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PROFESSOR: So
today, before we get
00:00:23.650 --> 00:00:27.420
into the meat of
today's lecture,
00:00:27.420 --> 00:00:29.880
Matt has very kindly--
Professor Evans
00:00:29.880 --> 00:00:36.090
has very kindly agreed
to do an experiment.
00:00:36.090 --> 00:00:38.240
Yeah, so for those
of you all who
00:00:38.240 --> 00:00:40.100
are in recitations
both he and Barton
00:00:40.100 --> 00:00:42.630
talked about polarization
in recitation last week.
00:00:42.630 --> 00:00:44.550
And Matt will pick
it up from there.
00:00:44.550 --> 00:00:47.350
MATTHEW EVANS: So back
to the ancient past--
00:00:47.350 --> 00:00:48.870
this was a week ago.
00:00:48.870 --> 00:00:50.410
We had our
hyper-intelligent monkeys
00:00:50.410 --> 00:00:51.451
that were sorting things.
00:00:51.451 --> 00:00:53.370
It all seemed very theoretical.
00:00:53.370 --> 00:00:56.820
And in recitation, I said
things about polarizers.
00:00:56.820 --> 00:00:58.470
And I said, look, if
we use polarizers,
00:00:58.470 --> 00:01:00.990
we can do exactly the same
thing as these monkeys.
00:01:00.990 --> 00:01:05.300
We just need to set up a
little polarization experiment
00:01:05.300 --> 00:01:06.840
and the results are identical.
00:01:06.840 --> 00:01:08.680
You can use the one
figure out the other.
00:01:08.680 --> 00:01:12.015
But I didn't have this or
a nice polarizer ready then
00:01:12.015 --> 00:01:14.580
to give a demo, so here we go.
00:01:14.580 --> 00:01:16.900
What I'm going to show
you is that, if we start
00:01:16.900 --> 00:01:20.860
with something polarized here
with all white-- and right
00:01:20.860 --> 00:01:23.750
now I have all vertical
polarization here--
00:01:23.750 --> 00:01:25.650
and if I just put
on this other box
00:01:25.650 --> 00:01:28.230
there, which is going to be
another polarizer, if I put it
00:01:28.230 --> 00:01:31.060
the same way, this is all
of our white electrons
00:01:31.060 --> 00:01:32.130
coming through all white.
00:01:32.130 --> 00:01:32.629
See?
00:01:32.629 --> 00:01:34.770
It doesn't really do much.
00:01:34.770 --> 00:01:38.110
And if I look at the
black output over here
00:01:38.110 --> 00:01:40.050
of the second Keller
sorting box, that's
00:01:40.050 --> 00:01:42.620
the same as turning my
polarizer 90 degrees,
00:01:42.620 --> 00:01:44.550
so nothing comes out black.
00:01:44.550 --> 00:01:46.715
So if we remove this
guy from the middle,
00:01:46.715 --> 00:01:48.400
you have just exactly
what you'd expect.
00:01:48.400 --> 00:01:51.390
You sort here, you have white,
and you get all white out.
00:01:51.390 --> 00:01:52.110
Great.
00:01:52.110 --> 00:01:53.401
Everyone thought that was easy.
00:01:53.401 --> 00:01:54.860
We all had that figured out.
00:01:54.860 --> 00:01:56.690
This box got thrown
in the center here
00:01:56.690 --> 00:02:00.720
and it became sort of confusing,
because you thought, well,
00:02:00.720 --> 00:02:02.220
they were white--
I'm going to throw
00:02:02.220 --> 00:02:08.070
my box in the middle here--
that's this guy at 45 degrees.
00:02:08.070 --> 00:02:10.570
And then if I throw this guy
on the end again, the idea was,
00:02:10.570 --> 00:02:14.166
well, they were all white
here, so maybe this guy
00:02:14.166 --> 00:02:16.040
identified the soft ones
from the white ones.
00:02:16.040 --> 00:02:17.331
And now we have white and soft.
00:02:17.331 --> 00:02:19.410
And it should still
all be white, right?
00:02:19.410 --> 00:02:21.610
So I put this guy on up here.
00:02:21.610 --> 00:02:25.270
They should all come out
but they sort of don't.
00:02:25.270 --> 00:02:26.840
And if you say,
well, are they black?
00:02:26.840 --> 00:02:28.631
Well, no, they're not
really black, either.
00:02:28.631 --> 00:02:31.360
They're some sort of strange
combination of the two.
00:02:31.360 --> 00:02:34.780
All right, so that's this
experiment done in polarizers.
00:02:34.780 --> 00:02:37.170
But let me just play the
polarizer trick a little bit,
00:02:37.170 --> 00:02:39.190
because it's fun.
00:02:39.190 --> 00:02:42.592
So this is if I say,
vertical polarization
00:02:42.592 --> 00:02:44.300
and how many of them
come out horizontal?
00:02:44.300 --> 00:02:46.040
So here I'm saying, white,
and how many of them
00:02:46.040 --> 00:02:46.873
will come out black?
00:02:46.873 --> 00:02:47.965
That's the analogy.
00:02:47.965 --> 00:02:50.000
The answer is none of them.
00:02:50.000 --> 00:02:51.950
And strangely, if I
take this thing, which
00:02:51.950 --> 00:02:55.190
seems to just attenuate--
this is our middle box here--
00:02:55.190 --> 00:02:56.910
and I just stuff
it in between them,
00:02:56.910 --> 00:03:01.170
I can get something to come
out even though I still
00:03:01.170 --> 00:03:04.080
have crossed
polarizers on the side.
00:03:04.080 --> 00:03:06.080
So you can see the middle
region is now brighter
00:03:06.080 --> 00:03:07.913
and you can still see
the dark corners there
00:03:07.913 --> 00:03:09.900
of the crossed polarizers.
00:03:09.900 --> 00:03:13.290
And as I turn this guy around,
I can make that better or worse.
00:03:13.290 --> 00:03:15.566
The maximum is
somewhere right there,
00:03:15.566 --> 00:03:17.270
and then it goes off again.
00:03:17.270 --> 00:03:21.150
So this is a way
of understanding
00:03:21.150 --> 00:03:25.020
our electron-sorting,
hyper-intelligent monkeys
00:03:25.020 --> 00:03:26.170
in terms of polarizations.
00:03:26.170 --> 00:03:28.085
And here it's just
a vector projected
00:03:28.085 --> 00:03:30.460
on another vector projected
on another vector-- something
00:03:30.460 --> 00:03:32.620
everybody knows how to do.
00:03:32.620 --> 00:03:35.310
So here's the
polarization analogy
00:03:35.310 --> 00:03:36.760
of the Stern-Gerlach experiment.
00:03:45.130 --> 00:03:46.080
PROFESSOR: Awesome.
00:03:46.080 --> 00:03:51.130
So the polarization analogy
for interference effects
00:03:51.130 --> 00:03:54.730
in quantum mechanics
is a canonical one
00:03:54.730 --> 00:03:58.810
in the texts of
quantum mechanics.
00:03:58.810 --> 00:04:01.096
So you'll find lots of
books talking about this.
00:04:01.096 --> 00:04:02.470
It's a very useful
analogy, and I
00:04:02.470 --> 00:04:03.910
encourage you to
read more about it.
00:04:03.910 --> 00:04:05.576
We won't talk about
it a whole lot more,
00:04:05.576 --> 00:04:07.874
but it's a useful one.
00:04:07.874 --> 00:04:09.290
All right, before
I get going, any
00:04:09.290 --> 00:04:12.587
questions from last lecture?
00:04:12.587 --> 00:04:14.420
Last lecture was pretty
much self-contained.
00:04:14.420 --> 00:04:17.530
It was experimental results.
00:04:17.530 --> 00:04:18.589
No, nothing?
00:04:18.589 --> 00:04:21.380
All right.
00:04:21.380 --> 00:04:24.800
The one thing that I
want to add to the last
00:04:24.800 --> 00:04:26.700
lecture-- one last
experimental observation.
00:04:26.700 --> 00:04:29.020
I glossed over
something that's kind
00:04:29.020 --> 00:04:31.040
of important, which
is the following.
00:04:31.040 --> 00:04:33.760
So we started off
by saying, look,
00:04:33.760 --> 00:04:37.150
we know that if I
have a ray of light,
00:04:37.150 --> 00:04:39.410
it's an electromagnetic
wave, and it
00:04:39.410 --> 00:04:40.535
has some wavelength lambda.
00:04:44.500 --> 00:04:49.760
And yet the photoelectric
effect tells us
00:04:49.760 --> 00:04:54.900
that, in addition to having the
wavelength lambda, the energy--
00:04:54.900 --> 00:04:57.900
it has a frequency, as
well, a frequency in time.
00:04:57.900 --> 00:04:59.690
And the photoelectric
effect suggested
00:04:59.690 --> 00:05:07.650
that the energy is
proportional to the frequency.
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
00:05:17.230 --> 00:05:21.310
and omega is equal to 2 pi nu.
00:05:21.310 --> 00:05:23.810
So this is just the angular
frequency, rather than
00:05:23.810 --> 00:05:27.010
the number-per-time frequency.
00:05:27.010 --> 00:05:29.600
And h bar is the
reduced Planck constant.
00:05:29.600 --> 00:05:31.910
So I'll typically write h
bar omega rather than h nu,
00:05:31.910 --> 00:05:35.432
because these two pi's will
just cause us endless pain if we
00:05:35.432 --> 00:05:37.890
don't use the bar.
00:05:37.890 --> 00:05:39.520
Anyway, so to an
electromagnetic wave,
00:05:39.520 --> 00:05:40.840
we have a wavelength
and a frequency
00:05:40.840 --> 00:05:42.362
and the photoelectric
effect led us
00:05:42.362 --> 00:05:44.130
to predict that the
energy is linearly
00:05:44.130 --> 00:05:46.713
proportional to the frequency,
with the linear proportionality
00:05:46.713 --> 00:05:50.210
coefficient h bar-- Planck
constant-- and the momentum
00:05:50.210 --> 00:05:55.800
is equal to h upon
lambda, also known
00:05:55.800 --> 00:06:00.280
as-- I'm going to write
this as h bar k, which
00:06:00.280 --> 00:06:04.740
is equal to h upon
lambda, where here, again,
00:06:04.740 --> 00:06:07.750
h bar is h upon 2 pi.
00:06:07.750 --> 00:06:11.020
And so k is equal
to 2 pi upon lambda.
00:06:16.147 --> 00:06:17.730
So k is called the
wave number and you
00:06:17.730 --> 00:06:20.920
should have seen this in 8.03.
00:06:20.920 --> 00:06:23.592
So these are our basic
relations for light.
00:06:23.592 --> 00:06:25.550
We know that light, as
an electromagnetic wave,
00:06:25.550 --> 00:06:26.970
has a frequency
and a wavelength--
00:06:26.970 --> 00:06:30.170
or a wave number, an
inverse wavelength.
00:06:30.170 --> 00:06:32.550
And the claim of the
photoelectric effect
00:06:32.550 --> 00:06:35.030
is that the energy and
the momenta of that light
00:06:35.030 --> 00:06:38.550
are thus quantized, that
light comes in chunks.
00:06:38.550 --> 00:06:40.270
So it has a wave-like
aspect and it also
00:06:40.270 --> 00:06:43.420
has properties that are more
familiar from particles.
00:06:43.420 --> 00:06:47.195
Now, early on shortly after
Einstein proposed this,
00:06:47.195 --> 00:06:51.800
a young French physicist named
de Broglie said, well, look,
00:06:51.800 --> 00:06:53.400
OK, this is true of light.
00:06:53.400 --> 00:06:59.050
Light has both wave-like and
particle-like properties.
00:06:59.050 --> 00:07:00.002
Why is it just light?
00:07:00.002 --> 00:07:01.710
The world would be
much more parsimonious
00:07:01.710 --> 00:07:03.959
if this relation were true
not just of light, but also
00:07:03.959 --> 00:07:04.840
of all particles.
00:07:04.840 --> 00:07:09.370
I am thus conjecturing,
with no evidence whatsoever,
00:07:09.370 --> 00:07:11.055
that, in fact,
this relation holds
00:07:11.055 --> 00:07:12.680
not just for light,
but for any object.
00:07:12.680 --> 00:07:17.490
Any object with momentum p has
associated to it a wavelength
00:07:17.490 --> 00:07:21.190
or a wave number,
which is p upon h bar.
00:07:21.190 --> 00:07:25.300
Every object that has energy
E has associated with it
00:07:25.300 --> 00:07:29.990
a wave with frequency omega.
00:07:29.990 --> 00:07:34.510
To those electrons that we send
through the Davisson-Germer
00:07:34.510 --> 00:07:40.050
experiment apparatus, which are
sent in with definite energy,
00:07:40.050 --> 00:07:42.390
there must be a frequency
associated with it, omega
00:07:42.390 --> 00:07:44.300
and a wavelength lambda
associated with it.
00:07:44.300 --> 00:07:46.440
And what we saw from the
Davisson-Germer experiment
00:07:46.440 --> 00:07:49.380
was experimental confirmation
of that prediction--
00:07:49.380 --> 00:07:53.200
that electrons have both
particulate and wave-like
00:07:53.200 --> 00:07:57.010
features simultaneously.
00:07:57.010 --> 00:08:00.770
So these relations are called
the de Broglie relations or "de
00:08:00.770 --> 00:08:05.900
BROG-lee"-- I leave it up to
you to decide how to pronounce
00:08:05.900 --> 00:08:08.317
that.
00:08:08.317 --> 00:08:10.900
And those relations are going
to play an important role for us
00:08:10.900 --> 00:08:11.941
in the next few lectures.
00:08:11.941 --> 00:08:15.680
I just wanted to give them
a name and a little context.
00:08:15.680 --> 00:08:19.240
This is a good example of
parsimony and elegance--
00:08:19.240 --> 00:08:21.810
the theoretical
elegance leading you
00:08:21.810 --> 00:08:24.210
to an idea that turns out
to be true of the world.
00:08:24.210 --> 00:08:28.210
Now, that's a dangerous
strategy for finding truth.
00:08:28.210 --> 00:08:31.780
Boy, wouldn't it be nice if--?
00:08:31.780 --> 00:08:34.243
Wouldn't it be nice if we
didn't have to pay taxes
00:08:34.243 --> 00:08:35.284
but we also had Medicare?
00:08:38.679 --> 00:08:41.883
So it's not a terribly
useful guide all the time,
00:08:41.883 --> 00:08:43.424
but sometimes it
really does lead you
00:08:43.424 --> 00:08:44.382
in the right direction.
00:08:44.382 --> 00:08:47.060
And this is a great example
of physical intuition,
00:08:47.060 --> 00:08:49.080
wildly divorced from
experiment, pushing you
00:08:49.080 --> 00:08:50.489
in the right direction.
00:08:50.489 --> 00:08:52.780
I'm making it sound a little
more shocking than-- well,
00:08:52.780 --> 00:08:53.446
it was shocking.
00:08:53.446 --> 00:08:54.580
It was just shocking.
00:08:54.580 --> 00:09:01.210
OK, so with that said,
let me introduce the moves
00:09:01.210 --> 00:09:02.330
for the next few lectures.
00:09:02.330 --> 00:09:04.670
For the next several lectures,
here's what we're going to do.
00:09:04.670 --> 00:09:06.690
I am not going to give you
experimental motivation.
00:09:06.690 --> 00:09:08.314
I've given you
experimental motivation.
00:09:08.314 --> 00:09:12.382
I'm going to give you a set
of rules, a set of postulates.
00:09:12.382 --> 00:09:14.590
These are going to be the
rules of quantum mechanics.
00:09:14.590 --> 00:09:17.480
And what quantum
mechanics is for us
00:09:17.480 --> 00:09:20.250
is a set of rules to allow
us to make predictions
00:09:20.250 --> 00:09:22.200
about the world.
00:09:22.200 --> 00:09:26.460
And these rules will be awesome
if their predictions are good.
00:09:26.460 --> 00:09:29.690
And if their predictions are
bad, these rules will suck.
00:09:29.690 --> 00:09:31.894
We will avoid bad rules
to the degree possible.
00:09:31.894 --> 00:09:34.310
I'm going to give you what
we've learned over the past 100
00:09:34.310 --> 00:09:38.750
years-- wow-- of developing
quantum mechanics.
00:09:38.750 --> 00:09:40.390
That is amazing.
00:09:40.390 --> 00:09:40.890
Wow.
00:09:40.890 --> 00:09:42.264
OK, yeah, over
the past 100 years
00:09:42.264 --> 00:09:44.142
of developing quantum mechanics.
00:09:44.142 --> 00:09:46.600
And I'm going to give them to
you as a series of postulates
00:09:46.600 --> 00:09:49.070
and then we're going to work
through the consequences,
00:09:49.070 --> 00:09:50.680
and then we're going to spend
the rest of the semester
00:09:50.680 --> 00:09:53.160
studying examples to develop
an understanding for what
00:09:53.160 --> 00:09:56.360
the rules of quantum
mechanics are giving you.
00:09:56.360 --> 00:09:59.390
So we're just going to
scrap classical mechanics
00:09:59.390 --> 00:10:00.580
and start over from scratch.
00:10:00.580 --> 00:10:01.330
So let me do that.
00:10:04.720 --> 00:10:11.240
And to begin, let me start with
the definition of a system.
00:10:11.240 --> 00:10:13.120
And to understand
that definition,
00:10:13.120 --> 00:10:15.990
I want to start with classical
mechanics as a guide.
00:10:21.000 --> 00:10:23.560
So in classical mechanics--
let's think about the easiest
00:10:23.560 --> 00:10:25.930
classical system you can--
just a single particle
00:10:25.930 --> 00:10:26.930
sitting somewhere.
00:10:26.930 --> 00:10:29.530
In classical mechanics
of a single particle,
00:10:29.530 --> 00:10:32.920
how do you specify the
configuration, or the state--
00:10:32.920 --> 00:10:34.730
just different words
for the same thing--
00:10:34.730 --> 00:10:36.400
how do you specify
the configuration
00:10:36.400 --> 00:10:39.932
or state of the system?
00:10:39.932 --> 00:10:41.390
AUDIENCE: By position
and momentum.
00:10:41.390 --> 00:10:43.640
PROFESSOR: Specify the
position and momentum, exactly.
00:10:43.640 --> 00:10:48.579
So in classical mechanics, if
you want to completely specify
00:10:48.579 --> 00:10:50.620
the configuration of a
system, all you have to do
00:10:50.620 --> 00:10:55.160
is give me x and
p for my particle.
00:10:55.160 --> 00:10:57.690
And if you tell me
this, I know everything.
00:10:57.690 --> 00:11:00.040
If you know these numbers,
you know everything.
00:11:00.040 --> 00:11:01.630
In particular, what
I mean by saying
00:11:01.630 --> 00:11:03.060
you know everything is that,
if there's anything else
00:11:03.060 --> 00:11:05.530
you want to measure--
the energy, for example.
00:11:05.530 --> 00:11:08.230
The energy is just some function
of the position and momentum.
00:11:08.230 --> 00:11:09.760
And if you know the
position and momentum,
00:11:09.760 --> 00:11:11.640
you can unambiguously
calculate the energy.
00:11:11.640 --> 00:11:14.184
Similarly, the angular
momentum, which is a vector,
00:11:14.184 --> 00:11:15.600
you can calculate
it if you know x
00:11:15.600 --> 00:11:20.250
and p-- which is just r cross p.
00:11:20.250 --> 00:11:22.470
So this gives you complete
knowledge of the system.
00:11:29.030 --> 00:11:32.600
There's nothing more to
know if you know that data.
00:11:32.600 --> 00:11:34.340
Now, there are certainly
still questions
00:11:34.340 --> 00:11:38.800
that you can't answer
given knowledge of x and p.
00:11:38.800 --> 00:11:42.090
For example, are there
14 invisible monkeys
00:11:42.090 --> 00:11:44.040
standing behind me?
00:11:44.040 --> 00:11:44.740
I'm here.
00:11:44.740 --> 00:11:45.890
I'm not moving.
00:11:45.890 --> 00:11:48.064
Are there 14 invisible
monkeys standing behind me?
00:11:48.064 --> 00:11:48.980
You can't answer that.
00:11:48.980 --> 00:11:51.200
It's a stupid question, right?
00:11:51.200 --> 00:11:52.830
OK, let me give you
another example.
00:11:52.830 --> 00:11:55.140
The electron is x
and p, some position.
00:11:55.140 --> 00:11:57.860
Is it happy?
00:11:57.860 --> 00:12:00.280
Right, so there are still
questions you can't answer.
00:12:00.280 --> 00:12:02.420
The point is, complete
knowledge of the system
00:12:02.420 --> 00:12:04.810
to answer any physically
observable question--
00:12:04.810 --> 00:12:06.820
any question that could
be meaningfully turned
00:12:06.820 --> 00:12:10.480
into an experiment,
the answer is
00:12:10.480 --> 00:12:12.420
contained in knowing
the state of the system.
00:12:17.520 --> 00:12:20.057
But this can't possibly be
true in quantum mechanics,
00:12:20.057 --> 00:12:21.640
because, as you saw
in the problem set
00:12:21.640 --> 00:12:23.710
and as we've
discussed previously,
00:12:23.710 --> 00:12:25.460
there's an uncertainty
relation which
00:12:25.460 --> 00:12:27.850
says that your knowledge--
or your uncertainty, rather,
00:12:27.850 --> 00:12:29.550
in the position of
a particle, quantum
00:12:29.550 --> 00:12:33.434
mechanically--
I'm not even going
00:12:33.434 --> 00:12:34.600
to say quantum mechanically.
00:12:34.600 --> 00:12:37.820
I'm just going to
say the real world.
00:12:37.820 --> 00:12:40.960
So in the real world,
our uncertainty
00:12:40.960 --> 00:12:43.610
in the position of
our point-like object
00:12:43.610 --> 00:12:45.210
and our uncertainty
in the momentum
00:12:45.210 --> 00:12:48.500
is always greater than or
roughly equal to something
00:12:48.500 --> 00:12:51.000
that's proportional
to Planck's constant.
00:12:51.000 --> 00:12:53.920
You can't be arbitrarily
confident of the position
00:12:53.920 --> 00:12:55.532
and of the momentum
simultaneously.
00:12:55.532 --> 00:12:57.240
You worked through a
good example of this
00:12:57.240 --> 00:12:58.187
on the problem set.
00:12:58.187 --> 00:12:59.770
We saw this in the
two-slit experiment
00:12:59.770 --> 00:13:01.470
and the interference
of electrons.
00:13:01.470 --> 00:13:03.595
This is something we're
going to have to deal with.
00:13:03.595 --> 00:13:05.890
So as a consequence,
you can't possibly
00:13:05.890 --> 00:13:08.160
specify the position
and the momentum
00:13:08.160 --> 00:13:09.390
with confidence of a system.
00:13:09.390 --> 00:13:11.830
You can't do it.
00:13:11.830 --> 00:13:12.740
This was a myth.
00:13:12.740 --> 00:13:16.060
It was a good approximation--
turned out to be false.
00:13:16.060 --> 00:13:20.430
So the first thing
we need is to specify
00:13:20.430 --> 00:13:22.300
the state, the
configuration of a system.
00:13:22.300 --> 00:13:25.910
So what specifies the
configuration of a system?
00:13:25.910 --> 00:13:29.870
And so this brings us
to the first postulate.
00:13:33.390 --> 00:13:37.900
The configuration, or
state, of a system--
00:13:37.900 --> 00:13:39.470
and here again,
just for simplicity,
00:13:39.470 --> 00:13:48.880
I'm going to talk about a single
object-- of a quantum object
00:13:48.880 --> 00:14:01.550
is completely specified by
a single function, a wave
00:14:01.550 --> 00:14:06.920
function, which I
will denote generally
00:14:06.920 --> 00:14:12.870
psi of x, which is
a complex function.
00:14:23.690 --> 00:14:26.440
The state of the quantum
object is completely
00:14:26.440 --> 00:14:30.640
specified once you know the wave
function of the system, which
00:14:30.640 --> 00:14:34.200
is a function of position.
00:14:34.200 --> 00:14:37.860
Let me emphasize that this is
a first pass at the postulates.
00:14:37.860 --> 00:14:40.760
What we're going to do is go
through the basic postulates
00:14:40.760 --> 00:14:43.640
of quantum mechanics, then we'll
go through them again and give
00:14:43.640 --> 00:14:44.990
them a little more generality.
00:14:44.990 --> 00:14:46.970
And then we'll go through
them again and give them
00:14:46.970 --> 00:14:47.640
full generality.
00:14:47.640 --> 00:14:51.401
That last pass is 8.05.
00:14:51.401 --> 00:14:52.775
So let me give
you some examples.
00:14:56.060 --> 00:14:59.634
Let me just draw some
characteristic wave functions.
00:14:59.634 --> 00:15:01.800
And these are going to turn
out to be useful for us.
00:15:08.550 --> 00:15:11.750
So for example, consider
the following function.
00:15:14.600 --> 00:15:17.970
So here is 0 and we're
plotting as a function of x.
00:15:17.970 --> 00:15:21.710
And then plotting the
real part of psi of x.
00:15:25.690 --> 00:15:29.417
So first consider a very
narrowly supported function.
00:15:29.417 --> 00:15:31.000
It's basically 0
everywhere, except it
00:15:31.000 --> 00:15:33.145
has some particular spot
at what I'll call x1.
00:15:37.040 --> 00:15:39.500
Here's another
wave function-- 0.
00:15:39.500 --> 00:15:50.825
It's basically 0 except for
some special spot at x2.
00:15:50.825 --> 00:15:52.700
And again, I'm plotting
the real part of psi.
00:15:52.700 --> 00:15:54.500
And I'm plotting
the real part of psi
00:15:54.500 --> 00:15:57.720
because A, psi is a complex
function-- at every point it
00:15:57.720 --> 00:15:59.430
specifies a complex number.
00:15:59.430 --> 00:16:01.854
And B, I can't draw
complex numbers.
00:16:01.854 --> 00:16:03.270
So to keep my head
from exploding,
00:16:03.270 --> 00:16:05.635
I'm just plotting the real
part of the wave function.
00:16:05.635 --> 00:16:09.030
But you should never forget that
the wave function is complex.
00:16:12.687 --> 00:16:14.270
So for the moment,
I'm going to assume
00:16:14.270 --> 00:16:15.620
that the imaginary part is 0.
00:16:15.620 --> 00:16:17.207
I'm just going to
draw the real parts.
00:16:17.207 --> 00:16:18.790
So let me draw a
couple more examples.
00:16:18.790 --> 00:16:20.730
What else could be a
good wave function?
00:16:20.730 --> 00:16:22.260
Well, those are fine.
00:16:22.260 --> 00:16:24.804
What about-- again, we want
a function of x and I'm
00:16:24.804 --> 00:16:25.970
going to draw the real part.
00:16:33.850 --> 00:16:34.955
And another one.
00:16:34.955 --> 00:16:37.205
So this is going to be a
perfectly good wave function.
00:16:45.900 --> 00:16:48.980
And let me draw two more.
00:16:48.980 --> 00:16:53.080
So what else could be a
reasonable wave function?
00:16:53.080 --> 00:17:15.525
Well-- this is harder
than you'd think.
00:17:18.079 --> 00:17:18.579
Oh, God.
00:17:21.990 --> 00:17:27.430
OK, so that could be the
wave function, I don't know.
00:17:33.940 --> 00:17:35.260
That is actually my signature.
00:17:38.890 --> 00:17:41.900
My wife calls it a
little [INAUDIBLE].
00:17:41.900 --> 00:17:45.850
OK, so here's the deal.
00:17:45.850 --> 00:17:48.350
Psi is a complex function.
00:17:48.350 --> 00:17:51.820
Psi also needs to not
be a stupid function.
00:17:51.820 --> 00:17:54.280
OK so you have to ask me--
look, could it be any function?
00:17:54.280 --> 00:17:55.560
Any arbitrary function?
00:17:55.560 --> 00:17:57.060
So this is going
to be a job for us.
00:17:57.060 --> 00:17:58.518
We're going to
define what it means
00:17:58.518 --> 00:17:59.910
to be not-stupid function.
00:17:59.910 --> 00:18:01.909
Well, this is a completely
reasonable function--
00:18:01.909 --> 00:18:02.620
it's fine.
00:18:02.620 --> 00:18:03.600
This is a reasonable function.
00:18:03.600 --> 00:18:04.766
Another reasonable function.
00:18:04.766 --> 00:18:05.350
Reasonable.
00:18:05.350 --> 00:18:09.320
That's a little weird,
but it's not horrible.
00:18:09.320 --> 00:18:11.390
That's stupid.
00:18:11.390 --> 00:18:13.060
So we're going to
have to come up
00:18:13.060 --> 00:18:16.350
with a good definition
of what not stupid means.
00:18:16.350 --> 00:18:18.280
So fine, these
are all functions.
00:18:18.280 --> 00:18:20.780
One of them is multivalued and
that looks a little worrying,
00:18:20.780 --> 00:18:22.330
but they're all functions.
00:18:22.330 --> 00:18:23.350
So here's the problem.
00:18:23.350 --> 00:18:24.260
What does it mean?
00:18:29.870 --> 00:18:38.170
So postulate 2-- The
meaning of the wave function
00:18:38.170 --> 00:18:43.190
is that the probability
that upon measurement
00:18:43.190 --> 00:18:45.690
the object is found
at the position x
00:18:45.690 --> 00:18:48.375
is equal to the norm
squared of psi of x.
00:18:51.599 --> 00:18:53.890
If you know the system is
ascribed to the wave function
00:18:53.890 --> 00:18:55.430
psi, and you want
to look at point x,
00:18:55.430 --> 00:18:57.304
you want to know with
what probability will I
00:18:57.304 --> 00:19:00.680
find the particle there,
the answer is psi squared.
00:19:00.680 --> 00:19:03.255
Notice that this is a complex
number, but absolute value
00:19:03.255 --> 00:19:05.130
squared, or norm squared,
of a complex number
00:19:05.130 --> 00:19:07.150
is always a real,
non-negative number.
00:19:07.150 --> 00:19:09.400
And that's important because
we want our probabilities
00:19:09.400 --> 00:19:10.890
to be real,
non-negative numbers.
00:19:10.890 --> 00:19:12.119
Could be 0, right?
00:19:12.119 --> 00:19:13.410
Could be 0 chance of something.
00:19:13.410 --> 00:19:16.460
Can't be negative 7 chance.
00:19:16.460 --> 00:19:20.510
Incidentally, there also
can't be probability 2.
00:19:20.510 --> 00:19:22.810
So that means that the
total probability had better
00:19:22.810 --> 00:19:25.350
be normalized.
00:19:25.350 --> 00:19:27.350
So let me just say this
in words, though, first.
00:19:27.350 --> 00:19:30.150
So P, which is the
norm squared of psi,
00:19:30.150 --> 00:19:39.400
determines the probability--
and, in particular,
00:19:39.400 --> 00:19:53.120
the probability density--
that the object in state psi,
00:19:53.120 --> 00:19:58.170
in the state given by the
wave function psi of x,
00:19:58.170 --> 00:20:05.260
will be found at x.
00:20:09.260 --> 00:20:11.322
So there's the second postulate.
00:20:11.322 --> 00:20:13.280
So in particular, when
I say it's a probability
00:20:13.280 --> 00:20:17.610
density, what I mean is the
probability that it is found
00:20:17.610 --> 00:20:22.830
between the position
x and x plus dx
00:20:22.830 --> 00:20:37.577
is equal to P of x dx, which is
equal to psi of x squared dx.
00:20:41.926 --> 00:20:42.800
Does that make sense?
00:20:42.800 --> 00:20:44.216
So the probability
that it's found
00:20:44.216 --> 00:20:46.160
in this infinitesimal
interval is
00:20:46.160 --> 00:20:51.010
equal to this density
times dx or psi squared dx.
00:20:51.010 --> 00:20:53.520
Now again, it's crucial
that the wave function
00:20:53.520 --> 00:20:55.430
is in fact properly normalized.
00:20:55.430 --> 00:20:57.940
Because if I say, look,
something could either be here
00:20:57.940 --> 00:20:59.675
or it could be
here, what's the sum
00:20:59.675 --> 00:21:01.050
of the probability
that it's here
00:21:01.050 --> 00:21:04.310
plus the probability
that it's here?
00:21:04.310 --> 00:21:07.500
It had better be 1, or there's
some other possibility.
00:21:07.500 --> 00:21:09.330
So probabilities
have to sum to 1.
00:21:09.330 --> 00:21:12.540
Total probability that you find
something somewhere must be 1.
00:21:12.540 --> 00:21:15.650
So what that tells you is
that total probability, which
00:21:15.650 --> 00:21:19.270
is equal to the integral over
all possible values of x-- so
00:21:19.270 --> 00:21:25.240
if I sum over all possible
values of P of x--
00:21:25.240 --> 00:21:28.860
all values-- should
be equal to 1.
00:21:33.030 --> 00:21:37.690
And we can write
this as integral dx
00:21:37.690 --> 00:21:38.845
over all values of x.
00:21:38.845 --> 00:21:41.220
And I write "all" here rather
than putting minus infinity
00:21:41.220 --> 00:21:43.780
to infinity because some
systems will be defined
00:21:43.780 --> 00:21:46.370
from 1 to minus 1, some systems
will be defined from minus
00:21:46.370 --> 00:21:48.580
infinity to infinity-- all
just means integrate over
00:21:48.580 --> 00:21:52.036
all possible values-- hold
on one sec-- of psi squared.
00:21:55.396 --> 00:21:57.437
AUDIENCE: Are you going
to use different notation
00:21:57.437 --> 00:21:59.684
for probability density
than probability?
00:21:59.684 --> 00:22:00.850
PROFESSOR: I'm not going to.
00:22:00.850 --> 00:22:03.800
Probability density is going
to have just one argument,
00:22:03.800 --> 00:22:05.456
and total probability
is going to have
00:22:05.456 --> 00:22:06.580
an interval as an argument.
00:22:09.460 --> 00:22:13.260
So they're distinct and this
is just the notation I like.
00:22:13.260 --> 00:22:13.950
Other questions?
00:22:18.550 --> 00:22:21.736
Just as a side note,
what are the dimensions
00:22:21.736 --> 00:22:22.610
of the wave function?
00:22:29.927 --> 00:22:31.760
So everyone think about
this one for second.
00:22:39.920 --> 00:22:40.920
What are the dimensions?
00:22:44.608 --> 00:22:46.510
AUDIENCE: Is it 1 over
square root length
00:22:46.510 --> 00:22:46.910
PROFESSOR: Awesome.
00:22:46.910 --> 00:22:47.409
Yes.
00:22:47.409 --> 00:22:49.442
It's 1 over root length.
00:22:49.442 --> 00:22:52.210
The dimensions of psi
are 1 over root length.
00:22:55.670 --> 00:22:59.480
And the way to see that is
that this should be equal to 1.
00:22:59.480 --> 00:23:01.400
It's a total probability.
00:23:01.400 --> 00:23:03.150
This is an infinitesimal
length, so this
00:23:03.150 --> 00:23:04.670
has dimensions of length.
00:23:04.670 --> 00:23:07.640
This has no dimension, so
this must have dimensions of 1
00:23:07.640 --> 00:23:09.590
over length.
00:23:09.590 --> 00:23:13.692
And so psi itself of x most have
dimensions of 1 over length.
00:23:13.692 --> 00:23:15.150
Now, something I
want to emphasize,
00:23:15.150 --> 00:23:17.274
I'm going to emphasize,
over and over in this class
00:23:17.274 --> 00:23:18.430
is dimensional analysis.
00:23:18.430 --> 00:23:21.330
You need to become comfortable
with dimensional analysis.
00:23:21.330 --> 00:23:23.200
It's absolutely essential.
00:23:23.200 --> 00:23:24.660
It's essential for two reasons.
00:23:24.660 --> 00:23:26.201
First off, it's
essential because I'm
00:23:26.201 --> 00:23:28.040
going to be merciless
in taking off points
00:23:28.040 --> 00:23:29.990
if you do write down a
dimensionally false thing.
00:23:29.990 --> 00:23:32.573
If you write down something on
a problem set or an exam that's
00:23:32.573 --> 00:23:36.300
like, a length is equal to
a velocity-- ooh, not good.
00:23:36.300 --> 00:23:38.800
But the second thing
is, forget the fact
00:23:38.800 --> 00:23:40.496
that I'm going to
take off points.
00:23:40.496 --> 00:23:42.620
Dimensional analysis is an
incredibly powerful tool
00:23:42.620 --> 00:23:43.095
for you.
00:23:43.095 --> 00:23:45.040
You can check something
that you've just calculated
00:23:45.040 --> 00:23:46.664
and, better yet,
sometimes you can just
00:23:46.664 --> 00:23:49.420
avoid a calculation entirely
by doing a dimensional analysis
00:23:49.420 --> 00:23:52.086
and seeing that there's only one
possible way to build something
00:23:52.086 --> 00:23:53.749
of dimensions length
in your system.
00:23:53.749 --> 00:23:55.290
So we'll do that
over and over again.
00:23:55.290 --> 00:23:57.340
But this is a question
I want you guys to start
00:23:57.340 --> 00:23:59.290
asking yourselves at
every step along the way
00:23:59.290 --> 00:24:01.331
of a calculation-- what
are the dimensions of all
00:24:01.331 --> 00:24:02.900
the objects in my system?
00:24:05.691 --> 00:24:06.857
Something smells like smoke.
00:24:10.370 --> 00:24:15.930
So with that said, if that's the
meaning of the wave function,
00:24:15.930 --> 00:24:22.165
what physically can we take
away from knowing these wave
00:24:22.165 --> 00:24:22.665
functions?
00:24:26.130 --> 00:24:27.780
Well, if this is
the wave function,
00:24:27.780 --> 00:24:29.740
let's draw the
probability distribution.
00:24:29.740 --> 00:24:31.240
What's the probability
distribution?
00:24:31.240 --> 00:24:34.430
P of x.
00:24:34.430 --> 00:24:37.830
And the probability distribution
here is really very simple.
00:24:37.830 --> 00:24:42.300
It's again 0 squared is
still 0 so it's still
00:24:42.300 --> 00:24:50.880
just a big spike at x1 and
this one is a big spike at x2.
00:24:59.100 --> 00:25:00.920
Everyone cool with that?
00:25:00.920 --> 00:25:03.700
So what do you know
when I tell you
00:25:03.700 --> 00:25:05.950
that this is the wave function
describing your system?
00:25:05.950 --> 00:25:07.642
You know that with
great confidence,
00:25:07.642 --> 00:25:10.100
you will find the particle to
be sitting at x1 if you look.
00:25:12.700 --> 00:25:16.580
So what this is telling you
is you expect x is roughly x1
00:25:16.580 --> 00:25:19.200
and our uncertainty
in x is small.
00:25:23.000 --> 00:25:25.840
Everyone cool with that?
00:25:25.840 --> 00:25:31.600
Similarly, here you see that
the position is likely to be x2,
00:25:31.600 --> 00:25:34.630
and your uncertainty
in your measurement--
00:25:34.630 --> 00:25:36.080
your confidence
in your prediction
00:25:36.080 --> 00:25:38.362
is another way to say
it-- is quite good,
00:25:38.362 --> 00:25:39.570
so your uncertainty is small.
00:25:44.947 --> 00:25:46.030
Now what about these guys?
00:25:46.030 --> 00:25:47.440
Well, now it's norm squared.
00:25:47.440 --> 00:25:48.850
I need to tell you what
the wave function is.
00:25:48.850 --> 00:25:50.391
Here, the wave
function that I want--
00:25:50.391 --> 00:25:56.740
so here is 0-- is
e to the i k1 x.
00:25:56.740 --> 00:26:02.480
And here the wave function
is equal to e to the i k2 x.
00:26:02.480 --> 00:26:04.300
And remember, I'm
drawing the real part
00:26:04.300 --> 00:26:06.740
because of practical
limitations.
00:26:06.740 --> 00:26:09.350
So the real part is just
a sinusoid-- or, in fact,
00:26:09.350 --> 00:26:12.070
the cosine-- and similarly,
here, the real part
00:26:12.070 --> 00:26:14.610
is a cosine.
00:26:14.610 --> 00:26:16.735
And I really should put 0
in the appropriate place,
00:26:16.735 --> 00:26:22.450
but-- that worked out well.
00:26:22.450 --> 00:26:25.745
So now the question is, what's
the probability distribution,
00:26:25.745 --> 00:26:30.440
P of x, associated to
these wave functions?
00:26:30.440 --> 00:26:33.260
So what's the norm squared
of minus e to the i k1 x?
00:26:36.680 --> 00:26:42.760
If I have a complex number
of phase e to the i alpha,
00:26:42.760 --> 00:26:46.850
and I take its norm
squared, what do I get?
00:26:46.850 --> 00:26:47.980
1.
00:26:47.980 --> 00:26:48.480
Right?
00:26:48.480 --> 00:26:50.570
But remember complex numbers.
00:26:50.570 --> 00:26:52.460
If we have a complex
number alpha-- or sorry,
00:26:52.460 --> 00:26:54.430
if we have a
complex number beta,
00:26:54.430 --> 00:26:57.020
then beta squared
is by definition
00:26:57.020 --> 00:27:00.290
beta complex
conjugate times beta.
00:27:00.290 --> 00:27:02.330
So e to the i alpha, if
the complex conjugate
00:27:02.330 --> 00:27:04.460
is e to the minus i alpha,
e to the i alpha times
00:27:04.460 --> 00:27:06.750
e to the minus i alpha,
they cancel out-- that's 1.
00:27:09.420 --> 00:27:11.072
So if this is the
wave function, what's
00:27:11.072 --> 00:27:12.280
the probability distribution?
00:27:15.940 --> 00:27:16.710
Well, it's 1.
00:27:16.710 --> 00:27:19.800
It's independent of x.
00:27:19.800 --> 00:27:21.934
So from this we've learned
two important things.
00:27:21.934 --> 00:27:23.850
The first is, this is
not properly normalized.
00:27:27.190 --> 00:27:27.970
That's not so key.
00:27:27.970 --> 00:27:32.140
But the most important thing is,
if this is our wave function,
00:27:32.140 --> 00:27:34.826
and we subsequently measure the
position of the particle-- we
00:27:34.826 --> 00:27:36.700
look at it, we say ah,
there's the particle--
00:27:36.700 --> 00:27:39.782
where are we likely to find it?
00:27:39.782 --> 00:27:42.110
Yeah, it could be anywhere.
00:27:42.110 --> 00:27:46.670
So what's the value of x
you expect-- typical x?
00:27:46.670 --> 00:27:50.830
I have no idea, no
information whatsoever.
00:27:50.830 --> 00:27:52.100
None.
00:27:52.100 --> 00:27:55.560
But and correspondingly,
what is our uncertainty
00:27:55.560 --> 00:27:58.800
in the position of x
that we'll measure?
00:27:58.800 --> 00:28:01.079
It's very large, exactly.
00:28:01.079 --> 00:28:03.120
Now, in order to tell you
it's actually infinite,
00:28:03.120 --> 00:28:05.839
I need to stretch this off and
tell you that it's actually
00:28:05.839 --> 00:28:08.130
constant off to infinity,
and my arms aren't that long,
00:28:08.130 --> 00:28:09.840
so I'll just say large.
00:28:09.840 --> 00:28:11.700
Similarly here, if
our wave function
00:28:11.700 --> 00:28:14.020
is e to the i k2 x--
here k2 is larger,
00:28:14.020 --> 00:28:18.760
the wavelength is
shorter-- what's
00:28:18.760 --> 00:28:21.790
the probability distribution?
00:28:21.790 --> 00:28:22.870
It's, again, constant.
00:28:26.890 --> 00:28:30.770
So-- this is 0, 0.
00:28:30.770 --> 00:28:36.850
So again, x-- we have no idea,
and our uncertainty in the x
00:28:36.850 --> 00:28:38.264
is large.
00:28:38.264 --> 00:28:39.430
And in fact it's very large.
00:28:43.880 --> 00:28:45.604
Questions?
00:28:45.604 --> 00:28:46.520
What about these guys?
00:28:49.350 --> 00:28:51.230
OK, this is the real challenge.
00:28:51.230 --> 00:28:52.730
OK, so if this is
our wave function,
00:28:52.730 --> 00:28:56.070
and let's just say that
it's real-- hard as it
00:28:56.070 --> 00:28:59.720
is to believe that-- then what's
our probably distribution?
00:28:59.720 --> 00:29:07.780
Well, something
like-- I don't know,
00:29:07.780 --> 00:29:15.950
something-- you get the point.
00:29:15.950 --> 00:29:20.430
OK, so if this is our
probability distribution,
00:29:20.430 --> 00:29:22.654
where are we likely
to find the particle?
00:29:22.654 --> 00:29:24.570
Well, now it's a little
more difficult, right?
00:29:24.570 --> 00:29:27.060
Because we're unlikely
to find it here,
00:29:27.060 --> 00:29:29.450
while it's reasonably likely
to find here, unlikely here,
00:29:29.450 --> 00:29:33.130
reasonably likely, unlikely,
like-- you know, it's a mess.
00:29:33.130 --> 00:29:34.750
So where is this?
00:29:34.750 --> 00:29:36.280
I'm not really sure.
00:29:36.280 --> 00:29:38.610
What's our uncertainty?
00:29:38.610 --> 00:29:41.850
Well, our uncertainty
is not infinite
00:29:41.850 --> 00:29:44.324
because-- OK, my name
ends at some point.
00:29:44.324 --> 00:29:45.490
So this is going to go to 0.
00:29:47.790 --> 00:29:50.040
So whatever else we know,
we know it's in this region.
00:29:50.040 --> 00:29:55.830
So it's not infinite,
it's not small, we'll say.
00:29:55.830 --> 00:29:58.900
But it's not arbitrarily
small-- it's not tiny.
00:29:58.900 --> 00:30:00.782
Or sorry, it's not
gigantic is what I meant.
00:30:00.782 --> 00:30:02.115
Our uncertainty is not gigantic.
00:30:06.615 --> 00:30:07.990
But it's still
pretty nontrivial,
00:30:07.990 --> 00:30:09.495
because I can say
with some confidence
00:30:09.495 --> 00:30:11.090
that it's more likely
to be here than here,
00:30:11.090 --> 00:30:12.923
but I really don't know
which of those peaks
00:30:12.923 --> 00:30:15.460
it's going to be found.
00:30:15.460 --> 00:30:18.150
OK, now what about this guy?
00:30:18.150 --> 00:30:20.050
What's the probability
distribution well now
00:30:20.050 --> 00:30:22.660
you see why this is a
stupid wave function,
00:30:22.660 --> 00:30:24.302
because it's multiply valued.
00:30:24.302 --> 00:30:26.510
It has multiple different
values at every value of x.
00:30:26.510 --> 00:30:27.593
So what's the probability?
00:30:27.593 --> 00:30:30.470
Well, it might be root 2,
maybe it's 1 over root 3.
00:30:30.470 --> 00:30:32.230
I'm really not sure.
00:30:32.230 --> 00:30:37.210
So this tells us an important
lesson-- this is stupid.
00:30:37.210 --> 00:30:41.027
And what I mean by stupid
is, it is multiply valued.
00:30:41.027 --> 00:30:42.610
So the wave function--
we just learned
00:30:42.610 --> 00:30:44.330
a lesson-- should
be single valued.
00:30:49.620 --> 00:30:53.652
And we will explore some more
on your problem set, which
00:30:53.652 --> 00:30:55.360
will be posted
immediately after lecture.
00:30:55.360 --> 00:30:57.443
There are problems that
walk you through a variety
00:30:57.443 --> 00:31:00.880
of other potential pathologies
of the wave function
00:31:00.880 --> 00:31:02.770
and guide you to
some more intuition.
00:31:02.770 --> 00:31:04.311
For example, the
wave function really
00:31:04.311 --> 00:31:05.780
needs to be continuous as well.
00:31:05.780 --> 00:31:06.405
You'll see why.
00:31:09.230 --> 00:31:10.485
All right.
00:31:10.485 --> 00:31:11.485
Questions at this point?
00:31:16.070 --> 00:31:16.570
No?
00:31:16.570 --> 00:31:17.620
OK.
00:31:17.620 --> 00:31:20.472
So these look like pretty
useful wave functions,
00:31:20.472 --> 00:31:22.180
because they corresponded
to the particle
00:31:22.180 --> 00:31:23.346
being at some definite spot.
00:31:23.346 --> 00:31:26.890
And I, for example, am at
a reasonably definite spot.
00:31:26.890 --> 00:31:28.340
These two wave
functions, though,
00:31:28.340 --> 00:31:30.760
look pretty much useless,
because they give us
00:31:30.760 --> 00:31:34.130
no information whatsoever
about what the position is.
00:31:34.130 --> 00:31:36.390
Everyone agree with that?
00:31:36.390 --> 00:31:42.000
Except-- remember the
de Broglie relations.
00:31:42.000 --> 00:31:45.650
The de Broglie relations say
that associated to a particle
00:31:45.650 --> 00:31:47.420
is also some wave.
00:31:47.420 --> 00:31:49.727
And the momentum
of that particle
00:31:49.727 --> 00:31:51.060
is determined by the wavelength.
00:31:51.060 --> 00:31:52.205
It's inversely related
to the wavelength.
00:31:52.205 --> 00:31:53.810
It's proportional
to the wave number.
00:31:53.810 --> 00:31:56.980
Any energy is proportional
to the frequency.
00:31:56.980 --> 00:32:01.280
Now, look at those
wave functions.
00:32:01.280 --> 00:32:04.180
Those wave functions give
us no position information
00:32:04.180 --> 00:32:07.090
whatsoever, but they have
very definite wavelengths.
00:32:07.090 --> 00:32:10.250
Those are periodic functions
with definite wavelengths.
00:32:10.250 --> 00:32:15.601
In particular, this guy has
a wavelength of from here
00:32:15.601 --> 00:32:16.100
to here.
00:32:18.930 --> 00:32:20.213
It has a wave number k1.
00:32:22.790 --> 00:32:24.940
So that tells us
that if we measure
00:32:24.940 --> 00:32:27.412
the momentum of
this particle, we
00:32:27.412 --> 00:32:28.870
can be pretty
confident, because it
00:32:28.870 --> 00:32:31.250
has a reasonably
well-defined wavelength
00:32:31.250 --> 00:32:33.850
corresponding to some
wave number k-- 2 pi
00:32:33.850 --> 00:32:35.359
upon the wavelength.
00:32:35.359 --> 00:32:37.150
It has some momentum,
and if we measure it,
00:32:37.150 --> 00:32:39.316
we should be pretty confident
that the momentum will
00:32:39.316 --> 00:32:39.980
be h-bar k1.
00:32:43.560 --> 00:32:45.040
Everybody agree with that?
00:32:45.040 --> 00:32:46.762
Looks like a sine wave.
00:32:46.762 --> 00:32:48.220
And de Broglie
tells us that if you
00:32:48.220 --> 00:32:51.240
have a wave of
wavelength lambda,
00:32:51.240 --> 00:32:55.294
that corresponds to a
particle having momentum p.
00:32:55.294 --> 00:32:58.370
Now, how confident can
we be in that estimation
00:32:58.370 --> 00:33:00.760
of the momentum?
00:33:00.760 --> 00:33:03.200
Well, if I tell you it's e
to the i k x, that's exactly
00:33:03.200 --> 00:33:07.260
a periodic function with
wavelength lambda 2 pi upon k.
00:33:07.260 --> 00:33:08.427
So how confident are we?
00:33:08.427 --> 00:33:09.135
Pretty confident.
00:33:09.135 --> 00:33:12.320
So our uncertainty in
the momentum is tiny.
00:33:15.260 --> 00:33:17.330
Everyone agree?
00:33:17.330 --> 00:33:21.070
Similarly, for
this wave, again we
00:33:21.070 --> 00:33:23.540
have a wavelength-- it's
a periodic function,
00:33:23.540 --> 00:33:26.730
but the wavelength
is much shorter.
00:33:26.730 --> 00:33:28.550
If the wavelength
is much shorter,
00:33:28.550 --> 00:33:31.330
then k is much larger-- the
momentum is much larger.
00:33:31.330 --> 00:33:33.010
So the momentum we
expect to measure,
00:33:33.010 --> 00:33:38.555
which is roughly h-bar k2,
is going to be much larger.
00:33:38.555 --> 00:33:39.680
What about our uncertainty?
00:33:39.680 --> 00:33:42.040
Again, it's a perfect
periodic function
00:33:42.040 --> 00:33:43.990
so our uncertainty in
the momentum is small.
00:33:46.840 --> 00:33:47.840
Everyone cool with that?
00:33:53.950 --> 00:33:57.100
And that comes, again, from
the de Broglie relations.
00:34:04.170 --> 00:34:09.920
So questions at this point?
00:34:13.220 --> 00:34:14.469
You guys are real quiet today.
00:34:17.980 --> 00:34:20.480
Questions?
00:34:20.480 --> 00:34:23.971
AUDIENCE: So delta
P is 0, basically?
00:34:23.971 --> 00:34:25.179
PROFESSOR: It's pretty small.
00:34:25.179 --> 00:34:27.510
Now, again, I haven't
drawn this off to infinity,
00:34:27.510 --> 00:34:29.380
but if it's exactly
the i k x, then yeah,
00:34:29.380 --> 00:34:30.554
it turns out to be 0.
00:34:30.554 --> 00:34:32.929
Now, an important thing, so
let me rephrase your question
00:34:32.929 --> 00:34:33.429
slightly.
00:34:33.429 --> 00:34:35.330
So the question
was, is delta P 0?
00:34:35.330 --> 00:34:37.219
Is it really 0?
00:34:37.219 --> 00:34:38.860
So here's a problem
for us right now.
00:34:38.860 --> 00:34:41.520
We don't have a
definition for delta P.
00:34:41.520 --> 00:34:43.421
So what is the
definition of delta P?
00:34:43.421 --> 00:34:44.420
I haven't given you one.
00:34:44.420 --> 00:34:46.389
So here, when I said
delta P is small,
00:34:46.389 --> 00:34:48.830
what I mean is, intuitively,
just by eyeball,
00:34:48.830 --> 00:34:51.069
our confidence in that
momentum is pretty good,
00:34:51.069 --> 00:34:52.360
using the de Broglie relations.
00:34:52.360 --> 00:34:53.419
I have not given
you a definition,
00:34:53.419 --> 00:34:55.585
and that will be part of
my job over the next couple
00:34:55.585 --> 00:34:56.520
of lectures.
00:34:56.520 --> 00:34:57.330
Very good question.
00:34:57.330 --> 00:34:58.303
Yeah.
00:34:58.303 --> 00:35:00.730
AUDIENCE: How do you code
noise in that function?
00:35:00.730 --> 00:35:01.025
PROFESSOR: Awesome.
00:35:01.025 --> 00:35:02.670
AUDIENCE: Do you just
have different wavelengths
00:35:02.670 --> 00:35:03.025
PROFESSOR: Yeah
00:35:03.025 --> 00:35:03.920
AUDIENCE: As you go along?
00:35:03.920 --> 00:35:04.711
PROFESSOR: Awesome.
00:35:04.711 --> 00:35:07.780
So for example, this-- does
it have a definite wavelength?
00:35:07.780 --> 00:35:09.200
Not so much.
00:35:09.200 --> 00:35:14.270
So hold that question and wait
until you see the next examples
00:35:14.270 --> 00:35:16.520
that I put on this board,
and if that doesn't answer
00:35:16.520 --> 00:35:18.020
your question, ask
it again, becayse
00:35:18.020 --> 00:35:19.380
it's a very important question.
00:35:19.380 --> 00:35:20.464
OK.
00:35:20.464 --> 00:35:23.960
AUDIENCE: When you
talk about a photon,
00:35:23.960 --> 00:35:26.630
you always say a photon
has a certain frequency.
00:35:26.630 --> 00:35:29.440
Doesn't that mean that it
must be a wave because you
00:35:29.440 --> 00:35:32.324
have to fix the wave number k?
00:35:32.324 --> 00:35:33.490
PROFESSOR: Awesome question.
00:35:33.490 --> 00:35:37.790
Does every wave packet of
light that hits your eye,
00:35:37.790 --> 00:35:40.950
does it always have a
single, unique frequency?
00:35:40.950 --> 00:35:43.650
No, you can take multiple
frequency sources
00:35:43.650 --> 00:35:45.430
and superpose them.
00:35:45.430 --> 00:35:47.590
An interesting choice
of words I used there.
00:35:47.590 --> 00:35:49.620
All right, so the
question is, since light
00:35:49.620 --> 00:35:52.580
has some wavelength,
does every chunk of light
00:35:52.580 --> 00:35:54.670
have a definite-- this
is the question, roughly.
00:35:54.670 --> 00:35:56.279
Yeah, so and the
answer is, light
00:35:56.279 --> 00:35:57.695
doesn't always
have a single-- You
00:35:57.695 --> 00:35:59.644
can have light
coming at you that
00:35:59.644 --> 00:36:01.810
has many different wavelengths
and put it in a prism
00:36:01.810 --> 00:36:04.420
and break it up into
its various components.
00:36:04.420 --> 00:36:07.240
So you can have a superposition
of different frequencies
00:36:07.240 --> 00:36:08.740
of light.
00:36:08.740 --> 00:36:12.960
We'll see the same
effect happening for us.
00:36:12.960 --> 00:36:22.660
OK, so again, de Broglie
made this conjecture
00:36:22.660 --> 00:36:25.610
that E is h-bar omega
and P is h-bar k.
00:36:30.060 --> 00:36:32.240
This was verified in the
Davisson-Germer experiment
00:36:32.240 --> 00:36:33.460
that we ran.
00:36:33.460 --> 00:36:35.650
But here, one of
the things that's
00:36:35.650 --> 00:36:37.820
sort of latent in
this is, what he means
00:36:37.820 --> 00:36:41.060
is, look, associated to
every particle with energy N
00:36:41.060 --> 00:36:47.640
and momentum P is a plane
wave of the form e to the i kx
00:36:47.640 --> 00:36:50.100
minus omega t.
00:36:50.100 --> 00:36:53.430
And this, properly, in three
dimensions should be k dot x.
00:36:53.430 --> 00:36:58.070
But at this point, this is
an important simplification.
00:36:58.070 --> 00:37:04.411
For the rest of 8.04,
until otherwise specified,
00:37:04.411 --> 00:37:06.410
we are going to be doing
one-dimensional quantum
00:37:06.410 --> 00:37:07.280
mechanics.
00:37:07.280 --> 00:37:10.691
So I'm going to remove arrow
marks and dot products.
00:37:10.691 --> 00:37:12.940
There's going to be one
spatial dimension and one time
00:37:12.940 --> 00:37:14.155
dimension.
00:37:14.155 --> 00:37:16.280
We're always going to have
just one time dimension,
00:37:16.280 --> 00:37:18.030
but sometimes we'll have
more spatial dimensions.
00:37:18.030 --> 00:37:20.030
But it's going to be a
while until we get there.
00:37:20.030 --> 00:37:21.840
So for now, we're
just going to have kx.
00:37:21.840 --> 00:37:23.419
So this is a general plane wave.
00:37:23.419 --> 00:37:24.960
And what de Broglie
really was saying
00:37:24.960 --> 00:37:29.130
is that, somehow, associated
to the particle with energy E
00:37:29.130 --> 00:37:32.030
and momentum P should be
some wave, a plane wave,
00:37:32.030 --> 00:37:34.340
with wave number k
and frequency omega.
00:37:34.340 --> 00:37:38.460
And that's the wave
function associated to it.
00:37:38.460 --> 00:37:42.780
The thing is, not every wave
function is a plane wave.
00:37:42.780 --> 00:37:45.900
Some wave functions
are well localized.
00:37:45.900 --> 00:37:48.842
Some of them are just
complicated morasses.
00:37:48.842 --> 00:37:50.050
Some of them are just a mess.
00:37:59.850 --> 00:38:08.910
So now is the most important
postulate in quantum mechanics.
00:38:08.910 --> 00:38:11.867
I remember vividly,
vividly, when
00:38:11.867 --> 00:38:13.200
I took the analog of this class.
00:38:13.200 --> 00:38:16.530
It was called Physics
143A at Harvard.
00:38:16.530 --> 00:38:19.660
And the professor
at this point said--
00:38:19.660 --> 00:38:21.570
I know him well now,
he's a friend-- he said,
00:38:21.570 --> 00:38:23.996
this is what quantum
mechanics is all about.
00:38:23.996 --> 00:38:24.870
And I was so psyched.
00:38:24.870 --> 00:38:26.980
And then he told me And it
was like, that's ridiculous.
00:38:26.980 --> 00:38:27.460
Seriously?
00:38:27.460 --> 00:38:29.270
That's what quantum
mechanics is all about?
00:38:29.270 --> 00:38:31.478
So I always felt like this
is some weird thing, where
00:38:31.478 --> 00:38:33.100
old physicists go crazy.
00:38:33.100 --> 00:38:36.430
But it turns out I'm going to
say exactly the same thing.
00:38:36.430 --> 00:38:39.440
This is the most important thing
in all of quantum mechanics.
00:38:39.440 --> 00:38:43.330
It is all contained in
the following proposition.
00:38:43.330 --> 00:38:46.680
Everything-- the two
slit experiments, the box
00:38:46.680 --> 00:38:49.330
experiments, all the cool
stuff in quantum mechanics,
00:38:49.330 --> 00:38:51.230
all the strange and
counter intuitive stuff
00:38:51.230 --> 00:38:53.450
comes directly from
the next postulate.
00:38:53.450 --> 00:38:54.220
So here it is.
00:38:58.460 --> 00:39:01.300
I love this.
00:39:01.300 --> 00:39:05.240
Three-- put a star on it.
00:39:07.860 --> 00:39:16.450
Given two possible wave
functions or states--
00:39:16.450 --> 00:39:26.970
I'll say configurations--
of a quantum system--
00:39:26.970 --> 00:39:29.510
I wish there was "Ride
of the Valkyries"
00:39:29.510 --> 00:39:37.690
playing in the background--
corresponding to two
00:39:37.690 --> 00:39:45.240
distinct wave functions--
f with an upper ns
00:39:45.240 --> 00:39:47.600
is going to be my
notation for functions
00:39:47.600 --> 00:39:54.050
because I have to write
it a lot-- psi1 and psi2--
00:39:54.050 --> 00:40:05.770
and I'll say, of x--
the system-- is down--
00:40:05.770 --> 00:40:11.820
can also be in a superposition.
00:40:19.950 --> 00:40:40.140
of psi1 and psi2, where alpha
and beta are complex numbers.
00:40:47.750 --> 00:40:52.620
Given any two possible
configurations of the system,
00:40:52.620 --> 00:40:55.077
there is also an allowed
configuration of the system
00:40:55.077 --> 00:40:56.660
corresponding to
being in an arbitrary
00:40:56.660 --> 00:40:59.410
superposition of them.
00:40:59.410 --> 00:41:03.670
If an electron can be
hard and it can be soft,
00:41:03.670 --> 00:41:06.290
it can also be in an
arbitrary superposition
00:41:06.290 --> 00:41:08.160
of being hard and soft.
00:41:08.160 --> 00:41:11.010
And what I mean by that
is that hard corresponds
00:41:11.010 --> 00:41:13.160
to some particular
wave function,
00:41:13.160 --> 00:41:15.640
soft will correspond to some
particular wave function,
00:41:15.640 --> 00:41:18.580
and the superposition
corresponds to a different wave
00:41:18.580 --> 00:41:21.009
function which is a linear
combination of them.
00:41:25.230 --> 00:41:27.223
AUDIENCE: [INAUDIBLE]
combination also
00:41:27.223 --> 00:41:28.610
have to be normalized?
00:41:28.610 --> 00:41:30.651
PROFESSOR: Yeah, OK, that's
a very good question.
00:41:30.651 --> 00:41:32.960
So and alpha and beta
are some complex numbers
00:41:32.960 --> 00:41:35.370
subject to the
normalization condition.
00:41:39.640 --> 00:41:44.060
So indeed, this wave function
should be properly normalized.
00:41:44.060 --> 00:41:45.890
Now, let me step
back for second.
00:41:45.890 --> 00:41:48.180
There's an alternate way
to phrase the probability
00:41:48.180 --> 00:41:49.870
distribution here,
which goes like this,
00:41:49.870 --> 00:41:51.189
and I'm going to put it here.
00:41:51.189 --> 00:41:53.480
The alternate statement of
the probability distribution
00:41:53.480 --> 00:41:56.750
is that the probability
density at x
00:41:56.750 --> 00:42:03.270
is equal to psi of x
norm squared divided
00:42:03.270 --> 00:42:09.280
by the integral over
all x dx of psi squared.
00:42:11.810 --> 00:42:15.420
So notice that, if we properly
normalize the wave function,
00:42:15.420 --> 00:42:20.739
this denominator is equal to 1--
and so it's not there, right,
00:42:20.739 --> 00:42:21.780
and then it's equivalent.
00:42:21.780 --> 00:42:23.488
But if we haven't
properly normalized it,
00:42:23.488 --> 00:42:25.890
then this probability
distribution
00:42:25.890 --> 00:42:27.820
is automatically
properly normalized.
00:42:27.820 --> 00:42:30.361
Because this is a constant, when
we integrate the top, that's
00:42:30.361 --> 00:42:33.650
equal to the bottom,
it integrates to 1.
00:42:33.650 --> 00:42:37.220
So I prefer,
personally, in thinking
00:42:37.220 --> 00:42:39.720
about this for the first pass
to just require that we always
00:42:39.720 --> 00:42:41.970
be careful to choose
some normalization.
00:42:41.970 --> 00:42:44.260
That won't always be easy,
and so sometimes it's
00:42:44.260 --> 00:42:45.830
useful to forget
about normalizing
00:42:45.830 --> 00:42:49.649
and just define the probability
distribution that way.
00:42:49.649 --> 00:42:50.190
Is that cool?
00:42:53.270 --> 00:42:53.770
OK.
00:42:53.770 --> 00:42:57.029
This is the beating soul
of quantum mechanics.
00:42:57.029 --> 00:42:58.820
Everything in quantum
mechanics is in here.
00:42:58.820 --> 00:43:00.778
Everything in quantum
mechanics is forced on us
00:43:00.778 --> 00:43:03.520
from these few principles
and a couple of requirements
00:43:03.520 --> 00:43:05.704
of matching to reality.
00:43:05.704 --> 00:43:08.174
AUDIENCE: When
you do this-- some
00:43:08.174 --> 00:43:12.000
of linear, some of
two wave functions,
00:43:12.000 --> 00:43:14.280
can you get interference?
00:43:14.280 --> 00:43:14.950
PROFESSOR: Yes.
00:43:14.950 --> 00:43:15.450
Excellent.
00:43:15.450 --> 00:43:17.900
So the question is, when
you have a sum of two wave
00:43:17.900 --> 00:43:21.250
functions, can you get some
sort of interference effect?
00:43:21.250 --> 00:43:22.760
And the answer is, absolutely.
00:43:22.760 --> 00:43:24.510
And that's exactly
we're going to do next.
00:43:24.510 --> 00:43:27.572
So in particular, let me
look at a particular pair
00:43:27.572 --> 00:43:30.266
of superpositions.
00:43:30.266 --> 00:43:31.640
So let's swap
these boards around
00:43:31.640 --> 00:43:33.473
so the parallelism is
a little more obvious.
00:43:36.300 --> 00:43:38.945
So let's scrap these
rather silly wave functions
00:43:38.945 --> 00:43:40.320
and come up with
something that's
00:43:40.320 --> 00:43:41.670
a little more interesting.
00:43:41.670 --> 00:43:46.930
So instead of using those as
characteristic wave functions,
00:43:46.930 --> 00:43:48.670
I want to build superpositions.
00:43:48.670 --> 00:43:50.452
So in particular,
I want to start
00:43:50.452 --> 00:43:52.660
by taking an arbitrary--
both of these wave functions
00:43:52.660 --> 00:43:53.570
have a simple interpretation.
00:43:53.570 --> 00:43:55.430
This corresponds to a
particle being here.
00:43:55.430 --> 00:43:57.430
This corresponds to a
particle being here.
00:43:57.430 --> 00:43:59.400
I want to take a
superposition of them.
00:43:59.400 --> 00:44:00.525
So here's my superposition.
00:44:03.230 --> 00:44:05.610
Oops, let's try that again.
00:44:08.640 --> 00:44:13.100
And my superposition-- so here
is 0 and here is x1 and here is
00:44:13.100 --> 00:44:18.780
x2-- is going to be some
amount times the first one
00:44:18.780 --> 00:44:22.390
plus some amount
times the second one.
00:44:22.390 --> 00:44:25.450
There's a superposition.
00:44:25.450 --> 00:44:28.320
Similarly, I could have taken
a superposition of the two
00:44:28.320 --> 00:44:31.921
functions on the
second chalkboard.
00:44:31.921 --> 00:44:33.420
And again I'm taking
a superposition
00:44:33.420 --> 00:44:36.085
of the complex e to the
i k1 x and e to the i
00:44:36.085 --> 00:44:38.000
k2 x and then taking
the real part.
00:44:47.650 --> 00:44:49.420
So that's a particular
superposition,
00:44:49.420 --> 00:44:50.753
a particular linear combination.
00:44:56.010 --> 00:44:57.680
So now let's go back to this.
00:44:57.680 --> 00:44:59.544
This was a particle
that was here.
00:44:59.544 --> 00:45:00.960
This is a particle
that was there.
00:45:00.960 --> 00:45:03.630
When we take the
superposition, what
00:45:03.630 --> 00:45:05.620
is the probability distribution?
00:45:05.620 --> 00:45:07.170
Where is this particle?
00:45:07.170 --> 00:45:09.560
Well, there's some
amplitude that it's here,
00:45:09.560 --> 00:45:12.201
and there's some
amplitude that it's here.
00:45:12.201 --> 00:45:14.450
And there's rather more
amplitude that it's over here,
00:45:14.450 --> 00:45:15.866
but there's still
some probability
00:45:15.866 --> 00:45:17.600
that it's over here.
00:45:17.600 --> 00:45:19.976
Where am I going to
find the particle?
00:45:19.976 --> 00:45:22.250
I'm not so sure anymore.
00:45:22.250 --> 00:45:25.960
It's either going to be here
or here, but I'm not positive.
00:45:25.960 --> 00:45:27.450
It's more likely
to be here than it
00:45:27.450 --> 00:45:31.660
is to be here, but
not a whole lot more.
00:45:31.660 --> 00:45:33.594
So where am I going
to find the particle?
00:45:33.594 --> 00:45:35.010
Well, now we have
to define this--
00:45:35.010 --> 00:45:36.770
where am I going to
find the particle?
00:45:36.770 --> 00:45:39.030
Look, if I did this experiment
a whole bunch of times,
00:45:39.030 --> 00:45:41.700
it'd be over here more
than it would be over here.
00:45:41.700 --> 00:45:44.230
So the average will be
somewhere around here--
00:45:44.230 --> 00:45:45.860
it'll be in between the two.
00:45:45.860 --> 00:45:48.690
So x is somewhere in between.
00:45:48.690 --> 00:45:55.435
That's where we expect
to find it, on average.
00:45:59.260 --> 00:46:02.560
What's our uncertainty
in the position?
00:46:02.560 --> 00:46:04.340
Well, it's not
that small anymore.
00:46:04.340 --> 00:46:07.085
It's now of order x1 minus x2.
00:46:12.729 --> 00:46:13.770
Everyone agree with that?
00:46:21.979 --> 00:46:23.020
Now, what about this guy?
00:46:26.740 --> 00:46:32.500
Well, does this thing
have a single wavelength?
00:46:32.500 --> 00:46:33.430
No.
00:46:33.430 --> 00:46:35.560
This is like light that
comes at you from the sun.
00:46:35.560 --> 00:46:36.570
It has many wavelengths.
00:46:36.570 --> 00:46:39.153
In this case, it has just two--
I've added those two together.
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
00:46:45.060 --> 00:46:45.865
e to the i k2 x.
00:46:51.800 --> 00:46:54.850
So in fact, it has two
wavelengths associated with it.
00:46:54.850 --> 00:46:57.550
lambda1 lambda2.
00:46:57.550 --> 00:46:59.530
And so the probability
distribution now,
00:46:59.530 --> 00:47:03.470
if we take the norm squared
of this-- the probability
00:47:03.470 --> 00:47:11.170
distribution is the norm
squared of this guy--
00:47:11.170 --> 00:47:15.607
is no longer constant, but
there's an interference term.
00:47:15.607 --> 00:47:17.190
And let's just see
how that works out.
00:47:20.450 --> 00:47:23.730
Let me be very
explicit about this.
00:47:23.730 --> 00:47:27.380
Note that the probability
in our superposition of psi1
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
00:47:30.690 --> 00:47:35.520
x, is equal to the norm squared
of the wave function, which
00:47:35.520 --> 00:47:40.110
is the superposition psi1
plus beta psi2, which
00:47:40.110 --> 00:47:46.856
is equal to alpha squared
psi1 squared plus beta squared
00:47:46.856 --> 00:47:57.780
psi2 squared plus alpha
star psi1 star-- actually,
00:47:57.780 --> 00:48:07.230
let me write this over
here-- beta psi2 plus
00:48:07.230 --> 00:48:12.150
alpha psi1 beta star
psi2 star, where star
00:48:12.150 --> 00:48:15.010
means complex conjugation.
00:48:15.010 --> 00:48:19.480
But notice that this is
equal to-- that first term
00:48:19.480 --> 00:48:21.930
is alpha squared times
the first probability,
00:48:21.930 --> 00:48:25.180
or the probability of
this thing, of alpha psi1,
00:48:25.180 --> 00:48:28.300
is equal to probability 1.
00:48:28.300 --> 00:48:29.920
This term, beta
squared psi2 squared,
00:48:29.920 --> 00:48:31.961
is the probability that
the second thing happens.
00:48:35.340 --> 00:48:37.870
But these terms
can't be understood
00:48:37.870 --> 00:48:39.960
in terms of the
probabilities of psi1
00:48:39.960 --> 00:48:41.726
or the probability
of psi2 alone.
00:48:41.726 --> 00:48:42.850
They're interference terms.
00:48:49.290 --> 00:48:52.190
So the superposition
principle, together
00:48:52.190 --> 00:48:55.210
with the interpretation of
the probability as the norm
00:48:55.210 --> 00:49:00.700
squared of the wave function,
gives us a correction
00:49:00.700 --> 00:49:04.590
to the classical addition
of probabilities,
00:49:04.590 --> 00:49:07.430
which is these
interference terms.
00:49:07.430 --> 00:49:09.220
Everyone happy with that?
00:49:09.220 --> 00:49:11.770
Now, here's something very
important to keep in mind.
00:49:11.770 --> 00:49:14.000
These things are norms
squared of complex numbers.
00:49:14.000 --> 00:49:16.190
That means they're real,
but in particular, they're
00:49:16.190 --> 00:49:17.587
non-negative.
00:49:17.587 --> 00:49:19.420
So these two are both
real and non-negative.
00:49:19.420 --> 00:49:20.580
But what about this?
00:49:20.580 --> 00:49:22.725
This is not the norm
squared of anything.
00:49:22.725 --> 00:49:24.350
However, this is its
complex conjugate.
00:49:24.350 --> 00:49:25.570
When you take something
and its complex conjugate
00:49:25.570 --> 00:49:26.986
and you add them
together, you get
00:49:26.986 --> 00:49:28.510
something that's
necessarily real.
00:49:28.510 --> 00:49:32.039
But it's not
necessarily positive.
00:49:32.039 --> 00:49:33.080
So this is a funny thing.
00:49:33.080 --> 00:49:36.020
The probability that something
happens if we add together
00:49:36.020 --> 00:49:39.400
our two configurations, we
superpose two configurations,
00:49:39.400 --> 00:49:41.014
has a positive probability term.
00:49:41.014 --> 00:49:42.430
But it's also got
terms that don't
00:49:42.430 --> 00:49:44.480
have a definite sign,
that could be negative.
00:49:44.480 --> 00:49:45.750
It's always real.
00:49:45.750 --> 00:49:48.480
And you can check but
this quantity is always
00:49:48.480 --> 00:49:49.610
greater than or equal to 0.
00:49:49.610 --> 00:49:52.740
It's never negative,
the total quantity.
00:49:52.740 --> 00:49:55.890
So remember Bell's inequality
that we talked about?
00:49:55.890 --> 00:49:58.388
Bell's inequality
said, look, if we
00:49:58.388 --> 00:50:00.596
have the probability of one
thing happening being P1,
00:50:00.596 --> 00:50:02.596
and the probability of
the other thing happening
00:50:02.596 --> 00:50:05.470
being P2, the probability
of both things happening
00:50:05.470 --> 00:50:07.360
is just P1 plus P2.
00:50:07.360 --> 00:50:09.750
And here we see that,
in quantum mechanics,
00:50:09.750 --> 00:50:11.860
probabilities
don't add that way.
00:50:11.860 --> 00:50:13.970
The wave functions add--
and the probability
00:50:13.970 --> 00:50:18.700
is the norm squared
of the wave function.
00:50:18.700 --> 00:50:21.630
The wave functions add,
not the probabilities.
00:50:21.630 --> 00:50:24.630
And that is what underlies all
of the interference effects
00:50:24.630 --> 00:50:25.402
we've seen.
00:50:25.402 --> 00:50:27.610
And it's going to be the
heart of the rest of quantum
00:50:27.610 --> 00:50:29.730
mechanics.
00:50:29.730 --> 00:50:32.630
So you're probably all going,
in your head, more or less
00:50:32.630 --> 00:50:35.530
like I was when I took
Intro Quantum, like-- yeah,
00:50:35.530 --> 00:50:38.750
but I mean, it's just, you know,
you're adding complex numbers.
00:50:38.750 --> 00:50:40.631
But trust me on this one.
00:50:40.631 --> 00:50:42.630
This is where it's all starting.
00:50:42.630 --> 00:50:45.940
OK so let's go back to this.
00:50:45.940 --> 00:50:47.562
Similarly, let's
look at this example.
00:50:47.562 --> 00:50:48.770
We've taken the norm squared.
00:50:48.770 --> 00:50:50.395
And now we have an
interference effect.
00:50:50.395 --> 00:50:52.130
And now, our probability
distribution,
00:50:52.130 --> 00:50:55.900
instead of being totally trivial
and containing no information,
00:50:55.900 --> 00:50:58.400
our probability distribution
now contains some information
00:50:58.400 --> 00:51:00.140
about the position
of the object.
00:51:00.140 --> 00:51:01.150
It's likely to be here.
00:51:01.150 --> 00:51:03.660
It is unlikely to be
here, likely and unlikely.
00:51:03.660 --> 00:51:05.580
We now have some
position information.
00:51:05.580 --> 00:51:07.620
We don't have enough
to say where it is.
00:51:07.620 --> 00:51:09.810
But x is-- you have
some information.
00:51:13.920 --> 00:51:15.860
Now, our uncertainty
still gigantic.
00:51:15.860 --> 00:51:18.640
Delta x is still huge.
00:51:18.640 --> 00:51:20.840
But OK, we just added
together two plane waves.
00:51:24.360 --> 00:51:25.060
Yeah?
00:51:25.060 --> 00:51:28.210
AUDIENCE: Why is
the probability not
00:51:28.210 --> 00:51:30.516
big, small, small,
big, small, small?
00:51:30.516 --> 00:51:31.390
PROFESSOR: Excellent.
00:51:31.390 --> 00:51:35.320
This was the real part
of the wave function.
00:51:35.320 --> 00:51:38.111
And the wave function
is a complex quantity.
00:51:38.111 --> 00:51:39.610
When you take e to
the i k1, and let
00:51:39.610 --> 00:51:41.600
me do this on the chalkboard.
00:51:41.600 --> 00:51:44.430
When we take e to the i
k1 x plus e to the i k2
00:51:44.430 --> 00:51:48.270
x-- Let me write this
slightly differently--
00:51:48.270 --> 00:51:54.470
e to the i a plus e to the i
b and take its norm squared.
00:51:54.470 --> 00:51:56.190
So this is equal to--
I'm going to write
00:51:56.190 --> 00:51:57.856
this in a slightly
more suggestive way--
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
00:52:03.450 --> 00:52:09.350
minus a parentheses
norm squared.
00:52:09.350 --> 00:52:12.300
So first off, the norm
squared of a product of things
00:52:12.300 --> 00:52:14.700
is the product of
the norm squareds.
00:52:14.700 --> 00:52:16.280
So I can do that.
00:52:16.280 --> 00:52:21.500
And this overall phase, the norm
squared of a phase is just 1,
00:52:21.500 --> 00:52:22.340
so that's just 1.
00:52:25.320 --> 00:52:27.850
So now we have the norm squared
of 1 plus a complex number.
00:52:32.180 --> 00:52:34.459
And so the norm squared of
1 is going to give me 1.
00:52:34.459 --> 00:52:37.000
The norm squared of the complex
number is going to give me 1.
00:52:37.000 --> 00:52:39.333
And the cross terms are going
to give me the real part--
00:52:39.333 --> 00:52:41.435
twice the real part--
of e to the i b minus
00:52:41.435 --> 00:52:45.550
a, which is going to be
equal to cosine of b minus a.
00:52:47.940 --> 00:52:49.440
And so what you see
here is that you
00:52:49.440 --> 00:52:52.400
have a single frequency
in the superposition.
00:53:00.690 --> 00:53:02.850
So good, our
uncertainty is large.
00:53:02.850 --> 00:53:05.850
So let's look at
this second example
00:53:05.850 --> 00:53:08.010
in a little more detail.
00:53:08.010 --> 00:53:11.800
By superimposing two states with
wavelength lambda1 and lambda2
00:53:11.800 --> 00:53:22.754
or k1 and k2 we get
something that, OK,
00:53:22.754 --> 00:53:24.170
it's still not
well localized-- we
00:53:24.170 --> 00:53:25.753
don't know where the
particle is going
00:53:25.753 --> 00:53:28.360
to be-- but it's better
localized than it was before,
00:53:28.360 --> 00:53:29.710
right?
00:53:29.710 --> 00:53:33.610
What happens if we superpose
with three wavelengths,
00:53:33.610 --> 00:53:35.920
or four, or more?
00:53:35.920 --> 00:53:40.810
So for that, I want to pull
out a Mathematica package.
00:53:40.810 --> 00:53:43.410
You guys should all have
seen Fourier analysis
00:53:43.410 --> 00:53:58.480
in 18.03, but just in case,
I'm putting on the web page,
00:53:58.480 --> 00:54:01.000
on the Stellar page, a
notebook that walks you
00:54:01.000 --> 00:54:03.540
through the basics of Fourier
analysis in Mathematica.
00:54:03.540 --> 00:54:05.470
You should all be
fluent in Mathematica.
00:54:05.470 --> 00:54:07.782
If you're not, you
should probably
00:54:07.782 --> 00:54:08.740
come up to speed on it.
00:54:08.740 --> 00:54:11.580
That's not what we wanted.
00:54:11.580 --> 00:54:13.170
Let's try that again.
00:54:13.170 --> 00:54:15.680
There we go.
00:54:15.680 --> 00:54:23.680
Oh, that's awesome-- where
by awesome, I mean not.
00:54:23.680 --> 00:54:24.180
It's coming.
00:54:24.180 --> 00:54:24.680
OK, good.
00:54:32.790 --> 00:54:35.470
I'm not even going to mess with
the screens after last time.
00:54:35.470 --> 00:54:43.150
So I'm not going to go through
the details of this package,
00:54:43.150 --> 00:54:50.220
but what this does is walk
you through the superposition
00:54:50.220 --> 00:54:51.885
of wave packets.
00:54:51.885 --> 00:54:56.450
So here I'm looking at the
probability distribution coming
00:54:56.450 --> 00:54:59.090
from summing up a
bunch of plane waves
00:54:59.090 --> 00:55:00.490
with some definite frequency.
00:55:00.490 --> 00:55:03.600
So here it's just one.
00:55:03.600 --> 00:55:05.420
That's one wave,
so first we have--
00:55:05.420 --> 00:55:15.720
let me make this bigger-- yes,
stupid Mathematica tricks.
00:55:15.720 --> 00:55:19.470
So here we have
the wave function
00:55:19.470 --> 00:55:23.190
and here we have the probability
distribution, the norm squared.
00:55:23.190 --> 00:55:25.294
And it's sort of
badly normalized here.
00:55:25.294 --> 00:55:26.460
So that's for a single wave.
00:55:26.460 --> 00:55:30.550
And as you see, the probability
distribution is constant.
00:55:30.550 --> 00:55:32.760
And that's not 0,
that's 0.15, it's
00:55:32.760 --> 00:55:35.220
just that I arbitrarily
normalized this.
00:55:35.220 --> 00:55:37.790
So let's add two plane waves.
00:55:37.790 --> 00:55:40.400
And now what you see is the
same effect as we had here.
00:55:40.400 --> 00:55:42.625
You see a slightly more
localized wave function.
00:55:42.625 --> 00:55:45.000
Now you have a little bit of
structure in the probability
00:55:45.000 --> 00:55:45.870
distribution.
00:55:45.870 --> 00:55:47.590
So there's the structure in
the probability distribution.
00:55:47.590 --> 00:55:48.430
We have a little
more information
00:55:48.430 --> 00:55:50.513
about where the particle
is more likely to be here
00:55:50.513 --> 00:55:52.600
than it is to be here.
00:55:52.600 --> 00:55:54.540
Let's add one more.
00:55:54.540 --> 00:55:56.770
And as we keep adding
more and more plane waves
00:55:56.770 --> 00:55:59.290
to our superposition, the wave
function and the probability
00:55:59.290 --> 00:56:00.873
distribution associated
with it become
00:56:00.873 --> 00:56:04.820
more and more
well-localized until, as we
00:56:04.820 --> 00:56:07.960
go to very high numbers of plane
waves that we're superposing,
00:56:07.960 --> 00:56:11.870
we get an extremely narrow
probability distribution--
00:56:11.870 --> 00:56:14.280
and wave function, for that
matter-- extremely narrow
00:56:14.280 --> 00:56:16.655
corresponding to a particle
that's very likely to be here
00:56:16.655 --> 00:56:19.378
and unlikely to
be anywhere else.
00:56:19.378 --> 00:56:20.940
Everyone cool with that?
00:56:20.940 --> 00:56:22.080
What's the expense?
00:56:22.080 --> 00:56:24.324
Want have we lost
in the process?
00:56:24.324 --> 00:56:25.740
Well we know with
great confidence
00:56:25.740 --> 00:56:29.420
now that the particle will be
found here upon observation.
00:56:29.420 --> 00:56:30.790
But what will its momentum be?
00:56:33.899 --> 00:56:35.940
Yeah, now it's the
superposition of a whole bunch
00:56:35.940 --> 00:56:37.130
of different momenta.
00:56:37.130 --> 00:56:39.100
So if it's a superposition
of a whole bunch
00:56:39.100 --> 00:56:44.190
of different
momenta, here this is
00:56:44.190 --> 00:56:46.719
like superposition of a whole
bunch of different positions--
00:56:46.719 --> 00:56:48.260
likely to be here,
likely to be here,
00:56:48.260 --> 00:56:49.570
likely to be here,
likely to be here.
00:56:49.570 --> 00:56:51.111
What's our knowledge
of its position?
00:56:51.111 --> 00:56:52.480
It's not very good.
00:56:52.480 --> 00:56:54.420
Similarly, now that
we have superposed
00:56:54.420 --> 00:56:56.420
many different momenta
with comparable strength.
00:56:56.420 --> 00:56:58.540
In fact, here they were
all with unit strength.
00:56:58.540 --> 00:57:01.150
We now have no information about
what the momentum is anymore.
00:57:01.150 --> 00:57:02.941
It could be anything
in that superposition.
00:57:05.490 --> 00:57:10.320
So now we're seeing
quite sharply
00:57:10.320 --> 00:57:11.470
the uncertainty relation.
00:57:11.470 --> 00:57:12.320
And here it is.
00:57:15.234 --> 00:57:16.650
So the uncertainty
relation is now
00:57:16.650 --> 00:57:18.140
pretty clear from these guys.
00:57:20.722 --> 00:57:21.555
So that didn't work?
00:57:26.490 --> 00:57:27.870
And I'm going to leave it alone.
00:57:27.870 --> 00:57:29.536
This is enough for
the Fourier analysis,
00:57:29.536 --> 00:57:31.520
but that Fourier
package is available
00:57:31.520 --> 00:57:35.910
with extensive commentary
on the Stellar web page.
00:57:35.910 --> 00:57:38.012
AUDIENCE: Now is
that sharp definition
00:57:38.012 --> 00:57:41.460
in the position caused by
the interference between all
00:57:41.460 --> 00:57:43.040
those waves and all that--
00:57:43.040 --> 00:57:44.650
PROFESSOR: That's
exactly what it is.
00:57:44.650 --> 00:57:45.380
Precisely.
00:57:45.380 --> 00:57:48.060
It's precisely the interference
between the different momentum
00:57:48.060 --> 00:57:50.185
nodes that leads to
certainty in the position.
00:57:50.185 --> 00:57:51.060
That's exactly right.
00:57:51.060 --> 00:57:52.035
Yeah.
00:57:52.035 --> 00:57:53.951
AUDIENCE: So as we're
certain of the position,
00:57:53.951 --> 00:57:56.339
we will not be certain
of the momentum.
00:57:56.339 --> 00:57:57.130
PROFESSOR: Exactly.
00:57:57.130 --> 00:57:57.970
And here we are.
00:57:57.970 --> 00:58:00.915
So in this example, we have
no idea what the position is,
00:58:00.915 --> 00:58:03.200
but we're quite confident
of the momentum.
00:58:03.200 --> 00:58:05.950
Here we have no idea
what the position is,
00:58:05.950 --> 00:58:08.910
but we have great
confidence in the momentum.
00:58:08.910 --> 00:58:12.720
Similarly here, we have
less perfect confidence
00:58:12.720 --> 00:58:15.900
of the position, and here we
have less perfect confidence
00:58:15.900 --> 00:58:17.335
in the momentum.
00:58:17.335 --> 00:58:18.960
It would be nice to
be able to estimate
00:58:18.960 --> 00:58:20.793
what our uncertainty
is in the momentum here
00:58:20.793 --> 00:58:23.580
and what our uncertainty
is in the position here.
00:58:23.580 --> 00:58:25.470
So we're going to
have to do that.
00:58:25.470 --> 00:58:27.178
That's going to be
one of our next tasks.
00:58:30.480 --> 00:58:31.160
Other questions?
00:58:31.160 --> 00:58:32.404
Yeah.
00:58:32.404 --> 00:58:35.830
AUDIENCE: In this half
of the blackboard,
00:58:35.830 --> 00:58:37.871
you said, obviously, if
we do it a bunch of times
00:58:37.871 --> 00:58:39.859
it'll have more in
the x2 than in the x1.
00:58:39.859 --> 00:58:40.853
PROFESSOR: Yes.
00:58:40.853 --> 00:58:44.129
AUDIENCE: The average, it will
never physically be at that--
00:58:44.129 --> 00:58:46.670
PROFESSOR: Yeah, that's right,
so, because it's a probability
00:58:46.670 --> 00:58:48.810
distribution, it won't
be exactly at that point.
00:58:48.810 --> 00:58:50.800
But it'll be nearby.
00:58:50.800 --> 00:58:56.180
OK, so in order to be more
precise-- And so for example
00:58:56.180 --> 00:58:59.530
for this what we do
here's a quick question.
00:58:59.530 --> 00:59:02.629
How well do you know the
position of this particle?
00:59:02.629 --> 00:59:03.420
Pretty well, right?
00:59:03.420 --> 00:59:06.700
But how well do you
know its momentum?
00:59:06.700 --> 00:59:10.735
Well, we'd all like to say
not very, but tell me why.
00:59:10.735 --> 00:59:15.024
Why is your uncertainty in the
momentum of the particle large?
00:59:15.024 --> 00:59:16.980
AUDIENCE: Heisenberg's
uncertainty principle.
00:59:16.980 --> 00:59:17.860
PROFESSOR: Yeah,
but that's a cheat
00:59:17.860 --> 00:59:19.550
because we haven't actually
proved Heisenberg's uncertainty
00:59:19.550 --> 00:59:19.860
principle.
00:59:19.860 --> 00:59:20.630
It's just something
we're inheriting.
00:59:20.630 --> 00:59:21.820
AUDIENCE: I believe it.
00:59:21.820 --> 00:59:22.730
PROFESSOR: I believe it, too.
00:59:22.730 --> 00:59:24.370
But I want a better
argument because I
00:59:24.370 --> 00:59:25.745
believe all sorts
of crazy stuff.
00:59:25.745 --> 00:59:28.880
So-- I really do.
00:59:28.880 --> 00:59:31.850
Black holes, fluids, I mean
look, don't get me started.
00:59:31.850 --> 00:59:32.395
Yeah.
00:59:32.395 --> 00:59:34.520
AUDIENCE: You can take the
Fourier transform of it.
00:59:34.520 --> 00:59:35.190
PROFESSOR: Yeah, excellent.
00:59:35.190 --> 00:59:36.650
OK, we'll get to
that in just one sec.
00:59:36.650 --> 00:59:38.525
So before taking the
Fourier transform, which
00:59:38.525 --> 00:59:40.330
is an excellent--
so the answer was,
00:59:40.330 --> 00:59:41.323
just take a Fourier
transform, that's
00:59:41.323 --> 00:59:42.180
going to give you
some information.
00:59:42.180 --> 00:59:43.700
We're going to do
that in just a moment.
00:59:43.700 --> 00:59:45.158
But before we do
Fourier transform,
00:59:45.158 --> 00:59:48.490
just intuitively, why would
de Broglie look at this
00:59:48.490 --> 00:59:50.869
and say, no, that doesn't
have a definite momentum.
00:59:50.869 --> 00:59:52.452
AUDIENCE: There's
no clear wavelength.
00:59:52.452 --> 00:59:54.060
PROFESSOR: Yeah, there's
no wavelength, right?
00:59:54.060 --> 00:59:56.237
It's not periodic by any
stretch of the imagination.
00:59:56.237 --> 00:59:58.570
It doesn't look like a thing
with a definite wavelength.
00:59:58.570 --> 01:00:01.774
And de Broglie said, look, if
you have a definite wavelength
01:00:01.774 --> 01:00:03.190
then you have a
definite momentum.
01:00:03.190 --> 01:00:04.130
And if you have a
definite momentum,
01:00:04.130 --> 01:00:05.820
you have a definite wavelength.
01:00:05.820 --> 01:00:07.780
This is not a wave with
a definite wavelength,
01:00:07.780 --> 01:00:10.440
so it is not corresponding
to the wave function
01:00:10.440 --> 01:00:12.720
for a particle with
a definite momentum.
01:00:12.720 --> 01:00:17.570
So our momentum is
unknown-- so this is large.
01:00:17.570 --> 01:00:20.170
And similarly, here, our
uncertainty in the momentum
01:00:20.170 --> 01:00:20.760
is large.
01:00:24.820 --> 01:00:26.480
So to do better
than this, we need
01:00:26.480 --> 01:00:28.400
to introduce the
Fourier transform,
01:00:28.400 --> 01:00:30.530
and I want to do that now.
01:00:30.530 --> 01:00:34.207
So you should all have seen
Fourier series in 8.03.
01:00:34.207 --> 01:00:36.040
Now we're going to do
the Fourier transform.
01:00:36.040 --> 01:00:37.623
And I'm going to
introduce this to you
01:00:37.623 --> 01:00:40.831
in 8.04 conventions
in the following way.
01:00:40.831 --> 01:00:42.330
And the theorem
says the following--
01:00:42.330 --> 01:00:44.913
we're not going to prove it by
any stretch of the imagination,
01:00:44.913 --> 01:00:48.900
but the theorem
says-- any function
01:00:48.900 --> 01:00:52.070
f of x that is sufficiently
well-behaved-- it shouldn't be
01:00:52.070 --> 01:00:55.630
discontinuous, it
shouldn't be singular--
01:00:55.630 --> 01:00:57.980
any reasonably well-behaved,
non-stupid function f
01:00:57.980 --> 01:01:19.574
of x can be built by superposing
enough plane waves of the form
01:01:19.574 --> 01:01:20.115
e to the ikx.
01:01:23.950 --> 01:01:26.720
Enough may be infinite.
01:01:26.720 --> 01:01:33.100
So any function f of x can be
expressed as 1 over root 2 pi,
01:01:33.100 --> 01:01:35.740
and this root 2 pi is a choice
of normalization-- everyone
01:01:35.740 --> 01:01:37.240
has their own
conventions, and these
01:01:37.240 --> 01:01:39.530
are the ones we'll be
using in 8.04 throughout--
01:01:39.530 --> 01:01:48.480
minus infinity to infinity
dk f tilde of k e to the ikx.
01:01:56.060 --> 01:01:58.980
So here, what we're
doing is, we're
01:01:58.980 --> 01:02:03.480
summing over plane waves
of the form e to the ikx.
01:02:03.480 --> 01:02:06.500
These are modes with a definite
wavelength 2 pi upon k.
01:02:11.920 --> 01:02:18.430
f tilde of k is telling us
the amplitude of the wave
01:02:18.430 --> 01:02:22.500
with wavelengths lambda
or wave number k.
01:02:22.500 --> 01:02:26.150
And we sum over all
possible values.
01:02:26.150 --> 01:02:28.360
And the claim is,
any function can
01:02:28.360 --> 01:02:31.840
be expressed as a superposition
of plane waves in this form.
01:02:31.840 --> 01:02:33.192
Cool?
01:02:33.192 --> 01:02:34.650
And this is for
functions which are
01:02:34.650 --> 01:02:37.960
non-periodic on the real line,
rather than periodic functions
01:02:37.960 --> 01:02:40.530
on the interval, which is what
you should've seen in 8.03.
01:02:40.530 --> 01:02:44.400
Now, conveniently, if
you know f tilde of k,
01:02:44.400 --> 01:02:46.350
you can compute f of
x by doing the sum.
01:02:46.350 --> 01:02:47.809
But suppose you
know f of x and you
01:02:47.809 --> 01:02:49.641
want to determine what
the coefficients are,
01:02:49.641 --> 01:02:50.800
the expansion coefficients.
01:02:50.800 --> 01:02:52.920
That's the inverse
Fourier transform.
01:02:52.920 --> 01:02:56.720
And the statement for
that is that f tilde of k
01:02:56.720 --> 01:03:02.410
is equal to 1 over root 2 pi
integral from minus infinity
01:03:02.410 --> 01:03:08.620
to infinity dx f of
x e to the minus ikx.
01:03:11.249 --> 01:03:13.540
OK, that's sometimes referred
to as the inverse Fourier
01:03:13.540 --> 01:03:14.040
transform.
01:03:16.620 --> 01:03:20.080
And here's something
absolutely essential.
01:03:20.080 --> 01:03:22.600
f tilde of k, the Fourier
transform coefficients
01:03:22.600 --> 01:03:26.070
of f of x, are
completely equivalent.
01:03:26.070 --> 01:03:29.670
If you know f of x, you
can determine f tilde of k.
01:03:29.670 --> 01:03:31.180
And if you know
f tilde of k, you
01:03:31.180 --> 01:03:33.762
can determine f of x
by just doing a sum,
01:03:33.762 --> 01:03:34.720
by just adding them up.
01:03:39.230 --> 01:03:42.030
So now here's the physical
version of this-- oh,
01:03:42.030 --> 01:03:47.786
I can't slide that out--
I'm now going to put here.
01:03:50.400 --> 01:03:50.900
Oh.
01:03:50.900 --> 01:03:51.600
No, I'm not.
01:03:51.600 --> 01:03:54.300
I'm going to put that down here.
01:03:54.300 --> 01:03:57.000
So the physical
version of this is
01:03:57.000 --> 01:04:26.270
that any wave function
psi of x can be expressed
01:04:26.270 --> 01:04:44.640
as the superposition
in the form psi of x
01:04:44.640 --> 01:04:50.120
is equal to 1 over root 2 pi
integral from minus infinity
01:04:50.120 --> 01:05:13.120
to infinity dk psi tilde of k
e to the ikx of states, or wave
01:05:13.120 --> 01:05:26.060
functions, with a definite
momentum p is equal to h bar k.
01:05:39.060 --> 01:05:47.380
And so now, it's useful
to sketch the Fourier
01:05:47.380 --> 01:05:52.030
transforms of each
of these functions.
01:05:52.030 --> 01:05:54.790
In fact, we want this up here.
01:06:05.330 --> 01:06:08.200
So here we have the function and
its probability distribution.
01:06:08.200 --> 01:06:14.210
Now I want to draw the Fourier
transforms of these guys.
01:06:14.210 --> 01:06:18.750
So here's psi tilde
of k, a function
01:06:18.750 --> 01:06:20.254
of a different
variable than of x,
01:06:20.254 --> 01:06:21.670
but nonetheless,
it's illuminating
01:06:21.670 --> 01:06:23.003
to draw them next to each other.
01:06:33.179 --> 01:06:34.720
And again, I'm
drawing the real part.
01:06:39.030 --> 01:06:43.010
And here, x2-- had I had
my druthers about me,
01:06:43.010 --> 01:06:45.605
I would have put x2
at a larger value.
01:06:51.070 --> 01:06:53.190
Good, so it's further
off to the right there.
01:07:06.210 --> 01:07:10.890
I'm so loathe to erase the
superposition principle.
01:07:10.890 --> 01:07:15.020
But fortunately,
I'm not there yet.
01:07:15.020 --> 01:07:20.200
Let's look at the Fourier
transform of these guys.
01:07:26.730 --> 01:07:39.490
The Fourier transform
of this guy-- this is k.
01:07:39.490 --> 01:07:43.640
Psi tilde of k well,
that's something
01:07:43.640 --> 01:07:45.840
with a definite value of k.
01:07:45.840 --> 01:07:55.965
And it's Fourier transform--
this is 0-- there's k1.
01:07:55.965 --> 01:08:05.690
And for this guy--
there's 0-- k2.
01:08:13.770 --> 01:08:16.240
And now if we look at
the Fourier transforms
01:08:16.240 --> 01:08:22.880
of these guys, see,
this way I don't
01:08:22.880 --> 01:08:36.420
have to erase the superposition
principle-- and the Fourier
01:08:36.420 --> 01:09:10.130
transform of this guy,
so note that there's
01:09:10.130 --> 01:09:13.529
a sort of pleasing
symmetry here.
01:09:13.529 --> 01:09:15.800
If your wave function
is well localized,
01:09:15.800 --> 01:09:18.660
corresponding to a reasonably
well-defined position,
01:09:18.660 --> 01:09:22.080
then your Fourier transform
is not well localized,
01:09:22.080 --> 01:09:26.479
corresponding to not
having a definite momentum.
01:09:26.479 --> 01:09:30.580
On the other hand, if you
have definite momentum,
01:09:30.580 --> 01:09:32.819
your position is
not well defined,
01:09:32.819 --> 01:09:40.160
but the Fourier transform
has a single peak
01:09:40.160 --> 01:09:43.939
at the value of k corresponding
to the momentum of your wave
01:09:43.939 --> 01:09:44.460
function.
01:09:44.460 --> 01:09:46.590
Everyone cool with that?
01:09:46.590 --> 01:09:48.840
So here's a question-- sorry,
there was a raised hand.
01:09:48.840 --> 01:09:50.120
Yeah?
01:09:50.120 --> 01:09:54.245
AUDIENCE: Are we going
to learn in this class
01:09:54.245 --> 01:09:56.472
how to determine the
Fourier transforms
01:09:56.472 --> 01:09:58.679
of these non-stupid functions?
01:09:58.679 --> 01:10:00.470
PROFESSOR: Yes, that
will be your homework.
01:10:00.470 --> 01:10:02.060
On your homework is
an extensive list
01:10:02.060 --> 01:10:06.210
of functions for you to
compute Fourier transforms of.
01:10:06.210 --> 01:10:11.620
And that will be the job of
problem sets and recitation.
01:10:11.620 --> 01:10:13.880
So Fourier series
and computing-- yeah,
01:10:13.880 --> 01:10:16.970
you know what's
coming-- Fourier series
01:10:16.970 --> 01:10:21.540
are assumed to have been covered
for everyone in 8.03 and 18.03
01:10:21.540 --> 01:10:23.691
in some linear
combination thereof.
01:10:23.691 --> 01:10:24.690
And Fourier transforms--
01:10:24.690 --> 01:10:27.899
[LAUGHTER AND GROANS]
01:10:27.899 --> 01:10:28.690
I couldn't help it.
01:10:28.690 --> 01:10:33.390
So Fourier transforms
are a slight embiggening
01:10:33.390 --> 01:10:36.090
of the space of Fourier
series, because we're not
01:10:36.090 --> 01:10:37.430
looking at periodic functions.
01:10:37.430 --> 01:10:39.636
AUDIENCE: So when we're
doing the Fourier transforms
01:10:39.636 --> 01:10:41.552
of a wave function, we're
basically writing it
01:10:41.552 --> 01:10:44.500
as a continuous set
of different waves.
01:10:44.500 --> 01:10:46.050
Can we write it
as a discrete set?
01:10:46.050 --> 01:10:47.230
So as a Fourier series?
01:10:47.230 --> 01:10:49.130
PROFESSOR: Absolutely,
so, however,
01:10:49.130 --> 01:10:51.470
what is true about
Fourier series?
01:10:51.470 --> 01:10:53.190
When you use a discrete
set of momenta,
01:10:53.190 --> 01:10:55.980
which are linear, which are--
It must be a periodic function,
01:10:55.980 --> 01:10:56.580
exactly.
01:10:56.580 --> 01:10:59.690
So here what we've
done is, we've said,
01:10:59.690 --> 01:11:01.410
look, we're writing
our wave function,
01:11:01.410 --> 01:11:05.350
our arbitrary wave function,
as a continuous superposition
01:11:05.350 --> 01:11:08.059
of a continuous value
of possible momenta.
01:11:08.059 --> 01:11:09.183
This is absolutely correct.
01:11:09.183 --> 01:11:10.490
This is exactly
what we're doing.
01:11:10.490 --> 01:11:11.880
However, that's
kind of annoying,
01:11:11.880 --> 01:11:13.840
because maybe you
just want one momentum
01:11:13.840 --> 01:11:15.257
and two momenta
and three momenta.
01:11:15.257 --> 01:11:16.714
What if you want
a discrete series?
01:11:16.714 --> 01:11:17.600
So discrete is fine.
01:11:17.600 --> 01:11:19.196
But if you make
that discrete series
01:11:19.196 --> 01:11:20.695
integer-related to
each other, which
01:11:20.695 --> 01:11:22.200
is what you do with
Fourier series,
01:11:22.200 --> 01:11:25.120
you force the function
f of x to be periodic.
01:11:25.120 --> 01:11:26.840
And we don't want
that, in general,
01:11:26.840 --> 01:11:29.100
because life isn't periodic.
01:11:29.100 --> 01:11:31.290
Thank goodness, right?
01:11:31.290 --> 01:11:35.710
I mean, there's like one film
in which it-- but so-- it's
01:11:35.710 --> 01:11:36.720
a good movie.
01:11:36.720 --> 01:11:38.274
So that's the
essential difference
01:11:38.274 --> 01:11:40.190
between Fourier series
and Fourier transforms.
01:11:40.190 --> 01:11:42.710
Fourier transforms
are continuous in k
01:11:42.710 --> 01:11:47.500
and do not assume
periodicity of the function.
01:11:47.500 --> 01:11:48.360
Other questions?
01:11:48.360 --> 01:11:49.070
Yeah.
01:11:49.070 --> 01:11:50.570
AUDIENCE: So
basically, the Fourier
01:11:50.570 --> 01:11:54.486
transform associates
an amplitude
01:11:54.486 --> 01:11:58.230
and a phase for each of
the individual momenta.
01:11:58.230 --> 01:11:59.351
PROFESSOR: Precisely.
01:11:59.351 --> 01:12:00.100
Precisely correct.
01:12:00.100 --> 01:12:01.750
So let me say that again.
01:12:01.750 --> 01:12:04.110
So the question was--
so a Fourier transform
01:12:04.110 --> 01:12:06.050
effectively
associates a magnitude
01:12:06.050 --> 01:12:09.890
and a phase for each
possible wave vector.
01:12:09.890 --> 01:12:10.940
And that's exactly right.
01:12:10.940 --> 01:12:13.317
So here there's some
amplitude and phase--
01:12:13.317 --> 01:12:14.900
this is a complex
number, because this
01:12:14.900 --> 01:12:17.274
is a complex function-- there's
some complex number which
01:12:17.274 --> 01:12:19.800
is an amplitude and
a phase associated
01:12:19.800 --> 01:12:22.830
to every possible momentum
going into the superposition.
01:12:22.830 --> 01:12:24.280
That amplitude may be 0.
01:12:24.280 --> 01:12:27.700
There may be no contribution
for a large number of momenta,
01:12:27.700 --> 01:12:29.330
or maybe insignificantly small.
01:12:29.330 --> 01:12:31.330
But it is indeed
doing precisely that.
01:12:31.330 --> 01:12:34.930
It is associating an amplitude
and a phase for every plane
01:12:34.930 --> 01:12:37.950
wave, with every different
value of momentum.
01:12:37.950 --> 01:12:44.346
And you can compute,
before panicking,
01:12:44.346 --> 01:12:45.970
precisely what that
amplitude and phase
01:12:45.970 --> 01:12:49.239
is by using the inverse
Fourier transform.
01:12:49.239 --> 01:12:50.280
So there's no magic here.
01:12:50.280 --> 01:12:51.280
You just calculate.
01:12:51.280 --> 01:12:57.450
You can use your calculator,
literally-- I hate that word.
01:12:57.450 --> 01:13:00.320
OK so now, here's
a natural question.
01:13:00.320 --> 01:13:04.340
So if this is the Fourier
transform of our wave function,
01:13:04.340 --> 01:13:06.770
we already knew that this
wave function corresponded
01:13:06.770 --> 01:13:08.710
to having a definite--
from de Broglie,
01:13:08.710 --> 01:13:11.270
we know that it has
a definite momentum.
01:13:11.270 --> 01:13:14.650
We also see that its Fourier
transform looks like this.
01:13:14.650 --> 01:13:17.620
So that leads to a
reasonable guess.
01:13:17.620 --> 01:13:20.260
What do you think the
probability distribution P of k
01:13:20.260 --> 01:13:23.930
is-- the probability
density to find the momentum
01:13:23.930 --> 01:13:26.620
to have wave vector h-bar k?
01:13:26.620 --> 01:13:27.517
AUDIENCE: [INAUDIBLE]
01:13:27.517 --> 01:13:29.600
PROFESSOR: Yeah, that's a
pretty reasonable guess.
01:13:29.600 --> 01:13:31.390
So we're totally
pulling this out
01:13:31.390 --> 01:13:34.460
of the dark-- psi
of k norm squared.
01:13:34.460 --> 01:13:35.920
OK, well let's
see if that works.
01:13:35.920 --> 01:13:37.450
So psy of k norm
squared for this
01:13:37.450 --> 01:13:42.715
is going to give us a nice,
well localized function.
01:13:42.715 --> 01:13:44.090
And so that makes
a lot of sense.
01:13:44.090 --> 01:13:46.230
That's exactly what
we expected, right?
01:13:46.230 --> 01:13:48.950
Definite value of P with
very small uncertainty.
01:13:48.950 --> 01:13:50.810
Similarly here.
01:13:50.810 --> 01:13:54.950
Definite value of P, with
a very small uncertainty.
01:13:54.950 --> 01:13:57.340
Rock on.
01:13:57.340 --> 01:13:59.860
However, let's look at this guy.
01:13:59.860 --> 01:14:02.510
What is the expected value
of P if this is the Fourier
01:14:02.510 --> 01:14:05.050
transform?
01:14:05.050 --> 01:14:08.610
Well remember, we have to take
the norm squared, and psi of k
01:14:08.610 --> 01:14:12.050
was e to the i k x1--
the Fourier transform.
01:14:12.050 --> 01:14:15.130
You will do much
practice on taking
01:14:15.130 --> 01:14:17.360
Fourier transforms
on the problem set.
01:14:17.360 --> 01:14:19.970
Where did my eraser go?
01:14:19.970 --> 01:14:20.795
There it is.
01:14:24.260 --> 01:14:27.340
Farewell, principle one.
01:14:27.340 --> 01:14:30.300
So what does norm squared
of psi tilde look like?
01:14:30.300 --> 01:14:31.969
Well, just like before,
the norm squared
01:14:31.969 --> 01:14:34.510
is constant, because the norm
squared of a phase is constant.
01:14:34.510 --> 01:14:36.780
And again, the
norm squared-- this
01:14:36.780 --> 01:14:42.180
is psi tilde of k norm
squared-- we believe,
01:14:42.180 --> 01:14:45.410
we're conjecturing
this is P of k.
01:14:45.410 --> 01:14:47.930
You will prove this relation
on your problem set.
01:14:47.930 --> 01:14:51.260
You'll prove that it follow
from what we said before.
01:14:51.260 --> 01:14:53.610
And similarly, this is
constant-- e to the i k x2.
01:14:58.320 --> 01:15:02.160
So now we have no
knowledge of the momenta.
01:15:02.160 --> 01:15:04.320
So that also fits.
01:15:04.320 --> 01:15:06.534
The momenta is, we have no idea.
01:15:06.534 --> 01:15:07.575
And uncertainty is large.
01:15:07.575 --> 01:15:10.330
And the momenta is,
we have no idea.
01:15:10.330 --> 01:15:11.760
And the uncertainty is large.
01:15:11.760 --> 01:15:13.310
So in all these
cases, we see that we
01:15:13.310 --> 01:15:15.600
satisfy quite nicely the
uncertainty relation--
01:15:15.600 --> 01:15:18.610
small position momentum,
large momentum uncertainty.
01:15:18.610 --> 01:15:21.210
Large position
uncertainty, we're
01:15:21.210 --> 01:15:24.410
allowed to have small
momentum uncertainty.
01:15:24.410 --> 01:15:26.724
And here, it's a little
more complicated.
01:15:26.724 --> 01:15:28.640
We have a little bit of
knowledge of position,
01:15:28.640 --> 01:15:30.650
and we have a little bit of
knowledge of the momenta.
01:15:30.650 --> 01:15:31.900
We have a little bit of
knowledge of position,
01:15:31.900 --> 01:15:34.430
and we have a little bit
of knowledge of momenta.
01:15:34.430 --> 01:15:36.490
So we'll walk through
examples with superposition
01:15:36.490 --> 01:15:37.900
like this on the problem set.
01:15:40.732 --> 01:15:42.190
Last questions
before we get going?
01:15:45.020 --> 01:15:48.170
OK so I have two things
to do before we're done.
01:15:48.170 --> 01:15:51.280
The first is, after lecture
ends, I have clickers.
01:15:51.280 --> 01:15:53.120
And anyone who wants
to borrow clickers,
01:15:53.120 --> 01:15:55.340
you're welcome to come
down and pick them up
01:15:55.340 --> 01:15:58.500
on a first come
first served basis.
01:15:58.500 --> 01:16:01.190
I will start using the
clickers in the next lecture.
01:16:01.190 --> 01:16:03.230
So if you don't already
have one, get one now.
01:16:03.230 --> 01:16:05.240
But the second thing is--
don't get started yet.
01:16:05.240 --> 01:16:07.260
I have a demo to do.
01:16:07.260 --> 01:16:10.749
And last time I told
you-- this is awesome.
01:16:10.749 --> 01:16:12.540
It's like I'm an
experimentalist for a day.
01:16:12.540 --> 01:16:15.270
Last time I told you that one of
the experimental facts of lice
01:16:15.270 --> 01:16:16.400
is-- of lice.
01:16:16.400 --> 01:16:18.899
One of the experimental
facts of lice.
01:16:18.899 --> 01:16:20.440
One of the experimental
facts of life
01:16:20.440 --> 01:16:25.480
is that there is
uncertainty in the world
01:16:25.480 --> 01:16:26.730
and that there is probability.
01:16:26.730 --> 01:16:30.700
There are unlikely
events that happen
01:16:30.700 --> 01:16:32.660
with some probability,
some finite probability.
01:16:32.660 --> 01:16:35.510
And a good example of the
randomness of the real world
01:16:35.510 --> 01:16:37.730
involves radiation.
01:16:37.730 --> 01:16:41.440
So hopefully you can hear this.
01:16:41.440 --> 01:16:44.590
Apparently, I'm not
very radioactive.
01:16:44.590 --> 01:16:46.840
You'd be surprised at the
things that are radioactive.
01:16:50.140 --> 01:16:51.450
Ah, got a little tick.
01:16:51.450 --> 01:16:51.950
Shh.
01:16:55.780 --> 01:17:00.640
This is a plate sold at
an Amish county fair.
01:17:00.640 --> 01:17:05.777
It's called vaseline ware
and it's made of local clays.
01:17:05.777 --> 01:17:09.120
[GEIGER COUNTER CLICKS]
01:17:09.120 --> 01:17:10.230
It's got uranium in it.
01:17:13.620 --> 01:17:17.040
But
01:17:17.040 --> 01:17:20.600
I want to emphasize-- exactly
when something goes click,
01:17:20.600 --> 01:17:21.720
it sounds pretty random.
01:17:21.720 --> 01:17:23.840
And it's actually a better
random number generator
01:17:23.840 --> 01:17:29.790
than anything you'll find in
Mathematica or C. In fact,
01:17:29.790 --> 01:17:32.520
for some purposes, the decay
of radioactive isotopes
01:17:32.520 --> 01:17:35.642
is used as the perfect
random number generator.
01:17:35.642 --> 01:17:37.850
Because it really is totally
random, as far as anyone
01:17:37.850 --> 01:17:38.605
can tell.
01:17:38.605 --> 01:17:39.670
But here's my favorite.
01:17:42.550 --> 01:17:43.970
People used to eat off these.
01:17:43.970 --> 01:17:46.870
[MUCH LOUDER, DENSER CLICKS]
01:17:46.870 --> 01:17:48.750
See you next time.