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PROFESSOR: OK, last time we
were talking about uncertainty.

9
00:00:26,300 --> 00:00:28,590
We gave a picture
for uncertainty--

10
00:00:28,590 --> 00:00:34,320
it was a neat picture, I
think of the uncertainty,

11
00:00:34,320 --> 00:00:40,200
refer to the uncertainty
measuring an operator

12
00:00:40,200 --> 00:00:43,540
A that was a Hermitian operator.

13
00:00:43,540 --> 00:00:47,360
And that uncertainty
depended on the state

14
00:00:47,360 --> 00:00:49,550
that you were measuring.

15
00:00:49,550 --> 00:00:52,590
If the state was
an eigenstate of A,

16
00:00:52,590 --> 00:00:54,320
there would be no uncertainty.

17
00:00:54,320 --> 00:00:56,300
If the state is not
an eigenstate of A,

18
00:00:56,300 --> 00:00:58,680
there was an uncertainty.

19
00:00:58,680 --> 00:01:02,640
And this uncertainty
was defined as the norm

20
00:01:02,640 --> 00:01:10,390
of A minus the expectation
value of A acting on psi.

21
00:01:14,060 --> 00:01:18,330
So that was our
definition of uncertainty.

22
00:01:18,330 --> 00:01:20,790
And it had nice properties.

23
00:01:20,790 --> 00:01:24,540
In fact, it was zero if
and only if the state was

24
00:01:24,540 --> 00:01:27,400
an eigenstate of the operator.

25
00:01:27,400 --> 00:01:30,710
We proved a couple
of things as well--

26
00:01:30,710 --> 00:01:34,880
that, in particular, one
that is kind of practical

27
00:01:34,880 --> 00:01:43,010
is that delta A of psi
squared is the expectation

28
00:01:43,010 --> 00:01:48,290
value of A squared on the
state psi minus the expectation

29
00:01:48,290 --> 00:01:52,055
value of A on the
state psi squared.

30
00:01:55,590 --> 00:02:01,590
So that was also proven,
which, since this number is

31
00:02:01,590 --> 00:02:07,130
greater than or equal to 0, this
is greater than or equal to 0.

32
00:02:07,130 --> 00:02:10,460
And in particular, the
expectation value of A squared

33
00:02:10,460 --> 00:02:15,686
is bigger than the
expectation of A squared.

34
00:02:19,650 --> 00:02:23,740
So let's do a trivial
example for a computation.

35
00:02:23,740 --> 00:02:30,750
Suppose somebody tells
you in an example

36
00:02:30,750 --> 00:02:36,010
that the spin is in
an eigenstate of Sz.

37
00:02:36,010 --> 00:02:45,470
So the state psi it's what we
called the plus state, or the z

38
00:02:45,470 --> 00:02:48,890
plus state.

39
00:02:48,890 --> 00:02:56,125
And you want to know what
is uncertainty delta of Sx.

40
00:03:02,890 --> 00:03:07,220
So you know if you're
in an eigenstate of z,

41
00:03:07,220 --> 00:03:09,650
you are not in an
eigenstate of x-- in fact,

42
00:03:09,650 --> 00:03:13,870
you're in a superposition
of two eigenstates of Sx.

43
00:03:13,870 --> 00:03:17,250
Therefore, there should
be some uncertainty here.

44
00:03:17,250 --> 00:03:19,760
And the question is, what is
the quickest way in which you

45
00:03:19,760 --> 00:03:25,080
compute this uncertainty,
and how much is it?

46
00:03:25,080 --> 00:03:30,280
So many times, the simplest way
is to just use this formula.

47
00:03:33,500 --> 00:03:40,200
So let's do that.

48
00:03:40,200 --> 00:03:48,990
So what is the expectation
value of Sx in that state?

49
00:03:48,990 --> 00:03:53,510
So it's Sx expectation
value would

50
00:03:53,510 --> 00:03:57,530
be given by Sx on this thing.

51
00:03:57,530 --> 00:04:03,290
Now, actually, it's
relatively clear

52
00:04:03,290 --> 00:04:08,070
to see that this expectation
value is going to be 0,

53
00:04:08,070 --> 00:04:14,180
because Sx really
in the state plus

54
00:04:14,180 --> 00:04:18,480
is equal amplitude to be
Sx equal plus h bar over 2,

55
00:04:18,480 --> 00:04:20,740
or minus h bar over 2.

56
00:04:20,740 --> 00:04:24,340
But suppose you
don't remember that.

57
00:04:24,340 --> 00:04:26,790
In order to compute
this, it may come

58
00:04:26,790 --> 00:04:32,720
handy to recall the matrix
presentation of Sx, which

59
00:04:32,720 --> 00:04:35,280
you don't need to know by heart.

60
00:04:35,280 --> 00:04:41,190
So this state plus
is the first state,

61
00:04:41,190 --> 00:04:44,780
and the basis state
is the state 1 0.

62
00:04:44,780 --> 00:04:51,960
And then we have Sx on plus
is equal to h bar over 2 0

63
00:04:51,960 --> 00:04:55,440
1 1 0, acting on 1 0.

64
00:04:55,440 --> 00:05:00,880
Zero and that's equal
to h bar over 2.

65
00:05:00,880 --> 00:05:05,810
The first thing gives you 0,
and the second one gives you 1.

66
00:05:05,810 --> 00:05:10,090
So that's, in fact, equal to h
bar over 2, the state of minus.

67
00:05:13,060 --> 00:05:18,990
So here you go to h
bar over 2 plus minus,

68
00:05:18,990 --> 00:05:26,470
and you know plus and minus are
orthogonal, so 0 is expected.

69
00:05:26,470 --> 00:05:28,550
Well, are we going to
get zero uncertainty?

70
00:05:28,550 --> 00:05:33,660
No, because Sx
squared, however, does

71
00:05:33,660 --> 00:05:35,540
have some expectation value.

72
00:05:35,540 --> 00:05:38,313
So what is the expectation
value of Sx squared?

73
00:05:42,930 --> 00:05:45,650
Well, there's an advantage here.

74
00:05:45,650 --> 00:05:50,640
You may remember that this
Sx squared is a funny matrix.

75
00:05:50,640 --> 00:05:54,110
It's a multiple of the
identity, because if you square

76
00:05:54,110 --> 00:05:56,810
this matrix, you get the
multiple of the identity.

77
00:05:56,810 --> 00:06:02,780
So Sx squared is h over 2
squared times the identity

78
00:06:02,780 --> 00:06:07,740
matrix-- the two by
two identity matrix.

79
00:06:07,740 --> 00:06:10,620
So the expectation
value of Sx squared

80
00:06:10,620 --> 00:06:15,340
is h bar over 2 squared
times expectation value

81
00:06:15,340 --> 00:06:17,832
of the identity.

82
00:06:17,832 --> 00:06:20,450
And on any state,
the expectation value

83
00:06:20,450 --> 00:06:23,350
on any normalized state,
the expectation value

84
00:06:23,350 --> 00:06:26,770
of the identity
will be equal to 1.

85
00:06:26,770 --> 00:06:31,210
So this is just h
squared over 2 squared.

86
00:06:31,210 --> 00:06:37,310
So back to our uncertainty,
delta Sx squared

87
00:06:37,310 --> 00:06:41,350
would be equal to the
expectation value of Sx squared

88
00:06:41,350 --> 00:06:44,800
minus the expectation
value of Sx squared.

89
00:06:44,800 --> 00:06:46,720
This was 0.

90
00:06:46,720 --> 00:06:52,990
This thing was equal to
h bar over 2 squared,

91
00:06:52,990 --> 00:06:58,450
and therefore, delta Sx
is equal to h bar over 2.

92
00:07:04,760 --> 00:07:11,500
So just I wanted to make
you familiar with that.

93
00:07:11,500 --> 00:07:14,840
You can compute these
things-- these norms and all

94
00:07:14,840 --> 00:07:20,770
these equations are pretty
practical, and easy to use.

95
00:07:20,770 --> 00:07:24,600
So today what we have
to do is the following--

96
00:07:24,600 --> 00:07:28,740
we're going to establish
the uncertainty principle.

97
00:07:28,740 --> 00:07:31,390
We're going to just prove it.

98
00:07:31,390 --> 00:07:35,930
And then, once we have
the uncertainty principle,

99
00:07:35,930 --> 00:07:40,670
we'll try to find some
applications for it.

100
00:07:40,670 --> 00:07:43,610
So before doing
an application, we

101
00:07:43,610 --> 00:07:47,130
will discuss the case of
the energy time uncertainty

102
00:07:47,130 --> 00:07:51,530
principle, which is
slightly more subtle

103
00:07:51,530 --> 00:07:55,310
and has interestingly
connotations that we

104
00:07:55,310 --> 00:07:56,480
will develop today.

105
00:07:56,480 --> 00:07:59,650
And finally, we'll use
the uncertainty principle

106
00:07:59,650 --> 00:08:04,970
to learn how to find bounds
for energies of ground states.

107
00:08:04,970 --> 00:08:08,400
So we might make a rigorous
application of the uncertainty

108
00:08:08,400 --> 00:08:10,090
principle.

109
00:08:10,090 --> 00:08:16,410
So the uncertainty principle
talks about two operators

110
00:08:16,410 --> 00:08:21,870
that are both
Hermitian, and states

111
00:08:21,870 --> 00:08:32,070
the following-- so given
the theorem, or uncertainty

112
00:08:32,070 --> 00:08:47,610
principle, given two
Hermitian operators A and B,

113
00:08:47,610 --> 00:08:59,970
and a state psi normalized, then
the following inequality holds.

114
00:08:59,970 --> 00:09:03,950
And we're going to write it in
one way, then in another way.

115
00:09:03,950 --> 00:09:12,910
Delta A psi squared times
delta B-- sometimes people

116
00:09:12,910 --> 00:09:15,870
in order to avoid cluttering
don't put the psi.

117
00:09:15,870 --> 00:09:17,910
I don't know whether
to put it or not.

118
00:09:17,910 --> 00:09:21,130
It does look a little more
messy with the psi there,

119
00:09:21,130 --> 00:09:24,420
but it's something you
have to keep in mind.

120
00:09:24,420 --> 00:09:26,940
Each time you have
an uncertainty,

121
00:09:26,940 --> 00:09:29,550
you are talking about
some specific state

122
00:09:29,550 --> 00:09:31,380
that should not be forgotten.

123
00:09:31,380 --> 00:09:35,940
So maybe I'll erase it to
make it look a little nicer.

124
00:09:35,940 --> 00:09:41,140
Delta B squared-- now
it's an inequality.

125
00:09:41,140 --> 00:09:44,260
So not just equality,
but inequality.

126
00:09:44,260 --> 00:09:48,270
That product of uncertainties
must exceed a number--

127
00:09:48,270 --> 00:09:52,410
a computable number-- which is
given by the following thing.

128
00:10:03,030 --> 00:10:04,695
OK, so here it is.

129
00:10:07,530 --> 00:10:11,220
This is a number, is
the expectation value

130
00:10:11,220 --> 00:10:17,236
of this strange operator
in the state psi squared.

131
00:10:22,700 --> 00:10:27,490
So even such a statement is
somewhat quite confusing,

132
00:10:27,490 --> 00:10:34,690
because you wish to know
what kind of number is this.

133
00:10:34,690 --> 00:10:36,430
Could this be a complex number?

134
00:10:36,430 --> 00:10:42,320
If it were a complex
number, why am I squaring?

135
00:10:42,320 --> 00:10:44,870
That doesn't make any sense.

136
00:10:44,870 --> 00:10:47,720
Inequalities-- these
are real numbers.

137
00:10:47,720 --> 00:10:50,550
Deltas are defined
to be real numbers.

138
00:10:50,550 --> 00:10:52,010
They're the norms.

139
00:10:52,010 --> 00:10:55,020
So this is real positive.

140
00:10:55,020 --> 00:11:00,260
This would make no sense if
this would be a complex number.

141
00:11:00,260 --> 00:11:03,410
So this number better be real.

142
00:11:03,410 --> 00:11:06,200
And the way it's
written, it seems

143
00:11:06,200 --> 00:11:08,490
to be particularly
confusing, because there

144
00:11:08,490 --> 00:11:10,660
seems to be an i here.

145
00:11:10,660 --> 00:11:16,530
So at first sight, you might
say, well, can it be real?

146
00:11:16,530 --> 00:11:19,060
But the thing that you
should really focus here

147
00:11:19,060 --> 00:11:20,930
is this whole thing.

148
00:11:20,930 --> 00:11:22,115
This is some operator.

149
00:11:27,010 --> 00:11:32,890
And against all first
impressions, this operator

150
00:11:32,890 --> 00:11:37,700
formed by taking the
commutator of A and B--

151
00:11:37,700 --> 00:11:40,850
this is the
commutator A B minus B

152
00:11:40,850 --> 00:11:46,580
A-- is Hermitian,
because, in fact,

153
00:11:46,580 --> 00:11:50,640
if you have two
operators, and you take

154
00:11:50,640 --> 00:11:54,880
the commutator, if the
two of them are Hermitian,

155
00:11:54,880 --> 00:11:58,080
the answer is not Hermitian.

156
00:11:58,080 --> 00:12:04,850
And that you know already--
x with p is equal to i h bar.

157
00:12:04,850 --> 00:12:09,500
These are Hermitian operators,
and suddenly the commutator

158
00:12:09,500 --> 00:12:11,630
is not a Hermitian operator.

159
00:12:11,630 --> 00:12:13,050
You have the unit here.

160
00:12:13,050 --> 00:12:16,090
A Hermitian operator
with a number

161
00:12:16,090 --> 00:12:18,210
here would have to
be a real things.

162
00:12:18,210 --> 00:12:21,380
So there's an extra i,
that's your first hint

163
00:12:21,380 --> 00:12:22,905
that this i is important.

164
00:12:25,480 --> 00:12:31,450
So the fact is that
this operator as defind

165
00:12:31,450 --> 00:12:40,980
here is Hermitian, because
if you take 1 over i A B--

166
00:12:40,980 --> 00:12:46,980
and we're going to try to
take its Hermitian conjugate--

167
00:12:46,980 --> 00:12:54,790
we have 1 over i A B
minus B A. And we're

168
00:12:54,790 --> 00:12:57,320
taking the Hermitian conjugate.

169
00:12:57,320 --> 00:13:01,940
Now, the i is going to
get complex conjugated,

170
00:13:01,940 --> 00:13:05,930
so you're going to
get 1 over minus i.

171
00:13:05,930 --> 00:13:09,810
The Hermitian
conjugate of a product

172
00:13:09,810 --> 00:13:12,850
is the Hermitian conjugate
in opposite order.

173
00:13:12,850 --> 00:13:19,065
So it would be B dagger A
dagger minus A dagger B dagger.

174
00:13:23,340 --> 00:13:26,950
And of course, these
operators are Hermitian,

175
00:13:26,950 --> 00:13:33,600
so 1 over minus i
is minus 1 over i.

176
00:13:33,600 --> 00:13:40,450
And here I get B A minus
A B. So with a minus sign,

177
00:13:40,450 --> 00:13:44,310
this is 1 over i A B again.

178
00:13:47,030 --> 00:13:52,720
So the operator is equal to
its dagger-- its adjoint.

179
00:13:52,720 --> 00:13:55,170
And therefore, this
operator is Hermitian.

180
00:14:02,380 --> 00:14:06,850
And as we proved,
the expectation value

181
00:14:06,850 --> 00:14:12,920
of any Hermitian
operator is real.

182
00:14:12,920 --> 00:14:15,500
And we're in good shape.

183
00:14:15,500 --> 00:14:17,040
We have a real number.

184
00:14:17,040 --> 00:14:19,950
This could be negative.

185
00:14:19,950 --> 00:14:22,150
And a number, when
you square it,

186
00:14:22,150 --> 00:14:23,910
is going to be a
positive number.

187
00:14:23,910 --> 00:14:25,650
So this makes sense.

188
00:14:25,650 --> 00:14:29,740
We're writing something
that at least makes sense.

189
00:14:29,740 --> 00:14:32,370
Another way, of course,
to write this equation,

190
00:14:32,370 --> 00:14:35,065
if you prefer--
this inequality, I

191
00:14:35,065 --> 00:14:38,750
mean-- is to take
the square root.

192
00:14:38,750 --> 00:14:43,040
So you could write it
delta A times delta

193
00:14:43,040 --> 00:14:50,460
B. Since this is a real number,
I can take the square root

194
00:14:50,460 --> 00:14:57,300
and write just this as absolute
value of psi, 1 over 2i i

195
00:14:57,300 --> 00:15:02,050
A B psi.

196
00:15:02,050 --> 00:15:05,685
And these bars here
are absolute value.

197
00:15:09,060 --> 00:15:12,880
They're not norm of a vector.

198
00:15:12,880 --> 00:15:16,080
They are not norm
of a complex number.

199
00:15:16,080 --> 00:15:20,460
They are just absolute value,
because the thing inside

200
00:15:20,460 --> 00:15:22,400
is a real thing.

201
00:15:22,400 --> 00:15:27,160
So if you prefer,
whatever you like better,

202
00:15:27,160 --> 00:15:32,070
you've got here the statement
of the uncertainty principle.

203
00:15:32,070 --> 00:15:36,450
So the good thing about
this uncertainty principle

204
00:15:36,450 --> 00:15:41,160
formulated this way is that
it's completely precise,

205
00:15:41,160 --> 00:15:44,700
because you've defined
uncertainties precisely.

206
00:15:44,700 --> 00:15:48,250
Many times, when you first
study the uncertainty principle,

207
00:15:48,250 --> 00:15:51,020
you don't define
uncertainties precisely,

208
00:15:51,020 --> 00:15:53,050
and the uncertainty
principle is something

209
00:15:53,050 --> 00:15:59,600
that goes with [? sim ?] is
approximately equal to this.

210
00:15:59,600 --> 00:16:02,830
And you make statements that
are intuitively interesting,

211
00:16:02,830 --> 00:16:04,670
but are not thoroughly precise.

212
00:16:04,670 --> 00:16:07,060
Yes, question, yes.

213
00:16:07,060 --> 00:16:08,810
AUDIENCE: Should that
be greater or equal?

214
00:16:08,810 --> 00:16:15,020
PROFESSOR: Greater than or equal
to, yes-- no miracles here.

215
00:16:15,020 --> 00:16:18,000
Other question?

216
00:16:18,000 --> 00:16:18,630
Other question?

217
00:16:25,850 --> 00:16:29,960
So we have to prove this.

218
00:16:29,960 --> 00:16:31,920
And why do you
have to prove this?

219
00:16:31,920 --> 00:16:34,640
This is a case,
actually, in which

220
00:16:34,640 --> 00:16:39,020
many interesting questions
are based on the proof.

221
00:16:39,020 --> 00:16:42,100
Why would that be the case?

222
00:16:42,100 --> 00:16:47,630
Well, a question that is
always of great interest

223
00:16:47,630 --> 00:16:50,080
is reducing uncertainties.

224
00:16:50,080 --> 00:16:58,140
Now, if two operators commute,
this right-hand side is 0

225
00:16:58,140 --> 00:17:00,420
and it just says
that the uncertainty

226
00:17:00,420 --> 00:17:03,830
could be made
perhaps equal to 0.

227
00:17:03,830 --> 00:17:06,839
It doesn't mean that
the uncertainty is 0.

228
00:17:06,839 --> 00:17:10,010
It may depend on the state,
even if the operators commute.

229
00:17:10,010 --> 00:17:13,190
This is just telling
you it's bigger than 0,

230
00:17:13,190 --> 00:17:18,579
and perhaps by being clever,
you can make it equal to 0.

231
00:17:18,579 --> 00:17:21,230
Similarly, when you
have two operators that

232
00:17:21,230 --> 00:17:24,670
just don't commute, it
is of great importance

233
00:17:24,670 --> 00:17:28,920
to try to figure out if there
is some states for which

234
00:17:28,920 --> 00:17:32,690
the uncertainty
relation is saturated.

235
00:17:32,690 --> 00:17:36,390
So this is the question
that, in fact, you could not

236
00:17:36,390 --> 00:17:42,490
answer if you just know this
theorem written like this,

237
00:17:42,490 --> 00:17:44,940
because there's
no statement here

238
00:17:44,940 --> 00:17:50,430
of what are the conditions
for which this inequality is

239
00:17:50,430 --> 00:17:51,960
saturated.

240
00:17:51,960 --> 00:17:55,410
So as we'll do the proof,
we'll find those conditions.

241
00:17:55,410 --> 00:17:59,350
And in fact, they go
a little beyond what

242
00:17:59,350 --> 00:18:02,690
the Schwarz
inequality would say.

243
00:18:02,690 --> 00:18:07,180
I mentioned last time that
this is a classic example

244
00:18:07,180 --> 00:18:09,880
of something that looks
like the Schwarz inequality,

245
00:18:09,880 --> 00:18:12,020
and indeed, that will
be the central part

246
00:18:12,020 --> 00:18:13,540
of the demonstration.

247
00:18:13,540 --> 00:18:18,570
But there's one extra step
there that we will have to do.

248
00:18:18,570 --> 00:18:23,230
And therefore, if you
want to understand

249
00:18:23,230 --> 00:18:28,020
when this is saturated, when
do you have minimum uncertainty

250
00:18:28,020 --> 00:18:33,230
states, then you need
to know the proof.

251
00:18:33,230 --> 00:18:37,830
So before we do, of
course, even the proof,

252
00:18:37,830 --> 00:18:40,750
there's an example--
the classic illustration

253
00:18:40,750 --> 00:18:50,509
that should be mentioned--
A equal x and B equals p,

254
00:18:50,509 --> 00:18:53,880
xp equal i h bar.

255
00:18:53,880 --> 00:18:55,960
That's the identity.

256
00:18:55,960 --> 00:19:02,930
So delta x squared
delta p squared

257
00:19:02,930 --> 00:19:09,965
is greater or equal than psi 1
over 2i-- the commutator-- i h

258
00:19:09,965 --> 00:19:16,940
bar 1 psi squared.

259
00:19:16,940 --> 00:19:18,740
And what do we get here?

260
00:19:18,740 --> 00:19:24,580
We get the i's cancel, the h bar
over 2 goes out, gets squared,

261
00:19:24,580 --> 00:19:28,356
and everything else is equal
to 1, because h is normalized.

262
00:19:31,130 --> 00:19:36,860
So the precise version of
the uncertainty principle

263
00:19:36,860 --> 00:19:43,340
is this one for x and p.

264
00:19:46,060 --> 00:19:49,240
And we will, of course,
try to figure out

265
00:19:49,240 --> 00:19:50,970
when we can saturate this.

266
00:19:50,970 --> 00:19:53,320
What kind of wave
functions saturate them?

267
00:19:53,320 --> 00:19:59,015
You know the ones that are
just sort of strange-- if x

268
00:19:59,015 --> 00:20:02,100
is totally localized, the
uncertainty of momentum

269
00:20:02,100 --> 00:20:07,210
must be infinite, because
if delta x is 0, well,

270
00:20:07,210 --> 00:20:09,850
to make this something that
at least doesn't contradict

271
00:20:09,850 --> 00:20:12,970
the identity, delta
p better be infinite.

272
00:20:12,970 --> 00:20:15,450
Similarly, if you
have an eigenstate

273
00:20:15,450 --> 00:20:19,420
of p, which is a wave,
is totally delocalized,

274
00:20:19,420 --> 00:20:22,270
and you have infinite
here and 0 here.

275
00:20:22,270 --> 00:20:25,370
Well, they're interesting
states that have both,

276
00:20:25,370 --> 00:20:28,580
and we're going to
try to find the ones

277
00:20:28,580 --> 00:20:30,960
of minimum uncertainty.

278
00:20:30,960 --> 00:20:34,810
So OK, we've stated
the principle.

279
00:20:34,810 --> 00:20:36,860
We've given an example.

280
00:20:36,860 --> 00:20:38,620
We've calculated an uncertainty.

281
00:20:38,620 --> 00:20:43,320
Let us prove the theorem.

282
00:20:43,320 --> 00:20:49,580
So as we mentioned before, this
idea that the uncertainty is

283
00:20:49,580 --> 00:20:51,810
a norm, is a good one.

284
00:20:51,810 --> 00:20:55,240
So let's define two
auxilliary variables--

285
00:20:55,240 --> 00:21:03,300
f, a state f, which is going
to be A minus the expectation

286
00:21:03,300 --> 00:21:08,180
value of A on psi.

287
00:21:08,180 --> 00:21:10,810
And we can put the ket here.

288
00:21:10,810 --> 00:21:15,960
And g, which is going to
be B minus the expectation

289
00:21:15,960 --> 00:21:20,505
value of B, psi.

290
00:21:23,330 --> 00:21:28,100
Now what do we know about this?

291
00:21:28,100 --> 00:21:32,700
Well the uncertainties are
the norms of these states,

292
00:21:32,700 --> 00:21:35,250
so the norm squared
of these states

293
00:21:35,250 --> 00:21:36,940
are the uncertainty squared.

294
00:21:36,940 --> 00:21:46,580
So delta A squared is
f f, the norm squared.

295
00:21:46,580 --> 00:21:51,712
And delta B squared is g g.

296
00:21:55,100 --> 00:22:02,490
And Schwarz' inequality
says that the norm

297
00:22:02,490 --> 00:22:06,000
of f times the normal
of g is greater than

298
00:22:06,000 --> 00:22:10,460
or equal than the absolute
value of the inner product of f

299
00:22:10,460 --> 00:22:12,520
with g.

300
00:22:12,520 --> 00:22:17,870
So squaring this thing,
which is convenient perhaps

301
00:22:17,870 --> 00:22:24,950
at this moment, we have f
f-- norm squared of f-- norm

302
00:22:24,950 --> 00:22:34,800
squared of g must be greater
than or equal than f g squared,

303
00:22:34,800 --> 00:22:36,170
absolute value squared.

304
00:22:40,490 --> 00:22:41,920
So this is Schwarz.

305
00:22:49,820 --> 00:22:55,540
And this is going to
just make a note-- here

306
00:22:55,540 --> 00:22:58,360
we know when this is saturated.

307
00:22:58,360 --> 00:23:02,400
It will be saturated
if f is parallel to g.

308
00:23:02,400 --> 00:23:05,230
If these two vectors are
parallel to each other,

309
00:23:05,230 --> 00:23:07,250
the Schwarz inequality
is saturated.

310
00:23:07,250 --> 00:23:09,980
So that's something
to keep in mind.

311
00:23:09,980 --> 00:23:12,510
We'll use it soon enough.

312
00:23:12,510 --> 00:23:18,480
But at this moment, we can
simply rewrite this as delta

313
00:23:18,480 --> 00:23:24,280
A squared times delta B
squared-- after all, those

314
00:23:24,280 --> 00:23:29,520
were definitions-- are
greater than or equal--

315
00:23:29,520 --> 00:23:35,420
and this is going to be a
complex number in general,

316
00:23:35,420 --> 00:23:41,780
so f g in Schwarz' inequality
is just a complex number.

317
00:23:41,780 --> 00:23:53,540
So this is real of f g squared,
plus the imaginary part

318
00:23:53,540 --> 00:23:57,345
of f g squared.

319
00:24:02,300 --> 00:24:07,720
So that's what we have--
real and imaginary part.

320
00:24:07,720 --> 00:24:11,680
So let's try to get what f g is.

321
00:24:11,680 --> 00:24:16,610
So what is f g?

322
00:24:16,610 --> 00:24:18,800
Let's compute it.

323
00:24:18,800 --> 00:24:24,180
Well we must take the bra
corresponding to this,

324
00:24:24,180 --> 00:24:25,730
so this is psi.

325
00:24:25,730 --> 00:24:28,180
Since the operator
is Hermitian, you

326
00:24:28,180 --> 00:24:33,130
have A minus
expectation value of A,

327
00:24:33,130 --> 00:24:39,740
and here you have B minus
expectation value of B psi.

328
00:24:49,090 --> 00:24:53,650
Now we can expand this, and
it will be useful to expand.

329
00:24:53,650 --> 00:24:58,690
But at the same time, I will
invent a little notation here.

330
00:24:58,690 --> 00:25:05,390
I'll call this A check,
and this B check.

331
00:25:05,390 --> 00:25:14,360
And for reference, I'll put
that this is psi A check B check

332
00:25:14,360 --> 00:25:14,860
psi.

333
00:25:18,970 --> 00:25:23,000
On the other hand, let's
just compute what we get.

334
00:25:23,000 --> 00:25:26,190
So what do we get?

335
00:25:26,190 --> 00:25:31,010
Well, let's expand this.

336
00:25:31,010 --> 00:25:35,670
Well, the first term is
A times B on psi psi,

337
00:25:35,670 --> 00:25:37,610
and we're not going
to be able to do

338
00:25:37,610 --> 00:25:42,960
much about that-- A B psi.

339
00:25:42,960 --> 00:25:49,050
And then we start getting
funny terms-- A cross with B,

340
00:25:49,050 --> 00:25:51,910
and that's-- if you
think about it a second,

341
00:25:51,910 --> 00:25:55,655
this is just going to be equal
to the expectation value of A

342
00:25:55,655 --> 00:25:58,280
times the expectation of B,
because the expectation value

343
00:25:58,280 --> 00:26:03,080
of B is a number, and then A
is sandwich between two psi.

344
00:26:03,080 --> 00:26:07,770
So from this cross product,
you get expectation value

345
00:26:07,770 --> 00:26:11,400
of A, expectation value
of B, with a minus sign.

346
00:26:11,400 --> 00:26:14,450
From this cross product, you
get the expectation value of A

347
00:26:14,450 --> 00:26:18,030
and expectation value of B--
another one with a minus sign.

348
00:26:18,030 --> 00:26:20,500
And then one with a plus sign.

349
00:26:20,500 --> 00:26:27,790
So the end result is a
single one with a minus sign.

350
00:26:27,790 --> 00:26:42,740
So expectation value of A,
expectation value of B. Now,

351
00:26:42,740 --> 00:26:47,250
if I change f and
g, I would like

352
00:26:47,250 --> 00:26:52,220
to compute not only fg inner
product, but gf inner product.

353
00:26:52,220 --> 00:26:54,090
And you may say why?

354
00:26:54,090 --> 00:26:58,370
Well, I want it because
I need the real part

355
00:26:58,370 --> 00:27:04,390
and the imaginary parts, and
gf is the complex conjugate

356
00:27:04,390 --> 00:27:08,070
of f g, so might
as well compute it.

357
00:27:08,070 --> 00:27:11,230
So what is gf?

358
00:27:11,230 --> 00:27:14,120
Now you don't have to do
the calculation again,

359
00:27:14,120 --> 00:27:17,510
because basically you
change g to f or f

360
00:27:17,510 --> 00:27:20,990
to g by exchanging A
and B. So I can just

361
00:27:20,990 --> 00:27:32,360
say that this is psi
B A psi minus A B.

362
00:27:32,360 --> 00:27:35,060
And if I write it
this way, I say

363
00:27:35,060 --> 00:27:41,300
it's just psi B
check A check psi.

364
00:27:44,530 --> 00:27:49,670
OK so we've done some work, and
the reason we've done this work

365
00:27:49,670 --> 00:27:51,825
is because we
actually need to write

366
00:27:51,825 --> 00:27:57,880
the right-hand side
of the inequality.

367
00:27:57,880 --> 00:28:01,400
And let's, therefore,
explore what these ones are.

368
00:28:01,400 --> 00:28:09,200
So for example, the
imaginary part of f g

369
00:28:09,200 --> 00:28:19,682
is 1 over 2i f g minus its
complex conjugate-- gf.

370
00:28:27,590 --> 00:28:29,680
Imaginary part of
a complex number

371
00:28:29,680 --> 00:28:34,040
is z minus z star divided by 2i.

372
00:28:34,040 --> 00:28:38,740
now, fg minus gf
is actually simple,

373
00:28:38,740 --> 00:28:43,660
because this product of
expectation values cancel,

374
00:28:43,660 --> 00:28:48,210
and this gives me the
commutator of A with B.

375
00:28:48,210 --> 00:28:55,650
So this is 1 over 2i, and
you have psi expectation

376
00:28:55,650 --> 00:29:00,040
value of A B commutator.

377
00:29:00,040 --> 00:29:04,550
So actually, that looks
exactly like what we want.

378
00:29:04,550 --> 00:29:09,470
And we're not going to be
able to simplify it more.

379
00:29:09,470 --> 00:29:12,550
We can put the 1 over 2i inside.

380
00:29:12,550 --> 00:29:13,450
That fine.

381
00:29:13,450 --> 00:29:15,800
It's sort of in the operator.

382
00:29:15,800 --> 00:29:21,490
It can go out, but we're not
going to do better than that.

383
00:29:21,490 --> 00:29:24,820
You already recognize,
in some sense,

384
00:29:24,820 --> 00:29:29,390
the inequality we want
to prove, because if this

385
00:29:29,390 --> 00:29:33,340
is that, you could ignore
this and say, well,

386
00:29:33,340 --> 00:29:36,380
it's anyway greater
than this thing.

387
00:29:36,380 --> 00:29:39,770
And that's this term.

388
00:29:39,770 --> 00:29:43,040
But let's write the other one,
at least for a little while.

389
00:29:43,040 --> 00:29:57,490
Real of fg would be
1/2 of fg plus gf.

390
00:29:57,490 --> 00:30:00,110
And now it is your choice
how you write this.

391
00:30:00,110 --> 00:30:03,870
There's nothing great
that you can do.

392
00:30:03,870 --> 00:30:09,890
The sum of these two things
have AB plus BA and then twice

393
00:30:09,890 --> 00:30:13,000
of this expectation
value, so it's not

394
00:30:13,000 --> 00:30:17,080
nothing particularly inspiring.

395
00:30:17,080 --> 00:30:22,420
So you put these
two terms and just

396
00:30:22,420 --> 00:30:29,450
write it like this-- 1/2
of psi anti-commutator

397
00:30:29,450 --> 00:30:32,520
off A check with B check.

398
00:30:32,520 --> 00:30:37,710
Anti-commutator, remember,
is this combination

399
00:30:37,710 --> 00:30:40,310
of operators in which you
take the product in one way,

400
00:30:40,310 --> 00:30:42,450
and add the product
in the other way.

401
00:30:42,450 --> 00:30:45,870
So I've used this
formula to write this,

402
00:30:45,870 --> 00:30:49,280
and you could write it as an
anti-commutator of A and B

403
00:30:49,280 --> 00:30:54,200
minus 2 times the
expectation values,

404
00:30:54,200 --> 00:30:56,490
or whichever way you want it.

405
00:30:56,490 --> 00:31:00,830
But at the end of the
day, that's what it is.

406
00:31:00,830 --> 00:31:03,140
And you cannot simplify it much.

407
00:31:03,140 --> 00:31:08,850
So your uncertainty
principle has become delta

408
00:31:08,850 --> 00:31:12,960
A squared delta
B squared greater

409
00:31:12,960 --> 00:31:25,340
than or equal to expectation
value of psi 1 over 2i A B psi

410
00:31:25,340 --> 00:31:36,130
squared plus expectation
value of psi 1 over 2

411
00:31:36,130 --> 00:31:42,840
A check B check psi squared.

412
00:31:42,840 --> 00:31:46,840
And some people call this
the generalized uncertainty

413
00:31:46,840 --> 00:31:48,120
principle.

414
00:31:48,120 --> 00:31:50,940
You may find some
textbooks that tell you

415
00:31:50,940 --> 00:31:54,710
"Prove the generalized
uncertainty principle,"

416
00:31:54,710 --> 00:31:56,880
because that's
really what you get

417
00:31:56,880 --> 00:32:01,860
if you follow the rules
and Schwarz' inequality.

418
00:32:01,860 --> 00:32:03,570
So it is of some interest.

419
00:32:03,570 --> 00:32:09,200
It is conceivable that sometimes
you may want to use this.

420
00:32:09,200 --> 00:32:14,530
But the fact is that
this is a real number.

421
00:32:14,530 --> 00:32:17,800
This is a Hermitian
operator as well.

422
00:32:17,800 --> 00:32:19,420
This is a real number.

423
00:32:19,420 --> 00:32:21,250
This is a positive number.

424
00:32:21,250 --> 00:32:27,350
So if you ignore it, you still
have the inequality holding.

425
00:32:27,350 --> 00:32:31,020
And many times-- and that's
the interesting thing--

426
00:32:31,020 --> 00:32:33,650
you really are
justified to ignore it.

427
00:32:33,650 --> 00:32:36,900
In fact, I don't know
of a single example--

428
00:32:36,900 --> 00:32:40,640
perhaps somebody can tell me--
in which that second term is

429
00:32:40,640 --> 00:32:41,140
useful.

430
00:32:44,300 --> 00:32:53,300
So what you say at this moment
is go ahead, drop that term,

431
00:32:53,300 --> 00:32:56,900
and get an inequality.

432
00:32:56,900 --> 00:33:07,380
So it follows directly from
that, from this inequality,

433
00:33:07,380 --> 00:33:12,710
that delta A squared
delta B squared

434
00:33:12,710 --> 00:33:15,720
is greater than or equal--
you might say, well,

435
00:33:15,720 --> 00:33:16,930
how do you know it's equal?

436
00:33:16,930 --> 00:33:19,460
Maybe that thing cannot be 0.

437
00:33:19,460 --> 00:33:22,330
Well, it can be 0
in some examples.

438
00:33:22,330 --> 00:33:29,460
So it's still greater than
or equal to psi 1 over 2i

439
00:33:29,460 --> 00:33:36,080
A B psi squared.

440
00:33:39,430 --> 00:33:43,090
And that's by ignoring
the positive quantity.

441
00:33:43,090 --> 00:33:49,130
So that is really the proof
of the uncertainty principle.

442
00:33:49,130 --> 00:33:54,960
But now we can ask what
are the things that

443
00:33:54,960 --> 00:33:58,030
have to happen for the
uncertainty principle

444
00:33:58,030 --> 00:34:00,290
to be saturated?

445
00:34:00,290 --> 00:34:08,199
That you really have delta A
delta B equal to this quantity,

446
00:34:08,199 --> 00:34:09,649
so when can we saturate?

447
00:34:19,530 --> 00:34:23,280
OK, what do we need?

448
00:34:23,280 --> 00:34:28,350
First we need Schwarz
inequality saturation.

449
00:34:28,350 --> 00:34:33,730
So f and g must be states that
are proportional to each other.

450
00:34:33,730 --> 00:34:42,623
So we need one, that
Schwarz is saturated.

451
00:34:49,080 --> 00:34:55,679
Which means that g is
some number times f,

452
00:34:55,679 --> 00:35:00,660
where beta is a complex number.

453
00:35:00,660 --> 00:35:03,780
This is complex vector
space, so parallel

454
00:35:03,780 --> 00:35:06,100
means multiply by
a complex number.

455
00:35:06,100 --> 00:35:09,430
That's still a parallel vector.

456
00:35:09,430 --> 00:35:13,130
So this is the
saturation of Schwarz.

457
00:35:13,130 --> 00:35:15,920
Now, what else do we need?

458
00:35:15,920 --> 00:35:18,780
Well, we need that
this quantity be

459
00:35:18,780 --> 00:35:25,570
0 as well, that the real part
of this thing is equal to 0.

460
00:35:25,570 --> 00:35:28,130
Otherwise, you really
cannot reach it.

461
00:35:28,130 --> 00:35:32,010
The true inequality is this,
so if you have Schwarz,

462
00:35:32,010 --> 00:35:33,070
you've saturated.

463
00:35:33,070 --> 00:35:36,390
This thing is equal
to this thing.

464
00:35:36,390 --> 00:35:39,440
The left-hand side is equal
to this whole right-hand side.

465
00:35:39,440 --> 00:35:41,680
Schwarz buys you that.

466
00:35:41,680 --> 00:35:45,440
But now we want this to
be just equal to that.

467
00:35:45,440 --> 00:35:52,010
So this thing must be
0, so the real part of f

468
00:35:52,010 --> 00:36:00,020
overlap g-- of fg must be 0.

469
00:36:00,020 --> 00:36:01,160
What does that mean?

470
00:36:01,160 --> 00:36:11,180
It means that fg
plus gf has to be 0.

471
00:36:11,180 --> 00:36:16,600
But now we know what g is,
so we can plug it here.

472
00:36:16,600 --> 00:36:19,310
So g is beta times f.

473
00:36:19,310 --> 00:36:26,850
Beta goes out, and
you get beta f f.

474
00:36:26,850 --> 00:36:31,120
Now when you form the bra
g, beta becomes beta star.

475
00:36:31,120 --> 00:36:38,730
So you get beta
star f f equals 0.

476
00:36:38,730 --> 00:36:45,640
And since f need not have
zero norm, because there

477
00:36:45,640 --> 00:36:48,530
is some uncertainty
presumably, you

478
00:36:48,530 --> 00:36:59,460
have that beta plus beta star
is equal to 0, or real of beta

479
00:36:59,460 --> 00:37:02,470
is equal to 0.

480
00:37:02,470 --> 00:37:07,190
So that said, it's not that bad.

481
00:37:07,190 --> 00:37:12,410
You need two things--
that the f and g vectors

482
00:37:12,410 --> 00:37:16,920
be parallel with a
complex constant,

483
00:37:16,920 --> 00:37:21,500
but actually, that constant
must be purely imaginary.

484
00:37:21,500 --> 00:37:31,260
So beta is purely
imaginary-- that this beta

485
00:37:31,260 --> 00:37:35,395
is equal to i lambda,
with lambda real.

486
00:37:40,600 --> 00:37:44,560
And we then are in shape.

487
00:37:44,560 --> 00:37:53,410
So for saturation,
we need just g

488
00:37:53,410 --> 00:37:57,120
to be that, and g to be beta f.

489
00:37:57,120 --> 00:38:08,220
So let me write that
equation over here.

490
00:38:08,220 --> 00:38:11,730
So g-- what was g?

491
00:38:11,730 --> 00:38:31,960
It's B, B minus absolute
value of B on psi, which is g,

492
00:38:31,960 --> 00:38:39,320
must be equal to beta,
which is i lambda

493
00:38:39,320 --> 00:38:43,963
A minus absolute
value of A on psi.

494
00:38:50,320 --> 00:38:54,186
Condition-- so this is the
final condition for saturation.

495
00:39:05,190 --> 00:39:09,380
now, that's a
strange-looking equation.

496
00:39:09,380 --> 00:39:11,810
It's not all that
obvious how you're even

497
00:39:11,810 --> 00:39:15,450
supposed to begin solving it.

498
00:39:15,450 --> 00:39:16,390
Why is that?

499
00:39:16,390 --> 00:39:20,080
Well, you're trying
to look for a psi,

500
00:39:20,080 --> 00:39:22,360
and you have a
constraint on the psi.

501
00:39:22,360 --> 00:39:24,485
The psi must satisfy this.

502
00:39:27,990 --> 00:39:32,640
I actually will tell
both Arum and Will

503
00:39:32,640 --> 00:39:37,990
to discuss some of these
things in recitation--

504
00:39:37,990 --> 00:39:42,030
how to calculate minimum
uncertainty wave packets based

505
00:39:42,030 --> 00:39:44,460
on this equation,
and what it means.

506
00:39:44,460 --> 00:39:46,810
But in principle, what
do you have to do?

507
00:39:46,810 --> 00:39:50,000
You have some kind of
differential equation,

508
00:39:50,000 --> 00:39:53,280
because you have, say, x and
p, and you want to saturate.

509
00:39:53,280 --> 00:39:56,330
So this is x, and this is p.

510
00:39:56,330 --> 00:39:59,760
Since p, you want to use a
coordinate representation,

511
00:39:59,760 --> 00:40:03,110
this will be a derivative, and
this will be a multiplication,

512
00:40:03,110 --> 00:40:06,460
so you'll get a differential
equation on the wave function.

513
00:40:06,460 --> 00:40:10,260
So you write an answer
for the wave function.

514
00:40:10,260 --> 00:40:13,260
You must calculate the
expectation value of B.

515
00:40:13,260 --> 00:40:15,480
You must calculate the
expectation value of A,

516
00:40:15,480 --> 00:40:17,620
and then plug into
this equation,

517
00:40:17,620 --> 00:40:20,570
and try to see if
your answer allows

518
00:40:20,570 --> 00:40:26,410
a solution-- and a solution
with some number here, lambda.

519
00:40:26,410 --> 00:40:28,590
At least one thing
I can tell you

520
00:40:28,590 --> 00:40:32,600
before you try this too hard--
this lambda is essentially

521
00:40:32,600 --> 00:40:38,470
fixed, because we can take
the norm of this equation.

522
00:40:38,470 --> 00:40:41,275
And that's an interesting
fact-- take the norm.

523
00:40:45,330 --> 00:40:48,190
And what is the norm of this?

524
00:40:48,190 --> 00:40:55,210
This is delta B, the
norm of this state.

525
00:40:55,210 --> 00:40:59,880
And the norm of i lambda--,
well norm of i is 1.

526
00:40:59,880 --> 00:41:02,740
Norm of lambda is
absolute value of lambda,

527
00:41:02,740 --> 00:41:04,910
because lambda was real.

528
00:41:04,910 --> 00:41:12,720
And you have delta A
here of psi, of course.

529
00:41:12,720 --> 00:41:19,410
So lambda can be either
plus or minus delta B

530
00:41:19,410 --> 00:41:23,910
of psi over delta A of psi.

531
00:41:23,910 --> 00:41:26,200
So that's not an
arbitrary constant.

532
00:41:26,200 --> 00:41:28,590
It's fixed by the
equation already,

533
00:41:28,590 --> 00:41:30,345
in terms of things
that you know.

534
00:41:35,760 --> 00:41:38,980
And therefore, this
will be a subject

535
00:41:38,980 --> 00:41:43,850
of problems in a little bit of
your recitation, in which you,

536
00:41:43,850 --> 00:41:50,980
hopefully, discuss how to find
minimum uncertainty packets.

537
00:41:50,980 --> 00:41:56,310
All right, so that's
it for the proof

538
00:41:56,310 --> 00:41:58,920
of the uncertainty principle.

539
00:41:58,920 --> 00:42:02,830
And as I told you, the proof
is useful in particular

540
00:42:02,830 --> 00:42:06,520
to find those special states
of saturated uncertainty.

541
00:42:06,520 --> 00:42:09,550
We'll have a lot to say
about them for the harmonic

542
00:42:09,550 --> 00:42:15,100
oscillator later on, and in
fact throughout the course.

543
00:42:15,100 --> 00:42:18,710
So are there any questions?

544
00:42:18,710 --> 00:42:19,950
Yes.

545
00:42:19,950 --> 00:42:22,450
AUDIENCE: So if we have one of
the states and an eigenstate,

546
00:42:22,450 --> 00:42:26,960
we know that [INAUDIBLE]
is 0 and we then

547
00:42:26,960 --> 00:42:29,214
mandate that the uncertainty
of the other variable

548
00:42:29,214 --> 00:42:29,922
must be infinite.

549
00:42:32,700 --> 00:42:35,770
But is it even possible to
talk about the uncertainty?

550
00:42:35,770 --> 00:42:38,125
And if so, are we
still guaranteed--

551
00:42:38,125 --> 00:42:40,096
we know that it's
infinite, but it's

552
00:42:40,096 --> 00:42:44,710
possible for 0 and an infinite
number to multiply [INAUDIBLE]

553
00:42:44,710 --> 00:42:48,860
PROFESSOR: Right, so you're in a
somewhat uncomfortable position

554
00:42:48,860 --> 00:42:51,470
if you have zero uncertainty.

555
00:42:51,470 --> 00:42:53,630
Then you need the other
one to be infinite.

556
00:42:53,630 --> 00:42:57,760
So the way, presumably,
you should think of that,

557
00:42:57,760 --> 00:43:01,590
is that you should take limits
of sequences of wave functions

558
00:43:01,590 --> 00:43:04,640
in which the uncertainty
in x is going to 0,

559
00:43:04,640 --> 00:43:07,710
and you will find that
as you take the limit,

560
00:43:07,710 --> 00:43:11,510
and delta x is going to 0, and
delta p is going to infinity,

561
00:43:11,510 --> 00:43:12,773
you can still have that.

562
00:43:16,160 --> 00:43:16,833
Other questions?

563
00:43:23,800 --> 00:43:29,720
Well, having done this, let's
try the more subtle case

564
00:43:29,720 --> 00:43:35,700
of the uncertainty principle
for energy and time.

565
00:43:35,700 --> 00:43:40,880
So that is a pretty
interesting subject, actually.

566
00:43:40,880 --> 00:43:44,520
And should I erase here?

567
00:43:44,520 --> 00:43:45,675
Yes, I think so.

568
00:43:52,550 --> 00:43:56,100
Actually, [? Griffith ?]
says that it's usually

569
00:43:56,100 --> 00:44:00,430
badly misunderstood, this
energy-time uncertainty

570
00:44:00,430 --> 00:44:04,960
principle, but seldom
your misunderstanding

571
00:44:04,960 --> 00:44:07,860
leads to a serious mistake.

572
00:44:07,860 --> 00:44:10,390
So you're saved.

573
00:44:10,390 --> 00:44:17,150
It's used in a hand-wavy way,
and it's roughly correct,

574
00:44:17,150 --> 00:44:19,835
although people say all
kinds of funny things

575
00:44:19,835 --> 00:44:21,670
that are not exactly right.

576
00:44:21,670 --> 00:44:37,670
So energy time
uncertainty-- so let

577
00:44:37,670 --> 00:44:42,910
me give a small motivation--
a hand-wavy motivation,

578
00:44:42,910 --> 00:44:46,290
so it doesn't get us
very far, but at least it

579
00:44:46,290 --> 00:44:49,060
gives you a picture
of what's going on.

580
00:44:49,060 --> 00:44:55,910
And these uncertainty
relations, in some sense,

581
00:44:55,910 --> 00:45:02,260
have a basis on some
simple statements that

582
00:45:02,260 --> 00:45:06,790
are totally classical, and
maybe a little imprecise,

583
00:45:06,790 --> 00:45:11,690
but incontrovertible,
about looking at waveforms,

584
00:45:11,690 --> 00:45:14,060
and trying to figure
out what's going on.

585
00:45:14,060 --> 00:45:20,850
So for example, suppose in
time you detect a fluctuation

586
00:45:20,850 --> 00:45:27,250
that as time progresses,
just suddenly turns on.

587
00:45:27,250 --> 00:45:31,660
Some wave that just dies
off after a little while.

588
00:45:31,660 --> 00:45:34,120
And you have a
good understanding

589
00:45:34,120 --> 00:45:36,640
of when it started,
and when it ended.

590
00:45:36,640 --> 00:45:45,610
And there's a time T.

591
00:45:45,610 --> 00:45:49,480
So whenever you have
a situation like that,

592
00:45:49,480 --> 00:45:53,560
you can try to count the
number of waves-- full waves

593
00:45:53,560 --> 00:45:56,260
that you see here.

594
00:45:56,260 --> 00:46:04,990
So the number of waves
would be equal to--

595
00:46:04,990 --> 00:46:11,790
or periods, number
of full waves--

596
00:46:11,790 --> 00:46:22,670
would be the total time divided
by the period of this wave.

597
00:46:22,670 --> 00:46:25,830
So sometimes T is
called the period.

598
00:46:25,830 --> 00:46:28,240
But here, T is the
total time here,

599
00:46:28,240 --> 00:46:30,690
and the period is
2 pi over omega.

600
00:46:30,690 --> 00:46:39,540
So we say this is
omega t over 2 pi.

601
00:46:39,540 --> 00:46:45,510
Now, the problem with these
waves that begin and end,

602
00:46:45,510 --> 00:46:48,330
is that you can't
quite see or make

603
00:46:48,330 --> 00:46:50,950
sure that you've got
the full wave here.

604
00:46:50,950 --> 00:46:56,210
So in the hand-wavy
way, we say that even

605
00:46:56,210 --> 00:46:59,090
as we looked at the
perfectly well-defined,

606
00:46:59,090 --> 00:47:01,170
and you know the
shape exactly-- it's

607
00:47:01,170 --> 00:47:04,070
been measured-- you can't
quite tell whether you've

608
00:47:04,070 --> 00:47:07,840
got the full wave here or
a quarter of a wave more,

609
00:47:07,840 --> 00:47:14,890
so there's an uncertainty in
delta n which is of order 1.

610
00:47:14,890 --> 00:47:18,700
You miss half on one side,
and half on the other side.

611
00:47:18,700 --> 00:47:22,350
So if you have an
uncertainty here of order 1,

612
00:47:22,350 --> 00:47:25,760
and you have no
uncertainty in T,

613
00:47:25,760 --> 00:47:29,090
you would claim that
you have, actually,

614
00:47:29,090 --> 00:47:33,990
in some sense, an
uncertainty in what omega is.

615
00:47:33,990 --> 00:47:36,670
Omega might be well
measured here, but somehow

616
00:47:36,670 --> 00:47:39,480
towards the end you
can't quite see.

617
00:47:39,480 --> 00:47:45,260
T we said was precise, so
over 2 pi is equal to 1.

618
00:47:45,260 --> 00:47:49,700
I just took a delta of here,
and I said P is precise,

619
00:47:49,700 --> 00:47:52,050
so it's delta omega.

620
00:47:52,050 --> 00:47:55,610
So this is a
classical statement.

621
00:47:55,610 --> 00:47:59,100
An electrical engineer
would not need

622
00:47:59,100 --> 00:48:03,820
to know any quantum mechanics
to say that's about right,

623
00:48:03,820 --> 00:48:07,240
and you can make it
more or less precise.

624
00:48:07,240 --> 00:48:10,160
But that's a
classical statement.

625
00:48:10,160 --> 00:48:12,650
In quantum mechanics,
all that happens

626
00:48:12,650 --> 00:48:17,110
is that something has
become quantum, and the idea

627
00:48:17,110 --> 00:48:19,520
that you have
something like this,

628
00:48:19,520 --> 00:48:24,310
we can associate it with
a particle, a photon,

629
00:48:24,310 --> 00:48:28,630
and in which case, the
uncertainty in omega

630
00:48:28,630 --> 00:48:30,350
is uncertainty in energy.

631
00:48:30,350 --> 00:48:41,020
So for a photon, the uncertainty
is equal to h bar omega,

632
00:48:41,020 --> 00:48:48,970
so delta omega times h bar
is equal to the uncertainty

633
00:48:48,970 --> 00:48:49,470
in energy.

634
00:48:52,010 --> 00:49:00,150
So if you plug it in here,
you multiply it by h bar here,

635
00:49:00,150 --> 00:49:10,280
and you would get delta E
times T is equal to 2 pi h bar.

636
00:49:14,800 --> 00:49:17,350
And then you have to add words.

637
00:49:17,350 --> 00:49:19,140
What is T?

638
00:49:19,140 --> 00:49:22,430
Well, this T is the
time it takes the photon

639
00:49:22,430 --> 00:49:24,710
to go through your detector.

640
00:49:24,710 --> 00:49:26,520
You've been seeing it.

641
00:49:26,520 --> 00:49:27,560
You saw a wave.

642
00:49:27,560 --> 00:49:30,500
You recorded it, and took
a time T-- began, ended.

643
00:49:30,500 --> 00:49:34,260
And it so it's the
time it took you

644
00:49:34,260 --> 00:49:38,170
to have the pulse go through.

645
00:49:38,170 --> 00:49:42,600
And that time is related
to an uncertainty

646
00:49:42,600 --> 00:49:44,790
in the energy of the photon.

647
00:49:44,790 --> 00:49:48,500
And that's sort of the beginning
of a time energy uncertainty

648
00:49:48,500 --> 00:49:49,830
relationship.

649
00:49:49,830 --> 00:49:52,640
This is quantum,
because the idea

650
00:49:52,640 --> 00:49:56,110
that photons carry energies
and they're quantized--

651
00:49:56,110 --> 00:50:00,230
this is a single photon-- and
this connection with energy

652
00:50:00,230 --> 00:50:01,105
is quantum mechanics.

653
00:50:04,170 --> 00:50:08,110
So this is good and
reasonable intuition, perhaps.

654
00:50:08,110 --> 00:50:11,480
And it can be the basis
of all kinds of things.

655
00:50:11,480 --> 00:50:16,470
But it points out the fact that
the more delicate part here

656
00:50:16,470 --> 00:50:23,100
is T. How could I speak
of a time uncertainty?

657
00:50:23,100 --> 00:50:27,420
And the fact is that you can't
speak of a time uncertainty

658
00:50:27,420 --> 00:50:29,750
really precisely.

659
00:50:29,750 --> 00:50:31,830
And the reason is,
because there's

660
00:50:31,830 --> 00:50:34,950
no Hermitian operator
for which we could say,

661
00:50:34,950 --> 00:50:39,630
OK the eigenstates of this
Hermitian operator are times,

662
00:50:39,630 --> 00:50:44,100
and then you have a norm,
and it's an uncertainty.

663
00:50:44,100 --> 00:50:45,210
So you can't do it.

664
00:50:45,210 --> 00:50:49,530
So you have to do something
different this time.

665
00:50:49,530 --> 00:50:52,160
And happily, there's
something you

666
00:50:52,160 --> 00:50:55,360
can do that is precise
and makes sense.

667
00:50:55,360 --> 00:51:00,460
So we'll do it.

668
00:51:00,460 --> 00:51:04,330
So what we have to do is just
try to use the uncertainty

669
00:51:04,330 --> 00:51:08,730
principle that we have,
and at least one operator.

670
00:51:08,730 --> 00:51:11,370
We can use something
that is good for us.

671
00:51:11,370 --> 00:51:14,740
We want uncertainty in energy,
and we have the Hamiltonian.

672
00:51:14,740 --> 00:51:15,850
It's an operator.

673
00:51:15,850 --> 00:51:19,660
So for that one, we can use
it, and that's the clue.

674
00:51:19,660 --> 00:51:31,900
So you'll take A to be
the Hamiltonian, and B

675
00:51:31,900 --> 00:51:35,870
to be some operator
Q that may depend

676
00:51:35,870 --> 00:51:41,070
on some things-- for example,
x and p, or whatever you want.

677
00:51:41,070 --> 00:51:43,820
But the one thing I want
to ask from this operator

678
00:51:43,820 --> 00:51:58,210
is that Q has no explicit time
dependence-- no explicit time

679
00:51:58,210 --> 00:51:59,255
dependence whatsoever.

680
00:52:02,800 --> 00:52:08,620
So let's see what this gives us
as an uncertainty relationship.

681
00:52:08,620 --> 00:52:15,140
Well, it would give us
that delta H squared--

682
00:52:15,140 --> 00:52:20,770
that's delta Q
squared-- would be

683
00:52:20,770 --> 00:52:27,780
greater than or equal
to the square of psi 1

684
00:52:27,780 --> 00:52:36,780
over 2i H with Q psi.

685
00:52:46,070 --> 00:52:49,160
OK, that's it.

686
00:52:49,160 --> 00:52:54,520
Well, but in order to get
some intuition from here,

687
00:52:54,520 --> 00:52:59,170
we better be able
to interpret this.

688
00:52:59,170 --> 00:53:00,820
This doesn't seem
to have anything

689
00:53:00,820 --> 00:53:03,050
to do with energy and time.

690
00:53:03,050 --> 00:53:08,260
So is there something
to do with time here?

691
00:53:08,260 --> 00:53:15,500
That is, in fact, a
very well-known result

692
00:53:15,500 --> 00:53:20,220
in quantum mechanics--
that somehow commutators

693
00:53:20,220 --> 00:53:27,460
with the Hamiltonian test the
time derivative of operators.

694
00:53:27,460 --> 00:53:31,970
So whenever you see an
H with Q commutator,

695
00:53:31,970 --> 00:53:35,800
you think ah, that's
roughly dQ dt.

696
00:53:41,229 --> 00:53:42,770
And we'll see what
happens with that.

697
00:53:42,770 --> 00:53:45,157
And say, oh, dQ
dt, but it doesn't

698
00:53:45,157 --> 00:53:46,850
depend on T-- you said 0.

699
00:53:46,850 --> 00:53:50,080
No it's not 0.

700
00:53:50,080 --> 00:53:54,350
There's no explicit dependence,
but we'll see what happens.

701
00:53:54,350 --> 00:53:58,080
So at this moment, you really
have to stop for one second

702
00:53:58,080 --> 00:54:05,030
and derive a familiar
result-- that may or may not

703
00:54:05,030 --> 00:54:08,480
be that familiar
to you from 804.

704
00:54:08,480 --> 00:54:11,370
I don't think it was
all that emphasized.

705
00:54:11,370 --> 00:54:15,380
Consider expectation value of Q.

706
00:54:15,380 --> 00:54:17,780
And then the
expectation of Q-- let

707
00:54:17,780 --> 00:54:24,920
me write it as psi
Q psi, like this.

708
00:54:24,920 --> 00:54:29,890
Now let's try to take the
time derivative of this thing.

709
00:54:29,890 --> 00:54:34,960
So what is the time derivative
of the expectation value of q?

710
00:54:34,960 --> 00:54:38,440
And the idea being
that look, the operator

711
00:54:38,440 --> 00:54:42,900
depends on some
things, and it can

712
00:54:42,900 --> 00:54:45,850
have time-dependent
expectation value,

713
00:54:45,850 --> 00:54:48,980
because the state
is changing in time.

714
00:54:48,980 --> 00:54:52,360
So operators can have
time-dependent expectation

715
00:54:52,360 --> 00:54:56,260
values even though the
operators don't depend on time.

716
00:54:56,260 --> 00:54:58,900
So for example, this
depends on x and p,

717
00:54:58,900 --> 00:55:03,770
and the x and p in a harmonic
oscillator are time dependent.

718
00:55:03,770 --> 00:55:09,970
They're moving around, and this
could have time dependence.

719
00:55:09,970 --> 00:55:11,730
So what do we get from here?

720
00:55:11,730 --> 00:55:15,630
Well, if I have to take the
time derivative of this,

721
00:55:15,630 --> 00:55:27,280
I have d psi dt here, Q
psi, plus psi Q d psi dt.

722
00:55:27,280 --> 00:55:31,990
And in doing this, and not
differentiating Q itself,

723
00:55:31,990 --> 00:55:35,100
I've used the fact that
this is an operator

724
00:55:35,100 --> 00:55:37,710
and there's no time
anywhere there.

725
00:55:37,710 --> 00:55:54,530
I didn't have to
differentiate Q.

726
00:55:54,530 --> 00:55:59,370
So how do we evaluate this?

727
00:55:59,370 --> 00:56:01,890
Well, you remember the
Schrodinger equation.

728
00:56:01,890 --> 00:56:04,180
Here the Schrodinger
equation comes in,

729
00:56:04,180 --> 00:56:07,280
because you have time
derivatives of your state.

730
00:56:07,280 --> 00:56:15,590
So i d psi dt, i H bar d
psi dt is equal to H psi.

731
00:56:15,590 --> 00:56:20,450
That's a full time-dependent
Schrodinger equation.

732
00:56:20,450 --> 00:56:23,345
So here, maybe, I
should write this

733
00:56:23,345 --> 00:56:27,380
like that-- this is all
time-dependent stuff.

734
00:56:27,380 --> 00:56:31,780
At this moment, I don't
ignore the time dependence.

735
00:56:31,780 --> 00:56:34,670
The states are not
stationary states.

736
00:56:34,670 --> 00:56:36,770
If they would be
stationary states,

737
00:56:36,770 --> 00:56:40,030
there would be no
energy uncertainty.

738
00:56:40,030 --> 00:56:45,350
So I have this, and therefore,
I plug this in here,

739
00:56:45,350 --> 00:56:55,100
and what do we get? i H
bar h psi Q psi plus psi

740
00:56:55,100 --> 00:56:57,975
Q i H bar H psi.

741
00:57:01,690 --> 00:57:06,020
Now, I got the i H in the
wrong place-- sorry-- 1

742
00:57:06,020 --> 00:57:11,180
over i H bar, and
1 over i H bar.

743
00:57:14,710 --> 00:57:18,690
Now the first term--
this thing comes out

744
00:57:18,690 --> 00:57:22,280
as its complex
conjugate-- 1 minus i H

745
00:57:22,280 --> 00:57:25,470
bar, because it's
on the first input.

746
00:57:25,470 --> 00:57:29,040
H is Hermitian, so I can
send it to the other side,

747
00:57:29,040 --> 00:57:32,165
so psi, HQ psi.

748
00:57:35,480 --> 00:57:39,790
Second term-- the 1
over i H just goes out,

749
00:57:39,790 --> 00:57:42,140
and I don't have
to move anybody.

750
00:57:42,140 --> 00:57:44,635
QH is there, psi.

751
00:57:48,210 --> 00:57:52,120
So actually, this
is i over H bar,

752
00:57:52,120 --> 00:57:55,780
because minus i
down goes up with i.

753
00:57:55,780 --> 00:58:01,902
And I have here psi
HQ, and this is minus i

754
00:58:01,902 --> 00:58:09,570
over H bar, so I
get HQ minus QH psi.

755
00:58:09,570 --> 00:58:15,975
So this is your final result--
the expectation value d

756
00:58:15,975 --> 00:58:25,400
dt of the expectation value
of Q is equal to i over H bar,

757
00:58:25,400 --> 00:58:43,060
expectation value of the
commutator of H with Q.

758
00:58:43,060 --> 00:58:49,490
So this is neat, and it should
always stick in your mind.

759
00:58:49,490 --> 00:58:50,890
This is true.

760
00:58:50,890 --> 00:58:54,520
We will see the
Heisenberg way of writing

761
00:58:54,520 --> 00:58:57,500
this equation in a
little while-- not today,

762
00:58:57,500 --> 00:58:59,680
but in a couple of weeks.

763
00:58:59,680 --> 00:59:05,935
But maybe even write
it even more briefly

764
00:59:05,935 --> 00:59:10,386
as i over H bar
expectation value of HQ.

765
00:59:15,920 --> 00:59:23,790
So what do we get from here?

766
00:59:23,790 --> 00:59:28,250
Well, we can go back to
our uncertainty principle,

767
00:59:28,250 --> 00:59:33,710
and rewrite it, having learned
that we have time derivative.

768
00:59:33,710 --> 00:59:38,300
So time finally showed
up, and that's good news.

769
00:59:38,300 --> 00:59:42,110
So we're maybe not too far
from a clear interpretation

770
00:59:42,110 --> 00:59:44,050
of the uncertainty principle.

771
00:59:44,050 --> 00:59:48,920
So we're going back
to that top equation,

772
00:59:48,920 --> 00:59:52,900
so that what we
have now is delta

773
00:59:52,900 --> 01:00:00,090
H squared delta Q squared
is that thing over there,

774
01:00:00,090 --> 01:00:03,500
the expectation
value of 1 over 2i.

775
01:00:03,500 --> 01:00:05,550
There's some signs
there, so what

776
01:00:05,550 --> 01:00:24,690
do we have-- equals 1 over
2i H bar over i d dt of Q.

777
01:00:24,690 --> 01:00:30,960
So what I did here was to say
that this expectation value was

778
01:00:30,960 --> 01:00:35,250
H bar over i d dt of Q,
and I plugged it in there.

779
01:00:38,000 --> 01:00:44,290
So you square this thing, so
there's not too much really

780
01:00:44,290 --> 01:00:44,910
to be done.

781
01:00:44,910 --> 01:00:47,460
The i don't matter at
the end of the day.

782
01:00:47,460 --> 01:00:49,910
It's a minus 1
that gets squared.

783
01:00:49,910 --> 01:00:56,020
So the H bar over 2-- I'm
sorry-- the H bar over 2

784
01:00:56,020 --> 01:00:59,440
does remain here, squared.

785
01:00:59,440 --> 01:01:06,485
And you have dQ dt squared.

786
01:01:09,930 --> 01:01:12,790
Q is a Hermitian operator.

787
01:01:12,790 --> 01:01:14,310
B was supposed to be Hermitian.

788
01:01:14,310 --> 01:01:16,630
The expectation value is real.

789
01:01:16,630 --> 01:01:18,345
The time derivative is real.

790
01:01:18,345 --> 01:01:20,920
It could be going up or down.

791
01:01:20,920 --> 01:01:23,720
So at the end of
the day, you have

792
01:01:23,720 --> 01:01:30,620
delta H delta Q is greater
than or equal to H bar

793
01:01:30,620 --> 01:01:36,050
over 2, the absolute
value of dQ over dt.

794
01:01:43,910 --> 01:01:46,830
There we go.

795
01:01:46,830 --> 01:01:50,110
This is, in a sense,
the best you can do.

796
01:01:50,110 --> 01:01:55,550
Let's try to interpret
what we've got.

797
01:01:55,550 --> 01:01:59,580
Well, we've got something
that still doesn't quite

798
01:01:59,580 --> 01:02:02,720
look like a time
uncertainty relationship,

799
01:02:02,720 --> 01:02:04,980
but there's time in there.

800
01:02:04,980 --> 01:02:07,700
But it's a matter
of a definition now.

801
01:02:11,600 --> 01:02:22,410
You see, if you have delta Q,
and you divide it by dQ dt,

802
01:02:22,410 --> 01:02:26,030
first it is some sort of time.

803
01:02:26,030 --> 01:02:28,570
It has the units of time.

804
01:02:28,570 --> 01:02:32,900
And we can define it, if
you wish, to be sub delta t.

805
01:02:35,660 --> 01:02:41,620
And what physically, does
this delta t represent?

806
01:02:41,620 --> 01:02:46,190
Well, it's roughly-- you
see, things change in time.

807
01:02:46,190 --> 01:02:49,750
The rate of change of the
expectation value of Q

808
01:02:49,750 --> 01:02:51,090
may not be uniform.

809
01:02:51,090 --> 01:02:55,160
It make change fast, or
it may change slowly.

810
01:02:55,160 --> 01:02:58,990
But suppose it's changing.

811
01:02:58,990 --> 01:03:01,850
Roughly, this ratio, of
this would be constant,

812
01:03:01,850 --> 01:03:06,060
is the time it takes the
expectation value of Q

813
01:03:06,060 --> 01:03:12,730
to change by delta Q. It
is like a distance divided

814
01:03:12,730 --> 01:03:13,313
by a velocity.

815
01:03:16,710 --> 01:03:25,480
So this is roughly the time
needed for the expectation

816
01:03:25,480 --> 01:03:38,280
value of Q to change by
delta Q, by the uncertainty.

817
01:03:38,280 --> 01:03:40,740
So it's a measure
of the time needed

818
01:03:40,740 --> 01:03:48,930
for a significant change,
if the expectation

819
01:03:48,930 --> 01:03:53,180
value, if the uncertainty
of Q is significant, and is

820
01:03:53,180 --> 01:03:56,550
comparable to Q. Well,
this is the time needed

821
01:03:56,550 --> 01:03:59,220
for significant change.

822
01:03:59,220 --> 01:04:02,960
Now this is pretty much all you
can do, except that of course,

823
01:04:02,960 --> 01:04:05,980
once you write it like
that, you pull this down,

824
01:04:05,980 --> 01:04:10,670
and you go up now,
delta H delta t

825
01:04:10,670 --> 01:04:15,630
is greater or equal
than H bar over 2.

826
01:04:15,630 --> 01:04:23,530
And this is the best you can
do with this kind of approach.

827
01:04:23,530 --> 01:04:25,518
Yes?

828
01:04:25,518 --> 01:04:27,890
AUDIENCE: [INAUDIBLE]

829
01:04:27,890 --> 01:04:31,290
PROFESSOR: Yeah, I
simply define this,

830
01:04:31,290 --> 01:04:35,180
which is a time that has
some meaning if you know what

831
01:04:35,180 --> 01:04:38,330
the uncertainty of the operator
is and how fast it's changing--

832
01:04:38,330 --> 01:04:41,310
is the time needed for a change.

833
01:04:41,310 --> 01:04:47,160
Once I defined this, I simply
brought this factor down here,

834
01:04:47,160 --> 01:04:51,210
so that delta Q over this
derivative is delta t,

835
01:04:51,210 --> 01:04:55,110
and the equation just
became this equation.

836
01:05:02,410 --> 01:05:07,120
So we'll try to figure out a
little more of what this means

837
01:05:07,120 --> 01:05:12,390
right away, but you can
make a few criticisms

838
01:05:12,390 --> 01:05:14,510
about this thing.

839
01:05:14,510 --> 01:05:17,380
You can say, look,
this delta time

840
01:05:17,380 --> 01:05:19,290
uncertainty is not universal.

841
01:05:19,290 --> 01:05:23,090
It depends which
operator Q you took.

842
01:05:23,090 --> 01:05:23,750
True enough.

843
01:05:26,440 --> 01:05:30,480
I cannot prove that it's
independent of the operator Q,

844
01:05:30,480 --> 01:05:34,440
and many times I cannot even
tell you which operator Q is

845
01:05:34,440 --> 01:05:37,880
the best operator
to think about.

846
01:05:37,880 --> 01:05:39,570
But you can try.

847
01:05:39,570 --> 01:05:42,600
And it does give
you-- first, it's

848
01:05:42,600 --> 01:05:47,660
a mathematical statement about
how fast things can change.

849
01:05:47,660 --> 01:05:54,080
And that contains physics, and
it contains a very precise fact

850
01:05:54,080 --> 01:05:54,580
as well.

851
01:05:57,490 --> 01:06:01,570
Actually, there's a version
of the uncertainty principle

852
01:06:01,570 --> 01:06:06,580
that you will explore in
the homework that is, maybe,

853
01:06:06,580 --> 01:06:12,130
an alternative picture of this,
and asks the following thing--

854
01:06:12,130 --> 01:06:15,630
if you have a state
and a stationary state,

855
01:06:15,630 --> 01:06:18,160
nothing changes in the state.

856
01:06:18,160 --> 01:06:21,880
But if it's a stationary
state, the energy uncertainty

857
01:06:21,880 --> 01:06:26,070
is 0, because the energy is
an eigenstate of the energy.

858
01:06:26,070 --> 01:06:27,750
So nothing changes.

859
01:06:27,750 --> 01:06:30,780
So you have to wait infinite
time for there to be a change,

860
01:06:30,780 --> 01:06:34,090
and this makes sense.

861
01:06:34,090 --> 01:06:36,860
Now you can ask the
following question-- suppose

862
01:06:36,860 --> 01:06:43,220
I have a state that is not
an eigenstate of energy.

863
01:06:43,220 --> 01:06:45,330
So therefore, for example,
the simplest thing

864
01:06:45,330 --> 01:06:47,624
would be a superposition
of two eigenstates

865
01:06:47,624 --> 01:06:48,540
of different energies.

866
01:06:51,100 --> 01:06:53,980
You can ask, well, there
will be time evolution

867
01:06:53,980 --> 01:06:57,710
and this state will
change in time.

868
01:06:57,710 --> 01:07:05,180
So how can I get a
constraint on changes?

869
01:07:05,180 --> 01:07:06,870
How can I approach changes?

870
01:07:06,870 --> 01:07:11,050
And people discovered the
following interesting fact--

871
01:07:11,050 --> 01:07:18,120
that if you have a state, it has
unit norm, and if it evolves,

872
01:07:18,120 --> 01:07:20,530
it may happen that
at some stage,

873
01:07:20,530 --> 01:07:24,320
it becomes orthogonal to
itself-- to the original one.

874
01:07:24,320 --> 01:07:26,150
And that is a big change.

875
01:07:26,150 --> 01:07:29,000
You become orthogonal
to what you used to be.

876
01:07:29,000 --> 01:07:32,400
That's as big a
change as can happen.

877
01:07:32,400 --> 01:07:37,350
And then you can ask,
is there a minimum time

878
01:07:37,350 --> 01:07:40,720
for which this can happen?

879
01:07:40,720 --> 01:07:42,830
What is the minimum
time in which

880
01:07:42,830 --> 01:07:46,410
a state can change
so much that it

881
01:07:46,410 --> 01:07:49,090
becomes orthogonal to itself?

882
01:07:49,090 --> 01:07:51,860
And there is such an
uncertainty principle.

883
01:07:51,860 --> 01:07:54,690
It's derived a little
differently from that.

884
01:07:54,690 --> 01:08:03,120
And it says that if you
take delta t to be the time

885
01:08:03,120 --> 01:08:23,760
it takes psi of x and t to
become orthogonal to psi of x0,

886
01:08:23,760 --> 01:08:28,638
then this delta
t times delta E--

887
01:08:28,638 --> 01:08:32,630
the uncertainty of the energies
is the uncertainty in h--

888
01:08:32,630 --> 01:08:37,494
is greater than or
equal to h bar over 4.

889
01:08:46,660 --> 01:08:50,040
Now a state may never
become orthogonal to itself,

890
01:08:50,040 --> 01:08:51,120
but that's OK.

891
01:08:51,120 --> 01:08:54,970
Then it's a big number
on the left-hand side.

892
01:08:54,970 --> 01:08:58,029
But the quickest it
can do it is that.

893
01:08:58,029 --> 01:09:00,540
And that's an interesting thing.

894
01:09:00,540 --> 01:09:02,650
And it's a version of the
uncertainty principle.

895
01:09:05,960 --> 01:09:08,270
I want to make a
couple more remarks,

896
01:09:08,270 --> 01:09:10,640
because this thing
is mysterious enough

897
01:09:10,640 --> 01:09:15,520
that it requires thinking.

898
01:09:15,520 --> 01:09:21,970
So let's make some precise
claims about energy

899
01:09:21,970 --> 01:09:26,550
uncertainties and then
give an example of what's

900
01:09:26,550 --> 01:09:29,439
happening in the
physical situation.

901
01:09:29,439 --> 01:09:30,720
Was there a question?

902
01:09:30,720 --> 01:09:33,095
Yes.

903
01:09:33,095 --> 01:09:33,970
AUDIENCE: [INAUDIBLE]

904
01:09:33,970 --> 01:09:36,303
PROFESSOR: You're going to
explore that in the homework.

905
01:09:36,303 --> 01:09:39,752
Actually, I don't think
you're going to show it, but--

906
01:09:39,752 --> 01:09:41,642
AUDIENCE: [INAUDIBLE]
H bar [INAUDIBLE]

907
01:09:41,642 --> 01:09:43,600
it's even less than the
uncertainty [INAUDIBLE]

908
01:09:49,776 --> 01:09:51,359
PROFESSOR: It's a
different statement.

909
01:09:51,359 --> 01:09:56,110
It's a very precise way of
measuring, creating a time.

910
01:09:56,110 --> 01:09:58,480
It's a precise
definition of time,

911
01:09:58,480 --> 01:10:02,054
and therefore,
there's no reason why

912
01:10:02,054 --> 01:10:03,220
it would have been the same.

913
01:10:07,280 --> 01:10:12,120
So here is a statement
that is interesting-- is

914
01:10:12,120 --> 01:10:27,740
that the uncertainty delta
E in an isolated system

915
01:10:27,740 --> 01:10:33,680
is constant-- doesn't change.

916
01:10:36,370 --> 01:10:39,770
And by an isolated
system, a system

917
01:10:39,770 --> 01:10:42,040
in which there's no
influences on it,

918
01:10:42,040 --> 01:10:47,900
a system in which you have
actually time independent

919
01:10:47,900 --> 01:10:48,920
Hamiltonians.

920
01:10:48,920 --> 01:10:56,030
So H is a time
independent Hamiltonian.

921
01:11:01,000 --> 01:11:04,370
Now that, of course, doesn't
mean the physics is boring.

922
01:11:04,370 --> 01:11:07,090
Time- independent Hamiltonians
are quite interesting,

923
01:11:07,090 --> 01:11:09,110
but you have a whole system.

924
01:11:09,110 --> 01:11:11,250
Let's take it to be isolated.

925
01:11:11,250 --> 01:11:14,050
There's no time dependent
things acting on it,

926
01:11:14,050 --> 01:11:18,880
and H should be a time
independent Hamiltonian.

927
01:11:18,880 --> 01:11:28,180
So I want to use
this statement to say

928
01:11:28,180 --> 01:11:32,170
the following--
if I take Q equals

929
01:11:32,170 --> 01:11:36,020
H in that theorem
over there, I get

930
01:11:36,020 --> 01:11:42,970
that d dt of the expectation
value of H would be what?

931
01:11:42,970 --> 01:11:45,480
It would be i over H bar.

932
01:11:45,480 --> 01:11:48,420
Since H is time
independent-- the condition

933
01:11:48,420 --> 01:11:50,700
here was that Q had
no time dependence.

934
01:11:50,700 --> 01:11:54,640
But then I get H
commutator with H.

935
01:11:54,640 --> 01:12:01,980
So I get here H commutator with
H. And that commutator is 0.

936
01:12:01,980 --> 01:12:07,230
However complicated an operator
is, it commutes with itself.

937
01:12:07,230 --> 01:12:13,150
So the expectation value of
the energy doesn't change.

938
01:12:13,150 --> 01:12:15,920
We call that energy
conservation.

939
01:12:15,920 --> 01:12:21,610
But still, if you take Q
now equal to H squared,

940
01:12:21,610 --> 01:12:25,790
the time derivative of
the expectation value of H

941
01:12:25,790 --> 01:12:29,760
squared, you get i over H bar.

942
01:12:29,760 --> 01:12:33,030
You're supposed to be
H commutator with Q,

943
01:12:33,030 --> 01:12:35,370
which is H squared, now.

944
01:12:35,370 --> 01:12:36,555
And that's also 0.

945
01:12:40,290 --> 01:12:46,550
So no power of the expectation
value of H vanishes.

946
01:12:46,550 --> 01:12:52,310
And therefore, we have
that the time derivative

947
01:12:52,310 --> 01:12:56,580
of the uncertainty
of H squared--

948
01:12:56,580 --> 01:13:01,300
which is the time derivative
of the expectation value of H

949
01:13:01,300 --> 01:13:06,270
squared minus the expectation
value of H squared-- well,

950
01:13:06,270 --> 01:13:10,490
we've shown each one of the
things on the right-hand side

951
01:13:10,490 --> 01:13:14,670
are 0, so this is 0.

952
01:13:14,670 --> 01:13:18,070
So delta H is constant.

953
01:13:25,650 --> 01:13:32,910
So the uncertainty-- delta
E or delta H of the system

954
01:13:32,910 --> 01:13:35,340
is constant.

955
01:13:35,340 --> 01:13:38,310
So what do we do with that?

956
01:13:38,310 --> 01:13:44,480
Well it helps us think a little
about time dependent processes.

957
01:13:44,480 --> 01:13:48,420
And the example we
must have in mind

958
01:13:48,420 --> 01:13:52,220
is perhaps the one
of a decay that

959
01:13:52,220 --> 01:13:57,010
leads to a radiation of
a photon, so a transition

960
01:13:57,010 --> 01:14:00,140
that leads to a
photon radiation.

961
01:14:00,140 --> 01:14:03,096
So let's consider that example.

962
01:14:08,900 --> 01:14:12,460
So we have an atom in
some excited state,

963
01:14:12,460 --> 01:14:16,020
decays to the ground state
and shoots out the photon.

964
01:14:28,340 --> 01:14:41,320
Then it's an unstable state,
because if it would be stable,

965
01:14:41,320 --> 01:14:44,190
it wouldn't change in time.

966
01:14:44,190 --> 01:14:47,300
And the excited state of an
atom is an unstable state,

967
01:14:47,300 --> 01:14:49,505
decays into-- goes
into the ground state.

968
01:14:55,590 --> 01:14:56,680
And it makes a photon.

969
01:15:02,090 --> 01:15:07,190
Now this idea of
the conservation

970
01:15:07,190 --> 01:15:12,980
of energy uncertainty at least
helps you in this situation

971
01:15:12,980 --> 01:15:16,610
that you would typically do
it with a lot of hand-waving,

972
01:15:16,610 --> 01:15:18,740
organize your thoughts.

973
01:15:18,740 --> 01:15:23,230
So what happens in such decay?

974
01:15:23,230 --> 01:15:34,090
There's a lifetime, which
is a typical time you

975
01:15:34,090 --> 01:15:38,480
have to wait for that
excited state to decay.

976
01:15:38,480 --> 01:15:41,590
And these lifetime
is called tau.

977
01:15:41,590 --> 01:15:45,830
And certainly as the
lifetime goes through,

978
01:15:45,830 --> 01:15:51,250
and the decay happens, some
observable changes a lot.

979
01:15:51,250 --> 01:15:53,950
Some observable Q
must change a lot.

980
01:15:53,950 --> 01:15:57,940
Maybe a position of the
electron in an orbit,

981
01:15:57,940 --> 01:16:02,480
or the angular momentum
of it, or some squared

982
01:16:02,480 --> 01:16:05,330
of the momentum--
some observable

983
01:16:05,330 --> 01:16:10,030
that we could do an atomic
calculation in more detail

984
01:16:10,030 --> 01:16:11,460
must change a lot.

985
01:16:11,460 --> 01:16:15,280
So there will be associated
with some observable that

986
01:16:15,280 --> 01:16:18,680
changes a lot
during the lifetime,

987
01:16:18,680 --> 01:16:23,140
because it takes that long
for this thing to change.

988
01:16:23,140 --> 01:16:26,150
There will be an
energy uncertainty

989
01:16:26,150 --> 01:16:28,130
associated to a lifetime.

990
01:16:28,130 --> 01:16:32,340
So how does the energy
uncertainty reflect itself?

991
01:16:32,340 --> 01:16:33,975
Well, you have a ground state.

992
01:16:38,830 --> 01:16:40,830
And you have this excited state.

993
01:16:40,830 --> 01:16:44,650
But generally, when you
have an excited state

994
01:16:44,650 --> 01:16:48,320
due to some interactions
that produce instability,

995
01:16:48,320 --> 01:16:52,380
you actually have a
lot of states here

996
01:16:52,380 --> 01:16:55,040
that are part of
the excited state.

997
01:16:55,040 --> 01:17:00,370
So you have an excited
state, but you do have,

998
01:17:00,370 --> 01:17:03,290
typically, a lot of
uncertainty-- but not

999
01:17:03,290 --> 01:17:06,380
a lot-- some uncertainty
of the energy here.

1000
01:17:06,380 --> 01:17:09,140
The state is not
a particular one.

1001
01:17:09,140 --> 01:17:10,865
If it would be a
particular one, it

1002
01:17:10,865 --> 01:17:14,790
would be a stationary state--
would stay there forever.

1003
01:17:14,790 --> 01:17:18,280
Nevertheless, it's a
combination of some things,

1004
01:17:18,280 --> 01:17:20,630
so it's not quite
a stationary state.

1005
01:17:20,630 --> 01:17:22,440
It couldn't be a
stationary state,

1006
01:17:22,440 --> 01:17:24,160
because it would be eternal.

1007
01:17:24,160 --> 01:17:27,580
So somehow, the
dynamics of this atom

1008
01:17:27,580 --> 01:17:31,625
must be such that there's
interactions between, say,

1009
01:17:31,625 --> 01:17:36,850
the electron and the nucleus, or
possibly a radiation field that

1010
01:17:36,850 --> 01:17:43,660
makes the state of
this electron unstable,

1011
01:17:43,660 --> 01:17:47,680
and associated to it an
uncertainty in the energy.

1012
01:17:47,680 --> 01:17:52,510
So there's an uncertainty
here, and this particle--

1013
01:17:52,510 --> 01:17:56,040
this electron goes eventually
to the ground state,

1014
01:17:56,040 --> 01:17:57,270
and it meets a photon.

1015
01:18:03,130 --> 01:18:09,670
So there is, associated to this
lifetime, an uncertainty delta

1016
01:18:09,670 --> 01:18:18,390
E times tau, and I will put
similar to H bar over 2.

1017
01:18:18,390 --> 01:18:24,060
And this would be
the delta E here,

1018
01:18:24,060 --> 01:18:26,430
because your state
must be a superposition

1019
01:18:26,430 --> 01:18:28,880
of some states over there.

1020
01:18:28,880 --> 01:18:31,740
And then what happens later?

1021
01:18:31,740 --> 01:18:34,360
Well, this particle goes
to the ground state--

1022
01:18:34,360 --> 01:18:40,060
no uncertainty any more
about what its energy is.

1023
01:18:40,060 --> 01:18:42,670
So the only possibility
at this moment

1024
01:18:42,670 --> 01:18:47,320
consistent with the conservation
of uncertainty in the system

1025
01:18:47,320 --> 01:18:49,780
is that the photon
carries the uncertainty.

1026
01:18:49,780 --> 01:18:54,120
So that photon must have
an uncertainty as well.

1027
01:18:54,120 --> 01:19:00,020
So delta energy
of the photon will

1028
01:19:00,020 --> 01:19:09,470
be equal to h bar delta
omega, or h delta nu.

1029
01:19:17,220 --> 01:19:23,230
So the end result is that
in a physical decay process,

1030
01:19:23,230 --> 01:19:24,230
there are uncertainties.

1031
01:19:24,230 --> 01:19:27,700
And the uncertainty
gets carried out,

1032
01:19:27,700 --> 01:19:30,970
and it's always there--
the delta E here

1033
01:19:30,970 --> 01:19:34,170
and the photon having
some uncertainty.

1034
01:19:34,170 --> 01:19:38,820
Now one of the most famous
applications of this thing

1035
01:19:38,820 --> 01:19:42,570
is related to the hyperfine
transition of hydrogen.

1036
01:19:42,570 --> 01:19:45,750
And we're very lucky in physics.

1037
01:19:45,750 --> 01:19:47,210
Physicists are very lucky.

1038
01:19:47,210 --> 01:19:52,220
This is a great break for
astronomy and cosmology,

1039
01:19:52,220 --> 01:19:56,370
and it's all based on this
uncertainty principle.

1040
01:19:56,370 --> 01:20:06,440
You have the hyperfine
transition of hydrogen.

1041
01:20:09,510 --> 01:20:12,960
So we will study
later in this course

1042
01:20:12,960 --> 01:20:18,430
that because of the
proton and electron

1043
01:20:18,430 --> 01:20:21,300
spins in the hydrogen
atom, there's

1044
01:20:21,300 --> 01:20:24,330
a splitting of
energies having to do

1045
01:20:24,330 --> 01:20:26,280
with the hyperfine interaction.

1046
01:20:26,280 --> 01:20:29,120
It's a magnetic
dipole interaction

1047
01:20:29,120 --> 01:20:31,560
between the proton
and the electron.

1048
01:20:31,560 --> 01:20:34,380
And there's going
to be a splitting.

1049
01:20:34,380 --> 01:20:38,510
And there's a transition
associated with this splitting.

1050
01:20:38,510 --> 01:20:42,160
So there's a hyperfine
splitting-- the ground state

1051
01:20:42,160 --> 01:20:45,270
of the hyperfine
splitting of some states.

1052
01:20:45,270 --> 01:20:49,360
And it's the top state
and the bottom state.

1053
01:20:49,360 --> 01:20:54,940
And as the system decays,
it emits a photon.

1054
01:20:54,940 --> 01:21:03,220
This photon is approximately
a 21 centimeter wavelength--

1055
01:21:03,220 --> 01:21:07,140
is the famous 21 centimeter
line of hydrogen.

1056
01:21:07,140 --> 01:21:15,740
And it corresponds to
about 1420 megahertz.

1057
01:21:15,740 --> 01:21:20,150
So how about so far so good.

1058
01:21:20,150 --> 01:21:23,960
There's an energy
splitting here,

1059
01:21:23,960 --> 01:21:30,670
21 centimeters wavelength,
5.9 times 10 to the minus 6

1060
01:21:30,670 --> 01:21:34,110
eV in here.

1061
01:21:34,110 --> 01:21:37,150
But that's not the
energy difference

1062
01:21:37,150 --> 01:21:39,130
that matters for
the uncertainty,

1063
01:21:39,130 --> 01:21:41,580
just like this is not the
energy difference that

1064
01:21:41,580 --> 01:21:43,250
matters for the uncertainty.

1065
01:21:43,250 --> 01:21:47,370
What matters for the uncertainty
is how broad this state

1066
01:21:47,370 --> 01:21:51,630
is, due to interactions
that will produce the decay.

1067
01:21:51,630 --> 01:21:55,120
It's a very funny,
magnetic transition.

1068
01:21:55,120 --> 01:21:58,630
And how long is the
lifetime of this state?

1069
01:21:58,630 --> 01:22:02,018
Anybody know?

1070
01:22:02,018 --> 01:22:09,410
A second, a millisecond, a day?

1071
01:22:09,410 --> 01:22:09,910
Nobody?

1072
01:22:14,580 --> 01:22:26,230
Ten million years-- a long
time-- 10 million years--

1073
01:22:26,230 --> 01:22:27,930
lifetime tau.

1074
01:22:27,930 --> 01:22:33,510
A year is about pi times
10 to the 7 seconds

1075
01:22:33,510 --> 01:22:34,370
is pretty accurate.

1076
01:22:38,410 --> 01:22:42,150
Anyway, 10 million
years is a lot of time.

1077
01:22:42,150 --> 01:22:45,410
It's such a large time
that it corresponds

1078
01:22:45,410 --> 01:22:50,540
to an energy uncertainty
that is so extraordinarily

1079
01:22:50,540 --> 01:22:54,930
small, that the wavelength
uncertainty, or the frequency

1080
01:22:54,930 --> 01:23:00,880
uncertainty, is so small that
corresponding to this 1420,

1081
01:23:00,880 --> 01:23:04,770
it's I think, the uncertainty
in lambda-- and lambda

1082
01:23:04,770 --> 01:23:09,250
is of the order of
10 to the minus 8.

1083
01:23:09,250 --> 01:23:14,190
The line is extremely sharp,
so it's not a fussy line

1084
01:23:14,190 --> 01:23:16,050
that it's hard to measure.

1085
01:23:16,050 --> 01:23:19,280
It's the sharpest possible line.

1086
01:23:19,280 --> 01:23:24,260
And it's so sharp because of
this 10 million years lifetime,

1087
01:23:24,260 --> 01:23:27,360
and the energy time
uncertainty relationship.

1088
01:23:27,360 --> 01:23:29,590
That's it for today.