1
00:00:00,500 --> 00:00:01,488
PROFESSOR: Uncertainty.
2
00:00:06,944 --> 00:00:12,020
When you talk about random
variables, random variable Q,
3
00:00:12,020 --> 00:00:16,740
we've said that it has
values Q1 up to, say, Qn,
4
00:00:16,740 --> 00:00:22,532
and probabilities
P1 up to Pn, we
5
00:00:22,532 --> 00:00:28,090
speak of a standard
deviation, delta Q,
6
00:00:28,090 --> 00:00:32,910
as the uncertainty,
the standard deviation.
7
00:00:36,558 --> 00:00:40,030
And how is that standard
deviation defined?
8
00:00:40,030 --> 00:00:42,190
Well you begin by
making sure you
9
00:00:42,190 --> 00:00:45,500
know what is the
expectation value of the--
10
00:00:45,500 --> 00:00:49,420
or the average value of
this random variable,
11
00:00:49,420 --> 00:00:53,300
which was defined, last
time, I think I put braces,
12
00:00:53,300 --> 00:00:55,900
but bar is kind
of nice sometimes
13
00:00:55,900 --> 00:00:58,410
too, at least for
random variables,
14
00:00:58,410 --> 00:01:02,340
and it's the sum of
the Pi times the Qi.
15
00:01:05,292 --> 00:01:12,220
The uncertainty is also
some expectation value.
16
00:01:12,220 --> 00:01:17,520
And expectation
value of deviation.
17
00:01:17,520 --> 00:01:21,215
So the uncertainty squared
is the expectation value,
18
00:01:21,215 --> 00:01:31,530
sum over i, of deviations of the
random variable from the mean.
19
00:01:31,530 --> 00:01:34,710
So you calculate
the expected value
20
00:01:34,710 --> 00:01:41,460
of the difference of your random
variable and the mean squared,
21
00:01:41,460 --> 00:01:47,860
and that is the square of
the standard deviation.
22
00:01:47,860 --> 00:01:51,230
Now this is the definition.
23
00:01:51,230 --> 00:01:53,760
And it's a very nice
definition because it
24
00:01:53,760 --> 00:01:56,200
makes a few things clear.
25
00:01:56,200 --> 00:02:01,530
For example, the left hand
side is delta Q squared, which
26
00:02:01,530 --> 00:02:03,180
means it's a positive number.
27
00:02:03,180 --> 00:02:06,470
And the right hand side
is also a positive number,
28
00:02:06,470 --> 00:02:13,680
because you have probabilities
times differences of quantities
29
00:02:13,680 --> 00:02:15,100
squared.
30
00:02:15,100 --> 00:02:19,756
So this is all greater
and equal to zero.
31
00:02:19,756 --> 00:02:23,880
And moreover, you can
actually say the following.
32
00:02:23,880 --> 00:02:31,080
If the uncertainty, or the
standard deviation, is zero,
33
00:02:31,080 --> 00:02:34,210
the random variable
is not that random.
34
00:02:34,210 --> 00:02:39,240
Because if this whole thing
is 0, this delta squared,
35
00:02:39,240 --> 00:02:42,990
delta Q squared must be
0 and this must be 0.
36
00:02:42,990 --> 00:02:45,480
But each term here is positive.
37
00:02:45,480 --> 00:02:50,400
So each term must be 0,
because of any one of them
38
00:02:50,400 --> 00:02:54,420
was not equal to zero, you would
get a non-zero contribution.
39
00:02:54,420 --> 00:03:00,710
So any possible Qi that must
have a Pi different from 0
40
00:03:00,710 --> 00:03:02,480
must be equal to Qbar.
41
00:03:02,480 --> 00:03:06,090
So if delta cubed
is equal to 0, Qi
42
00:03:06,090 --> 00:03:12,164
is equal to Q as
not random anymore.
43
00:03:19,970 --> 00:03:26,070
OK, now we can simplify
this expression.
44
00:03:41,750 --> 00:03:43,220
Do the following.
45
00:03:43,220 --> 00:03:49,072
By simplifying, I mean
expand the right-hand side.
46
00:03:49,072 --> 00:04:02,360
So sum over i, Pi Qi
squared, minus 2 sum over i,
47
00:04:02,360 --> 00:04:15,710
Pi Qi Q bar plus sum
over i, Pi Q bar squared.
48
00:04:18,310 --> 00:04:21,040
This kind of thing
shows up all the time,
49
00:04:21,040 --> 00:04:25,870
shows up in quantum mechanic as
well, as we'll see in a second.
50
00:04:25,870 --> 00:04:29,720
And you need to be able
to see what's happenening.
51
00:04:29,720 --> 00:04:34,750
Here, you're having
the expectation value
52
00:04:34,750 --> 00:04:36,925
of Qi squared.
53
00:04:41,200 --> 00:04:45,280
That's the definition of
a bar of some variable,
54
00:04:45,280 --> 00:04:50,910
you'd multiply with variable
by the exponent of [INAUDIBLE].
55
00:04:50,910 --> 00:04:51,830
What is this?
56
00:04:51,830 --> 00:04:53,360
This a little more funny.
57
00:04:53,360 --> 00:04:55,910
First, you should know
that Q bar is a number,
58
00:04:55,910 --> 00:04:57,375
so it can go out.
59
00:04:57,375 --> 00:05:01,280
So it's minus 2 Q bar.
60
00:05:01,280 --> 00:05:06,580
And then all that is left is
this, but that's another Q bar.
61
00:05:06,580 --> 00:05:08,070
So it's another Q bar.
62
00:05:10,770 --> 00:05:15,760
And here, you take this one
out because it's a number,
63
00:05:15,760 --> 00:05:18,950
and the sum of the
probabilities is 1,
64
00:05:18,950 --> 00:05:23,620
so it's Q bar squared as well.
65
00:05:23,620 --> 00:05:29,240
And it always comes out
that way, this minus 2
66
00:05:29,240 --> 00:05:31,350
Q bar squared plus
Q bar squared.
67
00:05:31,350 --> 00:05:37,190
So at the end, Delta Q, it's
another famous property,
68
00:05:37,190 --> 00:05:44,190
is the mean of the square
minus the square of the mean.
69
00:05:51,854 --> 00:05:57,630
And from this, since this
is greater or equal than 0,
70
00:05:57,630 --> 00:06:02,370
you always conclude that
the mean of the square
71
00:06:02,370 --> 00:06:06,420
is always bigger than the--
72
00:06:09,384 --> 00:06:11,854
maybe I shouldn't
have the i here,
73
00:06:11,854 --> 00:06:17,850
I think it's a random
variable Q squared.
74
00:06:17,850 --> 00:06:22,050
So the mean, the square of
this is greater or equal
75
00:06:22,050 --> 00:06:24,522
than Q bar squared.
76
00:06:27,540 --> 00:06:28,040
OK.
77
00:06:30,660 --> 00:06:36,721
Well, what happens
in quantum mechanics,
78
00:06:36,721 --> 00:06:46,480
let give you the definition and
a couple of ways of writing it.
79
00:06:46,480 --> 00:06:48,730
So here comes the definition.
80
00:06:48,730 --> 00:06:50,660
It's inspired by this thing.
81
00:06:50,660 --> 00:07:01,210
So in quantum mechanics,
permission operator Q
82
00:07:01,210 --> 00:07:07,500
will define the uncertainty
of Q in the state,
83
00:07:07,500 --> 00:07:15,110
Psi O squared as the
expectation value
84
00:07:15,110 --> 00:07:29,890
of Q squared minus the
expectation value of Q squared.
85
00:07:29,890 --> 00:07:33,010
Those are things that you
know in quantum mechanics,
86
00:07:33,010 --> 00:07:35,145
how you're supposed to compute.
87
00:07:37,900 --> 00:07:40,760
Because you know what
an expectation value
88
00:07:40,760 --> 00:07:43,240
is in any state Psi.
89
00:07:43,240 --> 00:07:46,190
You so Psi star,
the operator, Psi.
90
00:07:46,190 --> 00:07:50,900
And here you do this
thing, so it's all clear.
91
00:07:50,900 --> 00:07:54,250
So it's a perfectly
good definition.
92
00:07:54,250 --> 00:07:58,990
Maybe it doesn't give
you too much insight yet,
93
00:07:58,990 --> 00:08:02,620
but let me say two
things, and we'll leave
94
00:08:02,620 --> 00:08:06,600
them to complete for next time.
95
00:08:06,600 --> 00:08:15,980
Which is claim one, one,
that Delta Q squared
96
00:08:15,980 --> 00:08:21,240
Psi can be written
as the expectation
97
00:08:21,240 --> 00:08:32,002
value of Q minus absolute
expectation value of Q squared.
98
00:08:32,002 --> 00:08:32,940
Like that.
99
00:08:32,940 --> 00:08:34,360
Look.
100
00:08:34,360 --> 00:08:37,780
It looks funny, and
we'll elaborate this,
101
00:08:37,780 --> 00:08:42,159
but the first claim is that
this is a possible re-writing.
102
00:08:42,159 --> 00:08:49,390
You can write this uncertainty
as a single expectation value.
103
00:08:49,390 --> 00:08:56,950
This is the analog of this
equation in quantum mechanics.
104
00:08:56,950 --> 00:09:03,750
Claim two is another re-writing.
105
00:09:03,750 --> 00:09:14,150
Delta Q squared on Psi
can be re-written as this.
106
00:09:14,150 --> 00:09:15,427
That's an integral.
107
00:09:22,041 --> 00:09:29,470
Q minus Q and Psi.
108
00:09:29,470 --> 00:09:32,750
Look at that.
109
00:09:32,750 --> 00:09:39,200
You act on Psi with the
operator, Q, and multiplication
110
00:09:39,200 --> 00:09:41,905
by the expectation value
of Q. This is an operator,
111
00:09:41,905 --> 00:09:44,280
this is a number
multiplied by Psi.
112
00:09:44,280 --> 00:09:46,670
You can add to this on
the [? wave ?] function,
113
00:09:46,670 --> 00:09:50,460
you can square it,
and then integrate.
114
00:09:50,460 --> 00:09:54,590
And that is also
the uncertainty.
115
00:09:57,560 --> 00:10:00,890
We'll show these
two things next time
116
00:10:00,890 --> 00:10:06,630
and show one more thing that
the uncertainty vanishes
117
00:10:06,630 --> 00:10:13,400
if and only if the state
is an ideal state of Q.
118
00:10:13,400 --> 00:10:15,681
So If the state that
you are looking for
119
00:10:15,681 --> 00:10:20,321
is an ideal state of Q,
you have no uncertainty.
120
00:10:20,321 --> 00:10:23,870
And if you have no
uncertainty, the state
121
00:10:23,870 --> 00:10:27,500
must be an ideal state
of Q. So those all things
122
00:10:27,500 --> 00:10:31,960
will come from this planes, that
we'll elaborate on next time.