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PROFESSOR: So
today, before we get

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00:00:23,650 --> 00:00:27,420
into the meat of
today's lecture,

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00:00:27,420 --> 00:00:29,880
Matt has very kindly--
Professor Evans

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00:00:29,880 --> 00:00:36,090
has very kindly agreed
to do an experiment.

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00:00:36,090 --> 00:00:38,240
Yeah, so for those
of you all who

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00:00:38,240 --> 00:00:40,100
are in recitations
both he and Barton

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00:00:40,100 --> 00:00:42,630
talked about polarization
in recitation last week.

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00:00:42,630 --> 00:00:44,550
And Matt will pick
it up from there.

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00:00:44,550 --> 00:00:47,350
MATTHEW EVANS: So back
to the ancient past--

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00:00:47,350 --> 00:00:48,870
this was a week ago.

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00:00:48,870 --> 00:00:50,410
We had our
hyper-intelligent monkeys

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00:00:50,410 --> 00:00:51,451
that were sorting things.

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00:00:51,451 --> 00:00:53,370
It all seemed very theoretical.

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00:00:53,370 --> 00:00:56,820
And in recitation, I said
things about polarizers.

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00:00:56,820 --> 00:00:58,470
And I said, look, if
we use polarizers,

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00:00:58,470 --> 00:01:00,990
we can do exactly the same
thing as these monkeys.

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00:01:00,990 --> 00:01:05,300
We just need to set up a
little polarization experiment

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00:01:05,300 --> 00:01:06,840
and the results are identical.

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00:01:06,840 --> 00:01:08,680
You can use the one
figure out the other.

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00:01:08,680 --> 00:01:12,015
But I didn't have this or
a nice polarizer ready then

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00:01:12,015 --> 00:01:14,580
to give a demo, so here we go.

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00:01:14,580 --> 00:01:16,900
What I'm going to show
you is that, if we start

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00:01:16,900 --> 00:01:20,860
with something polarized here
with all white-- and right

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00:01:20,860 --> 00:01:23,750
now I have all vertical
polarization here--

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00:01:23,750 --> 00:01:25,650
and if I just put
on this other box

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00:01:25,650 --> 00:01:28,230
there, which is going to be
another polarizer, if I put it

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00:01:28,230 --> 00:01:31,060
the same way, this is all
of our white electrons

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00:01:31,060 --> 00:01:32,130
coming through all white.

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00:01:32,130 --> 00:01:32,629
See?

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00:01:32,629 --> 00:01:34,770
It doesn't really do much.

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00:01:34,770 --> 00:01:38,110
And if I look at the
black output over here

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00:01:38,110 --> 00:01:40,050
of the second Keller
sorting box, that's

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00:01:40,050 --> 00:01:42,620
the same as turning my
polarizer 90 degrees,

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00:01:42,620 --> 00:01:44,550
so nothing comes out black.

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00:01:44,550 --> 00:01:46,715
So if we remove this
guy from the middle,

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00:01:46,715 --> 00:01:48,400
you have just exactly
what you'd expect.

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00:01:48,400 --> 00:01:51,390
You sort here, you have white,
and you get all white out.

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00:01:51,390 --> 00:01:52,110
Great.

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00:01:52,110 --> 00:01:53,401
Everyone thought that was easy.

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00:01:53,401 --> 00:01:54,860
We all had that figured out.

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00:01:54,860 --> 00:01:56,690
This box got thrown
in the center here

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00:01:56,690 --> 00:02:00,720
and it became sort of confusing,
because you thought, well,

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00:02:00,720 --> 00:02:02,220
they were white--
I'm going to throw

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00:02:02,220 --> 00:02:08,070
my box in the middle here--
that's this guy at 45 degrees.

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00:02:08,070 --> 00:02:10,570
And then if I throw this guy
on the end again, the idea was,

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00:02:10,570 --> 00:02:14,166
well, they were all white
here, so maybe this guy

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00:02:14,166 --> 00:02:16,040
identified the soft ones
from the white ones.

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00:02:16,040 --> 00:02:17,331
And now we have white and soft.

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00:02:17,331 --> 00:02:19,410
And it should still
all be white, right?

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00:02:19,410 --> 00:02:21,610
So I put this guy on up here.

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00:02:21,610 --> 00:02:25,270
They should all come out
but they sort of don't.

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00:02:25,270 --> 00:02:26,840
And if you say,
well, are they black?

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00:02:26,840 --> 00:02:28,631
Well, no, they're not
really black, either.

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00:02:28,631 --> 00:02:31,360
They're some sort of strange
combination of the two.

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All right, so that's this
experiment done in polarizers.

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But let me just play the
polarizer trick a little bit,

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because it's fun.

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00:02:39,190 --> 00:02:42,592
So this is if I say,
vertical polarization

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and how many of them
come out horizontal?

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So here I'm saying, white,
and how many of them

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00:02:46,040 --> 00:02:46,873
will come out black?

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That's the analogy.

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The answer is none of them.

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00:02:50,000 --> 00:02:51,950
And strangely, if I
take this thing, which

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00:02:51,950 --> 00:02:55,190
seems to just attenuate--
this is our middle box here--

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00:02:55,190 --> 00:02:56,910
and I just stuff
it in between them,

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00:02:56,910 --> 00:03:01,170
I can get something to come
out even though I still

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00:03:01,170 --> 00:03:04,080
have crossed
polarizers on the side.

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00:03:04,080 --> 00:03:06,080
So you can see the middle
region is now brighter

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and you can still see
the dark corners there

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00:03:07,913 --> 00:03:09,900
of the crossed polarizers.

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00:03:09,900 --> 00:03:13,290
And as I turn this guy around,
I can make that better or worse.

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00:03:13,290 --> 00:03:15,566
The maximum is
somewhere right there,

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00:03:15,566 --> 00:03:17,270
and then it goes off again.

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So this is a way
of understanding

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00:03:21,150 --> 00:03:25,020
our electron-sorting,
hyper-intelligent monkeys

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00:03:25,020 --> 00:03:26,170
in terms of polarizations.

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00:03:26,170 --> 00:03:28,085
And here it's just
a vector projected

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on another vector projected
on another vector-- something

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00:03:30,460 --> 00:03:32,620
everybody knows how to do.

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00:03:32,620 --> 00:03:35,310
So here's the
polarization analogy

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00:03:35,310 --> 00:03:36,760
of the Stern-Gerlach experiment.

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00:03:45,130 --> 00:03:46,080
PROFESSOR: Awesome.

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00:03:46,080 --> 00:03:51,130
So the polarization analogy
for interference effects

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00:03:51,130 --> 00:03:54,730
in quantum mechanics
is a canonical one

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00:03:54,730 --> 00:03:58,810
in the texts of
quantum mechanics.

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00:03:58,810 --> 00:04:01,096
So you'll find lots of
books talking about this.

95
00:04:01,096 --> 00:04:02,470
It's a very useful
analogy, and I

96
00:04:02,470 --> 00:04:03,910
encourage you to
read more about it.

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00:04:03,910 --> 00:04:05,576
We won't talk about
it a whole lot more,

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00:04:05,576 --> 00:04:07,874
but it's a useful one.

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00:04:07,874 --> 00:04:09,290
All right, before
I get going, any

100
00:04:09,290 --> 00:04:12,587
questions from last lecture?

101
00:04:12,587 --> 00:04:14,420
Last lecture was pretty
much self-contained.

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00:04:14,420 --> 00:04:17,530
It was experimental results.

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00:04:17,530 --> 00:04:18,589
No, nothing?

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00:04:18,589 --> 00:04:21,380
All right.

105
00:04:21,380 --> 00:04:24,800
The one thing that I
want to add to the last

106
00:04:24,800 --> 00:04:26,700
lecture-- one last
experimental observation.

107
00:04:26,700 --> 00:04:29,020
I glossed over
something that's kind

108
00:04:29,020 --> 00:04:31,040
of important, which
is the following.

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00:04:31,040 --> 00:04:33,760
So we started off
by saying, look,

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00:04:33,760 --> 00:04:37,150
we know that if I
have a ray of light,

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00:04:37,150 --> 00:04:39,410
it's an electromagnetic
wave, and it

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00:04:39,410 --> 00:04:40,535
has some wavelength lambda.

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00:04:44,500 --> 00:04:49,760
And yet the photoelectric
effect tells us

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00:04:49,760 --> 00:04:54,900
that, in addition to having the
wavelength lambda, the energy--

115
00:04:54,900 --> 00:04:57,900
it has a frequency, as
well, a frequency in time.

116
00:04:57,900 --> 00:04:59,690
And the photoelectric
effect suggested

117
00:04:59,690 --> 00:05:07,650
that the energy is
proportional to the frequency.

118
00:05:07,650 --> 00:05:17,230
And we write this as h nu and
h bar is equal to h upon 2 pi

119
00:05:17,230 --> 00:05:21,310
and omega is equal to 2 pi nu.

120
00:05:21,310 --> 00:05:23,810
So this is just the angular
frequency, rather than

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00:05:23,810 --> 00:05:27,010
the number-per-time frequency.

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00:05:27,010 --> 00:05:29,600
And h bar is the
reduced Planck constant.

123
00:05:29,600 --> 00:05:31,910
So I'll typically write h
bar omega rather than h nu,

124
00:05:31,910 --> 00:05:35,432
because these two pi's will
just cause us endless pain if we

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00:05:35,432 --> 00:05:37,890
don't use the bar.

126
00:05:37,890 --> 00:05:39,520
Anyway, so to an
electromagnetic wave,

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00:05:39,520 --> 00:05:40,840
we have a wavelength
and a frequency

128
00:05:40,840 --> 00:05:42,362
and the photoelectric
effect led us

129
00:05:42,362 --> 00:05:44,130
to predict that the
energy is linearly

130
00:05:44,130 --> 00:05:46,713
proportional to the frequency,
with the linear proportionality

131
00:05:46,713 --> 00:05:50,210
coefficient h bar-- Planck
constant-- and the momentum

132
00:05:50,210 --> 00:05:55,800
is equal to h upon
lambda, also known

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00:05:55,800 --> 00:06:00,280
as-- I'm going to write
this as h bar k, which

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00:06:00,280 --> 00:06:04,740
is equal to h upon
lambda, where here, again,

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00:06:04,740 --> 00:06:07,750
h bar is h upon 2 pi.

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00:06:07,750 --> 00:06:11,020
And so k is equal
to 2 pi upon lambda.

137
00:06:16,147 --> 00:06:17,730
So k is called the
wave number and you

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00:06:17,730 --> 00:06:20,920
should have seen this in 8.03.

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00:06:20,920 --> 00:06:23,592
So these are our basic
relations for light.

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00:06:23,592 --> 00:06:25,550
We know that light, as
an electromagnetic wave,

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00:06:25,550 --> 00:06:26,970
has a frequency
and a wavelength--

142
00:06:26,970 --> 00:06:30,170
or a wave number, an
inverse wavelength.

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00:06:30,170 --> 00:06:32,550
And the claim of the
photoelectric effect

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00:06:32,550 --> 00:06:35,030
is that the energy and
the momenta of that light

145
00:06:35,030 --> 00:06:38,550
are thus quantized, that
light comes in chunks.

146
00:06:38,550 --> 00:06:40,270
So it has a wave-like
aspect and it also

147
00:06:40,270 --> 00:06:43,420
has properties that are more
familiar from particles.

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00:06:43,420 --> 00:06:47,195
Now, early on shortly after
Einstein proposed this,

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00:06:47,195 --> 00:06:51,800
a young French physicist named
de Broglie said, well, look,

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00:06:51,800 --> 00:06:53,400
OK, this is true of light.

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00:06:53,400 --> 00:06:59,050
Light has both wave-like and
particle-like properties.

152
00:06:59,050 --> 00:07:00,002
Why is it just light?

153
00:07:00,002 --> 00:07:01,710
The world would be
much more parsimonious

154
00:07:01,710 --> 00:07:03,959
if this relation were true
not just of light, but also

155
00:07:03,959 --> 00:07:04,840
of all particles.

156
00:07:04,840 --> 00:07:09,370
I am thus conjecturing,
with no evidence whatsoever,

157
00:07:09,370 --> 00:07:11,055
that, in fact,
this relation holds

158
00:07:11,055 --> 00:07:12,680
not just for light,
but for any object.

159
00:07:12,680 --> 00:07:17,490
Any object with momentum p has
associated to it a wavelength

160
00:07:17,490 --> 00:07:21,190
or a wave number,
which is p upon h bar.

161
00:07:21,190 --> 00:07:25,300
Every object that has energy
E has associated with it

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00:07:25,300 --> 00:07:29,990
a wave with frequency omega.

163
00:07:29,990 --> 00:07:34,510
To those electrons that we send
through the Davisson-Germer

164
00:07:34,510 --> 00:07:40,050
experiment apparatus, which are
sent in with definite energy,

165
00:07:40,050 --> 00:07:42,390
there must be a frequency
associated with it, omega

166
00:07:42,390 --> 00:07:44,300
and a wavelength lambda
associated with it.

167
00:07:44,300 --> 00:07:46,440
And what we saw from the
Davisson-Germer experiment

168
00:07:46,440 --> 00:07:49,380
was experimental confirmation
of that prediction--

169
00:07:49,380 --> 00:07:53,200
that electrons have both
particulate and wave-like

170
00:07:53,200 --> 00:07:57,010
features simultaneously.

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00:07:57,010 --> 00:08:00,770
So these relations are called
the de Broglie relations or "de

172
00:08:00,770 --> 00:08:05,900
BROG-lee"-- I leave it up to
you to decide how to pronounce

173
00:08:05,900 --> 00:08:08,317
that.

174
00:08:08,317 --> 00:08:10,900
And those relations are going
to play an important role for us

175
00:08:10,900 --> 00:08:11,941
in the next few lectures.

176
00:08:11,941 --> 00:08:15,680
I just wanted to give them
a name and a little context.

177
00:08:15,680 --> 00:08:19,240
This is a good example of
parsimony and elegance--

178
00:08:19,240 --> 00:08:21,810
the theoretical
elegance leading you

179
00:08:21,810 --> 00:08:24,210
to an idea that turns out
to be true of the world.

180
00:08:24,210 --> 00:08:28,210
Now, that's a dangerous
strategy for finding truth.

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00:08:28,210 --> 00:08:31,780
Boy, wouldn't it be nice if--?

182
00:08:31,780 --> 00:08:34,243
Wouldn't it be nice if we
didn't have to pay taxes

183
00:08:34,243 --> 00:08:35,284
but we also had Medicare?

184
00:08:38,679 --> 00:08:41,883
So it's not a terribly
useful guide all the time,

185
00:08:41,883 --> 00:08:43,424
but sometimes it
really does lead you

186
00:08:43,424 --> 00:08:44,382
in the right direction.

187
00:08:44,382 --> 00:08:47,060
And this is a great example
of physical intuition,

188
00:08:47,060 --> 00:08:49,080
wildly divorced from
experiment, pushing you

189
00:08:49,080 --> 00:08:50,489
in the right direction.

190
00:08:50,489 --> 00:08:52,780
I'm making it sound a little
more shocking than-- well,

191
00:08:52,780 --> 00:08:53,446
it was shocking.

192
00:08:53,446 --> 00:08:54,580
It was just shocking.

193
00:08:54,580 --> 00:09:01,210
OK, so with that said,
let me introduce the moves

194
00:09:01,210 --> 00:09:02,330
for the next few lectures.

195
00:09:02,330 --> 00:09:04,670
For the next several lectures,
here's what we're going to do.

196
00:09:04,670 --> 00:09:06,690
I am not going to give you
experimental motivation.

197
00:09:06,690 --> 00:09:08,314
I've given you
experimental motivation.

198
00:09:08,314 --> 00:09:12,382
I'm going to give you a set
of rules, a set of postulates.

199
00:09:12,382 --> 00:09:14,590
These are going to be the
rules of quantum mechanics.

200
00:09:14,590 --> 00:09:17,480
And what quantum
mechanics is for us

201
00:09:17,480 --> 00:09:20,250
is a set of rules to allow
us to make predictions

202
00:09:20,250 --> 00:09:22,200
about the world.

203
00:09:22,200 --> 00:09:26,460
And these rules will be awesome
if their predictions are good.

204
00:09:26,460 --> 00:09:29,690
And if their predictions are
bad, these rules will suck.

205
00:09:29,690 --> 00:09:31,894
We will avoid bad rules
to the degree possible.

206
00:09:31,894 --> 00:09:34,310
I'm going to give you what
we've learned over the past 100

207
00:09:34,310 --> 00:09:38,750
years-- wow-- of developing
quantum mechanics.

208
00:09:38,750 --> 00:09:40,390
That is amazing.

209
00:09:40,390 --> 00:09:40,890
Wow.

210
00:09:40,890 --> 00:09:42,264
OK, yeah, over
the past 100 years

211
00:09:42,264 --> 00:09:44,142
of developing quantum mechanics.

212
00:09:44,142 --> 00:09:46,600
And I'm going to give them to
you as a series of postulates

213
00:09:46,600 --> 00:09:49,070
and then we're going to work
through the consequences,

214
00:09:49,070 --> 00:09:50,680
and then we're going to spend
the rest of the semester

215
00:09:50,680 --> 00:09:53,160
studying examples to develop
an understanding for what

216
00:09:53,160 --> 00:09:56,360
the rules of quantum
mechanics are giving you.

217
00:09:56,360 --> 00:09:59,390
So we're just going to
scrap classical mechanics

218
00:09:59,390 --> 00:10:00,580
and start over from scratch.

219
00:10:00,580 --> 00:10:01,330
So let me do that.

220
00:10:04,720 --> 00:10:11,240
And to begin, let me start with
the definition of a system.

221
00:10:11,240 --> 00:10:13,120
And to understand
that definition,

222
00:10:13,120 --> 00:10:15,990
I want to start with classical
mechanics as a guide.

223
00:10:21,000 --> 00:10:23,560
So in classical mechanics--
let's think about the easiest

224
00:10:23,560 --> 00:10:25,930
classical system you can--
just a single particle

225
00:10:25,930 --> 00:10:26,930
sitting somewhere.

226
00:10:26,930 --> 00:10:29,530
In classical mechanics
of a single particle,

227
00:10:29,530 --> 00:10:32,920
how do you specify the
configuration, or the state--

228
00:10:32,920 --> 00:10:34,730
just different words
for the same thing--

229
00:10:34,730 --> 00:10:36,400
how do you specify
the configuration

230
00:10:36,400 --> 00:10:39,932
or state of the system?

231
00:10:39,932 --> 00:10:41,390
AUDIENCE: By position
and momentum.

232
00:10:41,390 --> 00:10:43,640
PROFESSOR: Specify the
position and momentum, exactly.

233
00:10:43,640 --> 00:10:48,579
So in classical mechanics, if
you want to completely specify

234
00:10:48,579 --> 00:10:50,620
the configuration of a
system, all you have to do

235
00:10:50,620 --> 00:10:55,160
is give me x and
p for my particle.

236
00:10:55,160 --> 00:10:57,690
And if you tell me
this, I know everything.

237
00:10:57,690 --> 00:11:00,040
If you know these numbers,
you know everything.

238
00:11:00,040 --> 00:11:01,630
In particular, what
I mean by saying

239
00:11:01,630 --> 00:11:03,060
you know everything is that,
if there's anything else

240
00:11:03,060 --> 00:11:05,530
you want to measure--
the energy, for example.

241
00:11:05,530 --> 00:11:08,230
The energy is just some function
of the position and momentum.

242
00:11:08,230 --> 00:11:09,760
And if you know the
position and momentum,

243
00:11:09,760 --> 00:11:11,640
you can unambiguously
calculate the energy.

244
00:11:11,640 --> 00:11:14,184
Similarly, the angular
momentum, which is a vector,

245
00:11:14,184 --> 00:11:15,600
you can calculate
it if you know x

246
00:11:15,600 --> 00:11:20,250
and p-- which is just r cross p.

247
00:11:20,250 --> 00:11:22,470
So this gives you complete
knowledge of the system.

248
00:11:29,030 --> 00:11:32,600
There's nothing more to
know if you know that data.

249
00:11:32,600 --> 00:11:34,340
Now, there are certainly
still questions

250
00:11:34,340 --> 00:11:38,800
that you can't answer
given knowledge of x and p.

251
00:11:38,800 --> 00:11:42,090
For example, are there
14 invisible monkeys

252
00:11:42,090 --> 00:11:44,040
standing behind me?

253
00:11:44,040 --> 00:11:44,740
I'm here.

254
00:11:44,740 --> 00:11:45,890
I'm not moving.

255
00:11:45,890 --> 00:11:48,064
Are there 14 invisible
monkeys standing behind me?

256
00:11:48,064 --> 00:11:48,980
You can't answer that.

257
00:11:48,980 --> 00:11:51,200
It's a stupid question, right?

258
00:11:51,200 --> 00:11:52,830
OK, let me give you
another example.

259
00:11:52,830 --> 00:11:55,140
The electron is x
and p, some position.

260
00:11:55,140 --> 00:11:57,860
Is it happy?

261
00:11:57,860 --> 00:12:00,280
Right, so there are still
questions you can't answer.

262
00:12:00,280 --> 00:12:02,420
The point is, complete
knowledge of the system

263
00:12:02,420 --> 00:12:04,810
to answer any physically
observable question--

264
00:12:04,810 --> 00:12:06,820
any question that could
be meaningfully turned

265
00:12:06,820 --> 00:12:10,480
into an experiment,
the answer is

266
00:12:10,480 --> 00:12:12,420
contained in knowing
the state of the system.

267
00:12:17,520 --> 00:12:20,057
But this can't possibly be
true in quantum mechanics,

268
00:12:20,057 --> 00:12:21,640
because, as you saw
in the problem set

269
00:12:21,640 --> 00:12:23,710
and as we've
discussed previously,

270
00:12:23,710 --> 00:12:25,460
there's an uncertainty
relation which

271
00:12:25,460 --> 00:12:27,850
says that your knowledge--
or your uncertainty, rather,

272
00:12:27,850 --> 00:12:29,550
in the position of
a particle, quantum

273
00:12:29,550 --> 00:12:33,434
mechanically--
I'm not even going

274
00:12:33,434 --> 00:12:34,600
to say quantum mechanically.

275
00:12:34,600 --> 00:12:37,820
I'm just going to
say the real world.

276
00:12:37,820 --> 00:12:40,960
So in the real world,
our uncertainty

277
00:12:40,960 --> 00:12:43,610
in the position of
our point-like object

278
00:12:43,610 --> 00:12:45,210
and our uncertainty
in the momentum

279
00:12:45,210 --> 00:12:48,500
is always greater than or
roughly equal to something

280
00:12:48,500 --> 00:12:51,000
that's proportional
to Planck's constant.

281
00:12:51,000 --> 00:12:53,920
You can't be arbitrarily
confident of the position

282
00:12:53,920 --> 00:12:55,532
and of the momentum
simultaneously.

283
00:12:55,532 --> 00:12:57,240
You worked through a
good example of this

284
00:12:57,240 --> 00:12:58,187
on the problem set.

285
00:12:58,187 --> 00:12:59,770
We saw this in the
two-slit experiment

286
00:12:59,770 --> 00:13:01,470
and the interference
of electrons.

287
00:13:01,470 --> 00:13:03,595
This is something we're
going to have to deal with.

288
00:13:03,595 --> 00:13:05,890
So as a consequence,
you can't possibly

289
00:13:05,890 --> 00:13:08,160
specify the position
and the momentum

290
00:13:08,160 --> 00:13:09,390
with confidence of a system.

291
00:13:09,390 --> 00:13:11,830
You can't do it.

292
00:13:11,830 --> 00:13:12,740
This was a myth.

293
00:13:12,740 --> 00:13:16,060
It was a good approximation--
turned out to be false.

294
00:13:16,060 --> 00:13:20,430
So the first thing
we need is to specify

295
00:13:20,430 --> 00:13:22,300
the state, the
configuration of a system.

296
00:13:22,300 --> 00:13:25,910
So what specifies the
configuration of a system?

297
00:13:25,910 --> 00:13:29,870
And so this brings us
to the first postulate.

298
00:13:33,390 --> 00:13:37,900
The configuration, or
state, of a system--

299
00:13:37,900 --> 00:13:39,470
and here again,
just for simplicity,

300
00:13:39,470 --> 00:13:48,880
I'm going to talk about a single
object-- of a quantum object

301
00:13:48,880 --> 00:14:01,550
is completely specified by
a single function, a wave

302
00:14:01,550 --> 00:14:06,920
function, which I
will denote generally

303
00:14:06,920 --> 00:14:12,870
psi of x, which is
a complex function.

304
00:14:23,690 --> 00:14:26,440
The state of the quantum
object is completely

305
00:14:26,440 --> 00:14:30,640
specified once you know the wave
function of the system, which

306
00:14:30,640 --> 00:14:34,200
is a function of position.

307
00:14:34,200 --> 00:14:37,860
Let me emphasize that this is
a first pass at the postulates.

308
00:14:37,860 --> 00:14:40,760
What we're going to do is go
through the basic postulates

309
00:14:40,760 --> 00:14:43,640
of quantum mechanics, then we'll
go through them again and give

310
00:14:43,640 --> 00:14:44,990
them a little more generality.

311
00:14:44,990 --> 00:14:46,970
And then we'll go through
them again and give them

312
00:14:46,970 --> 00:14:47,640
full generality.

313
00:14:47,640 --> 00:14:51,401
That last pass is 8.05.

314
00:14:51,401 --> 00:14:52,775
So let me give
you some examples.

315
00:14:56,060 --> 00:14:59,634
Let me just draw some
characteristic wave functions.

316
00:14:59,634 --> 00:15:01,800
And these are going to turn
out to be useful for us.

317
00:15:08,550 --> 00:15:11,750
So for example, consider
the following function.

318
00:15:14,600 --> 00:15:17,970
So here is 0 and we're
plotting as a function of x.

319
00:15:17,970 --> 00:15:21,710
And then plotting the
real part of psi of x.

320
00:15:25,690 --> 00:15:29,417
So first consider a very
narrowly supported function.

321
00:15:29,417 --> 00:15:31,000
It's basically 0
everywhere, except it

322
00:15:31,000 --> 00:15:33,145
has some particular spot
at what I'll call x1.

323
00:15:37,040 --> 00:15:39,500
Here's another
wave function-- 0.

324
00:15:39,500 --> 00:15:50,825
It's basically 0 except for
some special spot at x2.

325
00:15:50,825 --> 00:15:52,700
And again, I'm plotting
the real part of psi.

326
00:15:52,700 --> 00:15:54,500
And I'm plotting
the real part of psi

327
00:15:54,500 --> 00:15:57,720
because A, psi is a complex
function-- at every point it

328
00:15:57,720 --> 00:15:59,430
specifies a complex number.

329
00:15:59,430 --> 00:16:01,854
And B, I can't draw
complex numbers.

330
00:16:01,854 --> 00:16:03,270
So to keep my head
from exploding,

331
00:16:03,270 --> 00:16:05,635
I'm just plotting the real
part of the wave function.

332
00:16:05,635 --> 00:16:09,030
But you should never forget that
the wave function is complex.

333
00:16:12,687 --> 00:16:14,270
So for the moment,
I'm going to assume

334
00:16:14,270 --> 00:16:15,620
that the imaginary part is 0.

335
00:16:15,620 --> 00:16:17,207
I'm just going to
draw the real parts.

336
00:16:17,207 --> 00:16:18,790
So let me draw a
couple more examples.

337
00:16:18,790 --> 00:16:20,730
What else could be a
good wave function?

338
00:16:20,730 --> 00:16:22,260
Well, those are fine.

339
00:16:22,260 --> 00:16:24,804
What about-- again, we want
a function of x and I'm

340
00:16:24,804 --> 00:16:25,970
going to draw the real part.

341
00:16:33,850 --> 00:16:34,955
And another one.

342
00:16:34,955 --> 00:16:37,205
So this is going to be a
perfectly good wave function.

343
00:16:45,900 --> 00:16:48,980
And let me draw two more.

344
00:16:48,980 --> 00:16:53,080
So what else could be a
reasonable wave function?

345
00:16:53,080 --> 00:17:15,525
Well-- this is harder
than you'd think.

346
00:17:18,079 --> 00:17:18,579
Oh, God.

347
00:17:21,990 --> 00:17:27,430
OK, so that could be the
wave function, I don't know.

348
00:17:33,940 --> 00:17:35,260
That is actually my signature.

349
00:17:38,890 --> 00:17:41,900
My wife calls it a
little [INAUDIBLE].

350
00:17:41,900 --> 00:17:45,850
OK, so here's the deal.

351
00:17:45,850 --> 00:17:48,350
Psi is a complex function.

352
00:17:48,350 --> 00:17:51,820
Psi also needs to not
be a stupid function.

353
00:17:51,820 --> 00:17:54,280
OK so you have to ask me--
look, could it be any function?

354
00:17:54,280 --> 00:17:55,560
Any arbitrary function?

355
00:17:55,560 --> 00:17:57,060
So this is going
to be a job for us.

356
00:17:57,060 --> 00:17:58,518
We're going to
define what it means

357
00:17:58,518 --> 00:17:59,910
to be not-stupid function.

358
00:17:59,910 --> 00:18:01,909
Well, this is a completely
reasonable function--

359
00:18:01,909 --> 00:18:02,620
it's fine.

360
00:18:02,620 --> 00:18:03,600
This is a reasonable function.

361
00:18:03,600 --> 00:18:04,766
Another reasonable function.

362
00:18:04,766 --> 00:18:05,350
Reasonable.

363
00:18:05,350 --> 00:18:09,320
That's a little weird,
but it's not horrible.

364
00:18:09,320 --> 00:18:11,390
That's stupid.

365
00:18:11,390 --> 00:18:13,060
So we're going to
have to come up

366
00:18:13,060 --> 00:18:16,350
with a good definition
of what not stupid means.

367
00:18:16,350 --> 00:18:18,280
So fine, these
are all functions.

368
00:18:18,280 --> 00:18:20,780
One of them is multivalued and
that looks a little worrying,

369
00:18:20,780 --> 00:18:22,330
but they're all functions.

370
00:18:22,330 --> 00:18:23,350
So here's the problem.

371
00:18:23,350 --> 00:18:24,260
What does it mean?

372
00:18:29,870 --> 00:18:38,170
So postulate 2-- The
meaning of the wave function

373
00:18:38,170 --> 00:18:43,190
is that the probability
that upon measurement

374
00:18:43,190 --> 00:18:45,690
the object is found
at the position x

375
00:18:45,690 --> 00:18:48,375
is equal to the norm
squared of psi of x.

376
00:18:51,599 --> 00:18:53,890
If you know the system is
ascribed to the wave function

377
00:18:53,890 --> 00:18:55,430
psi, and you want
to look at point x,

378
00:18:55,430 --> 00:18:57,304
you want to know with
what probability will I

379
00:18:57,304 --> 00:19:00,680
find the particle there,
the answer is psi squared.

380
00:19:00,680 --> 00:19:03,255
Notice that this is a complex
number, but absolute value

381
00:19:03,255 --> 00:19:05,130
squared, or norm squared,
of a complex number

382
00:19:05,130 --> 00:19:07,150
is always a real,
non-negative number.

383
00:19:07,150 --> 00:19:09,400
And that's important because
we want our probabilities

384
00:19:09,400 --> 00:19:10,890
to be real,
non-negative numbers.

385
00:19:10,890 --> 00:19:12,119
Could be 0, right?

386
00:19:12,119 --> 00:19:13,410
Could be 0 chance of something.

387
00:19:13,410 --> 00:19:16,460
Can't be negative 7 chance.

388
00:19:16,460 --> 00:19:20,510
Incidentally, there also
can't be probability 2.

389
00:19:20,510 --> 00:19:22,810
So that means that the
total probability had better

390
00:19:22,810 --> 00:19:25,350
be normalized.

391
00:19:25,350 --> 00:19:27,350
So let me just say this
in words, though, first.

392
00:19:27,350 --> 00:19:30,150
So P, which is the
norm squared of psi,

393
00:19:30,150 --> 00:19:39,400
determines the probability--
and, in particular,

394
00:19:39,400 --> 00:19:53,120
the probability density--
that the object in state psi,

395
00:19:53,120 --> 00:19:58,170
in the state given by the
wave function psi of x,

396
00:19:58,170 --> 00:20:05,260
will be found at x.

397
00:20:09,260 --> 00:20:11,322
So there's the second postulate.

398
00:20:11,322 --> 00:20:13,280
So in particular, when
I say it's a probability

399
00:20:13,280 --> 00:20:17,610
density, what I mean is the
probability that it is found

400
00:20:17,610 --> 00:20:22,830
between the position
x and x plus dx

401
00:20:22,830 --> 00:20:37,577
is equal to P of x dx, which is
equal to psi of x squared dx.

402
00:20:41,926 --> 00:20:42,800
Does that make sense?

403
00:20:42,800 --> 00:20:44,216
So the probability
that it's found

404
00:20:44,216 --> 00:20:46,160
in this infinitesimal
interval is

405
00:20:46,160 --> 00:20:51,010
equal to this density
times dx or psi squared dx.

406
00:20:51,010 --> 00:20:53,520
Now again, it's crucial
that the wave function

407
00:20:53,520 --> 00:20:55,430
is in fact properly normalized.

408
00:20:55,430 --> 00:20:57,940
Because if I say, look,
something could either be here

409
00:20:57,940 --> 00:20:59,675
or it could be
here, what's the sum

410
00:20:59,675 --> 00:21:01,050
of the probability
that it's here

411
00:21:01,050 --> 00:21:04,310
plus the probability
that it's here?

412
00:21:04,310 --> 00:21:07,500
It had better be 1, or there's
some other possibility.

413
00:21:07,500 --> 00:21:09,330
So probabilities
have to sum to 1.

414
00:21:09,330 --> 00:21:12,540
Total probability that you find
something somewhere must be 1.

415
00:21:12,540 --> 00:21:15,650
So what that tells you is
that total probability, which

416
00:21:15,650 --> 00:21:19,270
is equal to the integral over
all possible values of x-- so

417
00:21:19,270 --> 00:21:25,240
if I sum over all possible
values of P of x--

418
00:21:25,240 --> 00:21:28,860
all values-- should
be equal to 1.

419
00:21:33,030 --> 00:21:37,690
And we can write
this as integral dx

420
00:21:37,690 --> 00:21:38,845
over all values of x.

421
00:21:38,845 --> 00:21:41,220
And I write "all" here rather
than putting minus infinity

422
00:21:41,220 --> 00:21:43,780
to infinity because some
systems will be defined

423
00:21:43,780 --> 00:21:46,370
from 1 to minus 1, some systems
will be defined from minus

424
00:21:46,370 --> 00:21:48,580
infinity to infinity-- all
just means integrate over

425
00:21:48,580 --> 00:21:52,036
all possible values-- hold
on one sec-- of psi squared.

426
00:21:55,396 --> 00:21:57,437
AUDIENCE: Are you going
to use different notation

427
00:21:57,437 --> 00:21:59,684
for probability density
than probability?

428
00:21:59,684 --> 00:22:00,850
PROFESSOR: I'm not going to.

429
00:22:00,850 --> 00:22:03,800
Probability density is going
to have just one argument,

430
00:22:03,800 --> 00:22:05,456
and total probability
is going to have

431
00:22:05,456 --> 00:22:06,580
an interval as an argument.

432
00:22:09,460 --> 00:22:13,260
So they're distinct and this
is just the notation I like.

433
00:22:13,260 --> 00:22:13,950
Other questions?

434
00:22:18,550 --> 00:22:21,736
Just as a side note,
what are the dimensions

435
00:22:21,736 --> 00:22:22,610
of the wave function?

436
00:22:29,927 --> 00:22:31,760
So everyone think about
this one for second.

437
00:22:39,920 --> 00:22:40,920
What are the dimensions?

438
00:22:44,608 --> 00:22:46,510
AUDIENCE: Is it 1 over
square root length

439
00:22:46,510 --> 00:22:46,910
PROFESSOR: Awesome.

440
00:22:46,910 --> 00:22:47,409
Yes.

441
00:22:47,409 --> 00:22:49,442
It's 1 over root length.

442
00:22:49,442 --> 00:22:52,210
The dimensions of psi
are 1 over root length.

443
00:22:55,670 --> 00:22:59,480
And the way to see that is
that this should be equal to 1.

444
00:22:59,480 --> 00:23:01,400
It's a total probability.

445
00:23:01,400 --> 00:23:03,150
This is an infinitesimal
length, so this

446
00:23:03,150 --> 00:23:04,670
has dimensions of length.

447
00:23:04,670 --> 00:23:07,640
This has no dimension, so
this must have dimensions of 1

448
00:23:07,640 --> 00:23:09,590
over length.

449
00:23:09,590 --> 00:23:13,692
And so psi itself of x most have
dimensions of 1 over length.

450
00:23:13,692 --> 00:23:15,150
Now, something I
want to emphasize,

451
00:23:15,150 --> 00:23:17,274
I'm going to emphasize,
over and over in this class

452
00:23:17,274 --> 00:23:18,430
is dimensional analysis.

453
00:23:18,430 --> 00:23:21,330
You need to become comfortable
with dimensional analysis.

454
00:23:21,330 --> 00:23:23,200
It's absolutely essential.

455
00:23:23,200 --> 00:23:24,660
It's essential for two reasons.

456
00:23:24,660 --> 00:23:26,201
First off, it's
essential because I'm

457
00:23:26,201 --> 00:23:28,040
going to be merciless
in taking off points

458
00:23:28,040 --> 00:23:29,990
if you do write down a
dimensionally false thing.

459
00:23:29,990 --> 00:23:32,573
If you write down something on
a problem set or an exam that's

460
00:23:32,573 --> 00:23:36,300
like, a length is equal to
a velocity-- ooh, not good.

461
00:23:36,300 --> 00:23:38,800
But the second thing
is, forget the fact

462
00:23:38,800 --> 00:23:40,496
that I'm going to
take off points.

463
00:23:40,496 --> 00:23:42,620
Dimensional analysis is an
incredibly powerful tool

464
00:23:42,620 --> 00:23:43,095
for you.

465
00:23:43,095 --> 00:23:45,040
You can check something
that you've just calculated

466
00:23:45,040 --> 00:23:46,664
and, better yet,
sometimes you can just

467
00:23:46,664 --> 00:23:49,420
avoid a calculation entirely
by doing a dimensional analysis

468
00:23:49,420 --> 00:23:52,086
and seeing that there's only one
possible way to build something

469
00:23:52,086 --> 00:23:53,749
of dimensions length
in your system.

470
00:23:53,749 --> 00:23:55,290
So we'll do that
over and over again.

471
00:23:55,290 --> 00:23:57,340
But this is a question
I want you guys to start

472
00:23:57,340 --> 00:23:59,290
asking yourselves at
every step along the way

473
00:23:59,290 --> 00:24:01,331
of a calculation-- what
are the dimensions of all

474
00:24:01,331 --> 00:24:02,900
the objects in my system?

475
00:24:05,691 --> 00:24:06,857
Something smells like smoke.

476
00:24:10,370 --> 00:24:15,930
So with that said, if that's the
meaning of the wave function,

477
00:24:15,930 --> 00:24:22,165
what physically can we take
away from knowing these wave

478
00:24:22,165 --> 00:24:22,665
functions?

479
00:24:26,130 --> 00:24:27,780
Well, if this is
the wave function,

480
00:24:27,780 --> 00:24:29,740
let's draw the
probability distribution.

481
00:24:29,740 --> 00:24:31,240
What's the probability
distribution?

482
00:24:31,240 --> 00:24:34,430
P of x.

483
00:24:34,430 --> 00:24:37,830
And the probability distribution
here is really very simple.

484
00:24:37,830 --> 00:24:42,300
It's again 0 squared is
still 0 so it's still

485
00:24:42,300 --> 00:24:50,880
just a big spike at x1 and
this one is a big spike at x2.

486
00:24:59,100 --> 00:25:00,920
Everyone cool with that?

487
00:25:00,920 --> 00:25:03,700
So what do you know
when I tell you

488
00:25:03,700 --> 00:25:05,950
that this is the wave function
describing your system?

489
00:25:05,950 --> 00:25:07,642
You know that with
great confidence,

490
00:25:07,642 --> 00:25:10,100
you will find the particle to
be sitting at x1 if you look.

491
00:25:12,700 --> 00:25:16,580
So what this is telling you
is you expect x is roughly x1

492
00:25:16,580 --> 00:25:19,200
and our uncertainty
in x is small.

493
00:25:23,000 --> 00:25:25,840
Everyone cool with that?

494
00:25:25,840 --> 00:25:31,600
Similarly, here you see that
the position is likely to be x2,

495
00:25:31,600 --> 00:25:34,630
and your uncertainty
in your measurement--

496
00:25:34,630 --> 00:25:36,080
your confidence
in your prediction

497
00:25:36,080 --> 00:25:38,362
is another way to say
it-- is quite good,

498
00:25:38,362 --> 00:25:39,570
so your uncertainty is small.

499
00:25:44,947 --> 00:25:46,030
Now what about these guys?

500
00:25:46,030 --> 00:25:47,440
Well, now it's norm squared.

501
00:25:47,440 --> 00:25:48,850
I need to tell you what
the wave function is.

502
00:25:48,850 --> 00:25:50,391
Here, the wave
function that I want--

503
00:25:50,391 --> 00:25:56,740
so here is 0-- is
e to the i k1 x.

504
00:25:56,740 --> 00:26:02,480
And here the wave function
is equal to e to the i k2 x.

505
00:26:02,480 --> 00:26:04,300
And remember, I'm
drawing the real part

506
00:26:04,300 --> 00:26:06,740
because of practical
limitations.

507
00:26:06,740 --> 00:26:09,350
So the real part is just
a sinusoid-- or, in fact,

508
00:26:09,350 --> 00:26:12,070
the cosine-- and similarly,
here, the real part

509
00:26:12,070 --> 00:26:14,610
is a cosine.

510
00:26:14,610 --> 00:26:16,735
And I really should put 0
in the appropriate place,

511
00:26:16,735 --> 00:26:22,450
but-- that worked out well.

512
00:26:22,450 --> 00:26:25,745
So now the question is, what's
the probability distribution,

513
00:26:25,745 --> 00:26:30,440
P of x, associated to
these wave functions?

514
00:26:30,440 --> 00:26:33,260
So what's the norm squared
of minus e to the i k1 x?

515
00:26:36,680 --> 00:26:42,760
If I have a complex number
of phase e to the i alpha,

516
00:26:42,760 --> 00:26:46,850
and I take its norm
squared, what do I get?

517
00:26:46,850 --> 00:26:47,980
1.

518
00:26:47,980 --> 00:26:48,480
Right?

519
00:26:48,480 --> 00:26:50,570
But remember complex numbers.

520
00:26:50,570 --> 00:26:52,460
If we have a complex
number alpha-- or sorry,

521
00:26:52,460 --> 00:26:54,430
if we have a
complex number beta,

522
00:26:54,430 --> 00:26:57,020
then beta squared
is by definition

523
00:26:57,020 --> 00:27:00,290
beta complex
conjugate times beta.

524
00:27:00,290 --> 00:27:02,330
So e to the i alpha, if
the complex conjugate

525
00:27:02,330 --> 00:27:04,460
is e to the minus i alpha,
e to the i alpha times

526
00:27:04,460 --> 00:27:06,750
e to the minus i alpha,
they cancel out-- that's 1.

527
00:27:09,420 --> 00:27:11,072
So if this is the
wave function, what's

528
00:27:11,072 --> 00:27:12,280
the probability distribution?

529
00:27:15,940 --> 00:27:16,710
Well, it's 1.

530
00:27:16,710 --> 00:27:19,800
It's independent of x.

531
00:27:19,800 --> 00:27:21,934
So from this we've learned
two important things.

532
00:27:21,934 --> 00:27:23,850
The first is, this is
not properly normalized.

533
00:27:27,190 --> 00:27:27,970
That's not so key.

534
00:27:27,970 --> 00:27:32,140
But the most important thing is,
if this is our wave function,

535
00:27:32,140 --> 00:27:34,826
and we subsequently measure the
position of the particle-- we

536
00:27:34,826 --> 00:27:36,700
look at it, we say ah,
there's the particle--

537
00:27:36,700 --> 00:27:39,782
where are we likely to find it?

538
00:27:39,782 --> 00:27:42,110
Yeah, it could be anywhere.

539
00:27:42,110 --> 00:27:46,670
So what's the value of x
you expect-- typical x?

540
00:27:46,670 --> 00:27:50,830
I have no idea, no
information whatsoever.

541
00:27:50,830 --> 00:27:52,100
None.

542
00:27:52,100 --> 00:27:55,560
But and correspondingly,
what is our uncertainty

543
00:27:55,560 --> 00:27:58,800
in the position of x
that we'll measure?

544
00:27:58,800 --> 00:28:01,079
It's very large, exactly.

545
00:28:01,079 --> 00:28:03,120
Now, in order to tell you
it's actually infinite,

546
00:28:03,120 --> 00:28:05,839
I need to stretch this off and
tell you that it's actually

547
00:28:05,839 --> 00:28:08,130
constant off to infinity,
and my arms aren't that long,

548
00:28:08,130 --> 00:28:09,840
so I'll just say large.

549
00:28:09,840 --> 00:28:11,700
Similarly here, if
our wave function

550
00:28:11,700 --> 00:28:14,020
is e to the i k2 x--
here k2 is larger,

551
00:28:14,020 --> 00:28:18,760
the wavelength is
shorter-- what's

552
00:28:18,760 --> 00:28:21,790
the probability distribution?

553
00:28:21,790 --> 00:28:22,870
It's, again, constant.

554
00:28:26,890 --> 00:28:30,770
So-- this is 0, 0.

555
00:28:30,770 --> 00:28:36,850
So again, x-- we have no idea,
and our uncertainty in the x

556
00:28:36,850 --> 00:28:38,264
is large.

557
00:28:38,264 --> 00:28:39,430
And in fact it's very large.

558
00:28:43,880 --> 00:28:45,604
Questions?

559
00:28:45,604 --> 00:28:46,520
What about these guys?

560
00:28:49,350 --> 00:28:51,230
OK, this is the real challenge.

561
00:28:51,230 --> 00:28:52,730
OK, so if this is
our wave function,

562
00:28:52,730 --> 00:28:56,070
and let's just say that
it's real-- hard as it

563
00:28:56,070 --> 00:28:59,720
is to believe that-- then what's
our probably distribution?

564
00:28:59,720 --> 00:29:07,780
Well, something
like-- I don't know,

565
00:29:07,780 --> 00:29:15,950
something-- you get the point.

566
00:29:15,950 --> 00:29:20,430
OK, so if this is our
probability distribution,

567
00:29:20,430 --> 00:29:22,654
where are we likely
to find the particle?

568
00:29:22,654 --> 00:29:24,570
Well, now it's a little
more difficult, right?

569
00:29:24,570 --> 00:29:27,060
Because we're unlikely
to find it here,

570
00:29:27,060 --> 00:29:29,450
while it's reasonably likely
to find here, unlikely here,

571
00:29:29,450 --> 00:29:33,130
reasonably likely, unlikely,
like-- you know, it's a mess.

572
00:29:33,130 --> 00:29:34,750
So where is this?

573
00:29:34,750 --> 00:29:36,280
I'm not really sure.

574
00:29:36,280 --> 00:29:38,610
What's our uncertainty?

575
00:29:38,610 --> 00:29:41,850
Well, our uncertainty
is not infinite

576
00:29:41,850 --> 00:29:44,324
because-- OK, my name
ends at some point.

577
00:29:44,324 --> 00:29:45,490
So this is going to go to 0.

578
00:29:47,790 --> 00:29:50,040
So whatever else we know,
we know it's in this region.

579
00:29:50,040 --> 00:29:55,830
So it's not infinite,
it's not small, we'll say.

580
00:29:55,830 --> 00:29:58,900
But it's not arbitrarily
small-- it's not tiny.

581
00:29:58,900 --> 00:30:00,782
Or sorry, it's not
gigantic is what I meant.

582
00:30:00,782 --> 00:30:02,115
Our uncertainty is not gigantic.

583
00:30:06,615 --> 00:30:07,990
But it's still
pretty nontrivial,

584
00:30:07,990 --> 00:30:09,495
because I can say
with some confidence

585
00:30:09,495 --> 00:30:11,090
that it's more likely
to be here than here,

586
00:30:11,090 --> 00:30:12,923
but I really don't know
which of those peaks

587
00:30:12,923 --> 00:30:15,460
it's going to be found.

588
00:30:15,460 --> 00:30:18,150
OK, now what about this guy?

589
00:30:18,150 --> 00:30:20,050
What's the probability
distribution well now

590
00:30:20,050 --> 00:30:22,660
you see why this is a
stupid wave function,

591
00:30:22,660 --> 00:30:24,302
because it's multiply valued.

592
00:30:24,302 --> 00:30:26,510
It has multiple different
values at every value of x.

593
00:30:26,510 --> 00:30:27,593
So what's the probability?

594
00:30:27,593 --> 00:30:30,470
Well, it might be root 2,
maybe it's 1 over root 3.

595
00:30:30,470 --> 00:30:32,230
I'm really not sure.

596
00:30:32,230 --> 00:30:37,210
So this tells us an important
lesson-- this is stupid.

597
00:30:37,210 --> 00:30:41,027
And what I mean by stupid
is, it is multiply valued.

598
00:30:41,027 --> 00:30:42,610
So the wave function--
we just learned

599
00:30:42,610 --> 00:30:44,330
a lesson-- should
be single valued.

600
00:30:49,620 --> 00:30:53,652
And we will explore some more
on your problem set, which

601
00:30:53,652 --> 00:30:55,360
will be posted
immediately after lecture.

602
00:30:55,360 --> 00:30:57,443
There are problems that
walk you through a variety

603
00:30:57,443 --> 00:31:00,880
of other potential pathologies
of the wave function

604
00:31:00,880 --> 00:31:02,770
and guide you to
some more intuition.

605
00:31:02,770 --> 00:31:04,311
For example, the
wave function really

606
00:31:04,311 --> 00:31:05,780
needs to be continuous as well.

607
00:31:05,780 --> 00:31:06,405
You'll see why.

608
00:31:09,230 --> 00:31:10,485
All right.

609
00:31:10,485 --> 00:31:11,485
Questions at this point?

610
00:31:16,070 --> 00:31:16,570
No?

611
00:31:16,570 --> 00:31:17,620
OK.

612
00:31:17,620 --> 00:31:20,472
So these look like pretty
useful wave functions,

613
00:31:20,472 --> 00:31:22,180
because they corresponded
to the particle

614
00:31:22,180 --> 00:31:23,346
being at some definite spot.

615
00:31:23,346 --> 00:31:26,890
And I, for example, am at
a reasonably definite spot.

616
00:31:26,890 --> 00:31:28,340
These two wave
functions, though,

617
00:31:28,340 --> 00:31:30,760
look pretty much useless,
because they give us

618
00:31:30,760 --> 00:31:34,130
no information whatsoever
about what the position is.

619
00:31:34,130 --> 00:31:36,390
Everyone agree with that?

620
00:31:36,390 --> 00:31:42,000
Except-- remember the
de Broglie relations.

621
00:31:42,000 --> 00:31:45,650
The de Broglie relations say
that associated to a particle

622
00:31:45,650 --> 00:31:47,420
is also some wave.

623
00:31:47,420 --> 00:31:49,727
And the momentum
of that particle

624
00:31:49,727 --> 00:31:51,060
is determined by the wavelength.

625
00:31:51,060 --> 00:31:52,205
It's inversely related
to the wavelength.

626
00:31:52,205 --> 00:31:53,810
It's proportional
to the wave number.

627
00:31:53,810 --> 00:31:56,980
Any energy is proportional
to the frequency.

628
00:31:56,980 --> 00:32:01,280
Now, look at those
wave functions.

629
00:32:01,280 --> 00:32:04,180
Those wave functions give
us no position information

630
00:32:04,180 --> 00:32:07,090
whatsoever, but they have
very definite wavelengths.

631
00:32:07,090 --> 00:32:10,250
Those are periodic functions
with definite wavelengths.

632
00:32:10,250 --> 00:32:15,601
In particular, this guy has
a wavelength of from here

633
00:32:15,601 --> 00:32:16,100
to here.

634
00:32:18,930 --> 00:32:20,213
It has a wave number k1.

635
00:32:22,790 --> 00:32:24,940
So that tells us
that if we measure

636
00:32:24,940 --> 00:32:27,412
the momentum of
this particle, we

637
00:32:27,412 --> 00:32:28,870
can be pretty
confident, because it

638
00:32:28,870 --> 00:32:31,250
has a reasonably
well-defined wavelength

639
00:32:31,250 --> 00:32:33,850
corresponding to some
wave number k-- 2 pi

640
00:32:33,850 --> 00:32:35,359
upon the wavelength.

641
00:32:35,359 --> 00:32:37,150
It has some momentum,
and if we measure it,

642
00:32:37,150 --> 00:32:39,316
we should be pretty confident
that the momentum will

643
00:32:39,316 --> 00:32:39,980
be h-bar k1.

644
00:32:43,560 --> 00:32:45,040
Everybody agree with that?

645
00:32:45,040 --> 00:32:46,762
Looks like a sine wave.

646
00:32:46,762 --> 00:32:48,220
And de Broglie
tells us that if you

647
00:32:48,220 --> 00:32:51,240
have a wave of
wavelength lambda,

648
00:32:51,240 --> 00:32:55,294
that corresponds to a
particle having momentum p.

649
00:32:55,294 --> 00:32:58,370
Now, how confident can
we be in that estimation

650
00:32:58,370 --> 00:33:00,760
of the momentum?

651
00:33:00,760 --> 00:33:03,200
Well, if I tell you it's e
to the i k x, that's exactly

652
00:33:03,200 --> 00:33:07,260
a periodic function with
wavelength lambda 2 pi upon k.

653
00:33:07,260 --> 00:33:08,427
So how confident are we?

654
00:33:08,427 --> 00:33:09,135
Pretty confident.

655
00:33:09,135 --> 00:33:12,320
So our uncertainty in
the momentum is tiny.

656
00:33:15,260 --> 00:33:17,330
Everyone agree?

657
00:33:17,330 --> 00:33:21,070
Similarly, for
this wave, again we

658
00:33:21,070 --> 00:33:23,540
have a wavelength-- it's
a periodic function,

659
00:33:23,540 --> 00:33:26,730
but the wavelength
is much shorter.

660
00:33:26,730 --> 00:33:28,550
If the wavelength
is much shorter,

661
00:33:28,550 --> 00:33:31,330
then k is much larger-- the
momentum is much larger.

662
00:33:31,330 --> 00:33:33,010
So the momentum we
expect to measure,

663
00:33:33,010 --> 00:33:38,555
which is roughly h-bar k2,
is going to be much larger.

664
00:33:38,555 --> 00:33:39,680
What about our uncertainty?

665
00:33:39,680 --> 00:33:42,040
Again, it's a perfect
periodic function

666
00:33:42,040 --> 00:33:43,990
so our uncertainty in
the momentum is small.

667
00:33:46,840 --> 00:33:47,840
Everyone cool with that?

668
00:33:53,950 --> 00:33:57,100
And that comes, again, from
the de Broglie relations.

669
00:34:04,170 --> 00:34:09,920
So questions at this point?

670
00:34:13,220 --> 00:34:14,469
You guys are real quiet today.

671
00:34:17,980 --> 00:34:20,480
Questions?

672
00:34:20,480 --> 00:34:23,971
AUDIENCE: So delta
P is 0, basically?

673
00:34:23,971 --> 00:34:25,179
PROFESSOR: It's pretty small.

674
00:34:25,179 --> 00:34:27,510
Now, again, I haven't
drawn this off to infinity,

675
00:34:27,510 --> 00:34:29,380
but if it's exactly
the i k x, then yeah,

676
00:34:29,380 --> 00:34:30,554
it turns out to be 0.

677
00:34:30,554 --> 00:34:32,929
Now, an important thing, so
let me rephrase your question

678
00:34:32,929 --> 00:34:33,429
slightly.

679
00:34:33,429 --> 00:34:35,330
So the question
was, is delta P 0?

680
00:34:35,330 --> 00:34:37,219
Is it really 0?

681
00:34:37,219 --> 00:34:38,860
So here's a problem
for us right now.

682
00:34:38,860 --> 00:34:41,520
We don't have a
definition for delta P.

683
00:34:41,520 --> 00:34:43,421
So what is the
definition of delta P?

684
00:34:43,421 --> 00:34:44,420
I haven't given you one.

685
00:34:44,420 --> 00:34:46,389
So here, when I said
delta P is small,

686
00:34:46,389 --> 00:34:48,830
what I mean is, intuitively,
just by eyeball,

687
00:34:48,830 --> 00:34:51,069
our confidence in that
momentum is pretty good,

688
00:34:51,069 --> 00:34:52,360
using the de Broglie relations.

689
00:34:52,360 --> 00:34:53,419
I have not given
you a definition,

690
00:34:53,419 --> 00:34:55,585
and that will be part of
my job over the next couple

691
00:34:55,585 --> 00:34:56,520
of lectures.

692
00:34:56,520 --> 00:34:57,330
Very good question.

693
00:34:57,330 --> 00:34:58,303
Yeah.

694
00:34:58,303 --> 00:35:00,730
AUDIENCE: How do you code
noise in that function?

695
00:35:00,730 --> 00:35:01,025
PROFESSOR: Awesome.

696
00:35:01,025 --> 00:35:02,670
AUDIENCE: Do you just
have different wavelengths

697
00:35:02,670 --> 00:35:03,025
PROFESSOR: Yeah

698
00:35:03,025 --> 00:35:03,920
AUDIENCE: As you go along?

699
00:35:03,920 --> 00:35:04,711
PROFESSOR: Awesome.

700
00:35:04,711 --> 00:35:07,780
So for example, this-- does
it have a definite wavelength?

701
00:35:07,780 --> 00:35:09,200
Not so much.

702
00:35:09,200 --> 00:35:14,270
So hold that question and wait
until you see the next examples

703
00:35:14,270 --> 00:35:16,520
that I put on this board,
and if that doesn't answer

704
00:35:16,520 --> 00:35:18,020
your question, ask
it again, becayse

705
00:35:18,020 --> 00:35:19,380
it's a very important question.

706
00:35:19,380 --> 00:35:20,464
OK.

707
00:35:20,464 --> 00:35:23,960
AUDIENCE: When you
talk about a photon,

708
00:35:23,960 --> 00:35:26,630
you always say a photon
has a certain frequency.

709
00:35:26,630 --> 00:35:29,440
Doesn't that mean that it
must be a wave because you

710
00:35:29,440 --> 00:35:32,324
have to fix the wave number k?

711
00:35:32,324 --> 00:35:33,490
PROFESSOR: Awesome question.

712
00:35:33,490 --> 00:35:37,790
Does every wave packet of
light that hits your eye,

713
00:35:37,790 --> 00:35:40,950
does it always have a
single, unique frequency?

714
00:35:40,950 --> 00:35:43,650
No, you can take multiple
frequency sources

715
00:35:43,650 --> 00:35:45,430
and superpose them.

716
00:35:45,430 --> 00:35:47,590
An interesting choice
of words I used there.

717
00:35:47,590 --> 00:35:49,620
All right, so the
question is, since light

718
00:35:49,620 --> 00:35:52,580
has some wavelength,
does every chunk of light

719
00:35:52,580 --> 00:35:54,670
have a definite-- this
is the question, roughly.

720
00:35:54,670 --> 00:35:56,279
Yeah, so and the
answer is, light

721
00:35:56,279 --> 00:35:57,695
doesn't always
have a single-- You

722
00:35:57,695 --> 00:35:59,644
can have light
coming at you that

723
00:35:59,644 --> 00:36:01,810
has many different wavelengths
and put it in a prism

724
00:36:01,810 --> 00:36:04,420
and break it up into
its various components.

725
00:36:04,420 --> 00:36:07,240
So you can have a superposition
of different frequencies

726
00:36:07,240 --> 00:36:08,740
of light.

727
00:36:08,740 --> 00:36:12,960
We'll see the same
effect happening for us.

728
00:36:12,960 --> 00:36:22,660
OK, so again, de Broglie
made this conjecture

729
00:36:22,660 --> 00:36:25,610
that E is h-bar omega
and P is h-bar k.

730
00:36:30,060 --> 00:36:32,240
This was verified in the
Davisson-Germer experiment

731
00:36:32,240 --> 00:36:33,460
that we ran.

732
00:36:33,460 --> 00:36:35,650
But here, one of
the things that's

733
00:36:35,650 --> 00:36:37,820
sort of latent in
this is, what he means

734
00:36:37,820 --> 00:36:41,060
is, look, associated to
every particle with energy N

735
00:36:41,060 --> 00:36:47,640
and momentum P is a plane
wave of the form e to the i kx

736
00:36:47,640 --> 00:36:50,100
minus omega t.

737
00:36:50,100 --> 00:36:53,430
And this, properly, in three
dimensions should be k dot x.

738
00:36:53,430 --> 00:36:58,070
But at this point, this is
an important simplification.

739
00:36:58,070 --> 00:37:04,411
For the rest of 8.04,
until otherwise specified,

740
00:37:04,411 --> 00:37:06,410
we are going to be doing
one-dimensional quantum

741
00:37:06,410 --> 00:37:07,280
mechanics.

742
00:37:07,280 --> 00:37:10,691
So I'm going to remove arrow
marks and dot products.

743
00:37:10,691 --> 00:37:12,940
There's going to be one
spatial dimension and one time

744
00:37:12,940 --> 00:37:14,155
dimension.

745
00:37:14,155 --> 00:37:16,280
We're always going to have
just one time dimension,

746
00:37:16,280 --> 00:37:18,030
but sometimes we'll have
more spatial dimensions.

747
00:37:18,030 --> 00:37:20,030
But it's going to be a
while until we get there.

748
00:37:20,030 --> 00:37:21,840
So for now, we're
just going to have kx.

749
00:37:21,840 --> 00:37:23,419
So this is a general plane wave.

750
00:37:23,419 --> 00:37:24,960
And what de Broglie
really was saying

751
00:37:24,960 --> 00:37:29,130
is that, somehow, associated
to the particle with energy E

752
00:37:29,130 --> 00:37:32,030
and momentum P should be
some wave, a plane wave,

753
00:37:32,030 --> 00:37:34,340
with wave number k
and frequency omega.

754
00:37:34,340 --> 00:37:38,460
And that's the wave
function associated to it.

755
00:37:38,460 --> 00:37:42,780
The thing is, not every wave
function is a plane wave.

756
00:37:42,780 --> 00:37:45,900
Some wave functions
are well localized.

757
00:37:45,900 --> 00:37:48,842
Some of them are just
complicated morasses.

758
00:37:48,842 --> 00:37:50,050
Some of them are just a mess.

759
00:37:59,850 --> 00:38:08,910
So now is the most important
postulate in quantum mechanics.

760
00:38:08,910 --> 00:38:11,867
I remember vividly,
vividly, when

761
00:38:11,867 --> 00:38:13,200
I took the analog of this class.

762
00:38:13,200 --> 00:38:16,530
It was called Physics
143A at Harvard.

763
00:38:16,530 --> 00:38:19,660
And the professor
at this point said--

764
00:38:19,660 --> 00:38:21,570
I know him well now,
he's a friend-- he said,

765
00:38:21,570 --> 00:38:23,996
this is what quantum
mechanics is all about.

766
00:38:23,996 --> 00:38:24,870
And I was so psyched.

767
00:38:24,870 --> 00:38:26,980
And then he told me And it
was like, that's ridiculous.

768
00:38:26,980 --> 00:38:27,460
Seriously?

769
00:38:27,460 --> 00:38:29,270
That's what quantum
mechanics is all about?

770
00:38:29,270 --> 00:38:31,478
So I always felt like this
is some weird thing, where

771
00:38:31,478 --> 00:38:33,100
old physicists go crazy.

772
00:38:33,100 --> 00:38:36,430
But it turns out I'm going to
say exactly the same thing.

773
00:38:36,430 --> 00:38:39,440
This is the most important thing
in all of quantum mechanics.

774
00:38:39,440 --> 00:38:43,330
It is all contained in
the following proposition.

775
00:38:43,330 --> 00:38:46,680
Everything-- the two
slit experiments, the box

776
00:38:46,680 --> 00:38:49,330
experiments, all the cool
stuff in quantum mechanics,

777
00:38:49,330 --> 00:38:51,230
all the strange and
counter intuitive stuff

778
00:38:51,230 --> 00:38:53,450
comes directly from
the next postulate.

779
00:38:53,450 --> 00:38:54,220
So here it is.

780
00:38:58,460 --> 00:39:01,300
I love this.

781
00:39:01,300 --> 00:39:05,240
Three-- put a star on it.

782
00:39:07,860 --> 00:39:16,450
Given two possible wave
functions or states--

783
00:39:16,450 --> 00:39:26,970
I'll say configurations--
of a quantum system--

784
00:39:26,970 --> 00:39:29,510
I wish there was "Ride
of the Valkyries"

785
00:39:29,510 --> 00:39:37,690
playing in the background--
corresponding to two

786
00:39:37,690 --> 00:39:45,240
distinct wave functions--
f with an upper ns

787
00:39:45,240 --> 00:39:47,600
is going to be my
notation for functions

788
00:39:47,600 --> 00:39:54,050
because I have to write
it a lot-- psi1 and psi2--

789
00:39:54,050 --> 00:40:05,770
and I'll say, of x--
the system-- is down--

790
00:40:05,770 --> 00:40:11,820
can also be in a superposition.

791
00:40:19,950 --> 00:40:40,140
of psi1 and psi2, where alpha
and beta are complex numbers.

792
00:40:47,750 --> 00:40:52,620
Given any two possible
configurations of the system,

793
00:40:52,620 --> 00:40:55,077
there is also an allowed
configuration of the system

794
00:40:55,077 --> 00:40:56,660
corresponding to
being in an arbitrary

795
00:40:56,660 --> 00:40:59,410
superposition of them.

796
00:40:59,410 --> 00:41:03,670
If an electron can be
hard and it can be soft,

797
00:41:03,670 --> 00:41:06,290
it can also be in an
arbitrary superposition

798
00:41:06,290 --> 00:41:08,160
of being hard and soft.

799
00:41:08,160 --> 00:41:11,010
And what I mean by that
is that hard corresponds

800
00:41:11,010 --> 00:41:13,160
to some particular
wave function,

801
00:41:13,160 --> 00:41:15,640
soft will correspond to some
particular wave function,

802
00:41:15,640 --> 00:41:18,580
and the superposition
corresponds to a different wave

803
00:41:18,580 --> 00:41:21,009
function which is a linear
combination of them.

804
00:41:25,230 --> 00:41:27,223
AUDIENCE: [INAUDIBLE]
combination also

805
00:41:27,223 --> 00:41:28,610
have to be normalized?

806
00:41:28,610 --> 00:41:30,651
PROFESSOR: Yeah, OK, that's
a very good question.

807
00:41:30,651 --> 00:41:32,960
So and alpha and beta
are some complex numbers

808
00:41:32,960 --> 00:41:35,370
subject to the
normalization condition.

809
00:41:39,640 --> 00:41:44,060
So indeed, this wave function
should be properly normalized.

810
00:41:44,060 --> 00:41:45,890
Now, let me step
back for second.

811
00:41:45,890 --> 00:41:48,180
There's an alternate way
to phrase the probability

812
00:41:48,180 --> 00:41:49,870
distribution here,
which goes like this,

813
00:41:49,870 --> 00:41:51,189
and I'm going to put it here.

814
00:41:51,189 --> 00:41:53,480
The alternate statement of
the probability distribution

815
00:41:53,480 --> 00:41:56,750
is that the probability
density at x

816
00:41:56,750 --> 00:42:03,270
is equal to psi of x
norm squared divided

817
00:42:03,270 --> 00:42:09,280
by the integral over
all x dx of psi squared.

818
00:42:11,810 --> 00:42:15,420
So notice that, if we properly
normalize the wave function,

819
00:42:15,420 --> 00:42:20,739
this denominator is equal to 1--
and so it's not there, right,

820
00:42:20,739 --> 00:42:21,780
and then it's equivalent.

821
00:42:21,780 --> 00:42:23,488
But if we haven't
properly normalized it,

822
00:42:23,488 --> 00:42:25,890
then this probability
distribution

823
00:42:25,890 --> 00:42:27,820
is automatically
properly normalized.

824
00:42:27,820 --> 00:42:30,361
Because this is a constant, when
we integrate the top, that's

825
00:42:30,361 --> 00:42:33,650
equal to the bottom,
it integrates to 1.

826
00:42:33,650 --> 00:42:37,220
So I prefer,
personally, in thinking

827
00:42:37,220 --> 00:42:39,720
about this for the first pass
to just require that we always

828
00:42:39,720 --> 00:42:41,970
be careful to choose
some normalization.

829
00:42:41,970 --> 00:42:44,260
That won't always be easy,
and so sometimes it's

830
00:42:44,260 --> 00:42:45,830
useful to forget
about normalizing

831
00:42:45,830 --> 00:42:49,649
and just define the probability
distribution that way.

832
00:42:49,649 --> 00:42:50,190
Is that cool?

833
00:42:53,270 --> 00:42:53,770
OK.

834
00:42:53,770 --> 00:42:57,029
This is the beating soul
of quantum mechanics.

835
00:42:57,029 --> 00:42:58,820
Everything in quantum
mechanics is in here.

836
00:42:58,820 --> 00:43:00,778
Everything in quantum
mechanics is forced on us

837
00:43:00,778 --> 00:43:03,520
from these few principles
and a couple of requirements

838
00:43:03,520 --> 00:43:05,704
of matching to reality.

839
00:43:05,704 --> 00:43:08,174
AUDIENCE: When
you do this-- some

840
00:43:08,174 --> 00:43:12,000
of linear, some of
two wave functions,

841
00:43:12,000 --> 00:43:14,280
can you get interference?

842
00:43:14,280 --> 00:43:14,950
PROFESSOR: Yes.

843
00:43:14,950 --> 00:43:15,450
Excellent.

844
00:43:15,450 --> 00:43:17,900
So the question is, when
you have a sum of two wave

845
00:43:17,900 --> 00:43:21,250
functions, can you get some
sort of interference effect?

846
00:43:21,250 --> 00:43:22,760
And the answer is, absolutely.

847
00:43:22,760 --> 00:43:24,510
And that's exactly
we're going to do next.

848
00:43:24,510 --> 00:43:27,572
So in particular, let me
look at a particular pair

849
00:43:27,572 --> 00:43:30,266
of superpositions.

850
00:43:30,266 --> 00:43:31,640
So let's swap
these boards around

851
00:43:31,640 --> 00:43:33,473
so the parallelism is
a little more obvious.

852
00:43:36,300 --> 00:43:38,945
So let's scrap these
rather silly wave functions

853
00:43:38,945 --> 00:43:40,320
and come up with
something that's

854
00:43:40,320 --> 00:43:41,670
a little more interesting.

855
00:43:41,670 --> 00:43:46,930
So instead of using those as
characteristic wave functions,

856
00:43:46,930 --> 00:43:48,670
I want to build superpositions.

857
00:43:48,670 --> 00:43:50,452
So in particular,
I want to start

858
00:43:50,452 --> 00:43:52,660
by taking an arbitrary--
both of these wave functions

859
00:43:52,660 --> 00:43:53,570
have a simple interpretation.

860
00:43:53,570 --> 00:43:55,430
This corresponds to a
particle being here.

861
00:43:55,430 --> 00:43:57,430
This corresponds to a
particle being here.

862
00:43:57,430 --> 00:43:59,400
I want to take a
superposition of them.

863
00:43:59,400 --> 00:44:00,525
So here's my superposition.

864
00:44:03,230 --> 00:44:05,610
Oops, let's try that again.

865
00:44:08,640 --> 00:44:13,100
And my superposition-- so here
is 0 and here is x1 and here is

866
00:44:13,100 --> 00:44:18,780
x2-- is going to be some
amount times the first one

867
00:44:18,780 --> 00:44:22,390
plus some amount
times the second one.

868
00:44:22,390 --> 00:44:25,450
There's a superposition.

869
00:44:25,450 --> 00:44:28,320
Similarly, I could have taken
a superposition of the two

870
00:44:28,320 --> 00:44:31,921
functions on the
second chalkboard.

871
00:44:31,921 --> 00:44:33,420
And again I'm taking
a superposition

872
00:44:33,420 --> 00:44:36,085
of the complex e to the
i k1 x and e to the i

873
00:44:36,085 --> 00:44:38,000
k2 x and then taking
the real part.

874
00:44:47,650 --> 00:44:49,420
So that's a particular
superposition,

875
00:44:49,420 --> 00:44:50,753
a particular linear combination.

876
00:44:56,010 --> 00:44:57,680
So now let's go back to this.

877
00:44:57,680 --> 00:44:59,544
This was a particle
that was here.

878
00:44:59,544 --> 00:45:00,960
This is a particle
that was there.

879
00:45:00,960 --> 00:45:03,630
When we take the
superposition, what

880
00:45:03,630 --> 00:45:05,620
is the probability distribution?

881
00:45:05,620 --> 00:45:07,170
Where is this particle?

882
00:45:07,170 --> 00:45:09,560
Well, there's some
amplitude that it's here,

883
00:45:09,560 --> 00:45:12,201
and there's some
amplitude that it's here.

884
00:45:12,201 --> 00:45:14,450
And there's rather more
amplitude that it's over here,

885
00:45:14,450 --> 00:45:15,866
but there's still
some probability

886
00:45:15,866 --> 00:45:17,600
that it's over here.

887
00:45:17,600 --> 00:45:19,976
Where am I going to
find the particle?

888
00:45:19,976 --> 00:45:22,250
I'm not so sure anymore.

889
00:45:22,250 --> 00:45:25,960
It's either going to be here
or here, but I'm not positive.

890
00:45:25,960 --> 00:45:27,450
It's more likely
to be here than it

891
00:45:27,450 --> 00:45:31,660
is to be here, but
not a whole lot more.

892
00:45:31,660 --> 00:45:33,594
So where am I going
to find the particle?

893
00:45:33,594 --> 00:45:35,010
Well, now we have
to define this--

894
00:45:35,010 --> 00:45:36,770
where am I going to
find the particle?

895
00:45:36,770 --> 00:45:39,030
Look, if I did this experiment
a whole bunch of times,

896
00:45:39,030 --> 00:45:41,700
it'd be over here more
than it would be over here.

897
00:45:41,700 --> 00:45:44,230
So the average will be
somewhere around here--

898
00:45:44,230 --> 00:45:45,860
it'll be in between the two.

899
00:45:45,860 --> 00:45:48,690
So x is somewhere in between.

900
00:45:48,690 --> 00:45:55,435
That's where we expect
to find it, on average.

901
00:45:59,260 --> 00:46:02,560
What's our uncertainty
in the position?

902
00:46:02,560 --> 00:46:04,340
Well, it's not
that small anymore.

903
00:46:04,340 --> 00:46:07,085
It's now of order x1 minus x2.

904
00:46:12,729 --> 00:46:13,770
Everyone agree with that?

905
00:46:21,979 --> 00:46:23,020
Now, what about this guy?

906
00:46:26,740 --> 00:46:32,500
Well, does this thing
have a single wavelength?

907
00:46:32,500 --> 00:46:33,430
No.

908
00:46:33,430 --> 00:46:35,560
This is like light that
comes at you from the sun.

909
00:46:35,560 --> 00:46:36,570
It has many wavelengths.

910
00:46:36,570 --> 00:46:39,153
In this case, it has just two--
I've added those two together.

911
00:46:39,153 --> 00:46:45,060
So this is a plane wave which
is psi is e to the i k1 x plus

912
00:46:45,060 --> 00:46:45,865
e to the i k2 x.

913
00:46:51,800 --> 00:46:54,850
So in fact, it has two
wavelengths associated with it.

914
00:46:54,850 --> 00:46:57,550
lambda1 lambda2.

915
00:46:57,550 --> 00:46:59,530
And so the probability
distribution now,

916
00:46:59,530 --> 00:47:03,470
if we take the norm squared
of this-- the probability

917
00:47:03,470 --> 00:47:11,170
distribution is the norm
squared of this guy--

918
00:47:11,170 --> 00:47:15,607
is no longer constant, but
there's an interference term.

919
00:47:15,607 --> 00:47:17,190
And let's just see
how that works out.

920
00:47:20,450 --> 00:47:23,730
Let me be very
explicit about this.

921
00:47:23,730 --> 00:47:27,380
Note that the probability
in our superposition of psi1

922
00:47:27,380 --> 00:47:30,690
plus psi2, which I'll call e to
the i k1 x plus e to the i k2

923
00:47:30,690 --> 00:47:35,520
x, is equal to the norm squared
of the wave function, which

924
00:47:35,520 --> 00:47:40,110
is the superposition psi1
plus beta psi2, which

925
00:47:40,110 --> 00:47:46,856
is equal to alpha squared
psi1 squared plus beta squared

926
00:47:46,856 --> 00:47:57,780
psi2 squared plus alpha
star psi1 star-- actually,

927
00:47:57,780 --> 00:48:07,230
let me write this over
here-- beta psi2 plus

928
00:48:07,230 --> 00:48:12,150
alpha psi1 beta star
psi2 star, where star

929
00:48:12,150 --> 00:48:15,010
means complex conjugation.

930
00:48:15,010 --> 00:48:19,480
But notice that this is
equal to-- that first term

931
00:48:19,480 --> 00:48:21,930
is alpha squared times
the first probability,

932
00:48:21,930 --> 00:48:25,180
or the probability of
this thing, of alpha psi1,

933
00:48:25,180 --> 00:48:28,300
is equal to probability 1.

934
00:48:28,300 --> 00:48:29,920
This term, beta
squared psi2 squared,

935
00:48:29,920 --> 00:48:31,961
is the probability that
the second thing happens.

936
00:48:35,340 --> 00:48:37,870
But these terms
can't be understood

937
00:48:37,870 --> 00:48:39,960
in terms of the
probabilities of psi1

938
00:48:39,960 --> 00:48:41,726
or the probability
of psi2 alone.

939
00:48:41,726 --> 00:48:42,850
They're interference terms.

940
00:48:49,290 --> 00:48:52,190
So the superposition
principle, together

941
00:48:52,190 --> 00:48:55,210
with the interpretation of
the probability as the norm

942
00:48:55,210 --> 00:49:00,700
squared of the wave function,
gives us a correction

943
00:49:00,700 --> 00:49:04,590
to the classical addition
of probabilities,

944
00:49:04,590 --> 00:49:07,430
which is these
interference terms.

945
00:49:07,430 --> 00:49:09,220
Everyone happy with that?

946
00:49:09,220 --> 00:49:11,770
Now, here's something very
important to keep in mind.

947
00:49:11,770 --> 00:49:14,000
These things are norms
squared of complex numbers.

948
00:49:14,000 --> 00:49:16,190
That means they're real,
but in particular, they're

949
00:49:16,190 --> 00:49:17,587
non-negative.

950
00:49:17,587 --> 00:49:19,420
So these two are both
real and non-negative.

951
00:49:19,420 --> 00:49:20,580
But what about this?

952
00:49:20,580 --> 00:49:22,725
This is not the norm
squared of anything.

953
00:49:22,725 --> 00:49:24,350
However, this is its
complex conjugate.

954
00:49:24,350 --> 00:49:25,570
When you take something
and its complex conjugate

955
00:49:25,570 --> 00:49:26,986
and you add them
together, you get

956
00:49:26,986 --> 00:49:28,510
something that's
necessarily real.

957
00:49:28,510 --> 00:49:32,039
But it's not
necessarily positive.

958
00:49:32,039 --> 00:49:33,080
So this is a funny thing.

959
00:49:33,080 --> 00:49:36,020
The probability that something
happens if we add together

960
00:49:36,020 --> 00:49:39,400
our two configurations, we
superpose two configurations,

961
00:49:39,400 --> 00:49:41,014
has a positive probability term.

962
00:49:41,014 --> 00:49:42,430
But it's also got
terms that don't

963
00:49:42,430 --> 00:49:44,480
have a definite sign,
that could be negative.

964
00:49:44,480 --> 00:49:45,750
It's always real.

965
00:49:45,750 --> 00:49:48,480
And you can check but
this quantity is always

966
00:49:48,480 --> 00:49:49,610
greater than or equal to 0.

967
00:49:49,610 --> 00:49:52,740
It's never negative,
the total quantity.

968
00:49:52,740 --> 00:49:55,890
So remember Bell's inequality
that we talked about?

969
00:49:55,890 --> 00:49:58,388
Bell's inequality
said, look, if we

970
00:49:58,388 --> 00:50:00,596
have the probability of one
thing happening being P1,

971
00:50:00,596 --> 00:50:02,596
and the probability of
the other thing happening

972
00:50:02,596 --> 00:50:05,470
being P2, the probability
of both things happening

973
00:50:05,470 --> 00:50:07,360
is just P1 plus P2.

974
00:50:07,360 --> 00:50:09,750
And here we see that,
in quantum mechanics,

975
00:50:09,750 --> 00:50:11,860
probabilities
don't add that way.

976
00:50:11,860 --> 00:50:13,970
The wave functions add--
and the probability

977
00:50:13,970 --> 00:50:18,700
is the norm squared
of the wave function.

978
00:50:18,700 --> 00:50:21,630
The wave functions add,
not the probabilities.

979
00:50:21,630 --> 00:50:24,630
And that is what underlies all
of the interference effects

980
00:50:24,630 --> 00:50:25,402
we've seen.

981
00:50:25,402 --> 00:50:27,610
And it's going to be the
heart of the rest of quantum

982
00:50:27,610 --> 00:50:29,730
mechanics.

983
00:50:29,730 --> 00:50:32,630
So you're probably all going,
in your head, more or less

984
00:50:32,630 --> 00:50:35,530
like I was when I took
Intro Quantum, like-- yeah,

985
00:50:35,530 --> 00:50:38,750
but I mean, it's just, you know,
you're adding complex numbers.

986
00:50:38,750 --> 00:50:40,631
But trust me on this one.

987
00:50:40,631 --> 00:50:42,630
This is where it's all starting.

988
00:50:42,630 --> 00:50:45,940
OK so let's go back to this.

989
00:50:45,940 --> 00:50:47,562
Similarly, let's
look at this example.

990
00:50:47,562 --> 00:50:48,770
We've taken the norm squared.

991
00:50:48,770 --> 00:50:50,395
And now we have an
interference effect.

992
00:50:50,395 --> 00:50:52,130
And now, our probability
distribution,

993
00:50:52,130 --> 00:50:55,900
instead of being totally trivial
and containing no information,

994
00:50:55,900 --> 00:50:58,400
our probability distribution
now contains some information

995
00:50:58,400 --> 00:51:00,140
about the position
of the object.

996
00:51:00,140 --> 00:51:01,150
It's likely to be here.

997
00:51:01,150 --> 00:51:03,660
It is unlikely to be
here, likely and unlikely.

998
00:51:03,660 --> 00:51:05,580
We now have some
position information.

999
00:51:05,580 --> 00:51:07,620
We don't have enough
to say where it is.

1000
00:51:07,620 --> 00:51:09,810
But x is-- you have
some information.

1001
00:51:13,920 --> 00:51:15,860
Now, our uncertainty
still gigantic.

1002
00:51:15,860 --> 00:51:18,640
Delta x is still huge.

1003
00:51:18,640 --> 00:51:20,840
But OK, we just added
together two plane waves.

1004
00:51:24,360 --> 00:51:25,060
Yeah?

1005
00:51:25,060 --> 00:51:28,210
AUDIENCE: Why is
the probability not

1006
00:51:28,210 --> 00:51:30,516
big, small, small,
big, small, small?

1007
00:51:30,516 --> 00:51:31,390
PROFESSOR: Excellent.

1008
00:51:31,390 --> 00:51:35,320
This was the real part
of the wave function.

1009
00:51:35,320 --> 00:51:38,111
And the wave function
is a complex quantity.

1010
00:51:38,111 --> 00:51:39,610
When you take e to
the i k1, and let

1011
00:51:39,610 --> 00:51:41,600
me do this on the chalkboard.

1012
00:51:41,600 --> 00:51:44,430
When we take e to the i
k1 x plus e to the i k2

1013
00:51:44,430 --> 00:51:48,270
x-- Let me write this
slightly differently--

1014
00:51:48,270 --> 00:51:54,470
e to the i a plus e to the i
b and take its norm squared.

1015
00:51:54,470 --> 00:51:56,190
So this is equal to--
I'm going to write

1016
00:51:56,190 --> 00:51:57,856
this in a slightly
more suggestive way--

1017
00:51:57,856 --> 00:52:03,450
the norm squared of e to the
i a times 1 plus e to the i b

1018
00:52:03,450 --> 00:52:09,350
minus a parentheses
norm squared.

1019
00:52:09,350 --> 00:52:12,300
So first off, the norm
squared of a product of things

1020
00:52:12,300 --> 00:52:14,700
is the product of
the norm squareds.

1021
00:52:14,700 --> 00:52:16,280
So I can do that.

1022
00:52:16,280 --> 00:52:21,500
And this overall phase, the norm
squared of a phase is just 1,

1023
00:52:21,500 --> 00:52:22,340
so that's just 1.

1024
00:52:25,320 --> 00:52:27,850
So now we have the norm squared
of 1 plus a complex number.

1025
00:52:32,180 --> 00:52:34,459
And so the norm squared of
1 is going to give me 1.

1026
00:52:34,459 --> 00:52:37,000
The norm squared of the complex
number is going to give me 1.

1027
00:52:37,000 --> 00:52:39,333
And the cross terms are going
to give me the real part--

1028
00:52:39,333 --> 00:52:41,435
twice the real part--
of e to the i b minus

1029
00:52:41,435 --> 00:52:45,550
a, which is going to be
equal to cosine of b minus a.

1030
00:52:47,940 --> 00:52:49,440
And so what you see
here is that you

1031
00:52:49,440 --> 00:52:52,400
have a single frequency
in the superposition.

1032
00:53:00,690 --> 00:53:02,850
So good, our
uncertainty is large.

1033
00:53:02,850 --> 00:53:05,850
So let's look at
this second example

1034
00:53:05,850 --> 00:53:08,010
in a little more detail.

1035
00:53:08,010 --> 00:53:11,800
By superimposing two states with
wavelength lambda1 and lambda2

1036
00:53:11,800 --> 00:53:22,754
or k1 and k2 we get
something that, OK,

1037
00:53:22,754 --> 00:53:24,170
it's still not
well localized-- we

1038
00:53:24,170 --> 00:53:25,753
don't know where the
particle is going

1039
00:53:25,753 --> 00:53:28,360
to be-- but it's better
localized than it was before,

1040
00:53:28,360 --> 00:53:29,710
right?

1041
00:53:29,710 --> 00:53:33,610
What happens if we superpose
with three wavelengths,

1042
00:53:33,610 --> 00:53:35,920
or four, or more?

1043
00:53:35,920 --> 00:53:40,810
So for that, I want to pull
out a Mathematica package.

1044
00:53:40,810 --> 00:53:43,410
You guys should all have
seen Fourier analysis

1045
00:53:43,410 --> 00:53:58,480
in 18.03, but just in case,
I'm putting on the web page,

1046
00:53:58,480 --> 00:54:01,000
on the Stellar page, a
notebook that walks you

1047
00:54:01,000 --> 00:54:03,540
through the basics of Fourier
analysis in Mathematica.

1048
00:54:03,540 --> 00:54:05,470
You should all be
fluent in Mathematica.

1049
00:54:05,470 --> 00:54:07,782
If you're not, you
should probably

1050
00:54:07,782 --> 00:54:08,740
come up to speed on it.

1051
00:54:08,740 --> 00:54:11,580
That's not what we wanted.

1052
00:54:11,580 --> 00:54:13,170
Let's try that again.

1053
00:54:13,170 --> 00:54:15,680
There we go.

1054
00:54:15,680 --> 00:54:23,680
Oh, that's awesome-- where
by awesome, I mean not.

1055
00:54:23,680 --> 00:54:24,180
It's coming.

1056
00:54:24,180 --> 00:54:24,680
OK, good.

1057
00:54:32,790 --> 00:54:35,470
I'm not even going to mess with
the screens after last time.

1058
00:54:35,470 --> 00:54:43,150
So I'm not going to go through
the details of this package,

1059
00:54:43,150 --> 00:54:50,220
but what this does is walk
you through the superposition

1060
00:54:50,220 --> 00:54:51,885
of wave packets.

1061
00:54:51,885 --> 00:54:56,450
So here I'm looking at the
probability distribution coming

1062
00:54:56,450 --> 00:54:59,090
from summing up a
bunch of plane waves

1063
00:54:59,090 --> 00:55:00,490
with some definite frequency.

1064
00:55:00,490 --> 00:55:03,600
So here it's just one.

1065
00:55:03,600 --> 00:55:05,420
That's one wave,
so first we have--

1066
00:55:05,420 --> 00:55:15,720
let me make this bigger-- yes,
stupid Mathematica tricks.

1067
00:55:15,720 --> 00:55:19,470
So here we have
the wave function

1068
00:55:19,470 --> 00:55:23,190
and here we have the probability
distribution, the norm squared.

1069
00:55:23,190 --> 00:55:25,294
And it's sort of
badly normalized here.

1070
00:55:25,294 --> 00:55:26,460
So that's for a single wave.

1071
00:55:26,460 --> 00:55:30,550
And as you see, the probability
distribution is constant.

1072
00:55:30,550 --> 00:55:32,760
And that's not 0,
that's 0.15, it's

1073
00:55:32,760 --> 00:55:35,220
just that I arbitrarily
normalized this.

1074
00:55:35,220 --> 00:55:37,790
So let's add two plane waves.

1075
00:55:37,790 --> 00:55:40,400
And now what you see is the
same effect as we had here.

1076
00:55:40,400 --> 00:55:42,625
You see a slightly more
localized wave function.

1077
00:55:42,625 --> 00:55:45,000
Now you have a little bit of
structure in the probability

1078
00:55:45,000 --> 00:55:45,870
distribution.

1079
00:55:45,870 --> 00:55:47,590
So there's the structure in
the probability distribution.

1080
00:55:47,590 --> 00:55:48,430
We have a little
more information

1081
00:55:48,430 --> 00:55:50,513
about where the particle
is more likely to be here

1082
00:55:50,513 --> 00:55:52,600
than it is to be here.

1083
00:55:52,600 --> 00:55:54,540
Let's add one more.

1084
00:55:54,540 --> 00:55:56,770
And as we keep adding
more and more plane waves

1085
00:55:56,770 --> 00:55:59,290
to our superposition, the wave
function and the probability

1086
00:55:59,290 --> 00:56:00,873
distribution associated
with it become

1087
00:56:00,873 --> 00:56:04,820
more and more
well-localized until, as we

1088
00:56:04,820 --> 00:56:07,960
go to very high numbers of plane
waves that we're superposing,

1089
00:56:07,960 --> 00:56:11,870
we get an extremely narrow
probability distribution--

1090
00:56:11,870 --> 00:56:14,280
and wave function, for that
matter-- extremely narrow

1091
00:56:14,280 --> 00:56:16,655
corresponding to a particle
that's very likely to be here

1092
00:56:16,655 --> 00:56:19,378
and unlikely to
be anywhere else.

1093
00:56:19,378 --> 00:56:20,940
Everyone cool with that?

1094
00:56:20,940 --> 00:56:22,080
What's the expense?

1095
00:56:22,080 --> 00:56:24,324
Want have we lost
in the process?

1096
00:56:24,324 --> 00:56:25,740
Well we know with
great confidence

1097
00:56:25,740 --> 00:56:29,420
now that the particle will be
found here upon observation.

1098
00:56:29,420 --> 00:56:30,790
But what will its momentum be?

1099
00:56:33,899 --> 00:56:35,940
Yeah, now it's the
superposition of a whole bunch

1100
00:56:35,940 --> 00:56:37,130
of different momenta.

1101
00:56:37,130 --> 00:56:39,100
So if it's a superposition
of a whole bunch

1102
00:56:39,100 --> 00:56:44,190
of different
momenta, here this is

1103
00:56:44,190 --> 00:56:46,719
like superposition of a whole
bunch of different positions--

1104
00:56:46,719 --> 00:56:48,260
likely to be here,
likely to be here,

1105
00:56:48,260 --> 00:56:49,570
likely to be here,
likely to be here.

1106
00:56:49,570 --> 00:56:51,111
What's our knowledge
of its position?

1107
00:56:51,111 --> 00:56:52,480
It's not very good.

1108
00:56:52,480 --> 00:56:54,420
Similarly, now that
we have superposed

1109
00:56:54,420 --> 00:56:56,420
many different momenta
with comparable strength.

1110
00:56:56,420 --> 00:56:58,540
In fact, here they were
all with unit strength.

1111
00:56:58,540 --> 00:57:01,150
We now have no information about
what the momentum is anymore.

1112
00:57:01,150 --> 00:57:02,941
It could be anything
in that superposition.

1113
00:57:05,490 --> 00:57:10,320
So now we're seeing
quite sharply

1114
00:57:10,320 --> 00:57:11,470
the uncertainty relation.

1115
00:57:11,470 --> 00:57:12,320
And here it is.

1116
00:57:15,234 --> 00:57:16,650
So the uncertainty
relation is now

1117
00:57:16,650 --> 00:57:18,140
pretty clear from these guys.

1118
00:57:20,722 --> 00:57:21,555
So that didn't work?

1119
00:57:26,490 --> 00:57:27,870
And I'm going to leave it alone.

1120
00:57:27,870 --> 00:57:29,536
This is enough for
the Fourier analysis,

1121
00:57:29,536 --> 00:57:31,520
but that Fourier
package is available

1122
00:57:31,520 --> 00:57:35,910
with extensive commentary
on the Stellar web page.

1123
00:57:35,910 --> 00:57:38,012
AUDIENCE: Now is
that sharp definition

1124
00:57:38,012 --> 00:57:41,460
in the position caused by
the interference between all

1125
00:57:41,460 --> 00:57:43,040
those waves and all that--

1126
00:57:43,040 --> 00:57:44,650
PROFESSOR: That's
exactly what it is.

1127
00:57:44,650 --> 00:57:45,380
Precisely.

1128
00:57:45,380 --> 00:57:48,060
It's precisely the interference
between the different momentum

1129
00:57:48,060 --> 00:57:50,185
nodes that leads to
certainty in the position.

1130
00:57:50,185 --> 00:57:51,060
That's exactly right.

1131
00:57:51,060 --> 00:57:52,035
Yeah.

1132
00:57:52,035 --> 00:57:53,951
AUDIENCE: So as we're
certain of the position,

1133
00:57:53,951 --> 00:57:56,339
we will not be certain
of the momentum.

1134
00:57:56,339 --> 00:57:57,130
PROFESSOR: Exactly.

1135
00:57:57,130 --> 00:57:57,970
And here we are.

1136
00:57:57,970 --> 00:58:00,915
So in this example, we have
no idea what the position is,

1137
00:58:00,915 --> 00:58:03,200
but we're quite confident
of the momentum.

1138
00:58:03,200 --> 00:58:05,950
Here we have no idea
what the position is,

1139
00:58:05,950 --> 00:58:08,910
but we have great
confidence in the momentum.

1140
00:58:08,910 --> 00:58:12,720
Similarly here, we have
less perfect confidence

1141
00:58:12,720 --> 00:58:15,900
of the position, and here we
have less perfect confidence

1142
00:58:15,900 --> 00:58:17,335
in the momentum.

1143
00:58:17,335 --> 00:58:18,960
It would be nice to
be able to estimate

1144
00:58:18,960 --> 00:58:20,793
what our uncertainty
is in the momentum here

1145
00:58:20,793 --> 00:58:23,580
and what our uncertainty
is in the position here.

1146
00:58:23,580 --> 00:58:25,470
So we're going to
have to do that.

1147
00:58:25,470 --> 00:58:27,178
That's going to be
one of our next tasks.

1148
00:58:30,480 --> 00:58:31,160
Other questions?

1149
00:58:31,160 --> 00:58:32,404
Yeah.

1150
00:58:32,404 --> 00:58:35,830
AUDIENCE: In this half
of the blackboard,

1151
00:58:35,830 --> 00:58:37,871
you said, obviously, if
we do it a bunch of times

1152
00:58:37,871 --> 00:58:39,859
it'll have more in
the x2 than in the x1.

1153
00:58:39,859 --> 00:58:40,853
PROFESSOR: Yes.

1154
00:58:40,853 --> 00:58:44,129
AUDIENCE: The average, it will
never physically be at that--

1155
00:58:44,129 --> 00:58:46,670
PROFESSOR: Yeah, that's right,
so, because it's a probability

1156
00:58:46,670 --> 00:58:48,810
distribution, it won't
be exactly at that point.

1157
00:58:48,810 --> 00:58:50,800
But it'll be nearby.

1158
00:58:50,800 --> 00:58:56,180
OK, so in order to be more
precise-- And so for example

1159
00:58:56,180 --> 00:58:59,530
for this what we do
here's a quick question.

1160
00:58:59,530 --> 00:59:02,629
How well do you know the
position of this particle?

1161
00:59:02,629 --> 00:59:03,420
Pretty well, right?

1162
00:59:03,420 --> 00:59:06,700
But how well do you
know its momentum?

1163
00:59:06,700 --> 00:59:10,735
Well, we'd all like to say
not very, but tell me why.

1164
00:59:10,735 --> 00:59:15,024
Why is your uncertainty in the
momentum of the particle large?

1165
00:59:15,024 --> 00:59:16,980
AUDIENCE: Heisenberg's
uncertainty principle.

1166
00:59:16,980 --> 00:59:17,860
PROFESSOR: Yeah,
but that's a cheat

1167
00:59:17,860 --> 00:59:19,550
because we haven't actually
proved Heisenberg's uncertainty

1168
00:59:19,550 --> 00:59:19,860
principle.

1169
00:59:19,860 --> 00:59:20,630
It's just something
we're inheriting.

1170
00:59:20,630 --> 00:59:21,820
AUDIENCE: I believe it.

1171
00:59:21,820 --> 00:59:22,730
PROFESSOR: I believe it, too.

1172
00:59:22,730 --> 00:59:24,370
But I want a better
argument because I

1173
00:59:24,370 --> 00:59:25,745
believe all sorts
of crazy stuff.

1174
00:59:25,745 --> 00:59:28,880
So-- I really do.

1175
00:59:28,880 --> 00:59:31,850
Black holes, fluids, I mean
look, don't get me started.

1176
00:59:31,850 --> 00:59:32,395
Yeah.

1177
00:59:32,395 --> 00:59:34,520
AUDIENCE: You can take the
Fourier transform of it.

1178
00:59:34,520 --> 00:59:35,190
PROFESSOR: Yeah, excellent.

1179
00:59:35,190 --> 00:59:36,650
OK, we'll get to
that in just one sec.

1180
00:59:36,650 --> 00:59:38,525
So before taking the
Fourier transform, which

1181
00:59:38,525 --> 00:59:40,330
is an excellent--
so the answer was,

1182
00:59:40,330 --> 00:59:41,323
just take a Fourier
transform, that's

1183
00:59:41,323 --> 00:59:42,180
going to give you
some information.

1184
00:59:42,180 --> 00:59:43,700
We're going to do
that in just a moment.

1185
00:59:43,700 --> 00:59:45,158
But before we do
Fourier transform,

1186
00:59:45,158 --> 00:59:48,490
just intuitively, why would
de Broglie look at this

1187
00:59:48,490 --> 00:59:50,869
and say, no, that doesn't
have a definite momentum.

1188
00:59:50,869 --> 00:59:52,452
AUDIENCE: There's
no clear wavelength.

1189
00:59:52,452 --> 00:59:54,060
PROFESSOR: Yeah, there's
no wavelength, right?

1190
00:59:54,060 --> 00:59:56,237
It's not periodic by any
stretch of the imagination.

1191
00:59:56,237 --> 00:59:58,570
It doesn't look like a thing
with a definite wavelength.

1192
00:59:58,570 --> 01:00:01,774
And de Broglie said, look, if
you have a definite wavelength

1193
01:00:01,774 --> 01:00:03,190
then you have a
definite momentum.

1194
01:00:03,190 --> 01:00:04,130
And if you have a
definite momentum,

1195
01:00:04,130 --> 01:00:05,820
you have a definite wavelength.

1196
01:00:05,820 --> 01:00:07,780
This is not a wave with
a definite wavelength,

1197
01:00:07,780 --> 01:00:10,440
so it is not corresponding
to the wave function

1198
01:00:10,440 --> 01:00:12,720
for a particle with
a definite momentum.

1199
01:00:12,720 --> 01:00:17,570
So our momentum is
unknown-- so this is large.

1200
01:00:17,570 --> 01:00:20,170
And similarly, here, our
uncertainty in the momentum

1201
01:00:20,170 --> 01:00:20,760
is large.

1202
01:00:24,820 --> 01:00:26,480
So to do better
than this, we need

1203
01:00:26,480 --> 01:00:28,400
to introduce the
Fourier transform,

1204
01:00:28,400 --> 01:00:30,530
and I want to do that now.

1205
01:00:30,530 --> 01:00:34,207
So you should all have seen
Fourier series in 8.03.

1206
01:00:34,207 --> 01:00:36,040
Now we're going to do
the Fourier transform.

1207
01:00:36,040 --> 01:00:37,623
And I'm going to
introduce this to you

1208
01:00:37,623 --> 01:00:40,831
in 8.04 conventions
in the following way.

1209
01:00:40,831 --> 01:00:42,330
And the theorem
says the following--

1210
01:00:42,330 --> 01:00:44,913
we're not going to prove it by
any stretch of the imagination,

1211
01:00:44,913 --> 01:00:48,900
but the theorem
says-- any function

1212
01:00:48,900 --> 01:00:52,070
f of x that is sufficiently
well-behaved-- it shouldn't be

1213
01:00:52,070 --> 01:00:55,630
discontinuous, it
shouldn't be singular--

1214
01:00:55,630 --> 01:00:57,980
any reasonably well-behaved,
non-stupid function f

1215
01:00:57,980 --> 01:01:19,574
of x can be built by superposing
enough plane waves of the form

1216
01:01:19,574 --> 01:01:20,115
e to the ikx.

1217
01:01:23,950 --> 01:01:26,720
Enough may be infinite.

1218
01:01:26,720 --> 01:01:33,100
So any function f of x can be
expressed as 1 over root 2 pi,

1219
01:01:33,100 --> 01:01:35,740
and this root 2 pi is a choice
of normalization-- everyone

1220
01:01:35,740 --> 01:01:37,240
has their own
conventions, and these

1221
01:01:37,240 --> 01:01:39,530
are the ones we'll be
using in 8.04 throughout--

1222
01:01:39,530 --> 01:01:48,480
minus infinity to infinity
dk f tilde of k e to the ikx.

1223
01:01:56,060 --> 01:01:58,980
So here, what we're
doing is, we're

1224
01:01:58,980 --> 01:02:03,480
summing over plane waves
of the form e to the ikx.

1225
01:02:03,480 --> 01:02:06,500
These are modes with a definite
wavelength 2 pi upon k.

1226
01:02:11,920 --> 01:02:18,430
f tilde of k is telling us
the amplitude of the wave

1227
01:02:18,430 --> 01:02:22,500
with wavelengths lambda
or wave number k.

1228
01:02:22,500 --> 01:02:26,150
And we sum over all
possible values.

1229
01:02:26,150 --> 01:02:28,360
And the claim is,
any function can

1230
01:02:28,360 --> 01:02:31,840
be expressed as a superposition
of plane waves in this form.

1231
01:02:31,840 --> 01:02:33,192
Cool?

1232
01:02:33,192 --> 01:02:34,650
And this is for
functions which are

1233
01:02:34,650 --> 01:02:37,960
non-periodic on the real line,
rather than periodic functions

1234
01:02:37,960 --> 01:02:40,530
on the interval, which is what
you should've seen in 8.03.

1235
01:02:40,530 --> 01:02:44,400
Now, conveniently, if
you know f tilde of k,

1236
01:02:44,400 --> 01:02:46,350
you can compute f of
x by doing the sum.

1237
01:02:46,350 --> 01:02:47,809
But suppose you
know f of x and you

1238
01:02:47,809 --> 01:02:49,641
want to determine what
the coefficients are,

1239
01:02:49,641 --> 01:02:50,800
the expansion coefficients.

1240
01:02:50,800 --> 01:02:52,920
That's the inverse
Fourier transform.

1241
01:02:52,920 --> 01:02:56,720
And the statement for
that is that f tilde of k

1242
01:02:56,720 --> 01:03:02,410
is equal to 1 over root 2 pi
integral from minus infinity

1243
01:03:02,410 --> 01:03:08,620
to infinity dx f of
x e to the minus ikx.

1244
01:03:11,249 --> 01:03:13,540
OK, that's sometimes referred
to as the inverse Fourier

1245
01:03:13,540 --> 01:03:14,040
transform.

1246
01:03:16,620 --> 01:03:20,080
And here's something
absolutely essential.

1247
01:03:20,080 --> 01:03:22,600
f tilde of k, the Fourier
transform coefficients

1248
01:03:22,600 --> 01:03:26,070
of f of x, are
completely equivalent.

1249
01:03:26,070 --> 01:03:29,670
If you know f of x, you
can determine f tilde of k.

1250
01:03:29,670 --> 01:03:31,180
And if you know
f tilde of k, you

1251
01:03:31,180 --> 01:03:33,762
can determine f of x
by just doing a sum,

1252
01:03:33,762 --> 01:03:34,720
by just adding them up.

1253
01:03:39,230 --> 01:03:42,030
So now here's the physical
version of this-- oh,

1254
01:03:42,030 --> 01:03:47,786
I can't slide that out--
I'm now going to put here.

1255
01:03:50,400 --> 01:03:50,900
Oh.

1256
01:03:50,900 --> 01:03:51,600
No, I'm not.

1257
01:03:51,600 --> 01:03:54,300
I'm going to put that down here.

1258
01:03:54,300 --> 01:03:57,000
So the physical
version of this is

1259
01:03:57,000 --> 01:04:26,270
that any wave function
psi of x can be expressed

1260
01:04:26,270 --> 01:04:44,640
as the superposition
in the form psi of x

1261
01:04:44,640 --> 01:04:50,120
is equal to 1 over root 2 pi
integral from minus infinity

1262
01:04:50,120 --> 01:05:13,120
to infinity dk psi tilde of k
e to the ikx of states, or wave

1263
01:05:13,120 --> 01:05:26,060
functions, with a definite
momentum p is equal to h bar k.

1264
01:05:39,060 --> 01:05:47,380
And so now, it's useful
to sketch the Fourier

1265
01:05:47,380 --> 01:05:52,030
transforms of each
of these functions.

1266
01:05:52,030 --> 01:05:54,790
In fact, we want this up here.

1267
01:06:05,330 --> 01:06:08,200
So here we have the function and
its probability distribution.

1268
01:06:08,200 --> 01:06:14,210
Now I want to draw the Fourier
transforms of these guys.

1269
01:06:14,210 --> 01:06:18,750
So here's psi tilde
of k, a function

1270
01:06:18,750 --> 01:06:20,254
of a different
variable than of x,

1271
01:06:20,254 --> 01:06:21,670
but nonetheless,
it's illuminating

1272
01:06:21,670 --> 01:06:23,003
to draw them next to each other.

1273
01:06:33,179 --> 01:06:34,720
And again, I'm
drawing the real part.

1274
01:06:39,030 --> 01:06:43,010
And here, x2-- had I had
my druthers about me,

1275
01:06:43,010 --> 01:06:45,605
I would have put x2
at a larger value.

1276
01:06:51,070 --> 01:06:53,190
Good, so it's further
off to the right there.

1277
01:07:06,210 --> 01:07:10,890
I'm so loathe to erase the
superposition principle.

1278
01:07:10,890 --> 01:07:15,020
But fortunately,
I'm not there yet.

1279
01:07:15,020 --> 01:07:20,200
Let's look at the Fourier
transform of these guys.

1280
01:07:26,730 --> 01:07:39,490
The Fourier transform
of this guy-- this is k.

1281
01:07:39,490 --> 01:07:43,640
Psi tilde of k well,
that's something

1282
01:07:43,640 --> 01:07:45,840
with a definite value of k.

1283
01:07:45,840 --> 01:07:55,965
And it's Fourier transform--
this is 0-- there's k1.

1284
01:07:55,965 --> 01:08:05,690
And for this guy--
there's 0-- k2.

1285
01:08:13,770 --> 01:08:16,240
And now if we look at
the Fourier transforms

1286
01:08:16,240 --> 01:08:22,880
of these guys, see,
this way I don't

1287
01:08:22,880 --> 01:08:36,420
have to erase the superposition
principle-- and the Fourier

1288
01:08:36,420 --> 01:09:10,130
transform of this guy,
so note that there's

1289
01:09:10,130 --> 01:09:13,529
a sort of pleasing
symmetry here.

1290
01:09:13,529 --> 01:09:15,800
If your wave function
is well localized,

1291
01:09:15,800 --> 01:09:18,660
corresponding to a reasonably
well-defined position,

1292
01:09:18,660 --> 01:09:22,080
then your Fourier transform
is not well localized,

1293
01:09:22,080 --> 01:09:26,479
corresponding to not
having a definite momentum.

1294
01:09:26,479 --> 01:09:30,580
On the other hand, if you
have definite momentum,

1295
01:09:30,580 --> 01:09:32,819
your position is
not well defined,

1296
01:09:32,819 --> 01:09:40,160
but the Fourier transform
has a single peak

1297
01:09:40,160 --> 01:09:43,939
at the value of k corresponding
to the momentum of your wave

1298
01:09:43,939 --> 01:09:44,460
function.

1299
01:09:44,460 --> 01:09:46,590
Everyone cool with that?

1300
01:09:46,590 --> 01:09:48,840
So here's a question-- sorry,
there was a raised hand.

1301
01:09:48,840 --> 01:09:50,120
Yeah?

1302
01:09:50,120 --> 01:09:54,245
AUDIENCE: Are we going
to learn in this class

1303
01:09:54,245 --> 01:09:56,472
how to determine the
Fourier transforms

1304
01:09:56,472 --> 01:09:58,679
of these non-stupid functions?

1305
01:09:58,679 --> 01:10:00,470
PROFESSOR: Yes, that
will be your homework.

1306
01:10:00,470 --> 01:10:02,060
On your homework is
an extensive list

1307
01:10:02,060 --> 01:10:06,210
of functions for you to
compute Fourier transforms of.

1308
01:10:06,210 --> 01:10:11,620
And that will be the job of
problem sets and recitation.

1309
01:10:11,620 --> 01:10:13,880
So Fourier series
and computing-- yeah,

1310
01:10:13,880 --> 01:10:16,970
you know what's
coming-- Fourier series

1311
01:10:16,970 --> 01:10:21,540
are assumed to have been covered
for everyone in 8.03 and 18.03

1312
01:10:21,540 --> 01:10:23,691
in some linear
combination thereof.

1313
01:10:23,691 --> 01:10:24,690
And Fourier transforms--

1314
01:10:24,690 --> 01:10:27,899
[LAUGHTER AND GROANS]

1315
01:10:27,899 --> 01:10:28,690
I couldn't help it.

1316
01:10:28,690 --> 01:10:33,390
So Fourier transforms
are a slight embiggening

1317
01:10:33,390 --> 01:10:36,090
of the space of Fourier
series, because we're not

1318
01:10:36,090 --> 01:10:37,430
looking at periodic functions.

1319
01:10:37,430 --> 01:10:39,636
AUDIENCE: So when we're
doing the Fourier transforms

1320
01:10:39,636 --> 01:10:41,552
of a wave function, we're
basically writing it

1321
01:10:41,552 --> 01:10:44,500
as a continuous set
of different waves.

1322
01:10:44,500 --> 01:10:46,050
Can we write it
as a discrete set?

1323
01:10:46,050 --> 01:10:47,230
So as a Fourier series?

1324
01:10:47,230 --> 01:10:49,130
PROFESSOR: Absolutely,
so, however,

1325
01:10:49,130 --> 01:10:51,470
what is true about
Fourier series?

1326
01:10:51,470 --> 01:10:53,190
When you use a discrete
set of momenta,

1327
01:10:53,190 --> 01:10:55,980
which are linear, which are--
It must be a periodic function,

1328
01:10:55,980 --> 01:10:56,580
exactly.

1329
01:10:56,580 --> 01:10:59,690
So here what we've
done is, we've said,

1330
01:10:59,690 --> 01:11:01,410
look, we're writing
our wave function,

1331
01:11:01,410 --> 01:11:05,350
our arbitrary wave function,
as a continuous superposition

1332
01:11:05,350 --> 01:11:08,059
of a continuous value
of possible momenta.

1333
01:11:08,059 --> 01:11:09,183
This is absolutely correct.

1334
01:11:09,183 --> 01:11:10,490
This is exactly
what we're doing.

1335
01:11:10,490 --> 01:11:11,880
However, that's
kind of annoying,

1336
01:11:11,880 --> 01:11:13,840
because maybe you
just want one momentum

1337
01:11:13,840 --> 01:11:15,257
and two momenta
and three momenta.

1338
01:11:15,257 --> 01:11:16,714
What if you want
a discrete series?

1339
01:11:16,714 --> 01:11:17,600
So discrete is fine.

1340
01:11:17,600 --> 01:11:19,196
But if you make
that discrete series

1341
01:11:19,196 --> 01:11:20,695
integer-related to
each other, which

1342
01:11:20,695 --> 01:11:22,200
is what you do with
Fourier series,

1343
01:11:22,200 --> 01:11:25,120
you force the function
f of x to be periodic.

1344
01:11:25,120 --> 01:11:26,840
And we don't want
that, in general,

1345
01:11:26,840 --> 01:11:29,100
because life isn't periodic.

1346
01:11:29,100 --> 01:11:31,290
Thank goodness, right?

1347
01:11:31,290 --> 01:11:35,710
I mean, there's like one film
in which it-- but so-- it's

1348
01:11:35,710 --> 01:11:36,720
a good movie.

1349
01:11:36,720 --> 01:11:38,274
So that's the
essential difference

1350
01:11:38,274 --> 01:11:40,190
between Fourier series
and Fourier transforms.

1351
01:11:40,190 --> 01:11:42,710
Fourier transforms
are continuous in k

1352
01:11:42,710 --> 01:11:47,500
and do not assume
periodicity of the function.

1353
01:11:47,500 --> 01:11:48,360
Other questions?

1354
01:11:48,360 --> 01:11:49,070
Yeah.

1355
01:11:49,070 --> 01:11:50,570
AUDIENCE: So
basically, the Fourier

1356
01:11:50,570 --> 01:11:54,486
transform associates
an amplitude

1357
01:11:54,486 --> 01:11:58,230
and a phase for each of
the individual momenta.

1358
01:11:58,230 --> 01:11:59,351
PROFESSOR: Precisely.

1359
01:11:59,351 --> 01:12:00,100
Precisely correct.

1360
01:12:00,100 --> 01:12:01,750
So let me say that again.

1361
01:12:01,750 --> 01:12:04,110
So the question was--
so a Fourier transform

1362
01:12:04,110 --> 01:12:06,050
effectively
associates a magnitude

1363
01:12:06,050 --> 01:12:09,890
and a phase for each
possible wave vector.

1364
01:12:09,890 --> 01:12:10,940
And that's exactly right.

1365
01:12:10,940 --> 01:12:13,317
So here there's some
amplitude and phase--

1366
01:12:13,317 --> 01:12:14,900
this is a complex
number, because this

1367
01:12:14,900 --> 01:12:17,274
is a complex function-- there's
some complex number which

1368
01:12:17,274 --> 01:12:19,800
is an amplitude and
a phase associated

1369
01:12:19,800 --> 01:12:22,830
to every possible momentum
going into the superposition.

1370
01:12:22,830 --> 01:12:24,280
That amplitude may be 0.

1371
01:12:24,280 --> 01:12:27,700
There may be no contribution
for a large number of momenta,

1372
01:12:27,700 --> 01:12:29,330
or maybe insignificantly small.

1373
01:12:29,330 --> 01:12:31,330
But it is indeed
doing precisely that.

1374
01:12:31,330 --> 01:12:34,930
It is associating an amplitude
and a phase for every plane

1375
01:12:34,930 --> 01:12:37,950
wave, with every different
value of momentum.

1376
01:12:37,950 --> 01:12:44,346
And you can compute,
before panicking,

1377
01:12:44,346 --> 01:12:45,970
precisely what that
amplitude and phase

1378
01:12:45,970 --> 01:12:49,239
is by using the inverse
Fourier transform.

1379
01:12:49,239 --> 01:12:50,280
So there's no magic here.

1380
01:12:50,280 --> 01:12:51,280
You just calculate.

1381
01:12:51,280 --> 01:12:57,450
You can use your calculator,
literally-- I hate that word.

1382
01:12:57,450 --> 01:13:00,320
OK so now, here's
a natural question.

1383
01:13:00,320 --> 01:13:04,340
So if this is the Fourier
transform of our wave function,

1384
01:13:04,340 --> 01:13:06,770
we already knew that this
wave function corresponded

1385
01:13:06,770 --> 01:13:08,710
to having a definite--
from de Broglie,

1386
01:13:08,710 --> 01:13:11,270
we know that it has
a definite momentum.

1387
01:13:11,270 --> 01:13:14,650
We also see that its Fourier
transform looks like this.

1388
01:13:14,650 --> 01:13:17,620
So that leads to a
reasonable guess.

1389
01:13:17,620 --> 01:13:20,260
What do you think the
probability distribution P of k

1390
01:13:20,260 --> 01:13:23,930
is-- the probability
density to find the momentum

1391
01:13:23,930 --> 01:13:26,620
to have wave vector h-bar k?

1392
01:13:26,620 --> 01:13:27,517
AUDIENCE: [INAUDIBLE]

1393
01:13:27,517 --> 01:13:29,600
PROFESSOR: Yeah, that's a
pretty reasonable guess.

1394
01:13:29,600 --> 01:13:31,390
So we're totally
pulling this out

1395
01:13:31,390 --> 01:13:34,460
of the dark-- psi
of k norm squared.

1396
01:13:34,460 --> 01:13:35,920
OK, well let's
see if that works.

1397
01:13:35,920 --> 01:13:37,450
So psy of k norm
squared for this

1398
01:13:37,450 --> 01:13:42,715
is going to give us a nice,
well localized function.

1399
01:13:42,715 --> 01:13:44,090
And so that makes
a lot of sense.

1400
01:13:44,090 --> 01:13:46,230
That's exactly what
we expected, right?

1401
01:13:46,230 --> 01:13:48,950
Definite value of P with
very small uncertainty.

1402
01:13:48,950 --> 01:13:50,810
Similarly here.

1403
01:13:50,810 --> 01:13:54,950
Definite value of P, with
a very small uncertainty.

1404
01:13:54,950 --> 01:13:57,340
Rock on.

1405
01:13:57,340 --> 01:13:59,860
However, let's look at this guy.

1406
01:13:59,860 --> 01:14:02,510
What is the expected value
of P if this is the Fourier

1407
01:14:02,510 --> 01:14:05,050
transform?

1408
01:14:05,050 --> 01:14:08,610
Well remember, we have to take
the norm squared, and psi of k

1409
01:14:08,610 --> 01:14:12,050
was e to the i k x1--
the Fourier transform.

1410
01:14:12,050 --> 01:14:15,130
You will do much
practice on taking

1411
01:14:15,130 --> 01:14:17,360
Fourier transforms
on the problem set.

1412
01:14:17,360 --> 01:14:19,970
Where did my eraser go?

1413
01:14:19,970 --> 01:14:20,795
There it is.

1414
01:14:24,260 --> 01:14:27,340
Farewell, principle one.

1415
01:14:27,340 --> 01:14:30,300
So what does norm squared
of psi tilde look like?

1416
01:14:30,300 --> 01:14:31,969
Well, just like before,
the norm squared

1417
01:14:31,969 --> 01:14:34,510
is constant, because the norm
squared of a phase is constant.

1418
01:14:34,510 --> 01:14:36,780
And again, the
norm squared-- this

1419
01:14:36,780 --> 01:14:42,180
is psi tilde of k norm
squared-- we believe,

1420
01:14:42,180 --> 01:14:45,410
we're conjecturing
this is P of k.

1421
01:14:45,410 --> 01:14:47,930
You will prove this relation
on your problem set.

1422
01:14:47,930 --> 01:14:51,260
You'll prove that it follow
from what we said before.

1423
01:14:51,260 --> 01:14:53,610
And similarly, this is
constant-- e to the i k x2.

1424
01:14:58,320 --> 01:15:02,160
So now we have no
knowledge of the momenta.

1425
01:15:02,160 --> 01:15:04,320
So that also fits.

1426
01:15:04,320 --> 01:15:06,534
The momenta is, we have no idea.

1427
01:15:06,534 --> 01:15:07,575
And uncertainty is large.

1428
01:15:07,575 --> 01:15:10,330
And the momenta is,
we have no idea.

1429
01:15:10,330 --> 01:15:11,760
And the uncertainty is large.

1430
01:15:11,760 --> 01:15:13,310
So in all these
cases, we see that we

1431
01:15:13,310 --> 01:15:15,600
satisfy quite nicely the
uncertainty relation--

1432
01:15:15,600 --> 01:15:18,610
small position momentum,
large momentum uncertainty.

1433
01:15:18,610 --> 01:15:21,210
Large position
uncertainty, we're

1434
01:15:21,210 --> 01:15:24,410
allowed to have small
momentum uncertainty.

1435
01:15:24,410 --> 01:15:26,724
And here, it's a little
more complicated.

1436
01:15:26,724 --> 01:15:28,640
We have a little bit of
knowledge of position,

1437
01:15:28,640 --> 01:15:30,650
and we have a little bit of
knowledge of the momenta.

1438
01:15:30,650 --> 01:15:31,900
We have a little bit of
knowledge of position,

1439
01:15:31,900 --> 01:15:34,430
and we have a little bit
of knowledge of momenta.

1440
01:15:34,430 --> 01:15:36,490
So we'll walk through
examples with superposition

1441
01:15:36,490 --> 01:15:37,900
like this on the problem set.

1442
01:15:40,732 --> 01:15:42,190
Last questions
before we get going?

1443
01:15:45,020 --> 01:15:48,170
OK so I have two things
to do before we're done.

1444
01:15:48,170 --> 01:15:51,280
The first is, after lecture
ends, I have clickers.

1445
01:15:51,280 --> 01:15:53,120
And anyone who wants
to borrow clickers,

1446
01:15:53,120 --> 01:15:55,340
you're welcome to come
down and pick them up

1447
01:15:55,340 --> 01:15:58,500
on a first come
first served basis.

1448
01:15:58,500 --> 01:16:01,190
I will start using the
clickers in the next lecture.

1449
01:16:01,190 --> 01:16:03,230
So if you don't already
have one, get one now.

1450
01:16:03,230 --> 01:16:05,240
But the second thing is--
don't get started yet.

1451
01:16:05,240 --> 01:16:07,260
I have a demo to do.

1452
01:16:07,260 --> 01:16:10,749
And last time I told
you-- this is awesome.

1453
01:16:10,749 --> 01:16:12,540
It's like I'm an
experimentalist for a day.

1454
01:16:12,540 --> 01:16:15,270
Last time I told you that one of
the experimental facts of lice

1455
01:16:15,270 --> 01:16:16,400
is-- of lice.

1456
01:16:16,400 --> 01:16:18,899
One of the experimental
facts of lice.

1457
01:16:18,899 --> 01:16:20,440
One of the experimental
facts of life

1458
01:16:20,440 --> 01:16:25,480
is that there is
uncertainty in the world

1459
01:16:25,480 --> 01:16:26,730
and that there is probability.

1460
01:16:26,730 --> 01:16:30,700
There are unlikely
events that happen

1461
01:16:30,700 --> 01:16:32,660
with some probability,
some finite probability.

1462
01:16:32,660 --> 01:16:35,510
And a good example of the
randomness of the real world

1463
01:16:35,510 --> 01:16:37,730
involves radiation.

1464
01:16:37,730 --> 01:16:41,440
So hopefully you can hear this.

1465
01:16:41,440 --> 01:16:44,590
Apparently, I'm not
very radioactive.

1466
01:16:44,590 --> 01:16:46,840
You'd be surprised at the
things that are radioactive.

1467
01:16:50,140 --> 01:16:51,450
Ah, got a little tick.

1468
01:16:51,450 --> 01:16:51,950
Shh.

1469
01:16:55,780 --> 01:17:00,640
This is a plate sold at
an Amish county fair.

1470
01:17:00,640 --> 01:17:05,777
It's called vaseline ware
and it's made of local clays.

1471
01:17:05,777 --> 01:17:09,120
[GEIGER COUNTER CLICKS]

1472
01:17:09,120 --> 01:17:10,230
It's got uranium in it.

1473
01:17:13,620 --> 01:17:17,040
But

1474
01:17:17,040 --> 01:17:20,600
I want to emphasize-- exactly
when something goes click,

1475
01:17:20,600 --> 01:17:21,720
it sounds pretty random.

1476
01:17:21,720 --> 01:17:23,840
And it's actually a better
random number generator

1477
01:17:23,840 --> 01:17:29,790
than anything you'll find in
Mathematica or C. In fact,

1478
01:17:29,790 --> 01:17:32,520
for some purposes, the decay
of radioactive isotopes

1479
01:17:32,520 --> 01:17:35,642
is used as the perfect
random number generator.

1480
01:17:35,642 --> 01:17:37,850
Because it really is totally
random, as far as anyone

1481
01:17:37,850 --> 01:17:38,605
can tell.

1482
01:17:38,605 --> 01:17:39,670
But here's my favorite.

1483
01:17:42,550 --> 01:17:43,970
People used to eat off these.

1484
01:17:43,970 --> 01:17:46,870
[MUCH LOUDER, DENSER CLICKS]

1485
01:17:46,870 --> 01:17:48,750
See you next time.