiment.

451
00:32:57,070 --> 00:32:59,950
So the molecule more
or less tells you

452
00:32:59,950 --> 00:33:03,280
how to focus on the
part that you care about

453
00:33:03,280 --> 00:33:08,380
and to get rid of the stuff that
is of no trouble whatsoever.

454
00:33:08,380 --> 00:33:11,242
And it just contaminates the
wave functions a little bit.

455
00:33:11,242 --> 00:33:13,450
And if you wanted to know
what that contamination is,

456
00:33:13,450 --> 00:33:14,408
you could deal with it.

457
00:33:19,050 --> 00:33:23,160
So this is the tool that you
can use to solve, basically,

458
00:33:23,160 --> 00:33:29,790
any problem involving
molecules with a potential

459
00:33:29,790 --> 00:33:34,050
like a harmonic
oscillator at the bottom.

460
00:33:34,050 --> 00:33:37,120
But it's usable
for all problems,

461
00:33:37,120 --> 00:33:39,450
but there's a different
basis set rather than

462
00:33:39,450 --> 00:33:41,460
the harmonic
oscillator basis set.

463
00:33:41,460 --> 00:33:45,150
So this is your handy dandy key.

464
00:33:45,150 --> 00:33:47,910
And you don't need a
computer, although when

465
00:33:47,910 --> 00:33:49,950
you see the horrible
complexity that

466
00:33:49,950 --> 00:33:53,850
will result when you start
dealing with these sums,

467
00:33:53,850 --> 00:33:56,460
you will say, well, I do
want to use a computer.

468
00:33:56,460 --> 00:33:59,480
But now it's up to me
to organize the program

469
00:33:59,480 --> 00:34:01,380
so that you can ask
the computer to do

470
00:34:01,380 --> 00:34:05,340
what you need in a sensible way
and you still get good answers.

471
00:34:07,870 --> 00:34:15,850
So in the notes, I'm dealing
with a two-mode molecule.

472
00:34:18,500 --> 00:34:20,030
There are no two-mode molecules.

473
00:34:20,030 --> 00:34:22,190
There's one-mode
molecules, and there

474
00:34:22,190 --> 00:34:27,139
might be three or four
or six or whatever.

475
00:34:27,139 --> 00:34:33,739
But the complexity is the
interaction between two modes.

476
00:34:33,739 --> 00:34:35,540
And so we're going
to talk about that.

477
00:34:48,530 --> 00:34:53,110
So we have for a
two-mode molecule--

478
00:34:53,110 --> 00:34:56,199
the Hamiltonian will
consist of Hamiltonian

479
00:34:56,199 --> 00:35:00,820
for mode 1,
Hamiltonian for mode 2,

480
00:35:00,820 --> 00:35:03,460
and the Hamiltonian
for modes 1 and 2,

481
00:35:03,460 --> 00:35:06,960
interacting with each other.

482
00:35:06,960 --> 00:35:08,320
Well, we know these.

483
00:35:08,320 --> 00:35:12,130
These are just ordinary
harmonic oscillator.

484
00:35:12,130 --> 00:35:14,610
And we've got to do
some work on this.

485
00:35:14,610 --> 00:35:17,030
And that's where
perturbation theory comes in.

486
00:35:17,030 --> 00:35:21,050
So H12-- and we know,
if this weren't here,

487
00:35:21,050 --> 00:35:23,660
we know that the
energy levels are

488
00:35:23,660 --> 00:35:26,930
that some of the energy
levels for the two

489
00:35:26,930 --> 00:35:30,200
independent oscillators
and the wave functions

490
00:35:30,200 --> 00:35:31,820
are the products.

491
00:35:31,820 --> 00:35:36,260
So even though the energy levels
have two quantum numbers, V1

492
00:35:36,260 --> 00:35:40,520
and V2, and the wave functions
have two quantum numbers,

493
00:35:40,520 --> 00:35:43,850
V1 and V2, if it
were only this, we'd

494
00:35:43,850 --> 00:35:47,520
have completely
solved that problem.

495
00:35:47,520 --> 00:35:50,490
But because of this,
there's something else.

496
00:35:50,490 --> 00:35:54,830
So these then turn out
to be the zero order

497
00:35:54,830 --> 00:35:58,740
states that you use to evaluate
all the integrals here.

498
00:36:02,250 --> 00:36:07,100
And we have these and A
and A dagger operators,

499
00:36:07,100 --> 00:36:10,270
which are enabling us to--

500
00:36:10,270 --> 00:36:17,050
we have the operator
for coordinate

501
00:36:17,050 --> 00:36:19,300
x is proportional
to A plus A dagger.

502
00:36:21,890 --> 00:36:25,160
And this has a selection rule,
delta V of plus and minus 1.

503
00:36:28,430 --> 00:36:30,170
So we like these
things because we

504
00:36:30,170 --> 00:36:32,390
don't have to do any integrals.

505
00:36:32,390 --> 00:36:35,330
They're all done for you.

506
00:36:35,330 --> 00:36:42,200
And so we're then going to
use these sorts of things

507
00:36:42,200 --> 00:36:48,050
to deal with the most important
terms beyond harmonic.

508
00:36:48,050 --> 00:36:51,910
And so there's a cubic
and there is a quartic.

509
00:36:55,460 --> 00:37:01,290
And although you
haven't really explored

510
00:37:01,290 --> 00:37:08,220
this, what happens when you
make dimensionless coordinates?

511
00:37:08,220 --> 00:37:11,100
You factor out something.

512
00:37:11,100 --> 00:37:15,060
And if you have
cubic terms, they're

513
00:37:15,060 --> 00:37:17,690
100 times smaller
than quadratic terms.

514
00:37:17,690 --> 00:37:20,270
And the quartic terms
are 100 times smaller

515
00:37:20,270 --> 00:37:22,490
than the cubic terms.

516
00:37:22,490 --> 00:37:25,020
And so you don't need
to go much further.

517
00:37:25,020 --> 00:37:28,010
So the cubic terms for
the two oscillators

518
00:37:28,010 --> 00:37:49,340
would be 1/2 K 122 Q1 Q2 squared
plus 1/2 K 112 Q 1 squared Q2.

519
00:37:49,340 --> 00:37:53,360
And you could have Q1
cubed, but we already

520
00:37:53,360 --> 00:37:56,030
deal with that in the
single-mode problem.

521
00:37:56,030 --> 00:37:57,590
So we won't worry about that.

522
00:37:57,590 --> 00:38:01,220
So these are the couplings
between modes 1 and 2

523
00:38:01,220 --> 00:38:03,560
that are cubic.

524
00:38:03,560 --> 00:38:13,680
And then we have 1/4 k 112
Q 1 squared Q 2 squared.

525
00:38:13,680 --> 00:38:20,040
There's also a Q1 Q2 cubed,
and those terms usually

526
00:38:20,040 --> 00:38:22,050
are not important
because they mostly

527
00:38:22,050 --> 00:38:23,590
are dealt with under here.

528
00:38:23,590 --> 00:38:25,950
So anyway, these
are the parameters,

529
00:38:25,950 --> 00:38:30,750
and these are the things
that you need in order

530
00:38:30,750 --> 00:38:32,640
to understand what
this molecule is

531
00:38:32,640 --> 00:38:34,980
going to do when it's excited.

532
00:38:34,980 --> 00:38:37,680
Now when I was a
graduate student,

533
00:38:37,680 --> 00:38:40,890
there was a great deal of
excitement about doing what's

534
00:38:40,890 --> 00:38:44,520
called mode-specific chemistry.

535
00:38:44,520 --> 00:38:50,430
Ordinary compounds cost on
the order of $1 a kilogram.

536
00:38:50,430 --> 00:38:54,620
But if you could do
mode specific chemistry,

537
00:38:54,620 --> 00:38:56,290
you could make
things that aren't

538
00:38:56,290 --> 00:38:59,150
makeable by ordinary,
organic chemistry.

539
00:38:59,150 --> 00:39:01,910
And the organic
chemists would love this

540
00:39:01,910 --> 00:39:09,650
because it's a load of garbage
because these guys make

541
00:39:09,650 --> 00:39:12,150
the modes talk to each other.

542
00:39:12,150 --> 00:39:15,770
And so even if you
could excite a pure

543
00:39:15,770 --> 00:39:19,190
overtone or combinational
level, the energy

544
00:39:19,190 --> 00:39:21,620
moves around the molecule.

545
00:39:21,620 --> 00:39:24,100
And that's called
intermolecular, vibrational

546
00:39:24,100 --> 00:39:26,240
redistribution.

547
00:39:26,240 --> 00:39:29,510
And those there are
processes that you

548
00:39:29,510 --> 00:39:32,930
could talk about the
primary paths for the energy

549
00:39:32,930 --> 00:39:35,990
to flow and the rates.

550
00:39:35,990 --> 00:39:39,530
And that was a major area of
research for the last 30 years.

551
00:39:39,530 --> 00:39:41,090
And most people are tired of it.

552
00:39:41,090 --> 00:39:47,300
But it started out being IVR,
yeah, anything can happen.

553
00:39:47,300 --> 00:39:50,160
It's statistic or whatever.

554
00:39:50,160 --> 00:39:52,970
But no, only specific
things can happen,

555
00:39:52,970 --> 00:39:57,380
and they're controlled by
these specific coupling terms.

556
00:39:57,380 --> 00:39:59,990
And you could calculate them.

557
00:39:59,990 --> 00:40:03,310
Now there's an
interesting other thing.

558
00:40:03,310 --> 00:40:09,890
I told you that the first-order
correction to the energy

559
00:40:09,890 --> 00:40:17,000
is equal to a diagonal matrix
element of the correction term

560
00:40:17,000 --> 00:40:18,345
to the Hamiltonian.

561
00:40:21,260 --> 00:40:23,820
All of the second-order
terms involve

562
00:40:23,820 --> 00:40:27,250
squares of matrix elements.

563
00:40:27,250 --> 00:40:31,080
The second-order terms
you don't know the signs.

564
00:40:31,080 --> 00:40:37,360
If there is a first-order
correction, you get the sine.

565
00:40:37,360 --> 00:40:39,550
And sometimes you
know you want to know:

566
00:40:39,550 --> 00:40:43,780
is the perturbation like that
or the perturbation like that?

567
00:40:46,740 --> 00:40:49,950
Is there a barrier
or an extra well?

568
00:40:49,950 --> 00:40:52,754
And this enables
you to know that

569
00:40:52,754 --> 00:40:54,420
in second-order
perturbation theory when

570
00:40:54,420 --> 00:40:56,003
you have to square
the matrix element,

571
00:40:56,003 --> 00:40:59,470
all that information
is concealed.

572
00:40:59,470 --> 00:41:03,180
Well, anyway-- so there is
a huge amount of algebra

573
00:41:03,180 --> 00:41:07,400
that's involved in
using these equations.

574
00:41:07,400 --> 00:41:14,030
And I don't really want to
go through that algebra.

575
00:41:14,030 --> 00:41:15,650
You can read my notes.

576
00:41:15,650 --> 00:41:19,460
I think the chances of you
reading those notes are small.

577
00:41:19,460 --> 00:41:24,020
But if I don't lecture on them,
it would be slightly greater.

578
00:41:24,020 --> 00:41:28,430
It's a complete treatment
of a two-mode problem

579
00:41:28,430 --> 00:41:31,700
with all of the possible
and harmonic terms

580
00:41:31,700 --> 00:41:33,500
cubic and quartic.

581
00:41:33,500 --> 00:41:34,950
And everything worked out.

582
00:41:34,950 --> 00:41:38,040
Now the trick is,
you could easily say,

583
00:41:38,040 --> 00:41:40,070
well, the algebra
is just so horrible.

584
00:41:40,070 --> 00:41:40,910
Why would I bother?

585
00:41:44,840 --> 00:41:49,250
But what you do is
you take these terms

586
00:41:49,250 --> 00:41:57,980
and you sort according
to selection rule.

587
00:42:02,390 --> 00:42:09,860
So for example, here we
have a selection rule delta

588
00:42:09,860 --> 00:42:17,680
V1 is plus or minus 1 and delta
V2 is plus or minus 2 and 0.

589
00:42:17,680 --> 00:42:24,160
And so there are then six
possible selection rules

590
00:42:24,160 --> 00:42:26,360
associated with
these sorts of terms.

591
00:42:26,360 --> 00:42:28,840
And what you want to
do is do the algebra

592
00:42:28,840 --> 00:42:31,720
that combines all this
horrible stuff according

593
00:42:31,720 --> 00:42:36,400
to selection rule that leads to
simplification of the formulas.

594
00:42:36,400 --> 00:42:38,050
And then once you've
got everything

595
00:42:38,050 --> 00:42:39,820
sorted according
to selection rule,

596
00:42:39,820 --> 00:42:41,410
then you can calculate
what happens.

597
00:42:46,070 --> 00:42:48,210
I can't promise that
I will never give you

598
00:42:48,210 --> 00:42:50,940
a two-mode problem
on an exam, but I

599
00:42:50,940 --> 00:42:54,280
can promise I will give
you a one-mode problem.

600
00:42:54,280 --> 00:42:57,374
And so you want
to really know how

601
00:42:57,374 --> 00:42:58,540
to do these sorts of things.

602
00:43:10,170 --> 00:43:13,800
So non-degenerate
perturbation theory

603
00:43:13,800 --> 00:43:19,230
works when the energy
denominator is large compared

604
00:43:19,230 --> 00:43:22,790
to the coupling matrix element.

605
00:43:22,790 --> 00:43:28,010
And accidents occur
when you have, say,

606
00:43:28,010 --> 00:43:33,110
a situation where omega
1 is approximately

607
00:43:33,110 --> 00:43:36,420
equal to 2 omega 2.

608
00:43:36,420 --> 00:43:38,390
Now this isn't
just blowing smoke.

609
00:43:41,070 --> 00:43:44,850
This happens an
amazing number of times

610
00:43:44,850 --> 00:43:49,860
because stretches are
higher frequency than bends.

611
00:43:49,860 --> 00:43:52,380
And it's very common
for the bending modes

612
00:43:52,380 --> 00:43:56,070
to be roughly half or
one third the frequency

613
00:43:56,070 --> 00:43:57,736
of a stretching mode.

614
00:43:57,736 --> 00:43:58,860
And so you get a resonance.

615
00:44:02,420 --> 00:44:04,280
So this is special
because now it's

616
00:44:04,280 --> 00:44:09,140
violating the
fundamental approximation

617
00:44:09,140 --> 00:44:13,170
of non-degenerate
perturbation theory.

618
00:44:13,170 --> 00:44:15,080
But it's a two-level
interaction.

619
00:44:15,080 --> 00:44:17,930
And so you can say, I know
how to do a two-level problem.

620
00:44:17,930 --> 00:44:19,490
I can solve that.

621
00:44:19,490 --> 00:44:24,260
And these resonances have names.

622
00:44:24,260 --> 00:44:28,724
There is a Fermi and there
is a Darling-Dennison.

623
00:44:33,680 --> 00:44:39,350
The Fermi resonance was
discovered and understood

624
00:44:39,350 --> 00:44:40,490
by Fermi.

625
00:44:40,490 --> 00:44:41,955
And it has to do with CO2.

626
00:44:47,610 --> 00:44:54,110
Omega 1 in CO2 is
approximately twice omega 2.

627
00:44:54,110 --> 00:44:59,600
The symmetric stretch and the
bend are in Fermi resonance.

628
00:44:59,600 --> 00:45:01,940
So what happens then?

629
00:45:01,940 --> 00:45:12,575
Suppose we have a level
that involves V1, V2, V3.

630
00:45:15,080 --> 00:45:23,520
And nearby there is a level
V1 minus 1 V2 plus 2 V3.

631
00:45:26,050 --> 00:45:28,100
And because of the
energy denominators,

632
00:45:28,100 --> 00:45:31,710
these two guys are
nearly degenerate.

633
00:45:31,710 --> 00:45:34,730
So what happens is
one gets shifted up,

634
00:45:34,730 --> 00:45:37,910
one gets shifted
down a little bit.

635
00:45:37,910 --> 00:45:43,975
And they're out of the
expectation of smooth behavior.

636
00:45:46,970 --> 00:45:53,230
And it might also be that this
state is what we call bright

637
00:45:53,230 --> 00:45:56,310
and this is called dark.

638
00:45:56,310 --> 00:45:59,670
This state might be connected
by an allowed transition

639
00:45:59,670 --> 00:46:01,905
from a lower--

640
00:46:01,905 --> 00:46:06,300
an initial state
and this might not.

641
00:46:06,300 --> 00:46:11,500
So the levels repel
because they're interacting

642
00:46:11,500 --> 00:46:13,050
and they're out of position.

643
00:46:13,050 --> 00:46:15,390
And this guy is
supposed to be bright.

644
00:46:15,390 --> 00:46:18,210
We're supposed to see a
transition in the spectrum.

645
00:46:18,210 --> 00:46:20,880
And this one-- well, it should
have been somewhere else.

646
00:46:20,880 --> 00:46:23,400
But we shouldn't see
it because it's dark.

647
00:46:23,400 --> 00:46:24,690
It's forbidden.

648
00:46:24,690 --> 00:46:28,290
But because of the interaction
between these two levels,

649
00:46:28,290 --> 00:46:31,050
the eigenstates have
mixed character.

650
00:46:31,050 --> 00:46:34,740
And you get both level
shifts and extra lines.

651
00:46:34,740 --> 00:46:39,210
This is called a
spectroscopic perturbation.

652
00:46:39,210 --> 00:46:42,570
This is the core
of everything I've

653
00:46:42,570 --> 00:46:45,210
done for the last 45 years--

654
00:46:45,210 --> 00:46:47,850
spectroscopic perturbations.

655
00:46:47,850 --> 00:46:52,770
And so you can learn about
some of these coupling terms

656
00:46:52,770 --> 00:46:56,340
because, instead of
hiding in the forest

657
00:46:56,340 --> 00:47:02,460
of these small corrections,
you get a big effect.

658
00:47:02,460 --> 00:47:04,200
And it's easy to
observe and it's

659
00:47:04,200 --> 00:47:09,390
easy to determine each coupling
term from these resonances.

660
00:47:12,580 --> 00:47:14,500
And the last thing
I want to talk

661
00:47:14,500 --> 00:47:18,440
about today is a little
bit of philosophy.

662
00:47:18,440 --> 00:47:23,150
This is why, among
spectroscopists

663
00:47:23,150 --> 00:47:27,140
or physical chemists,
there are two communities--

664
00:47:27,140 --> 00:47:31,245
communities that like small
molecules and communities

665
00:47:31,245 --> 00:47:36,990
that like big molecules, because
they're really different.

666
00:47:36,990 --> 00:47:39,960
For the small
molecule community,

667
00:47:39,960 --> 00:47:42,380
you have this
accidental resonance

668
00:47:42,380 --> 00:47:45,120
and you get this sort of thing.

669
00:47:45,120 --> 00:47:53,800
For big molecules,
you have a resonance

670
00:47:53,800 --> 00:47:55,690
with a dense manifold of levels.

671
00:47:58,300 --> 00:48:01,020
All of these levels
share, say, the character

672
00:48:01,020 --> 00:48:02,880
of the dark state.

673
00:48:02,880 --> 00:48:06,750
And so they can all
interact with this guy.

674
00:48:06,750 --> 00:48:12,990
And so in this case, you
get the main transition

675
00:48:12,990 --> 00:48:14,670
in an extra line.

676
00:48:14,670 --> 00:48:18,789
In this case, if you
have low resolution,

677
00:48:18,789 --> 00:48:19,830
you get a broadened line.

678
00:48:22,470 --> 00:48:24,390
And if you have high
enough resolution,

679
00:48:24,390 --> 00:48:27,710
you see that there is a whole
bunch of eigenstates under it.

680
00:48:30,500 --> 00:48:32,000
And if it's a large
enough molecule,

681
00:48:32,000 --> 00:48:34,850
you couldn't
resolve them anyway.

682
00:48:34,850 --> 00:48:43,430
And so some people are
always dealing with fast IVR,

683
00:48:43,430 --> 00:48:47,600
fast dephasing of
the bright transition

684
00:48:47,600 --> 00:48:50,680
into a dark manifold.

685
00:48:50,680 --> 00:48:52,150
And other people
are always looking

686
00:48:52,150 --> 00:48:53,890
at these sorts of things.

687
00:48:53,890 --> 00:48:55,840
Now I like these
because they really

688
00:48:55,840 --> 00:48:57,590
give you a lot of information.

689
00:48:57,590 --> 00:49:01,440
Now you can get the
information you want from this

690
00:49:01,440 --> 00:49:03,630
because the width of
this thing is related

691
00:49:03,630 --> 00:49:09,770
to the number of dark states and
their average coupling matrix

692
00:49:09,770 --> 00:49:10,860
element.

693
00:49:10,860 --> 00:49:12,300
That's called
Fermi's golden rule.

694
00:49:15,250 --> 00:49:17,060
We'll talk about that later.

695
00:49:17,060 --> 00:49:19,030
So I think there's
a pretty good place

696
00:49:19,030 --> 00:49:22,930
to stop because what I
try to do is to show you,

697
00:49:22,930 --> 00:49:26,910
yes, it can be
really complicated.

698
00:49:26,910 --> 00:49:28,880
But it's something
that you can do,

699
00:49:28,880 --> 00:49:33,290
and you can project out
the coupling constants

700
00:49:33,290 --> 00:49:36,500
that you want in order to
determine the stuff that

701
00:49:36,500 --> 00:49:38,510
is relevant to your experiment.

702
00:49:38,510 --> 00:49:42,920
But then there is this dichotomy
between small molecules

703
00:49:42,920 --> 00:49:44,900
where the vibrational
density of states

704
00:49:44,900 --> 00:49:48,890
is always smaller until you
get to really high energy.

705
00:49:48,890 --> 00:49:52,650
And bigger molecules--
now they're not very big.

706
00:49:52,650 --> 00:49:57,422
Benzene is plenty big
for this sort of thing.

707
00:49:57,422 --> 00:49:58,750
And there is the question.

708
00:49:58,750 --> 00:50:01,450
For example, in
some big molecules,

709
00:50:01,450 --> 00:50:05,770
when you excite an electronic
transition from the ground

710
00:50:05,770 --> 00:50:07,520
state to some excited state--

711
00:50:07,520 --> 00:50:10,420
so here's S0, S1.

712
00:50:10,420 --> 00:50:15,730
Sometimes you can't see
any fluorescence from S1

713
00:50:15,730 --> 00:50:23,100
because the dephasing is so
fast that there's nothing.

714
00:50:23,100 --> 00:50:25,920
And so it says,
well, tough luck.

715
00:50:25,920 --> 00:50:28,810
You can't do
spectroscopy in emission.

716
00:50:28,810 --> 00:50:31,830
But you can still see
the absorption spectrum

717
00:50:31,830 --> 00:50:34,920
because then your signal
is the removal of photons

718
00:50:34,920 --> 00:50:37,980
from your beam as
opposed to fluorescence.

719
00:50:37,980 --> 00:50:41,430
So there's a huge amount of
photochemistry and interesting

720
00:50:41,430 --> 00:50:47,370
stuff connected with a
large density of states.

721
00:50:47,370 --> 00:50:49,950
And again, when I was
a graduate student,

722
00:50:49,950 --> 00:50:57,450
there was a huge controversy
about non-radiative transitions

723
00:50:57,450 --> 00:51:00,360
in medium sized molecules.

724
00:51:00,360 --> 00:51:07,270
And there was one
community that says,

725
00:51:07,270 --> 00:51:12,050
the collisions which transfer
population between levels

726
00:51:12,050 --> 00:51:17,060
are so fast that, in order
to turn off this broadening,

727
00:51:17,060 --> 00:51:20,360
you had to go to
really low pressure

728
00:51:20,360 --> 00:51:22,850
because then there
wouldn't be collisions.

729
00:51:22,850 --> 00:51:24,860
And the answer,
whenever somebody

730
00:51:24,860 --> 00:51:27,854
failed to see sharp
spectra, was, well,

731
00:51:27,854 --> 00:51:29,270
you're not at low
enough pressure.

732
00:51:31,820 --> 00:51:36,140
But this has nothing
to do with collisions.

733
00:51:36,140 --> 00:51:38,990
And that got resolved
by two gentlemen

734
00:51:38,990 --> 00:51:41,880
called Bixon and Jortner.

735
00:51:41,880 --> 00:51:46,820
Those are two names that any
educated physical chemist

736
00:51:46,820 --> 00:51:50,150
will know to say, oh,
that's the Bixon-Jortner.

737
00:51:50,150 --> 00:51:52,110
And they're still alive.

738
00:51:52,110 --> 00:51:54,010
They're still doing
beautiful stuff.

739
00:51:54,010 --> 00:51:56,510
But anyway, that's all
I want to say today.

740
00:51:56,510 --> 00:52:02,030
I will do details on one
mode and Morse oscillator

741
00:52:02,030 --> 00:52:05,440
in other sorts of
things next time.