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ROBERT FIELD: Last time, we
talked about photochemistry.

9
00:00:25,930 --> 00:00:29,710
And the crucial thing
in photochemistry

10
00:00:29,710 --> 00:00:35,320
is that the density
of vibrational states

11
00:00:35,320 --> 00:00:38,930
increases extremely rapidly--

12
00:00:38,930 --> 00:00:43,610
extremely, extremely--
and especially

13
00:00:43,610 --> 00:00:46,140
for larger molecules.

14
00:00:46,140 --> 00:00:50,910
And this enables you to
understand intermolecular

15
00:00:50,910 --> 00:00:54,180
vibrational redistribution,
intersystem crossing,

16
00:00:54,180 --> 00:00:55,700
internal conversion.

17
00:00:55,700 --> 00:00:57,960
The matrix elements
are different in each

18
00:00:57,960 --> 00:00:59,440
of these cases.

19
00:00:59,440 --> 00:01:04,860
But what happens when the
vibrational density of states

20
00:01:04,860 --> 00:01:12,610
gets large is that you
have a special state, which

21
00:01:12,610 --> 00:01:14,830
is a localized
state or something

22
00:01:14,830 --> 00:01:19,440
that you care about that
you think you understand--

23
00:01:19,440 --> 00:01:24,170
a bright state-- which is
different from the masses.

24
00:01:24,170 --> 00:01:29,390
And because of the density
of states, that bright state

25
00:01:29,390 --> 00:01:32,480
or that special
state, gets diluted

26
00:01:32,480 --> 00:01:40,470
into an extremely dense manifold
of uninteresting states.

27
00:01:40,470 --> 00:01:43,460
And as a result,
the system forgets

28
00:01:43,460 --> 00:01:47,550
whatever the bright state
wanted it to tell you.

29
00:01:47,550 --> 00:01:53,690
And so you get faster
decay of fluorescence,

30
00:01:53,690 --> 00:01:56,810
not because the
population goes away,

31
00:01:56,810 --> 00:02:01,180
but because the unique
thing-- the bright state,

32
00:02:01,180 --> 00:02:04,820
the localized state-- the
thing that you can prepare

33
00:02:04,820 --> 00:02:08,030
by excitation from the
ground state, which is always

34
00:02:08,030 --> 00:02:13,370
localized, its
character is dissipated.

35
00:02:13,370 --> 00:02:20,640
And so you get what's
called statistical behavior.

36
00:02:20,640 --> 00:02:25,270
Now I have a very strong feeling
about statistical behavior,

37
00:02:25,270 --> 00:02:29,430
and that it's mostly a fraud.

38
00:02:29,430 --> 00:02:32,820
But the current limit
for statistical behavior

39
00:02:32,820 --> 00:02:38,790
is when the density of
states is on the order of 100

40
00:02:38,790 --> 00:02:40,920
per wave number.

41
00:02:40,920 --> 00:02:45,660
That's the usual threshold
for statistical behavior.

42
00:02:45,660 --> 00:02:49,650
As we know more, as we develop
better experimental techniques,

43
00:02:49,650 --> 00:02:53,610
we're going to push that curtain
of statistical behavior back.

44
00:02:53,610 --> 00:02:58,020
And we'll understand more
about the short-time dynamics

45
00:02:58,020 --> 00:03:04,500
and whatever we want to do
about manipulating systems.

46
00:03:04,500 --> 00:03:10,860
So statistical just
means we don't know,

47
00:03:10,860 --> 00:03:13,360
but it's not what we expected.

48
00:03:13,360 --> 00:03:16,160
And it's usually boring.

49
00:03:16,160 --> 00:03:18,420
OK.

50
00:03:18,420 --> 00:03:22,110
Today I'm going to talk
about the discrete variable

51
00:03:22,110 --> 00:03:26,760
representation, which is really
weird and wonderful thing,

52
00:03:26,760 --> 00:03:30,270
which is entirely inappropriate
for a course at this level.

53
00:03:33,560 --> 00:03:37,240
But I think you'll like it.

54
00:03:37,240 --> 00:03:40,360
So in order to talk about
the discrete variable

55
00:03:40,360 --> 00:03:45,620
representation, I'll introduce
you to delta functions.

56
00:03:45,620 --> 00:03:48,730
And you sort of know
what delta functions are.

57
00:03:48,730 --> 00:03:53,440
And then I'm going to say,
well for any one-dimensional

58
00:03:53,440 --> 00:03:55,310
problem--

59
00:03:55,310 --> 00:03:58,210
and remember, you can have
a many-dimensional problem

60
00:03:58,210 --> 00:04:01,570
treated as a whole bunch of
one-dimensional problems.

61
00:04:01,570 --> 00:04:05,510
And so it really is
a general problem.

62
00:04:05,510 --> 00:04:09,840
But if you have a
one-dimensional potential,

63
00:04:09,840 --> 00:04:14,730
you can obtain the energy
levels and wave functions

64
00:04:14,730 --> 00:04:17,850
resulting from that
one-dimensional potential.

65
00:04:17,850 --> 00:04:20,459
Regardless of how
horrible it is,

66
00:04:20,459 --> 00:04:22,770
without using perturbation
theory, which is

67
00:04:22,770 --> 00:04:25,110
should make you
feel pretty good.

68
00:04:25,110 --> 00:04:28,260
Because perturbation
theory is labor intensive

69
00:04:28,260 --> 00:04:30,826
before you use the computer.

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00:04:30,826 --> 00:04:35,450
The computer helps you,
but this is labor free

71
00:04:35,450 --> 00:04:37,482
because the computer
does everything.

72
00:04:41,180 --> 00:04:43,490
At the beginning of
the course, I said,

73
00:04:43,490 --> 00:04:47,940
you cannot experimentally
measure a wave function.

74
00:04:47,940 --> 00:04:50,005
And that's true.

75
00:04:50,005 --> 00:04:57,330
but if you can deal
with any potential

76
00:04:57,330 --> 00:05:06,390
and have a set of experimental
observations, energy levels,

77
00:05:06,390 --> 00:05:11,250
which you fit to an
effective Hamiltonian,

78
00:05:11,250 --> 00:05:15,480
you can generate the
wave functions associated

79
00:05:15,480 --> 00:05:18,150
with the effective
Hamiltonian directly

80
00:05:18,150 --> 00:05:20,620
from experimental observations.

81
00:05:20,620 --> 00:05:23,400
So it's not a
direct observation,

82
00:05:23,400 --> 00:05:25,770
but you get wave functions.

83
00:05:25,770 --> 00:05:31,010
And you can have whatever
complexity you want.

84
00:05:31,010 --> 00:05:32,430
OK, so let's begin.

85
00:05:36,380 --> 00:05:37,350
Delta functions.

86
00:05:45,570 --> 00:05:48,140
So this is not the
Kronecker delta.

87
00:05:48,140 --> 00:05:51,800
This is the delta
function that is

88
00:05:51,800 --> 00:05:54,440
a useful computational trick.

89
00:05:54,440 --> 00:05:55,610
And it's more than that.

90
00:06:04,100 --> 00:06:05,775
We write something like this.

91
00:06:15,770 --> 00:06:18,480
OK, this is a delta function.

92
00:06:18,480 --> 00:06:28,460
And it says that this thing is
non-zero when x is equal to xi.

93
00:06:28,460 --> 00:06:30,170
And it's really big.

94
00:06:30,170 --> 00:06:33,000
And it's 0 everywhere else.

95
00:06:33,000 --> 00:06:42,380
And as a result, you can say
that we get xi delta x xi.

96
00:06:42,380 --> 00:06:46,040
So that's an
eigenvalue equation.

97
00:06:46,040 --> 00:06:51,960
So we have the operator, x,
operating on this function.

98
00:06:51,960 --> 00:06:54,270
And this function has
the magical properties

99
00:06:54,270 --> 00:06:58,190
that it returns an
eigenvalue and the function.

100
00:07:02,410 --> 00:07:06,770
So this is more than
a mathematical trick.

101
00:07:06,770 --> 00:07:13,510
It's an entry into a
form of quantum mechanics

102
00:07:13,510 --> 00:07:16,300
that is truly wonderful.

103
00:07:16,300 --> 00:07:17,890
So this is part one.

104
00:07:17,890 --> 00:07:20,800
Part two will be DVR--

105
00:07:20,800 --> 00:07:24,830
discrete variable
representation.

106
00:07:24,830 --> 00:07:26,780
And the issue varies.

107
00:07:26,780 --> 00:07:32,117
Suppose we have a matrix
representation of an operator.

108
00:07:35,940 --> 00:07:41,380
Well, suppose we wanted some
function of that operator.

109
00:07:41,380 --> 00:07:43,915
How do we generate that?

110
00:07:43,915 --> 00:07:45,290
And there are lots
of cases where

111
00:07:45,290 --> 00:07:46,890
you care about such a thing.

112
00:07:46,890 --> 00:07:50,690
For example, you
might want to know

113
00:07:50,690 --> 00:08:00,010
about something like this--
e to the ih t over h bar.

114
00:08:02,780 --> 00:08:07,370
This would tell you something
about how a system propagates

115
00:08:07,370 --> 00:08:09,710
under the influence
of a Hamiltonian,

116
00:08:09,710 --> 00:08:12,600
which is not diagonal.

117
00:08:12,600 --> 00:08:15,560
This is kind of important.

118
00:08:15,560 --> 00:08:20,600
Almost all of NMR is based
on this sort of thing.

119
00:08:20,600 --> 00:08:22,430
Also-- well, we'll get to it.

120
00:08:25,780 --> 00:08:39,339
So extend DVR to
include rotation.

121
00:08:41,929 --> 00:08:43,669
And then the last part will be--

122
00:08:48,640 --> 00:08:55,470
can be determined for even
the most horrible situations,

123
00:08:55,470 --> 00:08:59,730
like a potential that
does something like this,

124
00:08:59,730 --> 00:09:12,270
or something like that.

125
00:09:12,270 --> 00:09:15,700
Now we don't want
to have a continuum.

126
00:09:15,700 --> 00:09:18,300
So I put a wall here.

127
00:09:18,300 --> 00:09:23,610
But this is a summarization
from an unstable isomer

128
00:09:23,610 --> 00:09:25,350
to a stable isomer.

129
00:09:25,350 --> 00:09:27,690
This is multiple minima.

130
00:09:27,690 --> 00:09:29,400
Anything you want,
you would never

131
00:09:29,400 --> 00:09:32,175
want to perturbation
theory on it.

132
00:09:32,175 --> 00:09:38,770
And you can solve these problems
automatically and wonderfully.

133
00:09:38,770 --> 00:09:43,320
OK, so let's just play
with delta functions.

134
00:09:43,320 --> 00:09:47,190
And a very good section
on delta functions

135
00:09:47,190 --> 00:09:56,040
is in Cohen-Tannoudji, and
it's pages 1468 to 1472,

136
00:09:56,040 --> 00:09:58,410
right at the end of the book.

137
00:09:58,410 --> 00:09:59,960
But it's not because
it's so hard,

138
00:09:59,960 --> 00:10:04,190
it's just because they
decided to put it there.

139
00:10:04,190 --> 00:10:07,700
And so we have notation--

140
00:10:07,700 --> 00:10:10,430
x xi.

141
00:10:10,430 --> 00:10:16,410
It's the same thing
as x minus xi.

142
00:10:16,410 --> 00:10:24,920
So basically, if you see a
variable at a specified value,

143
00:10:24,920 --> 00:10:27,560
it's equivalent to
this, and this thing

144
00:10:27,560 --> 00:10:32,980
is 0 everywhere except
when x is equal to xi.

145
00:10:32,980 --> 00:10:39,110
So the thing in parentheses
is the critical thing.

146
00:10:39,110 --> 00:10:41,480
And it has the
property that if we

147
00:10:41,480 --> 00:10:45,500
do an integral from minus
infinity to infinity,

148
00:10:45,500 --> 00:10:47,523
some function of x--

149
00:10:47,523 --> 00:10:54,980
dx-- delta x xi dx.

150
00:10:54,980 --> 00:10:57,680
We get f of xi.

151
00:10:57,680 --> 00:10:58,886
Isn't that wonderful?

152
00:11:02,290 --> 00:11:07,910
I mean, it's a very
lovely mathematical trick.

153
00:11:07,910 --> 00:11:08,910
But it's more than that.

154
00:11:13,860 --> 00:11:19,090
It says this thing is big
when x is equal to xi.

155
00:11:19,090 --> 00:11:22,570
It's 0 everywhere else.

156
00:11:22,570 --> 00:11:24,760
And it's normalized
to 1 in the sense

157
00:11:24,760 --> 00:11:28,210
that, well you get
back the function

158
00:11:28,210 --> 00:11:31,300
that you started with,
but at a particular value.

159
00:11:31,300 --> 00:11:37,510
So it's infinite,
but it's normalized.

160
00:11:37,510 --> 00:11:40,530
It should bother you,
but it's fantastic

161
00:11:40,530 --> 00:11:42,790
and you can deal with this.

162
00:11:49,880 --> 00:11:52,094
It's an eigenvalue equation.

163
00:11:52,094 --> 00:11:53,510
I mean, you can
say, all right, we

164
00:11:53,510 --> 00:12:01,710
have delta x xi xi delta x xi.

165
00:12:07,040 --> 00:12:11,420
And we can have delta functions
in position, in momentum,

166
00:12:11,420 --> 00:12:14,590
in anything you want.

167
00:12:14,590 --> 00:12:15,820
And they're useful.

168
00:12:22,850 --> 00:12:31,300
So suppose we have some
function, psi of x,

169
00:12:31,300 --> 00:12:34,930
and we want to
find out something

170
00:12:34,930 --> 00:12:37,600
about how it's composed.

171
00:12:37,600 --> 00:12:51,000
And so we say, well, we have
c of xi j delta x minus xi dx.

172
00:12:51,000 --> 00:12:57,000
This is the standard method
for expanding a function.

173
00:12:57,000 --> 00:13:00,240
OK, what we want is these
expansion coefficients.

174
00:13:00,240 --> 00:13:03,180
And you get them in
the standard way.

175
00:13:03,180 --> 00:13:06,520
And that is by--

176
00:13:06,520 --> 00:13:10,030
normally, when you
have a function,

177
00:13:10,030 --> 00:13:15,120
you write it ck j the function--

178
00:13:17,670 --> 00:13:18,170
I'm sorry.

179
00:13:24,620 --> 00:13:28,160
So here we have an
expansion of members

180
00:13:28,160 --> 00:13:30,610
of a complete set of functions.

181
00:13:30,610 --> 00:13:34,940
And in order to get the
expansion coefficients,

182
00:13:34,940 --> 00:13:44,800
you do the standard trick of
integrating phi j star, psi j

183
00:13:44,800 --> 00:13:46,490
dx.

184
00:13:46,490 --> 00:13:48,810
OK, so this is familiar.

185
00:13:48,810 --> 00:13:50,470
This is not.

186
00:13:50,470 --> 00:13:53,720
But it's the same business.

187
00:13:53,720 --> 00:13:59,250
In fact, we've been using
a notation incorrectly--

188
00:13:59,250 --> 00:14:01,550
the Dirac notation.

189
00:14:01,550 --> 00:14:05,030
We normally think that
if we have something

190
00:14:05,030 --> 00:14:14,197
like k and psi k of x,
they're just different ways

191
00:14:14,197 --> 00:14:15,280
of writing the same thing.

192
00:14:21,030 --> 00:14:26,130
But the equivalent to
the Schrodinger picture

193
00:14:26,130 --> 00:14:31,730
is really x psi.

194
00:14:31,730 --> 00:14:34,170
This is a vector,
not a function.

195
00:14:34,170 --> 00:14:38,790
This is a set of vectors.

196
00:14:38,790 --> 00:14:43,850
And this is how you relate
vectors to functions.

197
00:14:43,850 --> 00:14:46,620
We just have suppressed that.

198
00:14:46,620 --> 00:14:49,620
Because it's easier to
think that the wave function

199
00:14:49,620 --> 00:14:53,601
is equivalent to
the symbol here.

200
00:14:53,601 --> 00:14:55,840
It's not.

201
00:14:55,840 --> 00:14:57,760
And if you're going to
be doing derivations

202
00:14:57,760 --> 00:15:00,450
where you flip back and forth
between representations,

203
00:15:00,450 --> 00:15:03,835
you better remember this,
otherwise it won't make sense.

204
00:15:09,290 --> 00:15:12,170
This is familiar
stuff, you just didn't

205
00:15:12,170 --> 00:15:15,230
realize you understood it.

206
00:15:15,230 --> 00:15:19,460
OK, and so now, often,
you want to have

207
00:15:19,460 --> 00:15:23,914
a mathematical representation
of a delta function.

208
00:15:23,914 --> 00:15:25,455
And so this thing
has to be localized

209
00:15:25,455 --> 00:15:28,124
and it has to be normalized.

210
00:15:28,124 --> 00:15:30,040
And there are a whole
bunch of representations

211
00:15:30,040 --> 00:15:31,660
that work really well.

212
00:15:31,660 --> 00:15:35,490
One of them is we
take the limit,

213
00:15:35,490 --> 00:15:43,710
epsilon goes to 0 from
the positive side,

214
00:15:43,710 --> 00:15:50,330
and 1 over 2 epsilon e to
the minus x over epsilon.

215
00:15:53,090 --> 00:16:05,340
Well this guy, as
epsilon goes to 0,

216
00:16:05,340 --> 00:16:07,860
this becomes minus infinity.

217
00:16:07,860 --> 00:16:15,520
And it's 0 everywhere
except for x is equal to 0.

218
00:16:15,520 --> 00:16:20,800
And so if we wanted to
put x minus xi, then fine.

219
00:16:20,800 --> 00:16:24,820
Then it would be 0 everywhere
except when x is equal to xi.

220
00:16:24,820 --> 00:16:25,680
So that's one.

221
00:16:25,680 --> 00:16:27,970
That's a simple one.

222
00:16:27,970 --> 00:16:34,120
Another one is limit
as epsilon goes

223
00:16:34,120 --> 00:16:42,070
to 0 from the positive side of
1 over pi times epsilon over x

224
00:16:42,070 --> 00:16:46,020
squared plus epsilon squared.

225
00:16:46,020 --> 00:16:53,610
This also has the properties
of being an infinite spike at x

226
00:16:53,610 --> 00:16:55,950
equals 0.

227
00:16:55,950 --> 00:16:58,807
And it's also Lorentzian.

228
00:16:58,807 --> 00:16:59,890
And it has the full width.

229
00:16:59,890 --> 00:17:02,340
It'd have maximum-- or a
half width it would have--

230
00:17:02,340 --> 00:17:06,417
a full width at half
maximum of epsilon.

231
00:17:06,417 --> 00:17:08,250
If you make the width
as narrow as you want,

232
00:17:08,250 --> 00:17:10,410
well that's what this is.

233
00:17:10,410 --> 00:17:12,510
OK, there are a whole
bunch of representations.

234
00:17:12,510 --> 00:17:14,640
I'm going to stop giving
them because there's

235
00:17:14,640 --> 00:17:17,040
a lot of stuff I
want to say, and you

236
00:17:17,040 --> 00:17:18,270
can see them in the notes.

237
00:17:22,460 --> 00:17:29,070
OK, so what if we have
something like xi,

238
00:17:29,070 --> 00:17:34,210
that just is the same thing
as saying x minus xi is 0.

239
00:17:40,500 --> 00:17:45,300
You can produce a delta
function localized at any point

240
00:17:45,300 --> 00:17:48,110
by using this trick.

241
00:17:48,110 --> 00:17:49,830
OK, there are some
other tricks that you

242
00:17:49,830 --> 00:17:55,440
can do with delta functions,
which is a little surprising.

243
00:17:55,440 --> 00:18:01,420
And that's really
for the future.

244
00:18:01,420 --> 00:18:08,424
Delta of minus x is
equal to delta of x.

245
00:18:08,424 --> 00:18:09,340
It's an even function.

246
00:18:12,610 --> 00:18:14,950
Derivative of a delta
function is an odd function.

247
00:18:19,140 --> 00:18:24,270
Delta-- a constant times x
is 1 over the absolute value

248
00:18:24,270 --> 00:18:26,550
of that constant, delta x.

249
00:18:29,070 --> 00:18:33,450
A little surprising, but
if you have x, which--

250
00:18:38,710 --> 00:18:39,710
I better not say that.

251
00:18:39,710 --> 00:18:45,970
OK, we have now another
fantastic thing-- g of x.

252
00:18:45,970 --> 00:18:49,780
So what is the delta
function of a function?

253
00:18:52,530 --> 00:19:07,670
Well it's a sum over j of the
g ex and x of j times delta

254
00:19:07,670 --> 00:19:08,776
x minus xj.

255
00:19:18,270 --> 00:19:22,670
This is something
where you're harvesting

256
00:19:22,670 --> 00:19:27,190
the zeros of this function.

257
00:19:27,190 --> 00:19:33,200
OK, when this is 0,
we get a big thing.

258
00:19:36,350 --> 00:19:40,130
So the delta function
of our function

259
00:19:40,130 --> 00:19:45,640
is a sum over the derivative--

260
00:19:45,640 --> 00:19:49,260
the zeros of g of x.

261
00:19:49,260 --> 00:19:50,990
And at times, it's kind of neat.

262
00:19:54,140 --> 00:19:58,210
OK, there's also stuff
in Cohen-Tannoudji

263
00:19:58,210 --> 00:20:01,870
on Fourier transforms
of the delta functions.

264
00:20:01,870 --> 00:20:04,530
I'm not going to
talk about that.

265
00:20:04,530 --> 00:20:08,340
But one of the things is, if
you have a delta function and x,

266
00:20:08,340 --> 00:20:10,340
and you take the Fourier
transformer with it,

267
00:20:10,340 --> 00:20:13,040
you get a delta
function of p, momentum.

268
00:20:15,620 --> 00:20:16,730
Kind of useful.

269
00:20:16,730 --> 00:20:21,040
It enables you to do a transform
between an x representation

270
00:20:21,040 --> 00:20:26,500
and a position representation
and a momentum representation.

271
00:20:26,500 --> 00:20:28,340
It's kind of useful.

272
00:20:28,340 --> 00:20:30,320
OK, now we're going to
get to the good stuff.

273
00:20:35,710 --> 00:20:40,900
So for every system
that has a potential,

274
00:20:40,900 --> 00:20:45,010
and where the
potential has minima,

275
00:20:45,010 --> 00:20:46,950
what is the minimum potential?

276
00:20:46,950 --> 00:20:50,370
What is the condition for
a minimum of the potential?

277
00:20:53,970 --> 00:20:54,590
Yes?

278
00:20:54,590 --> 00:20:57,600
AUDIENCE: The first derivative
has to be 0 with respect

279
00:20:57,600 --> 00:20:58,277
to coordinates.

280
00:20:58,277 --> 00:20:59,110
ROBERT FIELD: Right.

281
00:20:59,110 --> 00:21:03,450
And so, any time
you have a minimum,

282
00:21:03,450 --> 00:21:07,190
the first term in the potential
is the quadratic term.

283
00:21:07,190 --> 00:21:10,320
And that's the same thing
as a harmonic oscillator.

284
00:21:10,320 --> 00:21:12,645
The rest is just excess baggage.

285
00:21:15,280 --> 00:21:17,440
I mean, that's what we do
for perturbation theory.

286
00:21:17,440 --> 00:21:21,070
We say, OK, we're going to
represent some arbitrary

287
00:21:21,070 --> 00:21:25,250
potential as a
harmonic oscillator.

288
00:21:25,250 --> 00:21:28,390
And all of the bad stuff,
all of the higher powers

289
00:21:28,390 --> 00:21:31,720
of the coordinate get treated
by perturbation theory.

290
00:21:34,410 --> 00:21:35,530
And you know how to do it.

291
00:21:35,530 --> 00:21:37,980
And you're not
excited about doing it

292
00:21:37,980 --> 00:21:41,100
because it's kind of
algebraically horrible.

293
00:21:43,780 --> 00:21:46,410
And nobody is going
to check your algebra.

294
00:21:46,410 --> 00:21:51,100
And almost guaranteed, you're
going to make a mistake.

295
00:21:51,100 --> 00:21:55,300
(Exam 2!)

296
00:21:55,300 --> 00:21:58,420
So it would be nice
to be able to deal

297
00:21:58,420 --> 00:22:00,730
with arbitrary potentials--

298
00:22:00,730 --> 00:22:04,690
potentials that might have
multiple minima, or might

299
00:22:04,690 --> 00:22:09,890
have all sorts of strange
stuff without doing

300
00:22:09,890 --> 00:22:10,820
perturbation theory.

301
00:22:15,450 --> 00:22:17,000
And that's what DVR does.

302
00:22:21,320 --> 00:22:27,400
So for example, we
want to know how

303
00:22:27,400 --> 00:22:41,630
to derive a matrix
representation of a matrix.

304
00:22:41,630 --> 00:22:46,360
So often, we have operators
like the overlap integral

305
00:22:46,360 --> 00:22:48,280
or the Hamiltonian.

306
00:22:48,280 --> 00:22:51,880
So we have the
operator, the S matrix,

307
00:22:51,880 --> 00:22:53,890
or the Hamiltonian matrix.

308
00:22:53,890 --> 00:22:57,460
And often, we want
to have something

309
00:22:57,460 --> 00:23:01,300
that is a matrix
representation of a matrix.

310
00:23:04,300 --> 00:23:10,150
For example, if we're dealing
with a problem in quantum

311
00:23:10,150 --> 00:23:17,470
chemistry, where our basis
set is not orthonormal,

312
00:23:17,470 --> 00:23:22,120
there is a trick
using the S matrix

313
00:23:22,120 --> 00:23:26,520
to orthonormalize everything.

314
00:23:26,520 --> 00:23:29,970
And that's useful because
then your secular equation

315
00:23:29,970 --> 00:23:32,550
is the standard secular
equation, which computers just

316
00:23:32,550 --> 00:23:33,512
love.

317
00:23:33,512 --> 00:23:35,220
And so you just have
to tell the computer

318
00:23:35,220 --> 00:23:39,060
to do something special before,
and it orthonormalizes stuff.

319
00:23:39,060 --> 00:23:41,920
And you can do this
in matrix language.

320
00:23:41,920 --> 00:23:44,340
And if you're interested
in time evolution,

321
00:23:44,340 --> 00:23:50,960
you often want to have e to
the minus I h t over h bar.

322
00:23:50,960 --> 00:23:52,770
And that's horrible.

323
00:23:52,770 --> 00:23:58,600
But so we'd like to know how to
obtain a matrix representation

324
00:23:58,600 --> 00:23:59,965
of a function of a matrix.

325
00:24:02,870 --> 00:24:09,070
Well, suppose we
have some matrix,

326
00:24:09,070 --> 00:24:18,510
and we can transform
it to be a1, a n, 0, 0.

327
00:24:18,510 --> 00:24:20,070
We diagnose it.

328
00:24:26,880 --> 00:24:33,440
And if a is real and
symmetric, or Hermitian.

329
00:24:33,440 --> 00:24:35,960
We know that the
transformation that

330
00:24:35,960 --> 00:24:43,040
diagonalized it has the
property that t dagger

331
00:24:43,040 --> 00:24:46,550
is equal to the inverse.

332
00:24:46,550 --> 00:24:50,360
And we also know that
if we diagonalize

333
00:24:50,360 --> 00:24:56,830
a matrix the eigenvectors
that-- they say,

334
00:24:56,830 --> 00:25:01,630
if you want the first
eigenvector, the thing that

335
00:25:01,630 --> 00:25:04,300
belongs to this
eigenvalue, we want

336
00:25:04,300 --> 00:25:07,830
the first column of t dagger.

337
00:25:07,830 --> 00:25:11,370
And if you want to use
perturbation theory instead

338
00:25:11,370 --> 00:25:16,080
of the computer to calculate
t dagger, you can do that,

339
00:25:16,080 --> 00:25:18,180
and you have a
good approximation.

340
00:25:18,180 --> 00:25:23,250
And you can write the vector,
the linear combination

341
00:25:23,250 --> 00:25:27,180
of basis functions, that
corresponds to t dagger

342
00:25:27,180 --> 00:25:28,440
as fast as you can write.

343
00:25:31,344 --> 00:25:32,760
And the computer
can do it faster.

344
00:25:38,150 --> 00:25:42,380
So if the matrix is Hermitian,
then all of these guys

345
00:25:42,380 --> 00:25:43,130
are real.

346
00:25:43,130 --> 00:25:45,350
And it's just real
and symmetric.

347
00:25:45,350 --> 00:25:48,140
Well then these guys
are still numbers

348
00:25:48,140 --> 00:25:49,520
that you could
generate, but they

349
00:25:49,520 --> 00:25:51,710
might be imaginary or complex.

350
00:25:59,390 --> 00:26:02,550
Suppose we want some
function of a matrix.

351
00:26:02,550 --> 00:26:06,050
So this is a matrix, this
has to be a matrix too.

352
00:26:06,050 --> 00:26:09,590
It has to be the same dimension
as the original matrix.

353
00:26:12,390 --> 00:26:14,575
So you can do this.

354
00:26:23,790 --> 00:26:25,980
Well we can call that f twiddle.

355
00:26:30,530 --> 00:26:33,150
But we don't want f twiddle.

356
00:26:33,150 --> 00:26:44,000
And so if we do this, we have
the matrix representation

357
00:26:44,000 --> 00:26:46,570
of the function.

358
00:26:46,570 --> 00:26:50,540
So this is something that could
be proven in linear algebra,

359
00:26:50,540 --> 00:26:52,790
but not in 5.61.

360
00:26:52,790 --> 00:26:56,090
You can do power
series expansions,

361
00:26:56,090 --> 00:26:57,920
and you can show term
by term that this

362
00:26:57,920 --> 00:27:02,930
is true for small
matrices, but it's true.

363
00:27:02,930 --> 00:27:07,750
So if your computer
can diagonalize this,

364
00:27:07,750 --> 00:27:11,860
then what happens is
that we have this--

365
00:27:16,380 --> 00:27:26,400
we can write that
f twiddle is the--

366
00:27:44,300 --> 00:27:47,830
So this is the crucial thing.

367
00:27:47,830 --> 00:27:50,950
We've diagonalized
this, so we have

368
00:27:50,950 --> 00:27:55,060
a bunch of eigenvalues of a.

369
00:27:55,060 --> 00:27:59,720
So the f twiddle
is just the values

370
00:27:59,720 --> 00:28:04,910
of the function at each
of the eigenvalues.

371
00:28:10,270 --> 00:28:11,420
And we have zeros here.

372
00:28:14,590 --> 00:28:18,460
And now we don't like
this because this isn't

373
00:28:18,460 --> 00:28:20,920
the matrix representation of f.

374
00:28:20,920 --> 00:28:26,440
It's f in a different
representation.

375
00:28:26,440 --> 00:28:29,730
So we have to go back to
the original representation.

376
00:28:29,730 --> 00:28:31,320
And so that's another
transformation.

377
00:28:31,320 --> 00:28:33,600
But it uses the same matrix.

378
00:28:36,190 --> 00:28:41,800
So if you did the
work to diagonalize a,

379
00:28:41,800 --> 00:28:46,135
well then you can go back
and undiagonalize this

380
00:28:46,135 --> 00:28:49,120
f twiddle to make the
representation of f.

381
00:28:52,410 --> 00:28:55,720
And so the only work involved
is asking the computer

382
00:28:55,720 --> 00:29:01,450
to find t dagger
for the a matrix.

383
00:29:01,450 --> 00:29:05,650
And then you get the true
matrix representation

384
00:29:05,650 --> 00:29:07,810
of this function of a.

385
00:29:10,910 --> 00:29:11,600
Is it useful?

386
00:29:11,600 --> 00:29:12,100
You bet.

387
00:29:24,010 --> 00:29:27,795
Now suppose a is
infinite dimension.

388
00:29:32,920 --> 00:29:36,130
And we know that this
is a very common case.

389
00:29:36,130 --> 00:29:38,240
Because, even for the
harmonic oscillator,

390
00:29:38,240 --> 00:29:43,000
we have an infinite
number of basis functions.

391
00:29:43,000 --> 00:29:46,520
But what about using
the delta function?

392
00:29:46,520 --> 00:29:48,390
We also have an
infinite number of them.

393
00:29:52,430 --> 00:29:54,520
So what do we do?

394
00:29:54,520 --> 00:29:56,590
We can't diagonalize
an infinite matrix.

395
00:30:00,440 --> 00:30:04,160
So what we do is we truncate it.

396
00:30:09,820 --> 00:30:12,820
Now the computer is quite
happy to deal with matrices

397
00:30:12,820 --> 00:30:16,000
of dimension 1,000.

398
00:30:16,000 --> 00:30:19,060
Your computer can
diagonalize a 1,000

399
00:30:19,060 --> 00:30:25,900
by 1,000 matrix in a few
minutes, maybe a few seconds

400
00:30:25,900 --> 00:30:30,250
depending on how up
to date this thing is.

401
00:30:30,250 --> 00:30:39,280
And so what we do is we say, oh,
well let's just take a 1,000.

402
00:30:39,280 --> 00:30:42,100
So here is an infinite matrix.

403
00:30:42,100 --> 00:30:45,870
And here is a 1,000
by 1,000 block.

404
00:30:45,870 --> 00:30:48,980
That's a million elements.

405
00:30:48,980 --> 00:30:50,810
The computer doesn't care.

406
00:30:50,810 --> 00:30:53,210
And we're just going to
throw away everything else.

407
00:30:57,350 --> 00:30:59,100
We don't care.

408
00:30:59,100 --> 00:31:00,930
Now this is an approximation.

409
00:31:00,930 --> 00:31:03,440
Now you can truncate
in clever ways,

410
00:31:03,440 --> 00:31:06,500
or just tell the computer to
throw away everything above

411
00:31:06,500 --> 00:31:09,530
the 1,000th basis function.

412
00:31:09,530 --> 00:31:10,940
That's very convenient.

413
00:31:10,940 --> 00:31:13,780
The computer doesn't care.

414
00:31:13,780 --> 00:31:17,160
Now there are
transformations that say,

415
00:31:17,160 --> 00:31:20,760
well you can fold in the effects
of the remote basis states,

416
00:31:20,760 --> 00:31:25,050
and do an augmented
representation.

417
00:31:25,050 --> 00:31:28,020
But usually, you just
throw everything away.

418
00:31:28,020 --> 00:31:32,700
And now you look at this
1,000 by 1,000 matrix.

419
00:31:32,700 --> 00:31:36,510
And you get the eigenvalues.

420
00:31:36,510 --> 00:31:46,300
And so this might be
the matrix of x, or q

421
00:31:46,300 --> 00:31:48,760
if we're talking in
the usual notation.

422
00:31:48,760 --> 00:31:52,193
This is the displacement
from equilibrium.

423
00:31:56,140 --> 00:31:57,990
Well, it seems a little--

424
00:32:01,590 --> 00:32:04,721
so we would like to
find this matrix.

425
00:32:04,721 --> 00:32:05,845
Well, we know what that is.

426
00:32:10,670 --> 00:32:14,220
Because we know the relationship
between x and a plus

427
00:32:14,220 --> 00:32:17,320
a dagger are friends.

428
00:32:17,320 --> 00:32:22,360
And so we have a matrix, which
is zeros along the diagonal,

429
00:32:22,360 --> 00:32:26,230
and numbers here and here,
and zeros everywhere else.

430
00:32:26,230 --> 00:32:28,330
It doesn't take much
to program a computer

431
00:32:28,330 --> 00:32:33,520
to fill in as many one-off
the diagonal, especially

432
00:32:33,520 --> 00:32:37,090
because they're square
roots of integers.

433
00:32:37,090 --> 00:32:38,920
So that's just a
few lines of code,

434
00:32:38,920 --> 00:32:43,560
and you have the matrix
representation of x.

435
00:32:43,560 --> 00:32:45,570
Now that is infinite.

436
00:32:45,570 --> 00:32:48,180
And you're going to
say, well, I don't care.

437
00:32:48,180 --> 00:32:52,280
I'm going to just
keep the first 1,000.

438
00:32:52,280 --> 00:32:56,690
We know that we
can always write--

439
00:32:56,690 --> 00:33:03,150
so we have v of x, and
this is a matrix now.

440
00:33:03,150 --> 00:33:05,300
And it's an infinite matrix.

441
00:33:05,300 --> 00:33:12,870
But we say, oh, we just
want v of x to the 1000th,

442
00:33:12,870 --> 00:33:15,830
and we'll get v of 1,000.

443
00:33:15,830 --> 00:33:18,496
We have a 1,000
by 1,000 v matrix.

444
00:33:18,496 --> 00:33:19,620
And we know how to do this.

445
00:33:19,620 --> 00:33:21,690
We diagonalize that.

446
00:33:21,690 --> 00:33:29,670
Then we write at each
eigenvalue of x, what v of x

447
00:33:29,670 --> 00:33:31,070
is at that eigenvalue.

448
00:33:33,820 --> 00:33:39,480
And so now we have a v
matrix, which is diagonal,

449
00:33:39,480 --> 00:33:44,040
but in the wrong
representation for us.

450
00:33:44,040 --> 00:33:46,740
And then we transform back
to the harmonic oscillator

451
00:33:46,740 --> 00:33:49,180
representation.

452
00:33:49,180 --> 00:33:51,250
And so everything is fine.

453
00:33:51,250 --> 00:33:53,830
We've done this.

454
00:33:53,830 --> 00:33:55,930
And we don't know how
good it's going to be.

455
00:33:58,700 --> 00:34:02,990
But what we do is
we do this problem.

456
00:34:02,990 --> 00:34:05,020
So we have the
Hamiltonian, which

457
00:34:05,020 --> 00:34:06,910
is equal to the
kinetic energy, I'm

458
00:34:06,910 --> 00:34:11,630
going to call it
k, plus v. We know

459
00:34:11,630 --> 00:34:16,880
how to generate the
representation of v

460
00:34:16,880 --> 00:34:21,900
in the harmonic oscillator
basis by writing v

461
00:34:21,900 --> 00:34:27,570
of x, diagonlizing x, and
then undiagonalizing--

462
00:34:27,570 --> 00:34:33,300
or then writing v at each
of the eigenvalues of x,

463
00:34:33,300 --> 00:34:37,980
and then going back to the
dramatic oscillator basis.

464
00:34:37,980 --> 00:34:41,679
We know k in the harmonic
oscillator basis.

465
00:34:41,679 --> 00:34:43,830
It's just tri-diagonal,
and it has

466
00:34:43,830 --> 00:34:49,500
matrix elements delta v
equals 0 plus minus two.

467
00:34:49,500 --> 00:34:53,550
So now we have a
matrix representation

468
00:34:53,550 --> 00:34:58,320
of k, which is simple,
add a v which is--

469
00:35:00,950 --> 00:35:03,380
computer makes it simple.

470
00:35:03,380 --> 00:35:06,890
Add any v you want, there it is.

471
00:35:06,890 --> 00:35:09,140
So that's a matrix,
the Hamiltonian.

472
00:35:09,140 --> 00:35:10,910
You solve the
Schrodinger equation

473
00:35:10,910 --> 00:35:12,290
by diagonalizing this matrix.

474
00:35:15,880 --> 00:35:26,260
And so you have this h matrix,
and it's for the 1,000-member x

475
00:35:26,260 --> 00:35:30,023
matrix, and you get a
bunch of eigenvalues.

476
00:35:34,280 --> 00:35:36,950
And so then you do it again.

477
00:35:36,950 --> 00:35:42,640
And maybe use a 900 by 900,
or maybe you use a 1,100--

478
00:35:42,640 --> 00:35:45,780
you do it again.

479
00:35:45,780 --> 00:35:50,340
And then you look
at the eigenenergies

480
00:35:50,340 --> 00:35:55,710
of the Hamiltonian,
e1, say, up to e100.

481
00:35:55,710 --> 00:35:57,540
Now if you did a
1,000 by 1,000, you

482
00:35:57,540 --> 00:35:59,790
have reasonable expectation
that the first 100

483
00:35:59,790 --> 00:36:02,830
eigenvalues will be right.

484
00:36:02,830 --> 00:36:07,030
And so you compare the results
you get for the 1,000 by 1,000

485
00:36:07,030 --> 00:36:10,660
to the 900 by 900,
or 1,100 by 1,100.

486
00:36:10,660 --> 00:36:15,070
And you see how accurate
your representation

487
00:36:15,070 --> 00:36:17,067
is for the first 100.

488
00:36:17,067 --> 00:36:19,150
Normally, you don't even
care about the first 100.

489
00:36:19,150 --> 00:36:20,483
You might care about 10 of them.

490
00:36:23,200 --> 00:36:27,725
So the computer is happy to
deal with 1,000 by 1,000s.

491
00:36:27,725 --> 00:36:29,100
There's no least
squares fitting,

492
00:36:29,100 --> 00:36:30,180
so you only do it once.

493
00:36:33,050 --> 00:36:37,090
And all of a sudden,
you've got the eigenvalues,

494
00:36:37,090 --> 00:36:40,380
and you've demonstrated
how accurate they are.

495
00:36:40,380 --> 00:36:44,610
And so depending on what
precision you want--

496
00:36:44,610 --> 00:36:48,720
you can trust this up to the
100th, or maybe the 73rd,

497
00:36:48,720 --> 00:36:52,470
or whatever, to a part in
a million, or whatever.

498
00:36:52,470 --> 00:36:54,900
And so you know how
it's going to work.

499
00:36:54,900 --> 00:36:57,000
And you have a check
for convergence.

500
00:36:59,750 --> 00:37:01,520
So it doesn't matter.

501
00:37:01,520 --> 00:37:03,700
So the only thing
that you want to do

502
00:37:03,700 --> 00:37:16,620
is you want to choose a
basis set where we have--

503
00:37:16,620 --> 00:37:18,900
x is the displacement
from equilibrium.

504
00:37:25,650 --> 00:37:29,360
OK, so this is the
equilibrium value.

505
00:37:29,360 --> 00:37:33,410
This is the definition
of the displacement.

506
00:37:33,410 --> 00:37:36,920
And so you want to
be able to choose

507
00:37:36,920 --> 00:37:41,450
your basis set, which is
centered at the equilibrium

508
00:37:41,450 --> 00:37:43,040
value.

509
00:37:43,040 --> 00:37:45,260
You could do it somewhere
else, it would be stupid.

510
00:37:45,260 --> 00:37:47,290
It wouldn't converge so well.

511
00:37:47,290 --> 00:37:52,400
And you want to use the
harmonic oscillator, k over u.

512
00:37:58,360 --> 00:38:01,910
You have a couple of choices
before you start telling

513
00:38:01,910 --> 00:38:04,060
the computer to go to work.

514
00:38:04,060 --> 00:38:07,320
And you tell it, well
I think the best basis

515
00:38:07,320 --> 00:38:10,690
that will be what works
at the equilibrium--

516
00:38:10,690 --> 00:38:13,730
the lowest minimum
of the potential,

517
00:38:13,730 --> 00:38:18,410
and matches the curvature there.

518
00:38:18,410 --> 00:38:20,360
You don't have to do that.

519
00:38:20,360 --> 00:38:23,470
But it would be a
good idea to ask

520
00:38:23,470 --> 00:38:28,760
it to do a problem that's likely
to be a good representation.

521
00:38:28,760 --> 00:38:32,120
And all this you've done.

522
00:38:32,120 --> 00:38:33,200
You get the energy level.

523
00:38:33,200 --> 00:38:34,760
So what you end up getting--

524
00:38:38,980 --> 00:38:44,230
So you produced
your Hamiltonian.

525
00:38:44,230 --> 00:38:50,770
And since it's not an infinite
dimension Hamiltonian,

526
00:38:50,770 --> 00:38:54,030
we can call it an
effective Hamiltonian.

527
00:38:54,030 --> 00:38:56,370
It contains everything
that is going

528
00:38:56,370 --> 00:38:58,350
to generate the eigenvalues.

529
00:38:58,350 --> 00:39:04,620
And we get from that a
set of energy levels--

530
00:39:04,620 --> 00:39:10,050
e vi-- and a set of functions.

531
00:39:12,920 --> 00:39:17,100
So these guys are the
true energy levels.

532
00:39:17,100 --> 00:39:18,950
And these are the
linear combinations

533
00:39:18,950 --> 00:39:21,020
of harmonic oscillator
functions that

534
00:39:21,020 --> 00:39:23,640
correspond to each of them.

535
00:39:23,640 --> 00:39:24,684
Who gives that to you?

536
00:39:27,350 --> 00:39:31,020
And so then you
say, well, I want

537
00:39:31,020 --> 00:39:35,760
to represent the Hamiltonian
by a traditional thing.

538
00:39:35,760 --> 00:39:43,660
Like, I want to say that we have
omega v plus 1/2 minus omega x

539
00:39:43,660 --> 00:39:48,090
equals 1/2 squared, et cetera.

540
00:39:48,090 --> 00:39:52,830
Now we do a least squares
fit of molecular constants

541
00:39:52,830 --> 00:39:55,900
to the energy levels
of the Hamiltonian.

542
00:40:00,900 --> 00:40:04,230
And there's lots of other
things we could do, but--

543
00:40:04,230 --> 00:40:08,730
so we say, well in the spectrum,
we would observe these things,

544
00:40:08,730 --> 00:40:14,220
but we're representing them as
a power series in v plus 1/2.

545
00:40:14,220 --> 00:40:21,810
Or maybe in-- where we would
have not just the energy--

546
00:40:21,810 --> 00:40:23,670
the vibrational quantum number--

547
00:40:23,670 --> 00:40:27,110
but the rotational constant.

548
00:40:27,110 --> 00:40:41,530
We could say the potential is
v of 0 plus b x j j plus 1.

549
00:40:41,530 --> 00:40:46,410
Well, that means we could extend
this to allow the molecule

550
00:40:46,410 --> 00:40:47,580
to rotate.

551
00:40:47,580 --> 00:40:52,430
And we just need to evaluate
the rotational constant

552
00:40:52,430 --> 00:40:57,800
as a function of x, and just
add that to what we have here.

553
00:41:02,100 --> 00:41:10,130
Another thing, you have t
dagger and t for our problem.

554
00:41:10,130 --> 00:41:13,820
And you have, say,
the 1,000 by 1,000,

555
00:41:13,820 --> 00:41:18,660
and maybe the 900 by
900 representations.

556
00:41:18,660 --> 00:41:21,370
You keep them.

557
00:41:21,370 --> 00:41:25,000
Because any problem you would
have, you could use these for.

558
00:41:27,620 --> 00:41:29,660
So you still have to
do a diagonalization

559
00:41:29,660 --> 00:41:33,250
of the Hamiltonian,
but the other

560
00:41:33,250 --> 00:41:36,330
stuff you don't
have to do anymore.

561
00:41:36,330 --> 00:41:45,750
Now maybe it's too bothersome to
store a 1,000 by 1,000 t dagger

562
00:41:45,750 --> 00:41:46,360
matrix.

563
00:41:46,360 --> 00:41:48,752
It's a million elements.

564
00:41:48,752 --> 00:41:50,960
Maybe you don't, and you
can just calculate it again.

565
00:41:50,960 --> 00:41:54,660
It takes 20 minutes
or maybe less.

566
00:41:54,660 --> 00:41:58,630
And so this is a pretty good.

567
00:41:58,630 --> 00:42:02,020
OK I've skipped a lot of
stuff in the notes, because I

568
00:42:02,020 --> 00:42:04,090
wanted to get to the end.

569
00:42:04,090 --> 00:42:09,610
But the end is really
just a correction

570
00:42:09,610 --> 00:42:13,045
of what I said was impossible
at the beginning of the course.

571
00:42:18,624 --> 00:42:20,040
We have in the
Schrodinger picture

572
00:42:20,040 --> 00:42:24,060
H psi is equal to E psi, right?

573
00:42:24,060 --> 00:42:27,320
That's the Schrodinger equation.

574
00:42:27,320 --> 00:42:31,750
So this wave function
is the essential thing

575
00:42:31,750 --> 00:42:34,460
in quantum mechanics.

576
00:42:34,460 --> 00:42:38,840
And I also told you,
you can't observe this.

577
00:42:38,840 --> 00:42:41,500
So it's a very
strange theory where

578
00:42:41,500 --> 00:42:45,490
the central quality
in the theory

579
00:42:45,490 --> 00:42:50,450
is experimentally inaccessible.

580
00:42:50,450 --> 00:42:51,590
But the theory works.

581
00:42:51,590 --> 00:42:55,500
The theory gives you
everything you need.

582
00:42:55,500 --> 00:43:00,000
It enables you to find
the eigenfunctions,

583
00:43:00,000 --> 00:43:02,460
if you have the
exact Hamiltonian.

584
00:43:02,460 --> 00:43:06,660
Or it says, well we can
take a model problem

585
00:43:06,660 --> 00:43:12,240
and we can find the
eigenvalues and wave functions

586
00:43:12,240 --> 00:43:14,250
for the model problem.

587
00:43:18,810 --> 00:43:21,180
We can generate an
effective Hamiltonian,

588
00:43:21,180 --> 00:43:23,970
which is expressed in terms
of molecular constants.

589
00:43:35,170 --> 00:43:42,490
We can then determine
the potential.

590
00:43:42,490 --> 00:43:45,820
And we can also determine psi.

591
00:43:45,820 --> 00:43:46,330
All of them.

592
00:43:50,630 --> 00:43:57,250
So what I told you was
true, but only a little bit

593
00:43:57,250 --> 00:44:02,200
of a lie in the sense that
you can get as accurate

594
00:44:02,200 --> 00:44:06,220
as you want a representation
of the wave function,

595
00:44:06,220 --> 00:44:08,150
if you want it.

596
00:44:08,150 --> 00:44:12,690
And DVR gives it to
you without any effort.

597
00:44:12,690 --> 00:44:18,480
And so it doesn't matter
how terrible the potential

598
00:44:18,480 --> 00:44:26,010
is, as long as it is more
or less well behaved.

599
00:44:26,010 --> 00:44:28,290
I mean, if you had
a potential, which--

600
00:44:34,300 --> 00:44:39,070
Even if I had a v potential--
the discontinuity here--

601
00:44:39,070 --> 00:44:45,900
you would still get a
reasonable result from DVR.

602
00:44:45,900 --> 00:44:47,620
What it doesn't
like is something

603
00:44:47,620 --> 00:44:53,060
like that, because then you
have a continuum over here.

604
00:44:53,060 --> 00:44:59,440
And the continuum uses up your
basis functions pretty fast.

605
00:44:59,440 --> 00:45:01,650
I mean, yeah, it
will work for this.

606
00:45:01,650 --> 00:45:08,250
But you don't quite know
how it's going to work,

607
00:45:08,250 --> 00:45:10,680
and you have to do very
careful convergence tests,

608
00:45:10,680 --> 00:45:12,150
because this might be good.

609
00:45:12,150 --> 00:45:17,910
But up here, it's going to
use a lot of basis functions.

610
00:45:17,910 --> 00:45:20,480
So for the first
time in a long time,

611
00:45:20,480 --> 00:45:24,410
I'm finishing on time
or even a little early.

612
00:45:24,410 --> 00:45:28,730
But DVR is really a
powerful computational tool

613
00:45:28,730 --> 00:45:36,950
that, if you are doing any kind
of theoretical calculation,

614
00:45:36,950 --> 00:45:41,660
you may very well much want
to use something like this

615
00:45:41,660 --> 00:45:45,710
rather than an infinite
set of basis functions

616
00:45:45,710 --> 00:45:49,260
and perturbation theory.

617
00:45:49,260 --> 00:45:53,550
It's something where you
leave almost all of the work

618
00:45:53,550 --> 00:45:54,390
to the computer.

619
00:45:54,390 --> 00:45:57,480
You don't have to do
much besides say, well

620
00:45:57,480 --> 00:46:00,750
what is the equilibrium value?

621
00:46:00,750 --> 00:46:03,030
And what vibrational
frequency have

622
00:46:03,030 --> 00:46:06,120
I got to use for my basis set?

623
00:46:06,120 --> 00:46:10,320
And if you choose
something that's

624
00:46:10,320 --> 00:46:13,110
appropriate for the curvature
at the absolute minimum

625
00:46:13,110 --> 00:46:17,190
of the potential, you're
likely to be doing very well.

626
00:46:17,190 --> 00:46:22,720
Now other choices might mean
you don't get the first 100,

627
00:46:22,720 --> 00:46:23,940
you only get the first 50.

628
00:46:23,940 --> 00:46:27,520
But you might only care
about the first 10.

629
00:46:27,520 --> 00:46:29,590
Or you could say, I'm
going to choose something

630
00:46:29,590 --> 00:46:34,640
which is a compromise
between two minima,

631
00:46:34,640 --> 00:46:36,960
and maybe I'll do better.

632
00:46:36,960 --> 00:46:40,390
But it doesn't cost you
anything in the computer.

633
00:46:40,390 --> 00:46:43,650
Your computer is mostly
sitting idly on your desk,

634
00:46:43,650 --> 00:46:46,740
and you could have it
doing these calculations.

635
00:46:46,740 --> 00:46:54,540
And there is no problem
where you can't use DVR.

636
00:46:54,540 --> 00:46:57,460
Because if you have a
function of two variables,

637
00:46:57,460 --> 00:47:00,810
you do a two-variable DVR.

638
00:47:00,810 --> 00:47:02,970
It gets a little
bit more complicated

639
00:47:02,970 --> 00:47:05,747
because, if you
have two DVRs, now

640
00:47:05,747 --> 00:47:07,080
you're talking about a million--

641
00:47:10,150 --> 00:47:14,070
1,000 by 1,000-- two of them,
and couplings between them.

642
00:47:14,070 --> 00:47:17,600
And so maybe you have
to be a little bit more

643
00:47:17,600 --> 00:47:21,680
thoughtful about how
you employ this trick.

644
00:47:21,680 --> 00:47:23,810
But it's a very powerful trick.

645
00:47:23,810 --> 00:47:27,300
And there are other
powerful tricks

646
00:47:27,300 --> 00:47:31,260
that you can use in conjunction
with quantum chemistry that

647
00:47:31,260 --> 00:47:35,320
enable you to deal
with things like--

648
00:47:35,320 --> 00:47:41,900
I've chosen a basis set,
which is not orthonormal.

649
00:47:41,900 --> 00:47:47,130
And my computer only
knows how to diagonalize

650
00:47:47,130 --> 00:47:51,860
an ordinary Hamiltonian
without subtracting

651
00:47:51,860 --> 00:47:54,170
an overlap matrix from it.

652
00:47:54,170 --> 00:47:58,790
And so if I transform to
diagonalize the overlap matrix,

653
00:47:58,790 --> 00:48:02,220
well then I can fix the problem.

654
00:48:02,220 --> 00:48:06,080
And so there is a way of
using this kind of theory

655
00:48:06,080 --> 00:48:11,630
to fix the problem,
which is based

656
00:48:11,630 --> 00:48:15,650
on choosing a convenient way of
solving the problem, as opposed

657
00:48:15,650 --> 00:48:21,280
to the most rigorous
way of doing it.

658
00:48:21,280 --> 00:48:26,120
OK, so I'm hoping
that I will have

659
00:48:26,120 --> 00:48:28,310
a sensible lecture on
the two-level problem

660
00:48:28,310 --> 00:48:29,420
for Wednesday.

661
00:48:29,420 --> 00:48:32,000
I've been struggling with
this for a long time.

662
00:48:32,000 --> 00:48:33,840
And maybe I can do it.

663
00:48:33,840 --> 00:48:37,680
If not, I'll review
the whole course.

664
00:48:37,680 --> 00:48:40,709
OK, see you on Wednesday.