1
00:00:00,120 --> 00:00:02,430
The following content is
provided under a Creative

2
00:00:02,430 --> 00:00:03,820
Commons license.

3
00:00:03,820 --> 00:00:06,030
Your support will help
MIT OpenCourseWare

4
00:00:06,030 --> 00:00:10,120
continue to offer high quality
educational resources for free.

5
00:00:10,120 --> 00:00:12,660
To make a donation or to
view additional materials

6
00:00:12,660 --> 00:00:16,620
from hundreds of MIT courses,
visit MIT OpenCourseWare

7
00:00:16,620 --> 00:00:17,850
at ocw.mit.edu.

8
00:00:21,310 --> 00:00:26,332
PROFESSOR: The outline for today
is a little bit more review

9
00:00:26,332 --> 00:00:28,540
of feeling the power of the
creation and annihilation

10
00:00:28,540 --> 00:00:31,420
operators.

11
00:00:31,420 --> 00:00:35,320
They enable you to do anything
with harmonic oscillators,

12
00:00:35,320 --> 00:00:37,310
really fast.

13
00:00:37,310 --> 00:00:42,760
And so you don't
spend time thinking

14
00:00:42,760 --> 00:00:44,860
about how to do the math.

15
00:00:44,860 --> 00:00:48,760
You think about the meaning of
the problem and understanding

16
00:00:48,760 --> 00:00:51,940
how to manipulate,
or how to understand

17
00:00:51,940 --> 00:00:54,780
harmonic oscillators.

18
00:00:54,780 --> 00:00:56,740
And then the exciting thing.

19
00:00:56,740 --> 00:00:59,790
The real Schrodinger
equation is not the one

20
00:00:59,790 --> 00:01:01,160
we've been playing with.

21
00:01:01,160 --> 00:01:04,060
It's the time dependent
Schrodinger equation.

22
00:01:04,060 --> 00:01:09,330
And I like to introduce
the time independent one

23
00:01:09,330 --> 00:01:15,840
first, because it enables you
to sort of develop some insight

24
00:01:15,840 --> 00:01:18,030
and assemble some of
the tools before we

25
00:01:18,030 --> 00:01:21,450
hit the really serious stuff.

26
00:01:21,450 --> 00:01:25,970
And so I will talk about this
and use the time dependent

27
00:01:25,970 --> 00:01:28,740
Schrodinger equation
to show what

28
00:01:28,740 --> 00:01:37,600
it takes to get a motion of this
product of the psi star psi.

29
00:01:37,600 --> 00:01:38,350
What does it take?

30
00:01:38,350 --> 00:01:42,750
It takes a superposition
state consisting of at least

31
00:01:42,750 --> 00:01:45,660
two different energies.

32
00:01:45,660 --> 00:01:48,830
Then this normalization
integral--

33
00:01:48,830 --> 00:01:52,940
well, you'd expect
normalization, or something

34
00:01:52,940 --> 00:01:56,150
like this to be preserved.

35
00:01:56,150 --> 00:02:00,910
And in fact, the normalization
is time independent.

36
00:02:00,910 --> 00:02:04,910
Then, if we calculate--

37
00:02:04,910 --> 00:02:08,570
now, this is a vector rather
than just a simple coordinate.

38
00:02:08,570 --> 00:02:13,810
If we calculate the expectation
value of the position,

39
00:02:13,810 --> 00:02:19,610
and the expectation
value of the momentum,

40
00:02:19,610 --> 00:02:23,030
Ehrenfest's theorem
tells you that this

41
00:02:23,030 --> 00:02:25,820
is going to describe the motion
of the center of the wave

42
00:02:25,820 --> 00:02:27,170
packet.

43
00:02:27,170 --> 00:02:30,380
And the motion of the
center of the wave packet

44
00:02:30,380 --> 00:02:32,720
is following Newton's laws.

45
00:02:32,720 --> 00:02:34,050
No surprise.

46
00:02:34,050 --> 00:02:37,490
And so we're seeing
how classical mechanics

47
00:02:37,490 --> 00:02:40,680
is given back to
you once we start

48
00:02:40,680 --> 00:02:43,470
making particle-like states.

49
00:02:43,470 --> 00:02:46,380
We can see how these
particle-like states evolve.

50
00:02:46,380 --> 00:02:51,510
But there is a lot more than
just particle-like behavior.

51
00:02:51,510 --> 00:02:56,070
You would, perhaps, challenge
any outfielder for the Red Sox

52
00:02:56,070 --> 00:03:00,810
to go beyond calculating the
center of the wave packet.

53
00:03:00,810 --> 00:03:02,149
That's what they know to do.

54
00:03:02,149 --> 00:03:03,690
But they don't know
some other things

55
00:03:03,690 --> 00:03:06,280
like survival
probability, how fast

56
00:03:06,280 --> 00:03:09,870
does the wave packet move
away from its birthplace,

57
00:03:09,870 --> 00:03:15,360
or the wave packet does stuff
and wanders around and comes

58
00:03:15,360 --> 00:03:18,450
back and rephases sometimes.

59
00:03:18,450 --> 00:03:22,080
And sometimes it just dephases.

60
00:03:22,080 --> 00:03:25,260
And these are words that
you're going to want

61
00:03:25,260 --> 00:03:28,030
to put into your vocabulary.

62
00:03:28,030 --> 00:03:30,900
So there's a lot of
really beautiful stuff

63
00:03:30,900 --> 00:03:33,480
when we start looking at the
time dependent Schrodinger

64
00:03:33,480 --> 00:03:34,860
equation.

65
00:03:34,860 --> 00:03:40,770
And we are going to mostly
consider the time dependent

66
00:03:40,770 --> 00:03:43,950
Schrodinger equation
when the Hamiltonian is

67
00:03:43,950 --> 00:03:47,460
independent of time, because
we can get our arms around that

68
00:03:47,460 --> 00:03:48,800
really easily.

69
00:03:48,800 --> 00:03:51,390
But when the Hamiltonian
is dependent on time,

70
00:03:51,390 --> 00:03:54,450
then it opens up a
world of complexity

71
00:03:54,450 --> 00:03:59,400
that is really best left
to a more advanced course.

72
00:04:02,860 --> 00:04:09,210
But how do molecules get excited
from one state to another?

73
00:04:09,210 --> 00:04:11,300
A time dependent Hamiltonian.

74
00:04:11,300 --> 00:04:15,440
So we are going to have
to at least briefly talk

75
00:04:15,440 --> 00:04:18,800
about a time dependent
Hamiltonian and what it does.

76
00:04:18,800 --> 00:04:20,959
And you'll see that eventually.

77
00:04:26,480 --> 00:04:29,800
I'm first going to do
a little bit of review

78
00:04:29,800 --> 00:04:33,280
of the a's and a daggers.

79
00:04:33,280 --> 00:04:36,310
So the most important
thing is, we're

80
00:04:36,310 --> 00:04:41,860
going to have some problem
where we have this x operator

81
00:04:41,860 --> 00:04:44,810
to some integer power.

82
00:04:44,810 --> 00:04:48,220
And so we need to be
able to relate that

83
00:04:48,220 --> 00:04:49,840
to the a's and a daggers.

84
00:05:00,750 --> 00:05:04,800
I'll drop the hats, because
we know they're there.

85
00:05:04,800 --> 00:05:09,390
And one of the nice things
is, if we have this operator

86
00:05:09,390 --> 00:05:16,010
to the nth power, we have this
co-factor to the nth power.

87
00:05:16,010 --> 00:05:17,762
So we don't ever
have to deal with it

88
00:05:17,762 --> 00:05:19,970
until the end of the problem
when we say, oh, wow, we

89
00:05:19,970 --> 00:05:21,230
had x to the n.

90
00:05:21,230 --> 00:05:25,621
Well, we have then n h bar.

91
00:05:25,621 --> 00:05:26,120
I'm sorry.

92
00:05:26,120 --> 00:05:30,042
We have h bar over 2
mu n n over 2 power.

93
00:05:32,900 --> 00:05:34,940
So it really saves
a lot of writing.

94
00:05:34,940 --> 00:05:38,150
And it means you
solve one problem

95
00:05:38,150 --> 00:05:40,160
for a harmonic oscillator.

96
00:05:40,160 --> 00:05:43,110
And you've solved it for
all harmonic oscillators.

97
00:05:43,110 --> 00:05:45,990
And this is just details
about what is the mass

98
00:05:45,990 --> 00:05:50,830
and what is the force constant,
which you need to know.

99
00:05:50,830 --> 00:05:52,430
And there is a
similar thing for p.

100
00:05:54,990 --> 00:05:57,360
So that opens the door.

101
00:05:57,360 --> 00:06:00,720
It says, OK, we have
some problem involving

102
00:06:00,720 --> 00:06:03,765
the coordinate and the momentum,
some function of the coordinate

103
00:06:03,765 --> 00:06:08,130
and momentum, and we
want to know things

104
00:06:08,130 --> 00:06:10,290
about the coordinate
and momentum.

105
00:06:10,290 --> 00:06:15,070
And so we use this
operator algebra.

106
00:06:15,070 --> 00:06:27,760
And so a operating on psi v
gives v square root v minus 1.

107
00:06:27,760 --> 00:06:29,490
And a dagger, we know that.

108
00:06:29,490 --> 00:06:31,080
So that's hardwired.

109
00:06:31,080 --> 00:06:35,400
But suppose we
wanted a to the 5th,

110
00:06:35,400 --> 00:06:40,810
operating on psi v.
Well, what do we do?

111
00:06:40,810 --> 00:06:46,720
Well, we start, and we
start operating on...

112
00:06:46,720 --> 00:06:48,900
You know, if this were
a complicated operator,

113
00:06:48,900 --> 00:06:52,440
we would take the rightmost
piece of that operator

114
00:06:52,440 --> 00:06:54,430
and operate on
the wave function.

115
00:06:54,430 --> 00:07:02,260
And that will give us v and then
v minus 1 and then v minus 2,

116
00:07:02,260 --> 00:07:04,154
v minus 3.

117
00:07:04,154 --> 00:07:05,070
I've got five of them.

118
00:07:05,070 --> 00:07:06,495
1, 2, 3, 4, 5.

119
00:07:06,495 --> 00:07:08,640
v minus 4.

120
00:07:08,640 --> 00:07:12,080
Square root psi v minus 5.

121
00:07:15,080 --> 00:07:15,860
OK.

122
00:07:15,860 --> 00:07:17,450
It's mechanical.

123
00:07:17,450 --> 00:07:19,370
You don't remember this.

124
00:07:19,370 --> 00:07:23,720
You generate this
one step at a time.

125
00:07:23,720 --> 00:07:25,570
And it's automatic.

126
00:07:25,570 --> 00:07:27,110
And so it doesn't
stress your brain.

127
00:07:27,110 --> 00:07:28,430
You can be thinking
about the next thing

128
00:07:28,430 --> 00:07:29,846
while you're writing
that garbage.

129
00:07:37,290 --> 00:07:40,320
We have this number
operator, which

130
00:07:40,320 --> 00:07:43,920
is a friend, because
it enables you to just

131
00:07:43,920 --> 00:07:45,840
get rid of a bunch of terms.

132
00:07:45,840 --> 00:07:50,340
The number operator is a dagger
a, and the number operator

133
00:07:50,340 --> 00:07:58,660
operating on psi
v gives v psi v.

134
00:07:58,660 --> 00:08:03,790
If we want any harmonic
oscillator function,

135
00:08:03,790 --> 00:08:15,190
we can operate on psi 0 with
a dagger to the v power.

136
00:08:15,190 --> 00:08:18,580
And then we have to
correct, because you

137
00:08:18,580 --> 00:08:24,420
know that this operating v
times on this will give psi v.

138
00:08:24,420 --> 00:08:27,730
But it will also give a
bunch of garbage, right?

139
00:08:27,730 --> 00:08:29,800
And you want to
cancel that garbage.

140
00:08:29,800 --> 00:08:32,919
And so you write v
factorial minus 1/2.

141
00:08:36,980 --> 00:08:39,409
And so that gives you
the normalized function.

142
00:08:39,409 --> 00:08:43,100
So these are really,
really simple things.

143
00:08:43,100 --> 00:08:45,954
And most of them, once you've
thought about it a little bit,

144
00:08:45,954 --> 00:08:47,120
you can figure out yourself.

145
00:08:51,790 --> 00:08:53,580
OK.

146
00:08:53,580 --> 00:09:01,590
Now, many problems involve
x, x cubed, x to the 4th.

147
00:09:01,590 --> 00:09:05,340
And we know that x is this.

148
00:09:05,340 --> 00:09:08,046
And x squared is this squared.

149
00:09:08,046 --> 00:09:10,020
And x cubed is this cubed.

150
00:09:10,020 --> 00:09:12,580
And so we have to do a
little algebra to simplify.

151
00:09:12,580 --> 00:09:24,060
And what we want to do is
simplify to sum of terms

152
00:09:24,060 --> 00:09:31,000
according to delta
v selection rule.

153
00:09:35,790 --> 00:09:38,460
So there is some
algebra that we do,

154
00:09:38,460 --> 00:09:44,780
when we have, say, a dagger,
a dagger, a, a, a dagger.

155
00:09:44,780 --> 00:09:49,010
But we know that three a
daggers and two a's means

156
00:09:49,010 --> 00:09:50,330
delta v of plus 1.

157
00:09:54,780 --> 00:09:59,110
And that came from
x to the 5th power.

158
00:09:59,110 --> 00:10:02,290
But you get a lot of terms
from x to the 5th power.

159
00:10:02,290 --> 00:10:04,130
And you have to simplify them.

160
00:10:04,130 --> 00:10:08,260
And in order to do that, you
use this commutation rule--

161
00:10:08,260 --> 00:10:13,857
a, a dagger is equal to 1.

162
00:10:13,857 --> 00:10:14,940
And that rearranges terms.

163
00:10:14,940 --> 00:10:16,830
So all of the work
you do when you're

164
00:10:16,830 --> 00:10:20,820
faced with a problem
involving integrals

165
00:10:20,820 --> 00:10:24,990
of integer powers of x and
p for a harmonic oscillator

166
00:10:24,990 --> 00:10:29,700
is playing around with
moving the a's and a daggers

167
00:10:29,700 --> 00:10:32,220
around, so that you
have all the terms that

168
00:10:32,220 --> 00:10:37,680
have the same selection rule
compressed into one term.

169
00:10:37,680 --> 00:10:38,650
That's the work.

170
00:10:38,650 --> 00:10:39,990
It's not much.

171
00:10:39,990 --> 00:10:44,430
And once you've done it for
x squared, x cubed and x 4th,

172
00:10:44,430 --> 00:10:47,040
you've done it as much as
you'll ever need to do.

173
00:10:47,040 --> 00:10:48,220
And that's it.

174
00:10:48,220 --> 00:10:49,370
That's the end.

175
00:10:49,370 --> 00:10:49,870
OK.

176
00:10:54,390 --> 00:11:00,070
Now here's an
example of a problem

177
00:11:00,070 --> 00:11:01,850
that's a little bit tricky.

178
00:11:01,850 --> 00:11:06,350
And it's sort of right
at the borderline of what

179
00:11:06,350 --> 00:11:08,920
I might use on the exam.

180
00:11:08,920 --> 00:11:15,840
So we have this, we have
a dagger to the m power.

181
00:11:15,840 --> 00:11:18,430
A to the n power.

182
00:11:18,430 --> 00:11:20,330
Psi.

183
00:11:20,330 --> 00:11:20,910
OK.

184
00:11:20,910 --> 00:11:22,870
Now, here.

185
00:11:22,870 --> 00:11:26,290
What v is going to give
a non-zero integral?

186
00:11:26,290 --> 00:11:28,690
We have a dagger to the m.

187
00:11:28,690 --> 00:11:31,120
And we have a to the n.

188
00:11:31,120 --> 00:11:41,570
And so that's going to be
v minus n plus m, right?

189
00:11:41,570 --> 00:11:47,430
Because we lose n
quanta, because of this.

190
00:11:47,430 --> 00:11:49,440
And we gain m quanta
because of that.

191
00:11:52,180 --> 00:11:54,220
And now, well, that's good.

192
00:11:54,220 --> 00:11:55,810
You've used the selection rule.

193
00:11:55,810 --> 00:11:58,890
Now, how do we write
out this interval?

194
00:11:58,890 --> 00:12:01,960
And there is a little
bit of art there too.

195
00:12:01,960 --> 00:12:07,240
Because we have a
whole bunch of terms.

196
00:12:07,240 --> 00:12:11,360
We have n plus m terms
in the square root.

197
00:12:11,360 --> 00:12:15,250
And how do you generate
them without getting lost?

198
00:12:18,721 --> 00:12:19,220
Yes?

199
00:12:19,220 --> 00:12:20,886
AUDIENCE: I think you
have it backwards.

200
00:12:20,886 --> 00:12:24,512
Shouldn't it be plus n minus n?

201
00:12:24,512 --> 00:12:26,920
It means you're going
to lower it n times.

202
00:12:26,920 --> 00:12:27,950
PROFESSOR: Yes.

203
00:12:27,950 --> 00:12:30,410
Yes.

204
00:12:30,410 --> 00:12:32,190
I wonder what I had in my notes.

205
00:12:32,190 --> 00:12:33,290
This is wrong.

206
00:12:39,980 --> 00:12:41,690
OK.

207
00:12:41,690 --> 00:12:49,900
So this has to
withstand this and this.

208
00:12:49,900 --> 00:12:51,730
And so, yes.

209
00:12:51,730 --> 00:12:52,230
OK.

210
00:12:52,230 --> 00:12:54,810
However you remember it,
you've got to do it right.

211
00:12:54,810 --> 00:13:00,430
And now we have the
actual matrix element.

212
00:13:00,430 --> 00:13:07,950
And so the first term is going
to be what does a n do to that?

213
00:13:07,950 --> 00:13:10,080
So you start on the right.

214
00:13:10,080 --> 00:13:13,460
And you start
building up this way.

215
00:13:13,460 --> 00:13:19,830
And so the first term is going
to be, what does a do to this?

216
00:13:19,830 --> 00:13:24,950
Well, it's going
to leave it alone.

217
00:13:24,950 --> 00:13:27,440
But it's going to lower v.

218
00:13:27,440 --> 00:13:32,060
So we have v plus n minus m.

219
00:13:32,060 --> 00:13:37,790
And then we have v
plus n minus m minus 1,

220
00:13:37,790 --> 00:13:39,995
et cetera, until
we have n terms.

221
00:13:45,740 --> 00:13:49,940
And then we start going
back up, because we're now

222
00:13:49,940 --> 00:13:53,160
dealing with this.

223
00:13:53,160 --> 00:13:56,410
And so I'm not going to
write the rest of this.

224
00:13:56,410 --> 00:14:00,760
Maybe this is going to be a
problem I start the exam with.

225
00:14:04,180 --> 00:14:05,500
So you don't want to get lost.

226
00:14:05,500 --> 00:14:08,080
And the main thing is,
you have these operators.

227
00:14:08,080 --> 00:14:12,910
And you start operating on the
right, and one step at a time.

228
00:14:12,910 --> 00:14:16,060
And this is a little
tricky because you're

229
00:14:16,060 --> 00:14:18,130
changing the quantum
number, and you're

230
00:14:18,130 --> 00:14:20,100
changing the wave function.

231
00:14:20,100 --> 00:14:22,420
And you have to
keep both in mind,

232
00:14:22,420 --> 00:14:25,610
but you're only
writing down this.

233
00:14:25,610 --> 00:14:26,110
OK?

234
00:14:35,330 --> 00:14:38,330
I want to save enough time
so that we can actually

235
00:14:38,330 --> 00:14:41,510
do the time dependent
Schrodinger equation.

236
00:14:41,510 --> 00:14:43,470
The time dependent
Schrodinger equation.

237
00:14:43,470 --> 00:14:45,340
H psi.

238
00:14:55,250 --> 00:14:57,530
It still looks pretty simple.

239
00:14:57,530 --> 00:15:02,130
Instead of e psi here,
we have this thing.

240
00:15:02,130 --> 00:15:04,500
This is the time dependent
Schrodinger equation.

241
00:15:04,500 --> 00:15:05,460
This is it.

242
00:15:05,460 --> 00:15:07,170
This is quantum mechanics.

243
00:15:07,170 --> 00:15:09,900
Everything that comes
from quantum mechanics

244
00:15:09,900 --> 00:15:11,850
starts with this.

245
00:15:11,850 --> 00:15:15,270
When we don't care
about time, we

246
00:15:15,270 --> 00:15:17,760
can use the time independent
Schrodinger equation.

247
00:15:17,760 --> 00:15:19,950
But when we do
care about time, we

248
00:15:19,950 --> 00:15:22,480
have to be a little bit careful.

249
00:15:22,480 --> 00:15:25,030
So this is the real
Schrodinger equation.

250
00:15:25,030 --> 00:15:27,250
And notice that
I'm using a capital

251
00:15:27,250 --> 00:15:33,900
psi, rather than a lower
case, or less decorated psi.

252
00:15:33,900 --> 00:15:37,890
And so this is usually used
to indicate the time dependent

253
00:15:37,890 --> 00:15:38,910
Schrodinger equation.

254
00:15:38,910 --> 00:15:40,680
It's time dependent
wave function.

255
00:15:40,680 --> 00:15:46,200
This is used to indicate the
time independent equation.

256
00:15:46,200 --> 00:15:49,520
Now, if the Hamiltonian--

257
00:15:49,520 --> 00:15:51,180
and this is wonderful--

258
00:15:51,180 --> 00:15:54,940
if the Hamiltonian is
independent of time,

259
00:15:54,940 --> 00:16:03,010
then if we know the
solutions psi n en.

260
00:16:03,010 --> 00:16:05,950
If we know all of
these solutions,

261
00:16:05,950 --> 00:16:07,940
then there's nothing new.

262
00:16:07,940 --> 00:16:10,900
We just are
repackaging the stuff

263
00:16:10,900 --> 00:16:13,900
that we know from the time
independent Schrodinger

264
00:16:13,900 --> 00:16:15,430
equation.

265
00:16:15,430 --> 00:16:17,840
So the first thing I
want to do is to show you

266
00:16:17,840 --> 00:16:24,180
that if the Hamiltonian
is independent,

267
00:16:24,180 --> 00:16:30,090
we can always write a
solution to the time

268
00:16:30,090 --> 00:16:36,180
dependent Schrodinger
equation-- e to the minus i e n

269
00:16:36,180 --> 00:16:42,690
t over h bar psi n x.

270
00:16:45,330 --> 00:16:52,110
So this, for a time
independent Hamiltonian,

271
00:16:52,110 --> 00:16:55,661
is always the solution of the
time dependent Schrodinger

272
00:16:55,661 --> 00:16:56,160
equation.

273
00:17:02,010 --> 00:17:04,319
So we're just using
this stuff that we know,

274
00:17:04,319 --> 00:17:06,210
or at least we barely
know, because we just

275
00:17:06,210 --> 00:17:08,450
started playing the game.

276
00:17:08,450 --> 00:17:12,089
But we can manipulate to see
all sorts of useful stuff.

277
00:17:12,089 --> 00:17:12,589
OK.

278
00:17:12,589 --> 00:17:16,319
So let's show if this form
satisfies the Schrodinger

279
00:17:16,319 --> 00:17:17,359
equation.

280
00:17:17,359 --> 00:17:25,339
So we have ih bar partial
PSI with respect to t.

281
00:17:25,339 --> 00:17:28,339
So we get an ih bar.

282
00:17:28,339 --> 00:17:32,660
And then we take the
partial with respect to t.

283
00:17:32,660 --> 00:17:34,370
This is independent of time.

284
00:17:34,370 --> 00:17:36,290
This has time
[INAUDIBLE] And so we

285
00:17:36,290 --> 00:17:42,830
get a minus i e n over h bar.

286
00:17:42,830 --> 00:17:50,060
And then we get e to the minus
i e n t over h bar psi n of x.

287
00:17:56,090 --> 00:17:56,900
Well.

288
00:17:56,900 --> 00:17:59,040
So, let's put this together.

289
00:17:59,040 --> 00:18:01,520
We have an i times
i times minus 1.

290
00:18:01,520 --> 00:18:02,840
So that's plus 1.

291
00:18:02,840 --> 00:18:04,540
We have an h bar in
the numerator and h

292
00:18:04,540 --> 00:18:06,180
bar in the denominator.

293
00:18:06,180 --> 00:18:07,160
That's 1.

294
00:18:07,160 --> 00:18:14,690
And so what we end up getting
is e n e to the minus i e n

295
00:18:14,690 --> 00:18:17,090
t over h bar psi n.

296
00:18:20,054 --> 00:18:24,980
Well, this is psi.

297
00:18:28,320 --> 00:18:37,020
This function here is
the solution to the time

298
00:18:37,020 --> 00:18:38,850
dependent Schrodinger equation.

299
00:18:38,850 --> 00:18:40,080
How do we know that?

300
00:18:40,080 --> 00:18:44,700
Well, we have this
factor, h psi.

301
00:18:44,700 --> 00:18:50,200
Well, h doesn't operate
on either the i e n t.

302
00:18:50,200 --> 00:18:57,280
So what we have is that we
get E n e to the i omega t.

303
00:19:01,118 --> 00:19:01,618
Sorry.

304
00:19:01,618 --> 00:19:03,546
I'm jumping ahead.

305
00:19:03,546 --> 00:19:07,386
i e n t over h bar psi.

306
00:19:16,810 --> 00:19:18,190
All right.

307
00:19:18,190 --> 00:19:23,770
So, if we apply the
Hamiltonian to this function,

308
00:19:23,770 --> 00:19:26,500
we get just e n
times the function.

309
00:19:31,530 --> 00:19:35,240
And that's what we got when
we did i h bar times a partial

310
00:19:35,240 --> 00:19:37,040
with respect to t.

311
00:19:37,040 --> 00:19:41,900
So what that shows is
that this form always

312
00:19:41,900 --> 00:19:46,220
satisfies the time dependent
Schrodinger equation, provided

313
00:19:46,220 --> 00:19:49,630
that the Hamiltonian
is independent of time.

314
00:19:49,630 --> 00:19:53,400
Now, that's a large
range of problems, things

315
00:19:53,400 --> 00:19:55,630
that we need to understand.

316
00:19:55,630 --> 00:19:58,920
But it's not the whole potato,
because the Hamiltonian

317
00:19:58,920 --> 00:20:01,560
is often dependent on time.

318
00:20:01,560 --> 00:20:06,570
But it enables us to
build up insight, and then

319
00:20:06,570 --> 00:20:10,890
treat the time dependence
as a perturbation.

320
00:20:10,890 --> 00:20:13,860
And we're going to do
perturbation theory in the time

321
00:20:13,860 --> 00:20:15,994
independent world.

322
00:20:15,994 --> 00:20:17,910
And then we're going to
do perturbation theory

323
00:20:17,910 --> 00:20:21,470
a little bit in the
time dependent world.

324
00:20:21,470 --> 00:20:21,970
OK.

325
00:20:21,970 --> 00:20:26,440
So what we always do here is
we solve a familiar problem,

326
00:20:26,440 --> 00:20:29,050
and then we say, OK, well,
there's something more

327
00:20:29,050 --> 00:20:31,400
to this familiar problem.

328
00:20:31,400 --> 00:20:34,410
And so we treat that
as something extra.

329
00:20:34,410 --> 00:20:36,250
And we work out the
formalism for dealing

330
00:20:36,250 --> 00:20:38,210
with that extra thing.

331
00:20:38,210 --> 00:20:39,700
But before we do
the extra thing,

332
00:20:39,700 --> 00:20:42,460
we have to really
kill the problem that

333
00:20:42,460 --> 00:20:45,950
is within our grasp.

334
00:20:45,950 --> 00:20:46,450
OK.

335
00:20:49,470 --> 00:20:54,630
So now, our job
is to just explore

336
00:20:54,630 --> 00:20:55,920
what we've really got here.

337
00:20:58,840 --> 00:21:04,750
So the first problem is motion.

338
00:21:08,140 --> 00:21:16,975
So we have psi star
x and t psi x and t.

339
00:21:20,640 --> 00:21:22,350
So we have this
thing, which we're not

340
00:21:22,350 --> 00:21:23,640
going to integrate yet.

341
00:21:29,790 --> 00:21:32,220
Well, when is this
thing going to move?

342
00:21:35,000 --> 00:21:39,010
Well, the only way this
probability density

343
00:21:39,010 --> 00:21:42,190
is going to evolve
in time is going

344
00:21:42,190 --> 00:21:46,650
to be if we have a
wave function, psi,

345
00:21:46,650 --> 00:21:54,400
which is of x and t is equal to
c1 psi 1, e to the minus i e 1,

346
00:21:54,400 --> 00:22:00,720
e over h bar, plus c2,
psi 2, e to the minus i e

347
00:22:00,720 --> 00:22:10,290
2, t over h bar, where
e1 is not equal to e2.

348
00:22:10,290 --> 00:22:12,780
So this is the first,
most elementary step.

349
00:22:12,780 --> 00:22:18,040
And remember, we have this
notation e to the i something.

350
00:22:18,040 --> 00:22:20,140
And e to the minus i something.

351
00:22:20,140 --> 00:22:24,200
And when we take a plus 1 and
a minus and put them together,

352
00:22:24,200 --> 00:22:26,040
we get 1.

353
00:22:26,040 --> 00:22:28,820
So this notation, this
exponential notation,

354
00:22:28,820 --> 00:22:30,450
is really valuable.

355
00:22:30,450 --> 00:22:30,950
OK.

356
00:22:30,950 --> 00:22:34,910
So now let's just look at
this quantity, psi star psi.

357
00:22:39,050 --> 00:22:39,550
OK.

358
00:22:39,550 --> 00:22:40,870
Psi star psi.

359
00:22:44,980 --> 00:22:54,040
Well, it's going to be c1
squared, psi 1 squared--

360
00:22:54,040 --> 00:23:01,430
square modulus-- and we
get c2 square modulus,

361
00:23:01,430 --> 00:23:05,360
psi 2 square modulus.

362
00:23:05,360 --> 00:23:06,120
Are we done?

363
00:23:06,120 --> 00:23:07,280
No.

364
00:23:07,280 --> 00:23:14,360
So, what we did is, I put
in c1, psi 1 plus c2, psi 2.

365
00:23:14,360 --> 00:23:17,150
And I looked at the
easy terms, the terms

366
00:23:17,150 --> 00:23:20,980
where the exponential
factor goes away.

367
00:23:20,980 --> 00:23:22,690
And then there's
two cross terms.

368
00:23:22,690 --> 00:23:25,730
Those two cross
terms are c1 star--

369
00:23:25,730 --> 00:23:49,080
whoops-- c2 e to the minus
i e 2 minus e1 t over h bar

370
00:23:49,080 --> 00:23:54,370
psi 1 star, psi 2.

371
00:23:54,370 --> 00:24:04,780
And we have c1, c2 star e
to the plus i e2 minus e1

372
00:24:04,780 --> 00:24:10,930
t over h bar, psi 1 star.

373
00:24:10,930 --> 00:24:14,370
psi 1, psi 2 star.

374
00:24:14,370 --> 00:24:17,371
Now, this is just the
automatic writing.

375
00:24:17,371 --> 00:24:17,870
OK.

376
00:24:17,870 --> 00:24:23,760
So we have two terms that
are time independent.

377
00:24:23,760 --> 00:24:26,570
So, this is no big surprise.

378
00:24:26,570 --> 00:24:28,520
But then we have
this stuff here.

379
00:24:28,520 --> 00:24:38,970
And if we say e2 minus e1
over h bar is omega 2, 1,

380
00:24:38,970 --> 00:24:42,830
everything becomes
very transparent,

381
00:24:42,830 --> 00:24:44,630
because now we
have something that

382
00:24:44,630 --> 00:24:52,070
looks like it looks like it's
trying to be a cosine omega t.

383
00:24:52,070 --> 00:24:55,050
We have to be a
little bit careful.

384
00:24:55,050 --> 00:24:57,440
These are the two
time dependent terms.

385
00:24:57,440 --> 00:25:01,180
And they are the complex
conjugate of each other.

386
00:25:01,180 --> 00:25:12,690
And so we know that if we
have two complex numbers,

387
00:25:12,690 --> 00:25:21,840
c plus c star, we're going to
get twice the real part of c.

388
00:25:27,140 --> 00:25:31,460
So this enables us to take these
two terms and combine them.

389
00:25:31,460 --> 00:25:35,900
And we know it had
better be real,

390
00:25:35,900 --> 00:25:38,360
because we're talking
about a probability here.

391
00:25:38,360 --> 00:25:42,010
This probability
has got to be real.

392
00:25:42,010 --> 00:25:44,410
And it's got to be positive.

393
00:25:44,410 --> 00:25:48,825
It's got to be real and
positive everywhere and forever.

394
00:25:51,580 --> 00:25:55,410
Because there's no such thing
as a negative probability.

395
00:25:55,410 --> 00:25:57,650
There's no such thing as
a negative probability

396
00:25:57,650 --> 00:26:00,170
in a little region of
space, and we say, well,

397
00:26:00,170 --> 00:26:04,170
we integrate over all space,
and so that goes away.

398
00:26:04,170 --> 00:26:12,330
This psi star psi is going to
be real everywhere for all time.

399
00:26:12,330 --> 00:26:14,060
And so that's a good
thing, because we

400
00:26:14,060 --> 00:26:21,450
have a sum of a plus
its complex conjugate.

401
00:26:21,450 --> 00:26:23,270
And so this is real.

402
00:26:23,270 --> 00:26:32,900
And we can simplify everything,
and we can write it simply.

403
00:26:32,900 --> 00:26:35,400
But suppose we choose
a particular case--

404
00:26:35,400 --> 00:26:42,980
c1 star is equal to
c1, which is equal to 1

405
00:26:42,980 --> 00:26:45,560
over square root of 2.

406
00:26:45,560 --> 00:26:52,910
And we can say psi 1
and psi 2 are real.

407
00:26:52,910 --> 00:26:57,330
When we do that, then this
complicated-looking thing, psi

408
00:26:57,330 --> 00:27:03,760
star psi, becomes
1/2 psi 1 squared,

409
00:27:03,760 --> 00:27:15,430
plus 1/2 psi 2 squared, plus
cosine omega 1 2 t psi 1 psi 2.

410
00:27:20,950 --> 00:27:22,760
OK.

411
00:27:22,760 --> 00:27:25,490
Well, these two
guys aren't moving.

412
00:27:25,490 --> 00:27:27,540
And they're real.

413
00:27:27,540 --> 00:27:28,590
And positive.

414
00:27:28,590 --> 00:27:29,240
Yeah.

415
00:27:29,240 --> 00:27:32,160
AUDIENCE: Wait, is c2 also
1 over square root of 2?

416
00:27:35,932 --> 00:27:37,640
PROFESSOR: I had 1
over square root of 2,

417
00:27:37,640 --> 00:27:38,490
one over square root of 2.

418
00:27:38,490 --> 00:27:39,060
That's 1/2.

419
00:27:41,880 --> 00:27:46,560
But then when you
add the two terms,

420
00:27:46,560 --> 00:27:48,600
you get twice the real part.

421
00:27:48,600 --> 00:27:50,640
So we get a 1/2 times 2.

422
00:27:50,640 --> 00:27:54,000
And so this is not a mistake.

423
00:27:54,000 --> 00:27:56,730
It comes out this way, OK?

424
00:27:56,730 --> 00:27:58,800
AUDIENCE: I think you meant c2.

425
00:27:58,800 --> 00:28:05,340
You're only finding c1 star and
c1 to be 2 over square root--

426
00:28:05,340 --> 00:28:07,580
1 over square root of 2.

427
00:28:07,580 --> 00:28:09,467
You didn't say
anything about c2.

428
00:28:09,467 --> 00:28:10,050
PROFESSOR: Oh.

429
00:28:10,050 --> 00:28:10,860
OK.

430
00:28:10,860 --> 00:28:12,010
I want c1 and c2.

431
00:28:16,710 --> 00:28:18,640
That's what I wanted.

432
00:28:18,640 --> 00:28:19,140
OK.

433
00:28:24,450 --> 00:28:25,560
Thank you.

434
00:28:25,560 --> 00:28:27,360
That shows you're listening.

435
00:28:27,360 --> 00:28:30,570
And it shows that
I'm sufficiently here

436
00:28:30,570 --> 00:28:32,460
to understand your
questions, which

437
00:28:32,460 --> 00:28:34,350
is another wonderful thing.

438
00:28:34,350 --> 00:28:36,960
So this is now,
we have something

439
00:28:36,960 --> 00:28:40,460
that's positive everywhere.

440
00:28:40,460 --> 00:28:41,860
It's time independent.

441
00:28:41,860 --> 00:28:45,800
And we have something
that's oscillating.

442
00:28:45,800 --> 00:28:49,080
And so this term
can be negative.

443
00:28:49,080 --> 00:28:51,150
But you can show--

444
00:28:51,150 --> 00:28:52,750
I don't choose to do that.

445
00:28:52,750 --> 00:28:55,500
You can show that this
term is never larger

446
00:28:55,500 --> 00:28:57,480
than psi 1 squared
plus psi 2 squared.

447
00:29:00,770 --> 00:29:05,720
And so even though this term
can be negative at some points,

448
00:29:05,720 --> 00:29:10,970
it never is a negative enough
to make the evolving probability

449
00:29:10,970 --> 00:29:12,211
go negative.

450
00:29:15,880 --> 00:29:18,780
Now, you may want to play with
that just to convince yourself.

451
00:29:18,780 --> 00:29:19,780
And it's an easy proof.

452
00:29:19,780 --> 00:29:22,600
And I'm just not going to do it.

453
00:29:22,600 --> 00:29:25,080
OK.

454
00:29:25,080 --> 00:29:29,400
So we have something
that says, just

455
00:29:29,400 --> 00:29:35,270
like for the wave equation,
what does it take to get motion?

456
00:29:35,270 --> 00:29:37,040
And to get motion
you had to have

457
00:29:37,040 --> 00:29:41,810
a superposition of two
waves of different energy

458
00:29:41,810 --> 00:29:43,070
or different wave vector.

459
00:29:45,630 --> 00:29:51,180
And so here we have, if
e1 is not equal to e2,

460
00:29:51,180 --> 00:29:53,695
we have a non-zero omega 1, 2.

461
00:29:53,695 --> 00:29:56,340
It doesn't matter whether it's
omega 1, 2, or omega 2, 1,

462
00:29:56,340 --> 00:29:59,100
because it's cosine.

463
00:29:59,100 --> 00:30:02,700
And so that's motion.

464
00:30:02,700 --> 00:30:07,890
So we get a standing
wave sort of situation.

465
00:30:07,890 --> 00:30:10,950
And then we get this motion.

466
00:30:10,950 --> 00:30:14,430
OK, now, the fun begins.

467
00:30:14,430 --> 00:30:23,614
Suppose we want to calculate the
expectation value of x and p.

468
00:30:30,290 --> 00:30:30,790
OK.

469
00:30:30,790 --> 00:30:35,160
And let's just take
it as a 1D problem.

470
00:30:35,160 --> 00:30:37,300
And so x of t.

471
00:30:40,090 --> 00:30:41,640
No, there was no question.

472
00:30:41,640 --> 00:30:42,870
All right.

473
00:30:42,870 --> 00:30:43,410
We have--

474
00:30:59,122 --> 00:30:59,940
OK.

475
00:30:59,940 --> 00:31:02,100
And again, we take
this thing apart.

476
00:31:02,100 --> 00:31:05,460
And we say, all right, suppose
we have the same superposition

477
00:31:05,460 --> 00:31:09,160
of psi is equal to c1.

478
00:31:09,160 --> 00:31:10,330
psi 1.

479
00:31:21,640 --> 00:31:23,050
OK.

480
00:31:23,050 --> 00:31:30,650
Actually, these should
be the time independent.

481
00:31:30,650 --> 00:31:35,120
So what we get is, when we
do this integral, we get c1

482
00:31:35,120 --> 00:31:47,940
squared integral psi
star x, psi 1 dx.

483
00:31:47,940 --> 00:31:59,890
And we get c2 squared integral
psi 2 star x, psi 2 dx.

484
00:31:59,890 --> 00:32:03,260
And then we get cross
terms. c1 star, c2.

485
00:32:05,980 --> 00:32:18,970
e to the minus i omega 2, 1 t
times the integral psi 1 star

486
00:32:18,970 --> 00:32:23,170
x psi 2 dx.

487
00:32:23,170 --> 00:32:31,330
And we have c1, c2 star,
e to the plus i omega 2 1

488
00:32:31,330 --> 00:32:38,090
t integral psi 2
star x psi 1 dx.

489
00:32:40,741 --> 00:32:43,170
A lot of stuff here.

490
00:32:43,170 --> 00:32:47,810
Now, we're talking about
the harmonic oscillator.

491
00:32:47,810 --> 00:32:49,385
This integral is 0.

492
00:32:52,360 --> 00:32:55,340
Because x is a plus a dagger.

493
00:32:55,340 --> 00:32:58,750
The selection rule is delta
v of plus and minus 1.

494
00:32:58,750 --> 00:33:01,190
This one is 0.

495
00:33:01,190 --> 00:33:05,790
For the particle in
a box, one can also

496
00:33:05,790 --> 00:33:08,610
ask, what about this integral?

497
00:33:08,610 --> 00:33:11,880
And for the particle in a
box, it's a little bit more

498
00:33:11,880 --> 00:33:12,510
complicated.

499
00:33:12,510 --> 00:33:15,510
Because we've chosen a
mathematically simple way

500
00:33:15,510 --> 00:33:19,110
to solve the particle in a
box, with the box having a zero

501
00:33:19,110 --> 00:33:21,500
left edge.

502
00:33:21,500 --> 00:33:25,310
If we make the box symmetric,
then we can make judgments

503
00:33:25,310 --> 00:33:26,930
and say, oh, yeah.

504
00:33:26,930 --> 00:33:30,930
This integral is also 0
for the particle in a box.

505
00:33:36,010 --> 00:33:38,490
And that's a little bit more
complicated, the argument,

506
00:33:38,490 --> 00:33:40,200
than what I just did.

507
00:33:40,200 --> 00:33:45,070
So these two terms are 0.

508
00:33:45,070 --> 00:33:51,100
And now we have motion of the
expectation value, which is

509
00:33:51,100 --> 00:33:52,720
described by these two terms.

510
00:33:52,720 --> 00:33:56,180
And again, they're the complex
conjugate of each other.

511
00:33:56,180 --> 00:34:13,949
And so what we have is
x of t is equal to twice

512
00:34:13,949 --> 00:34:22,900
the real part of c1 star, c2
e to the minus i omega 2, 1 t.

513
00:34:28,500 --> 00:34:32,131
Times x 1, 2.

514
00:34:32,131 --> 00:34:32,630
OK.

515
00:34:32,630 --> 00:34:34,310
This x 1, 2 is an integral.

516
00:34:37,210 --> 00:34:42,969
So x 1, 2 if these are the
vibrational quantum numbers,

517
00:34:42,969 --> 00:34:46,310
well, then this is non-zero.

518
00:34:46,310 --> 00:34:52,370
This is just the square root
of 2, or square root of 1.

519
00:35:00,051 --> 00:35:00,550
OK.

520
00:35:00,550 --> 00:35:05,005
So we have motion
described by this.

521
00:35:07,990 --> 00:35:14,440
And so the only
time we get motion

522
00:35:14,440 --> 00:35:20,650
is if v1 is equal to
v2 plus or minus 1,

523
00:35:20,650 --> 00:35:22,065
for the harmonic oscillator.

524
00:35:25,320 --> 00:35:29,720
For the particle in a box,
there's different rules.

525
00:35:29,720 --> 00:35:32,010
And often, for these
simple problems,

526
00:35:32,010 --> 00:35:35,090
you want to go through in your
head all of the simple cases.

527
00:35:48,810 --> 00:35:52,630
Now, we already can see that--

528
00:35:59,950 --> 00:36:02,270
I don't want to talk about
the particle in a box.

529
00:36:02,270 --> 00:36:05,320
So now let's just
take another step.

530
00:36:05,320 --> 00:36:17,780
And we're going to have
Ehrenfest's theorem, which

531
00:36:17,780 --> 00:36:26,800
you can prove, says that m times
the derivative of vector p--

532
00:36:31,790 --> 00:36:35,210
so this is a time dependent
expectation value--

533
00:36:35,210 --> 00:36:37,681
is equal to--

534
00:36:37,681 --> 00:36:38,180
I'm sorry.

535
00:36:38,180 --> 00:36:39,200
This is not vector p.

536
00:36:39,200 --> 00:36:41,420
This is vector r--

537
00:36:41,420 --> 00:36:42,080
is equal to--

538
00:36:48,550 --> 00:36:49,720
So this is the vector p.

539
00:36:49,720 --> 00:36:53,500
So this is the derivative
of the coordinate,

540
00:36:53,500 --> 00:36:54,380
with respect to time.

541
00:36:54,380 --> 00:36:55,600
That's velocity.

542
00:36:55,600 --> 00:36:59,080
Velocity times mass is momentum.

543
00:36:59,080 --> 00:37:02,380
And so we have a relationship
between the expectation

544
00:37:02,380 --> 00:37:06,070
value of the position
and the expectation

545
00:37:06,070 --> 00:37:08,480
value of the momentum.

546
00:37:08,480 --> 00:37:10,520
And that's for
Newton's first law.

547
00:37:10,520 --> 00:37:14,170
And then there is another
Newton's equation translated

548
00:37:14,170 --> 00:37:16,580
into quantum mechanics.

549
00:37:16,580 --> 00:37:25,270
So we have the expectation
value of the momentum, dt,

550
00:37:25,270 --> 00:37:30,520
is equal to minus
the expectation

551
00:37:30,520 --> 00:37:33,250
value of the potential.

552
00:37:37,310 --> 00:37:43,540
While this is acceleration
times mass, and this is force--

553
00:37:43,540 --> 00:37:46,020
minus the gradient of a
potential is the force--

554
00:37:46,020 --> 00:37:47,860
these are Newton's
two equations.

555
00:37:47,860 --> 00:37:58,280
And what they're saying is,
if we know these things,

556
00:37:58,280 --> 00:38:01,520
we know something about the
center of the wave packet,

557
00:38:01,520 --> 00:38:03,570
and how the center of
the wave packet moves.

558
00:38:03,570 --> 00:38:06,540
Now, the wave packet might
be localized at one time.

559
00:38:06,540 --> 00:38:09,210
It might be mushed
out at another time.

560
00:38:09,210 --> 00:38:11,710
But you can always calculate
the center of the wave packet.

561
00:38:11,710 --> 00:38:13,501
It's just that there
are only certain times

562
00:38:13,501 --> 00:38:16,650
that it looks like a particle.

563
00:38:16,650 --> 00:38:19,830
But this thing,
these quantities,

564
00:38:19,830 --> 00:38:24,380
which you define by an integral,
they evolve classically.

565
00:38:24,380 --> 00:38:26,000
So I told you at
the beginning, you

566
00:38:26,000 --> 00:38:28,790
had to give up
classical mechanics.

567
00:38:28,790 --> 00:38:31,130
It's all coming back.

568
00:38:31,130 --> 00:38:34,320
But it's coming back in a
quantum mechanical framework,

569
00:38:34,320 --> 00:38:38,360
because we're talking now
about wave functions, which

570
00:38:38,360 --> 00:38:43,350
have amplitudes and phases,
and can do terrible things.

571
00:38:43,350 --> 00:38:49,110
But at certain limits, they're
going to act like particles.

572
00:38:49,110 --> 00:38:57,650
But if you were to
ask a question, well--

573
00:38:57,650 --> 00:39:00,230
suppose we do this experiment.

574
00:39:00,230 --> 00:39:06,770
And so here we have an
electronic ground state

575
00:39:06,770 --> 00:39:07,970
potential surface.

576
00:39:07,970 --> 00:39:10,610
Now, I'm jumping way ahead.

577
00:39:10,610 --> 00:39:16,730
But this is the wave function
for that vibrational level.

578
00:39:16,730 --> 00:39:21,780
And you excite the molecule with
a time dependent Hamiltonian,

579
00:39:21,780 --> 00:39:25,200
a time dependent
radiation field light.

580
00:39:25,200 --> 00:39:29,880
And you vertically
transport this wave function

581
00:39:29,880 --> 00:39:32,520
to the upper state,
until you get something

582
00:39:32,520 --> 00:39:35,690
which is not an eigenstate.

583
00:39:35,690 --> 00:39:37,470
It's a pluck.

584
00:39:37,470 --> 00:39:41,560
This pluck is a
superposition of eigenstates.

585
00:39:41,560 --> 00:39:43,360
And we can ask, how
does this evolve?

586
00:39:43,360 --> 00:39:45,130
And what it's going
to do is, it's

587
00:39:45,130 --> 00:39:47,910
going to start out localized.

588
00:39:47,910 --> 00:39:51,280
And at some point, it'll
do terrible things.

589
00:39:51,280 --> 00:39:53,140
And then at some
other time it'll

590
00:39:53,140 --> 00:39:55,380
be localized again at
the other turning point.

591
00:39:55,380 --> 00:39:57,780
And it will come back and forth.

592
00:39:57,780 --> 00:40:00,810
And now, if it's not
a harmonic oscillator,

593
00:40:00,810 --> 00:40:03,780
it won't quite relocalize.

594
00:40:03,780 --> 00:40:05,810
It'll mush out a little
bit, and it'll come back

595
00:40:05,810 --> 00:40:07,660
and it'll mush out more.

596
00:40:07,660 --> 00:40:11,900
And so again, we can use the
evolution of the wave packet

597
00:40:11,900 --> 00:40:15,530
to sample the shape
of the potential.

598
00:40:15,530 --> 00:40:17,940
We can measure
the anharmonicity.

599
00:40:17,940 --> 00:40:20,780
And so let's now
talk about that.

600
00:40:24,970 --> 00:40:27,400
What are other
quantities that we

601
00:40:27,400 --> 00:40:34,039
can calculate from the
wave function of the time

602
00:40:34,039 --> 00:40:35,330
dependent Schrodinger equation?

603
00:40:35,330 --> 00:40:38,960
So let's talk about the
survival probability.

604
00:40:38,960 --> 00:40:41,500
So this is a capital P.
Survival probability.

605
00:40:41,500 --> 00:40:45,790
It's going to be
the wave function.

606
00:40:45,790 --> 00:40:50,440
The time dependent wave function
is created at t equals 0.

607
00:40:50,440 --> 00:40:51,970
It has some shape.

608
00:40:51,970 --> 00:40:55,990
Now, we always like to have
a simple shape at t equals 0.

609
00:40:55,990 --> 00:41:00,670
And we'd like to know how
fast that thing moves away

610
00:41:00,670 --> 00:41:04,210
from its birthplace.

611
00:41:04,210 --> 00:41:10,900
So we have this
survival probability.

612
00:41:10,900 --> 00:41:12,280
It's a probability.

613
00:41:12,280 --> 00:41:25,640
So we're going to have
integral psi star xt psi xt dx.

614
00:41:25,640 --> 00:41:28,730
And this integral is
going to look like that.

615
00:41:32,490 --> 00:41:40,610
Now, whoops.

616
00:41:40,610 --> 00:41:42,800
I knew I was going to screw up.

617
00:41:42,800 --> 00:41:46,370
If I put a t here,
then it would just

618
00:41:46,370 --> 00:41:48,320
be the normalization interval.

619
00:41:48,320 --> 00:41:51,250
And we already
know what that is.

620
00:41:51,250 --> 00:41:52,690
But this is the birthplace.

621
00:41:52,690 --> 00:41:55,460
This is what was
created at t equals 0.

622
00:41:55,460 --> 00:41:58,300
And this is time evolving thing.

623
00:41:58,300 --> 00:42:01,150
And so we can calculate
how that behaves.

624
00:42:01,150 --> 00:42:04,790
And I'm just going to
write the solution.

625
00:42:09,770 --> 00:42:19,150
So the result is, if we have the
same kind of two state c1 psi

626
00:42:19,150 --> 00:42:27,530
1, e to the minus
i e1 t for h bar

627
00:42:27,530 --> 00:42:35,140
plus c2 psi 2 e to the
minus i e2 e over h bar,

628
00:42:35,140 --> 00:42:41,330
well then, what we get
is c1 to the 4th power

629
00:42:41,330 --> 00:42:55,380
plus c2 to the 4th power plus
c1 squared, c2 squared, times

630
00:42:55,380 --> 00:43:06,210
e to the i omega 2, 1 t plus
e to the minus i omega 2, 1 t.

631
00:43:10,390 --> 00:43:14,090
Now we're integrating.

632
00:43:14,090 --> 00:43:18,520
When we integrate, the
wave functions go away.

633
00:43:18,520 --> 00:43:23,010
The wave functions
become either 1 or 0.

634
00:43:23,010 --> 00:43:24,750
And so we're integrating.

635
00:43:24,750 --> 00:43:27,030
We're making the wave
functions go away.

636
00:43:27,030 --> 00:43:32,490
And we have some amplitudes
of the wave functions.

637
00:43:32,490 --> 00:43:36,210
And we have a time independent
part and a time dependent part.

638
00:43:36,210 --> 00:43:41,472
And this is 2
cosine omega 2, 1 t.

639
00:43:44,070 --> 00:43:49,530
So what we see is this survival
probability, the wave function,

640
00:43:49,530 --> 00:43:51,870
starts out at some place.

641
00:43:51,870 --> 00:43:53,560
And it goes away.

642
00:43:53,560 --> 00:43:54,700
And it comes back.

643
00:43:54,700 --> 00:43:55,800
And it goes away.

644
00:43:55,800 --> 00:43:59,060
And it comes back.

645
00:43:59,060 --> 00:43:59,930
And it does that.

646
00:43:59,930 --> 00:44:04,670
It completely rephases,
because there's only two terms.

647
00:44:04,670 --> 00:44:07,070
But if there were three
terms that are not

648
00:44:07,070 --> 00:44:09,400
satisfying a
certain requirement,

649
00:44:09,400 --> 00:44:11,870
then when it comes back,
it can't completely

650
00:44:11,870 --> 00:44:12,960
reconstruct itself.

651
00:44:17,080 --> 00:44:20,560
And this is the basis
for doing experiments.

652
00:44:20,560 --> 00:44:26,770
One can observe the
periodic rephasings

653
00:44:26,770 --> 00:44:29,500
of some initial pluck.

654
00:44:29,500 --> 00:44:31,840
And you can look at
the decay of them,

655
00:44:31,840 --> 00:44:33,850
and that gives you
something about the shape

656
00:44:33,850 --> 00:44:36,650
of the potential,
the anharmonicity.

657
00:44:36,650 --> 00:44:40,490
And you can do all sorts
of fantastic things.

658
00:44:40,490 --> 00:44:43,760
You can create a
wave packet, and you

659
00:44:43,760 --> 00:44:47,250
can wait until it reaches
the other turning point.

660
00:44:47,250 --> 00:44:48,920
And at the other
turning point, it's

661
00:44:48,920 --> 00:44:50,495
possible that you
could excite it

662
00:44:50,495 --> 00:44:53,900
to a different
higher excited state.

663
00:44:53,900 --> 00:44:55,770
Ahmed Zewail got the
Nobel Prize for that.

664
00:44:59,550 --> 00:45:06,690
So we're right at the
frontier of what you can do,

665
00:45:06,690 --> 00:45:08,430
and what you can
understand using

666
00:45:08,430 --> 00:45:11,810
this very simple problem of a
time independent Hamiltonian

667
00:45:11,810 --> 00:45:17,400
and a 2 or a 3
term superposition.

668
00:45:17,400 --> 00:45:17,900
OK.

669
00:45:27,850 --> 00:45:28,510
Recurrence.

670
00:45:35,910 --> 00:45:38,750
This is a special property.

671
00:45:38,750 --> 00:45:50,960
When all of the wave functions,
or all of the difference--

672
00:45:50,960 --> 00:45:54,410
all of the energy levels, or
differences in energy levels

673
00:45:54,410 --> 00:46:06,140
are integer multiples of
a common factor, well then

674
00:46:06,140 --> 00:46:09,830
any coherent superposition
state you make

675
00:46:09,830 --> 00:46:13,940
will rephase at a special time.

676
00:46:13,940 --> 00:46:19,440
And so what we can do is say,
OK, for a particle in a box,

677
00:46:19,440 --> 00:46:26,360
the energy levels are
e1 times n squared.

678
00:46:26,360 --> 00:46:29,450
For our harmonic oscillator,
the energy level difference

679
00:46:29,450 --> 00:46:39,360
is ev plus n minus
ev our n h bar omega.

680
00:46:39,360 --> 00:46:41,970
For a rigid rotor, which
we haven't seen yet,

681
00:46:41,970 --> 00:46:49,520
the energy levels are a function
of this quantum number j hcb.

682
00:46:49,520 --> 00:46:54,410
This is the rotational
constant-- times j plus 1.

683
00:46:56,940 --> 00:47:01,970
So all of the energy
levels are related.

684
00:47:01,970 --> 00:47:03,590
I'm jumping ahead.

685
00:47:03,590 --> 00:47:04,550
Sorry.

686
00:47:04,550 --> 00:47:12,380
So the energy levels
are j times j plus 1.

687
00:47:12,380 --> 00:47:18,980
And the differences,
ej plus 1 minus ej,

688
00:47:18,980 --> 00:47:27,620
are given by 2 hcb
times j plus 1.

689
00:47:27,620 --> 00:47:31,880
So for these three problems,
we have this perfect situation

690
00:47:31,880 --> 00:47:34,820
where one can have
these oscillating

691
00:47:34,820 --> 00:47:37,330
terms all have a common factor.

692
00:47:37,330 --> 00:47:42,130
And at certain times, the
oscillating terms are all 1.

693
00:47:42,130 --> 00:47:44,510
Or some are 1 and
some are minus 1.

694
00:47:44,510 --> 00:47:47,660
And we get a really
wonderful simplification.

695
00:47:47,660 --> 00:47:52,190
So at a time which
we call, or I call,

696
00:47:52,190 --> 00:47:59,860
the grand recurrence
time, where it's

697
00:47:59,860 --> 00:48:09,010
equal to h over e1 for this
case of the particle in a box,

698
00:48:09,010 --> 00:48:24,060
or h over h bar omega,
or h over 2 hcb,

699
00:48:24,060 --> 00:48:27,540
we get all of the phase
factors becoming 1.

700
00:48:33,040 --> 00:48:38,740
And sometimes, something special
happens at the grand recurrence

701
00:48:38,740 --> 00:48:41,170
time divided by 2.

702
00:48:41,170 --> 00:48:44,410
And that's a little
bit like this.

703
00:48:44,410 --> 00:48:47,900
We have a wave function here.

704
00:48:47,900 --> 00:48:50,010
And at half the recurrence
time, it's over here.

705
00:48:53,720 --> 00:48:56,050
And so you want to work
your way through the algebra

706
00:48:56,050 --> 00:49:00,590
to convince yourself that
that is in fact true.

707
00:49:00,590 --> 00:49:04,280
And in between, if
you have enough terms

708
00:49:04,280 --> 00:49:08,380
in your superposition,
which is usually the case,

709
00:49:08,380 --> 00:49:09,610
in between you get this.

710
00:49:09,610 --> 00:49:13,080
You get garbage looking.

711
00:49:13,080 --> 00:49:15,330
It's still moving.

712
00:49:15,330 --> 00:49:18,390
It's still satisfying
the Ehrenfest theorem,

713
00:49:18,390 --> 00:49:20,010
but it doesn't look
like a particle.

714
00:49:20,010 --> 00:49:23,470
It just looks like garbage.

715
00:49:23,470 --> 00:49:26,140
But you get these
wonderful things happening.

716
00:49:26,140 --> 00:49:28,480
And now, if they're
not perfectly

717
00:49:28,480 --> 00:49:32,350
satisfying this integer
rule, then each time you

718
00:49:32,350 --> 00:49:36,130
get to a turning
point, the amplitude

719
00:49:36,130 --> 00:49:40,440
has decreased and
decreased and decreased.

720
00:49:40,440 --> 00:49:46,830
And at infinite
time, it might recur.

721
00:49:46,830 --> 00:49:48,580
But nobody waits around
for infinite time,

722
00:49:48,580 --> 00:49:51,970
because other things happen and
destroy the coherence that you

723
00:49:51,970 --> 00:49:55,005
built. So this is a glimpse.

724
00:49:59,310 --> 00:50:01,020
Time independent
quantum mechanics

725
00:50:01,020 --> 00:50:03,980
is complicated enough.

726
00:50:03,980 --> 00:50:07,610
Well, we can embed
what we understand

727
00:50:07,610 --> 00:50:10,730
in the time dependent mechanics.

728
00:50:10,730 --> 00:50:14,560
And there are a lot of beautiful
things that we can anticipate.

729
00:50:14,560 --> 00:50:19,210
Now, we can use these
beautiful things

730
00:50:19,210 --> 00:50:22,420
to do experiments
which measure stuff

731
00:50:22,420 --> 00:50:27,020
that is related to, how does
energy move in the molecule?

732
00:50:27,020 --> 00:50:28,550
What is going on
in the molecule?

733
00:50:28,550 --> 00:50:31,964
What are the mechanisms
for stuff happening?

734
00:50:31,964 --> 00:50:33,880
And in magnetic resonance,
there are all sorts

735
00:50:33,880 --> 00:50:37,150
of pulse sequences that
interrogate distances

736
00:50:37,150 --> 00:50:39,830
and correlated motions.

737
00:50:39,830 --> 00:50:43,570
And it is really a laboratory
for time dependent quantum

738
00:50:43,570 --> 00:50:48,130
mechanics with a time
dependent Hamiltonian.

739
00:50:48,130 --> 00:50:54,700
But a lot of stuff that we do
with ordinary laser experiments

740
00:50:54,700 --> 00:51:01,880
are usually understandable
in a time independent way.

741
00:51:01,880 --> 00:51:07,590
Now, we want to get rid of
things that are complicated.

742
00:51:07,590 --> 00:51:11,220
And so one of the
things that we do

743
00:51:11,220 --> 00:51:14,970
is, we map a problem
onto something

744
00:51:14,970 --> 00:51:17,460
which is time independent.

745
00:51:17,460 --> 00:51:20,970
And so one of the
tricks that you will see

746
00:51:20,970 --> 00:51:25,890
is that if we go into what's
called a rotating coordinate

747
00:51:25,890 --> 00:51:28,890
system, this rotating
coordinate system

748
00:51:28,890 --> 00:51:36,330
is rotating at an energy level
different divided by h bar.

749
00:51:36,330 --> 00:51:39,984
And that converts the problem
into a time independent problem

750
00:51:39,984 --> 00:51:41,400
in the rotating
coordinate system.

751
00:51:41,400 --> 00:51:43,660
So that you're in a
rotating coordinate system

752
00:51:43,660 --> 00:51:47,610
and, basically, you use the
ordinary perturbation theory.

753
00:51:47,610 --> 00:51:50,670
And then you go back to the
non-rotating coordinate system.

754
00:51:50,670 --> 00:51:53,880
So there's all sorts of tricks
where we build on the stuff

755
00:51:53,880 --> 00:51:56,480
that we understood.

756
00:51:56,480 --> 00:52:00,560
And we can have a picture
which is intuitive,

757
00:52:00,560 --> 00:52:02,870
because we want to strip
away a lot of the mathematics

758
00:52:02,870 --> 00:52:05,540
and see the universal stuff.

759
00:52:05,540 --> 00:52:08,660
And so I'm going to try
to present as much of that

760
00:52:08,660 --> 00:52:10,950
as I can during this course.

761
00:52:10,950 --> 00:52:13,882
But a lot of the time
independent stuff

762
00:52:13,882 --> 00:52:16,340
is heavy lifting, and I'm going
to have to do a lot of that

763
00:52:16,340 --> 00:52:17,250
too.

764
00:52:17,250 --> 00:52:18,080
OK.

765
00:52:18,080 --> 00:52:20,600
Have a nice long weekend.

766
00:52:20,600 --> 00:52:22,250
I'll see you on Monday.