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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.