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PROFESSOR: Anything lingering
and disturbing or bewildering?
00:00:28.750 --> 00:00:29.420
No?
00:00:29.420 --> 00:00:30.570
Nothing?
00:00:30.570 --> 00:00:32.430
All right.
00:00:32.430 --> 00:00:43.285
OK, so the story so far is
basically three postulates.
00:00:46.140 --> 00:00:58.260
The first is that the
configuration of a particle
00:00:58.260 --> 00:01:06.410
is given by, or described
by, a wave function psi of x.
00:01:10.150 --> 00:01:11.310
Yeah?
00:01:11.310 --> 00:01:15.120
So in particular, just to
flesh this out a little more,
00:01:15.120 --> 00:01:17.650
if we were in 3D, for
example-- which we're not.
00:01:17.650 --> 00:01:19.490
We're currently in
our one dimensional
00:01:19.490 --> 00:01:20.710
tripped out tricycles.
00:01:20.710 --> 00:01:23.150
In 3D, the wave function
would be a function
00:01:23.150 --> 00:01:26.010
of all three
positions x, y and z.
00:01:29.860 --> 00:01:36.180
If we had two particles,
our wave function
00:01:36.180 --> 00:01:39.360
would be a function of the
position of each particle.
00:01:39.360 --> 00:01:42.790
x1, x2, and so on.
00:01:42.790 --> 00:01:46.566
So we'll go through lots of
details and examples later on.
00:01:46.566 --> 00:01:47.982
But for the most
part, we're going
00:01:47.982 --> 00:01:49.731
to be sticking with
single particle in one
00:01:49.731 --> 00:01:51.650
dimension for the
next few weeks.
00:01:51.650 --> 00:01:54.290
Now again, I want to emphasize
this is our first pass
00:01:54.290 --> 00:01:56.680
through our definition
of quantum mechanics.
00:01:56.680 --> 00:01:59.720
Once we use the language and
the machinery a little bit,
00:01:59.720 --> 00:02:04.460
we're going to develop a more
general, more coherent set
00:02:04.460 --> 00:02:06.850
of rules or definition
of quantum mechanics.
00:02:06.850 --> 00:02:09.210
But this is our first pass.
00:02:09.210 --> 00:02:12.950
Two, the meaning of
the wave function
00:02:12.950 --> 00:02:18.440
is that the norm squared psi of
x, norm squared, it's complex,
00:02:18.440 --> 00:02:29.450
dx is the probability of
finding the particle- There's
00:02:29.450 --> 00:02:30.320
an n in their.
00:02:30.320 --> 00:02:38.900
Finding the particle-- in the
region between x and x plus dx.
00:02:38.900 --> 00:02:40.690
So psi squared
itself, norm squared,
00:02:40.690 --> 00:02:44.430
is the probability density.
00:02:44.430 --> 00:02:46.550
OK?
00:02:46.550 --> 00:02:52.980
And third, the
superposition principle.
00:02:52.980 --> 00:02:55.130
If there are two
possible configurations
00:02:55.130 --> 00:02:58.250
the system can be in,
which in quantum mechanics
00:02:58.250 --> 00:03:00.570
means two different
wave functions that
00:03:00.570 --> 00:03:05.394
could describe the system
given psi 1 and psi 2, two
00:03:05.394 --> 00:03:06.810
wave functions
that could describe
00:03:06.810 --> 00:03:09.190
two different configurations
of the system.
00:03:09.190 --> 00:03:12.190
For example, the particles here
or the particles over here.
00:03:12.190 --> 00:03:21.360
It's also possible to find
the system in a superposition
00:03:21.360 --> 00:03:25.120
of those two psi is
equal to some arbitrary
00:03:25.120 --> 00:03:34.780
linear combination alpha
psi 1 plus beta psi 2 of x.
00:03:34.780 --> 00:03:35.280
OK?
00:03:41.490 --> 00:03:43.720
So some things to note--
so questions about those
00:03:43.720 --> 00:03:46.320
before we move on?
00:03:46.320 --> 00:03:49.030
No questions?
00:03:49.030 --> 00:03:50.330
Nothing?
00:03:50.330 --> 00:03:52.450
Really?
00:03:52.450 --> 00:03:55.105
You're going to make he
threaten you with something.
00:03:55.105 --> 00:03:56.230
I know there are questions.
00:03:56.230 --> 00:03:57.480
This is not trivial stuff.
00:04:00.560 --> 00:04:02.440
OK.
00:04:02.440 --> 00:04:06.409
So some things to note.
00:04:06.409 --> 00:04:07.825
The first is we
want to normalize.
00:04:11.455 --> 00:04:13.080
We will generally
normalize and require
00:04:13.080 --> 00:04:16.640
that the integral over
all possible positions
00:04:16.640 --> 00:04:19.839
of the probability density
psi of x norm squared
00:04:19.839 --> 00:04:20.839
is equal to 1.
00:04:20.839 --> 00:04:23.830
This is just saying that the
total probability that we find
00:04:23.830 --> 00:04:27.179
the particle somewhere
had better be one.
00:04:27.179 --> 00:04:28.970
This is like saying if
I know a particle is
00:04:28.970 --> 00:04:30.470
in one of two
boxes, because I've
00:04:30.470 --> 00:04:31.950
put a particle in
one of the boxes.
00:04:31.950 --> 00:04:33.510
I just don't remember which one.
00:04:33.510 --> 00:04:35.690
Then the probability that
it's in the first box
00:04:35.690 --> 00:04:37.523
plus probability that
it's in the second box
00:04:37.523 --> 00:04:38.720
must be 100% or one.
00:04:38.720 --> 00:04:42.020
If it's less, then the particle
has simply disappeared.
00:04:42.020 --> 00:04:45.140
And basic rule, things
don't just disappear.
00:04:45.140 --> 00:04:46.665
So probability
should be normalized.
00:04:49.430 --> 00:04:50.890
And this is our prescription.
00:04:50.890 --> 00:04:57.740
So a second thing to note is
that all reasonable, or non
00:04:57.740 --> 00:05:05.210
stupid, functions
psi of x are equally
00:05:05.210 --> 00:05:06.770
reasonable as wave functions.
00:05:16.510 --> 00:05:17.280
OK?
00:05:17.280 --> 00:05:21.300
So this is a very
reasonable function.
00:05:21.300 --> 00:05:22.420
It's nice and smooth.
00:05:22.420 --> 00:05:24.310
It converges to 0 infinity.
00:05:24.310 --> 00:05:27.140
It's got all the nice
properties you might want.
00:05:27.140 --> 00:05:30.005
This is also a
reasonable function.
00:05:30.005 --> 00:05:32.100
It's a little annoying,
but there it is.
00:05:32.100 --> 00:05:36.240
And they're both perfectly
reasonable as wave functions.
00:05:36.240 --> 00:05:38.940
This on the other
hand, not so much.
00:05:38.940 --> 00:05:39.740
So for two reasons.
00:05:39.740 --> 00:05:40.990
First off, it's discontinuous.
00:05:40.990 --> 00:05:43.390
And as you're going to
show in your problem set,
00:05:43.390 --> 00:05:45.462
discontinuities are very
bad for wave functions.
00:05:45.462 --> 00:05:47.420
So we need our wave
functions to be continuous.
00:05:47.420 --> 00:05:49.740
The second is over some
domain it's multi valued.
00:05:49.740 --> 00:05:51.230
There are two different
values of the function.
00:05:51.230 --> 00:05:52.780
That's also bad, because
what's the probability?
00:05:52.780 --> 00:05:54.500
It's the norm squared,
but if it two values,
00:05:54.500 --> 00:05:56.530
two values for the probability,
that doesn't make any sense.
00:05:56.530 --> 00:05:58.071
What's the probability
that I'm going
00:05:58.071 --> 00:06:00.150
to fall over in 10 seconds?
00:06:00.150 --> 00:06:05.000
Well, it's small, but it's not
actually equal to 1% or 3%.
00:06:05.000 --> 00:06:07.600
It's one of those.
00:06:07.600 --> 00:06:10.290
Hopefully is much
lower than that.
00:06:10.290 --> 00:06:14.640
So all reasonable
functions are equally
00:06:14.640 --> 00:06:17.270
reasonable as wave functions.
00:06:17.270 --> 00:06:19.710
And in particular, what
that means is all states
00:06:19.710 --> 00:06:21.520
corresponding to
reasonable wave functions
00:06:21.520 --> 00:06:25.890
are equally reasonable
as physical states.
00:06:25.890 --> 00:06:30.200
There's no primacy in wave
functions or in states.
00:06:30.200 --> 00:06:36.430
However, with that said,
some wave functions
00:06:36.430 --> 00:06:38.520
are more equal than others.
00:06:38.520 --> 00:06:40.120
OK?
00:06:40.120 --> 00:06:42.966
And this is important,
and coming up
00:06:42.966 --> 00:06:44.340
with a good
definition of this is
00:06:44.340 --> 00:06:45.696
going to be an important
challenge for us
00:06:45.696 --> 00:06:47.200
in the next couple of lectures.
00:06:47.200 --> 00:06:49.346
So in particular,
this wave function
00:06:49.346 --> 00:06:50.720
has a nice simple
interpretation.
00:06:50.720 --> 00:06:52.560
If I tell you this
is psi of x, then
00:06:52.560 --> 00:06:55.750
what can you tell me about the
particle whose wave function is
00:06:55.750 --> 00:06:56.560
the psi of x?
00:06:58.240 --> 00:06:59.490
What can you tell me about it?
00:06:59.490 --> 00:07:00.603
What do you know?
00:07:00.603 --> 00:07:02.700
AUDIENCE: [INAUDIBLE].
00:07:02.700 --> 00:07:04.330
PROFESSOR: It's here, right?
00:07:04.330 --> 00:07:05.166
It's not over here.
00:07:05.166 --> 00:07:06.290
Probability is basically 0.
00:07:06.290 --> 00:07:07.770
Probability is large.
00:07:07.770 --> 00:07:10.480
It's pretty much here with
this great confidence.
00:07:10.480 --> 00:07:12.940
What about this guy?
00:07:12.940 --> 00:07:15.190
Less informative, right?
00:07:15.190 --> 00:07:17.540
It's less obvious what this
wave function is telling me.
00:07:17.540 --> 00:07:19.623
So some wave functions are
more equal in the sense
00:07:19.623 --> 00:07:21.030
that they have-- i.e.
00:07:21.030 --> 00:07:22.405
they have simple
interpretations.
00:07:29.220 --> 00:07:35.650
So for example,
this wave function
00:07:35.650 --> 00:07:38.970
continuing on infinitely, this
wave function doesn't tell me
00:07:38.970 --> 00:07:41.116
where the particle is,
but what does it tell me?
00:07:41.116 --> 00:07:42.030
AUDIENCE: Momentum.
00:07:42.030 --> 00:07:43.180
PROFESSOR: The
momentum, exactly.
00:07:43.180 --> 00:07:45.000
So this is giving me
information about the momentum
00:07:45.000 --> 00:07:47.375
of the particle because it
has a well defined wavelength.
00:07:47.375 --> 00:07:50.810
So this one, I would also say
is more equal than this one.
00:07:50.810 --> 00:07:52.790
They're both perfectly
physical, but this one
00:07:52.790 --> 00:07:54.480
has a simple interpretation.
00:07:54.480 --> 00:07:57.670
And that's going to
be important for us.
00:07:57.670 --> 00:08:11.310
Related to that is that any
reasonable function psi of x
00:08:11.310 --> 00:08:24.980
can be expressed as a
superposition of more
00:08:24.980 --> 00:08:36.169
equal wave functions,
or more precisely easily
00:08:36.169 --> 00:08:46.056
interpretable wave functions.
00:08:46.056 --> 00:08:47.930
We saw this last time
in the Fourier theorem.
00:08:47.930 --> 00:08:50.764
The Fourier theorem said look,
take any wave function-- take
00:08:50.764 --> 00:08:52.430
any function, but I'm
going to interpret
00:08:52.430 --> 00:08:53.570
in the language of
quantum mechanics.
00:08:53.570 --> 00:08:56.069
Take any wave function which
is given by some complex valued
00:08:56.069 --> 00:08:57.460
function, and it
can be expressed
00:08:57.460 --> 00:09:00.530
as a superposition
of plane waves.
00:09:00.530 --> 00:09:07.810
1 over 2pi in our normalization
integral dk psi tilde of k,
00:09:07.810 --> 00:09:09.736
but this is a set
of coefficients.
00:09:09.736 --> 00:09:10.797
e to the ikx.
00:09:10.797 --> 00:09:11.880
So what are we doing here?
00:09:11.880 --> 00:09:14.251
We're saying pick a value of k.
00:09:14.251 --> 00:09:15.750
There's a number
associated with it,
00:09:15.750 --> 00:09:18.814
which is going to be an
a magnitude and a phase.
00:09:18.814 --> 00:09:20.230
And that's the
magnitude and phase
00:09:20.230 --> 00:09:22.380
of a plane wave, e to the ikx.
00:09:22.380 --> 00:09:25.980
Now remember that
e to the ikx is
00:09:25.980 --> 00:09:33.007
equal to cos kx plus i sin kx.
00:09:33.007 --> 00:09:35.090
Which you should all know,
but just to remind you.
00:09:35.090 --> 00:09:36.850
This is a periodic function.
00:09:36.850 --> 00:09:38.080
These are periodic functions.
00:09:38.080 --> 00:09:42.820
So this is a plane wave
with a definite wavelength,
00:09:42.820 --> 00:09:45.490
2pi upon k.
00:09:45.490 --> 00:09:47.990
So this is a more equal
wave function in the sense
00:09:47.990 --> 00:09:49.480
that it has a
definite wavelength.
00:09:49.480 --> 00:09:50.770
We know what its momentum is.
00:09:50.770 --> 00:09:52.840
Its momentum is h bar k.
00:09:52.840 --> 00:09:56.030
Any function, we're saying, can
be expressed as a superposition
00:09:56.030 --> 00:09:57.920
by summing over all
possible values of k,
00:09:57.920 --> 00:09:59.560
all possible
different wavelengths.
00:09:59.560 --> 00:10:03.030
Any function can be expressed
as a superposition of wave
00:10:03.030 --> 00:10:05.622
functions with a
definite momentum.
00:10:05.622 --> 00:10:07.300
That make sense?
00:10:07.300 --> 00:10:09.002
Fourier didn't think
about it that way,
00:10:09.002 --> 00:10:10.460
but from quantum
mechanics, this is
00:10:10.460 --> 00:10:11.960
the way we want
to think about it.
00:10:11.960 --> 00:10:12.950
It's just a true statement.
00:10:12.950 --> 00:10:13.991
It's a mathematical fact.
00:10:17.206 --> 00:10:18.080
Questions about that?
00:10:20.750 --> 00:10:24.820
Similarly, I claim that I can
expand the very same function,
00:10:24.820 --> 00:10:28.107
psi of x, as a
superposition of states,
00:10:28.107 --> 00:10:29.815
not with definite
momentum, but of states
00:10:29.815 --> 00:10:30.773
with definite position.
00:10:34.029 --> 00:10:35.820
So what's a state with
a definite position?
00:10:35.820 --> 00:10:36.929
AUDIENCE: Delta.
00:10:36.929 --> 00:10:38.470
PROFESSOR: A delta
function, exactly.
00:10:38.470 --> 00:10:40.510
So I claim that any
function psi of x
00:10:40.510 --> 00:10:45.569
can be expanded a
sum over all states
00:10:45.569 --> 00:10:46.610
with a definite position.
00:10:46.610 --> 00:10:49.470
So delta of-- well, what's a
state with a definite position?
00:10:49.470 --> 00:10:50.670
x0.
00:10:50.670 --> 00:10:53.810
Delta of x minus x0.
00:10:53.810 --> 00:10:54.420
OK?
00:10:54.420 --> 00:10:56.795
This goes bing when
x0 is equal to x.
00:10:56.795 --> 00:10:58.920
But I want a sum over all
possible delta functions.
00:10:58.920 --> 00:11:00.920
That means all
possible positions.
00:11:00.920 --> 00:11:04.930
That means all possible
values of x0, dx0.
00:11:04.930 --> 00:11:06.900
And I need some
coefficient function here.
00:11:06.900 --> 00:11:09.490
Well, the coefficient function
I'm going to call psi of x0.
00:11:13.210 --> 00:11:16.015
So is this true?
00:11:16.015 --> 00:11:17.640
Is it true that I
can take any function
00:11:17.640 --> 00:11:21.730
and expand it in a superposition
of delta functions?
00:11:21.730 --> 00:11:22.410
Absolutely.
00:11:22.410 --> 00:11:24.364
Because look at what
this equation does.
00:11:24.364 --> 00:11:26.030
Remember, delta
function is your friend.
00:11:26.030 --> 00:11:29.210
It's a map from integrals
to numbers or functions.
00:11:29.210 --> 00:11:31.280
So this integral, is
an integral over x0.
00:11:31.280 --> 00:11:33.000
Here we have a
delta of x minus x0.
00:11:33.000 --> 00:11:35.946
So this basically says
the value of this integral
00:11:35.946 --> 00:11:37.570
is what you get by
taking the integrand
00:11:37.570 --> 00:11:39.040
and replacing x by x0.
00:11:39.040 --> 00:11:41.250
Set x equals x0, that's
when delta equals 0.
00:11:41.250 --> 00:11:45.020
So this is equal to the argument
evaluated at x0 is equal to x.
00:11:45.020 --> 00:11:47.620
That's your psi of x.
00:11:47.620 --> 00:11:48.670
OK?
00:11:48.670 --> 00:11:52.310
Any arbitrarily ugly function
can be expressed either
00:11:52.310 --> 00:11:55.650
as a superposition of states
with definite momentum
00:11:55.650 --> 00:11:59.280
or a superposition of states
with definite position.
00:11:59.280 --> 00:12:00.022
OK?
00:12:00.022 --> 00:12:01.230
And this is going to be true.
00:12:01.230 --> 00:12:03.188
We're going to find this
is a general statement
00:12:03.188 --> 00:12:06.780
that any state can be expressed
as a superposition of states
00:12:06.780 --> 00:12:10.780
with a well defined
observable quantity
00:12:10.780 --> 00:12:12.457
for any observable
quantity you want.
00:12:12.457 --> 00:12:14.790
So let me give you just a
quick little bit of intuition.
00:12:14.790 --> 00:12:19.644
In 2D, this is a perfectly
good vector, right?
00:12:19.644 --> 00:12:21.310
Now here's a question
I want to ask you.
00:12:21.310 --> 00:12:22.310
Is that a superposition?
00:12:25.730 --> 00:12:26.230
Yeah.
00:12:26.230 --> 00:12:28.070
I mean every vector
can be written
00:12:28.070 --> 00:12:30.900
as the sum of other
vectors, right?
00:12:30.900 --> 00:12:34.052
And it can be done in an
infinite number of ways, right?
00:12:34.052 --> 00:12:35.510
So there's no such
thing as a state
00:12:35.510 --> 00:12:38.410
which is not a superposition.
00:12:38.410 --> 00:12:40.740
Every vector is a
superposition of other vectors.
00:12:40.740 --> 00:12:43.980
It's a sum of other vector.
00:12:43.980 --> 00:12:48.380
So in particular we often
find it useful to pick a basis
00:12:48.380 --> 00:12:50.992
and say look, I know what
I mean by the vector y,
00:12:50.992 --> 00:12:52.700
y hat is a unit vector
in this direction.
00:12:52.700 --> 00:12:54.325
I know what I mean
by the vector x hat.
00:12:54.325 --> 00:12:55.990
It's a unit vector
in this direction.
00:12:55.990 --> 00:12:59.630
And now I can ask, given that
these are my natural guys,
00:12:59.630 --> 00:13:03.660
the guys I want to attend
to, is this a superposition
00:13:03.660 --> 00:13:04.510
of x and y?
00:13:04.510 --> 00:13:06.585
Or is it just x or y?
00:13:06.585 --> 00:13:09.990
Well, that's a superposition.
00:13:09.990 --> 00:13:11.950
Whereas x hat itself is not.
00:13:11.950 --> 00:13:16.760
So this somehow is about finding
convenient choice of basis.
00:13:16.760 --> 00:13:18.920
But any given vector
can be expressed
00:13:18.920 --> 00:13:21.730
as a superposition of
some pair of basis vectors
00:13:21.730 --> 00:13:24.900
or a different pair
of basis vectors.
00:13:24.900 --> 00:13:28.060
There's nothing hallowed
about your choice of basis.
00:13:28.060 --> 00:13:30.392
There's no God given
basis for the universe.
00:13:30.392 --> 00:13:32.600
We look out in the universe
in the Hubble deep field,
00:13:32.600 --> 00:13:34.766
and you don't see somewhere
in the Hubble deep field
00:13:34.766 --> 00:13:37.470
an arrow going x, right?
00:13:37.470 --> 00:13:39.060
So there's no natural
choice of basis,
00:13:39.060 --> 00:13:41.090
but it's sometimes
convenient to pick a basis.
00:13:41.090 --> 00:13:42.590
This is the direction of
the surface of the earth.
00:13:42.590 --> 00:13:44.640
This is the direction
perpendicular to it.
00:13:44.640 --> 00:13:46.900
So sometimes
particular basis sets
00:13:46.900 --> 00:13:49.730
have particular meanings to us.
00:13:49.730 --> 00:13:50.821
That's true in vectors.
00:13:50.821 --> 00:13:51.820
This is along the earth.
00:13:51.820 --> 00:13:52.890
This is perpendicular to it.
00:13:52.890 --> 00:13:54.260
This would be slightly strange.
00:13:54.260 --> 00:13:56.450
Maybe if you're leaning.
00:13:56.450 --> 00:13:58.910
And similarly, this
is an expansion
00:13:58.910 --> 00:14:01.790
of a function as a
sum, as a superposition
00:14:01.790 --> 00:14:03.140
of other functions.
00:14:03.140 --> 00:14:06.030
And you could have done this
in any good space of functions.
00:14:06.030 --> 00:14:07.291
We'll talk about that more.
00:14:07.291 --> 00:14:08.790
These are particularly
natural ones.
00:14:08.790 --> 00:14:09.581
They're more equal.
00:14:09.581 --> 00:14:12.590
These are ones with different
definite values of position,
00:14:12.590 --> 00:14:14.270
different definite
values of momentum.
00:14:14.270 --> 00:14:15.520
Everyone cool?
00:14:15.520 --> 00:14:17.936
Quickly what's the momentum
associated to the plane wave e
00:14:17.936 --> 00:14:18.724
to the ikx?
00:14:18.724 --> 00:14:19.640
AUDIENCE: [INAUDIBLE].
00:14:22.220 --> 00:14:23.100
PROFESSOR: h bar k.
00:14:23.100 --> 00:14:23.600
Good.
00:14:33.180 --> 00:14:35.140
So now I want to
just quickly run over
00:14:35.140 --> 00:14:37.850
some concept questions for you.
00:14:37.850 --> 00:14:39.390
So whip out your clickers.
00:14:39.390 --> 00:14:41.030
OK, we'll do this verbally.
00:14:49.052 --> 00:14:50.385
All right, let's try this again.
00:14:50.385 --> 00:14:53.200
So how would you interpret
this wave function?
00:14:53.200 --> 00:14:54.452
AUDIENCE: e.
00:14:54.452 --> 00:14:55.160
PROFESSOR: Solid.
00:14:58.580 --> 00:15:00.080
How do you know
whether the particle
00:15:00.080 --> 00:15:02.332
is big or small by looking
at the wave function?
00:15:02.332 --> 00:15:03.248
AUDIENCE: [INAUDIBLE].
00:15:05.640 --> 00:15:06.780
PROFESSOR: All right.
00:15:06.780 --> 00:15:09.306
Two particles described by
a plane wave of the form e
00:15:09.306 --> 00:15:10.820
to the ikx.
00:15:10.820 --> 00:15:13.979
Particle one is a smaller
wavelength than particle two.
00:15:13.979 --> 00:15:15.520
Which particle has
a larger momentum?
00:15:15.520 --> 00:15:17.803
Think about it, but
don't say it out loud.
00:15:22.230 --> 00:15:24.647
And this sort of defeats the
purpose of the clicker thing,
00:15:24.647 --> 00:15:27.146
because now I'm supposed to be
able to know without you guys
00:15:27.146 --> 00:15:28.040
saying anything.
00:15:28.040 --> 00:15:29.549
So instead of
saying it out loud,
00:15:29.549 --> 00:15:30.840
here's what I'd like you to do.
00:15:30.840 --> 00:15:33.350
Talk to the person
next to you and discuss
00:15:33.350 --> 00:15:37.271
which one has the larger
00:15:37.271 --> 00:15:38.104
AUDIENCE: [CHATTER].
00:16:00.550 --> 00:16:02.000
All right.
00:16:02.000 --> 00:16:07.350
Cool, so which one has
the larger momentum?
00:16:07.350 --> 00:16:09.460
AUDIENCE: A.
00:16:09.460 --> 00:16:10.485
PROFESSOR: How come?
00:16:10.485 --> 00:16:12.760
[INTERPOSING VOICES]
00:16:12.760 --> 00:16:15.540
PROFESSOR: RIght,
smaller wavelength.
00:16:15.540 --> 00:16:18.262
P equals h bar k.
00:16:18.262 --> 00:16:22.330
k equals 2pi over lambda.
00:16:22.330 --> 00:16:23.880
Solid?
00:16:23.880 --> 00:16:25.869
Smaller wavelength,
higher momentum.
00:16:25.869 --> 00:16:27.660
If it has higher
momentum, what do you just
00:16:27.660 --> 00:16:30.642
intuitively expect to
know about its energy?
00:16:30.642 --> 00:16:31.940
It's probably higher.
00:16:31.940 --> 00:16:33.310
Are you positive about that?
00:16:33.310 --> 00:16:35.810
No, you need to know how the
energy depends on the momentum,
00:16:35.810 --> 00:16:38.130
but it's probably higher.
00:16:38.130 --> 00:16:39.990
So this is an
important little lesson
00:16:39.990 --> 00:16:41.990
that you probably all
know from optics and maybe
00:16:41.990 --> 00:16:43.030
from core mechanics.
00:16:43.030 --> 00:16:45.670
Shorter wavelength
thing, higher energy.
00:16:45.670 --> 00:16:47.010
Higher momentum for sure.
00:16:47.010 --> 00:16:50.550
Usually higher energy as well.
00:16:50.550 --> 00:16:53.190
Very useful rule of
thumb to keep in mind.
00:16:53.190 --> 00:16:54.490
Indeed, it's particle one.
00:16:54.490 --> 00:16:55.730
OK next one.
00:16:55.730 --> 00:16:59.490
Compared to the wave function
psi of x, it's Fourier
00:16:59.490 --> 00:17:02.660
transform, psi tilde of x
contains more information,
00:17:02.660 --> 00:17:06.740
or less, or the
same, or something.
00:17:06.740 --> 00:17:08.569
Don't say it out loud.
00:17:08.569 --> 00:17:12.579
OK, so how many people
know the answer?
00:17:12.579 --> 00:17:13.079
Awesome.
00:17:13.079 --> 00:17:16.380
And how many people
are not sure.
00:17:16.380 --> 00:17:17.869
OK, good.
00:17:17.869 --> 00:17:22.053
So talk to the person next to
you and convince them briefly.
00:17:43.100 --> 00:17:43.640
All right.
00:17:48.800 --> 00:17:49.590
So let's vote.
00:17:49.590 --> 00:17:52.980
A, more information.
00:17:52.980 --> 00:17:55.350
B, less information.
00:17:55.350 --> 00:17:57.690
C, same.
00:17:57.690 --> 00:17:59.586
OK, good you got it.
00:17:59.586 --> 00:18:00.710
So these are not hard ones.
00:18:04.600 --> 00:18:08.530
This function, which is
a sine wave of length l,
00:18:08.530 --> 00:18:10.080
0 outside of that region.
00:18:10.080 --> 00:18:12.340
Which is closer to true?
00:18:12.340 --> 00:18:15.370
f has a single well defined
wavelength for the most part?
00:18:15.370 --> 00:18:16.280
It's closer to true.
00:18:16.280 --> 00:18:17.570
This doesn't have to be exact.
00:18:17.570 --> 00:18:19.440
f has a single well
defined wavelengths.
00:18:19.440 --> 00:18:21.996
Or f is made up of a wide
range of wavelengths?
00:18:26.350 --> 00:18:28.382
Think it to yourself.
00:18:28.382 --> 00:18:29.590
Ponder that one for a minute.
00:18:40.380 --> 00:18:42.430
OK, now before we
get talking about it.
00:18:42.430 --> 00:18:44.152
Hold on, hold on, hold on.
00:18:44.152 --> 00:18:45.610
Since we don't have
clickers, but I
00:18:45.610 --> 00:18:47.410
want to pull off
the same effect,
00:18:47.410 --> 00:18:50.170
and we can do this,
because it's binary here.
00:18:50.170 --> 00:18:53.780
I want everyone close your eyes.
00:18:53.780 --> 00:18:56.020
Just close your eyes,
just for a moment.
00:18:56.020 --> 00:18:56.777
Yeah.
00:18:56.777 --> 00:18:58.610
Or close the eyes of
the person next to you.
00:18:58.610 --> 00:18:59.720
That's fine.
00:18:59.720 --> 00:19:02.610
And now and I want you to vote.
00:19:02.610 --> 00:19:05.200
A is f has a single
well defined wavelength.
00:19:05.200 --> 00:19:07.830
B is f has a wide
range of wavelengths.
00:19:07.830 --> 00:19:10.840
So how many people think
A, a single wavelength?
00:19:10.840 --> 00:19:12.540
OK.
00:19:12.540 --> 00:19:14.556
Lower your hands, good.
00:19:14.556 --> 00:19:18.000
And how many people think B,
a wide range of wavelengths?
00:19:18.000 --> 00:19:18.550
Awesome.
00:19:18.550 --> 00:19:20.340
So this is exactly what happens
when we actually use clickers.
00:19:20.340 --> 00:19:21.470
It's 50/50.
00:19:21.470 --> 00:19:24.040
So now you guys need to talk
to the person next to you
00:19:24.040 --> 00:19:26.225
and convince each
other of the truth.
00:19:26.225 --> 00:19:27.058
AUDIENCE: [CHATTER].
00:20:23.840 --> 00:20:30.719
All right, so the volume sort of
tones down as people, I think,
00:20:30.719 --> 00:20:31.510
come to resolution.
00:20:31.510 --> 00:20:33.020
Close your eyes again.
00:20:33.020 --> 00:20:34.700
Once more into the
breach, my friends.
00:20:34.700 --> 00:20:37.980
So close your eyes, and
now let's vote again.
00:20:37.980 --> 00:20:41.530
f of x has a single,
well defined wavelength.
00:20:41.530 --> 00:20:45.370
And now f of x is made up
of a range of wavelengths?
00:20:45.370 --> 00:20:46.090
OK.
00:20:46.090 --> 00:20:48.940
There's a dramatic
shift in the field to B,
00:20:48.940 --> 00:20:51.600
it has a wide range
of wavelengths, not
00:20:51.600 --> 00:20:52.920
a single wavelength.
00:20:52.920 --> 00:20:55.350
And that is, in fact,
the correct answer.
00:20:55.350 --> 00:20:58.010
OK, so learning happens.
00:20:58.010 --> 00:21:00.810
That was an empirical test.
00:21:00.810 --> 00:21:02.700
So does anyone want
to defend this view
00:21:02.700 --> 00:21:04.900
that f is made of a wide
range of wavelengths?
00:21:04.900 --> 00:21:06.190
Sure, bring it.
00:21:06.190 --> 00:21:08.650
AUDIENCE: So, the sine
wave is an infinite,
00:21:08.650 --> 00:21:11.217
and it cancels out
past minus l over 2
00:21:11.217 --> 00:21:12.800
and positive l over
2, which means you
00:21:12.800 --> 00:21:13.830
need to add a bunch
of wavelengths
00:21:13.830 --> 00:21:15.464
to actually cancel it out there.
00:21:15.464 --> 00:21:16.630
PROFESSOR: Awesome, exactly.
00:21:16.630 --> 00:21:17.240
Exactly.
00:21:17.240 --> 00:21:19.470
If you only had the thing
of a single wavelength,
00:21:19.470 --> 00:21:21.720
it would continue with a single
wavelength all the way out.
00:21:21.720 --> 00:21:23.386
In fact, there's a
nice way to say this.
00:21:23.386 --> 00:21:28.441
When you have a sine wave, what
can you say about it's-- we
00:21:28.441 --> 00:21:29.940
know that a sine
wave is continuous,
00:21:29.940 --> 00:21:32.580
and it's continuous
everywhere, right?
00:21:32.580 --> 00:21:34.220
It's also differentiable
everywhere.
00:21:34.220 --> 00:21:37.210
Its derivative is continuous
and differentiable everywhere,
00:21:37.210 --> 00:21:39.510
because it's a cosine, right?
00:21:39.510 --> 00:21:42.050
So if yo you take a
superposition of sines
00:21:42.050 --> 00:21:46.180
and cosines, do you ever
get a discontinuity?
00:21:46.180 --> 00:21:47.090
No.
00:21:47.090 --> 00:21:49.990
Do you ever get something whose
derivative is discontinuous?
00:21:49.990 --> 00:21:50.690
No.
00:21:50.690 --> 00:21:53.630
So how would you ever
reproduce a thing
00:21:53.630 --> 00:21:57.320
with a discontinuity
using sines and cosines?
00:21:57.320 --> 00:22:00.390
Well, you'd need some infinite
sum of sines and cosines
00:22:00.390 --> 00:22:03.400
where there's some technicality
about the infinite limit being
00:22:03.400 --> 00:22:04.900
singular, because
you can't do it
00:22:04.900 --> 00:22:07.200
a finite number of
sines and cosines.
00:22:07.200 --> 00:22:10.390
That function is continuous,
but its derivative
00:22:10.390 --> 00:22:12.240
is discontinuous.
00:22:12.240 --> 00:22:13.409
Yeah?
00:22:13.409 --> 00:22:15.450
So it's going to take an
infinite number of sines
00:22:15.450 --> 00:22:18.940
and cosines to reproduce
that little kink at the edge.
00:22:18.940 --> 00:22:19.658
Yeah?
00:22:19.658 --> 00:22:21.898
AUDIENCE: So a finite
number of sines and cosines
00:22:21.898 --> 00:22:26.176
doesn't mean finding-- or
an infinite number of sines
00:22:26.176 --> 00:22:28.384
and cosines doesn't mean
infinite [? regular ?] sines
00:22:28.384 --> 00:22:29.204
and cosines, right?
00:22:29.204 --> 00:22:33.430
Because over a finite
region [INAUDIBLE].
00:22:33.430 --> 00:22:35.864
PROFESSOR: That's true, but
you need arbitrarily-- so
00:22:35.864 --> 00:22:36.780
let's talk about that.
00:22:36.780 --> 00:22:38.804
That's an excellent question.
00:22:38.804 --> 00:22:39.970
That's a very good question.
00:22:39.970 --> 00:22:41.780
The question here
is look, there's
00:22:41.780 --> 00:22:43.696
two different things you
can be talking about.
00:22:43.696 --> 00:22:46.280
One is arbitrarily large and
arbitrarily short wavelengths,
00:22:46.280 --> 00:22:47.640
so an arbitrary
range of wavelengths.
00:22:47.640 --> 00:22:49.170
And the other is
an infinite number.
00:22:49.170 --> 00:22:50.520
But an infinite number
is silly, because there's
00:22:50.520 --> 00:22:51.932
a continuous variable here k.
00:22:51.932 --> 00:22:53.640
You got an infinite
number of wavelengths
00:22:53.640 --> 00:22:56.170
between one and 1.2, right?
00:22:56.170 --> 00:22:56.970
It's continuous.
00:22:56.970 --> 00:22:58.190
So which one do you mean?
00:22:58.190 --> 00:23:00.470
So let's go back
to this connection
00:23:00.470 --> 00:23:02.950
that we got a minute
ago from short distance
00:23:02.950 --> 00:23:04.080
and high momentum.
00:23:07.499 --> 00:23:09.790
That thing looks like it has
one particular wavelength.
00:23:09.790 --> 00:23:11.540
But I claim, in
order to reproduce
00:23:11.540 --> 00:23:14.980
that as a superposition of
states with definite momentum,
00:23:14.980 --> 00:23:17.714
I need arbitrarily
high wavelength.
00:23:17.714 --> 00:23:19.880
And why do I need arbitrarily
high wavelength modes?
00:23:19.880 --> 00:23:22.290
Why do we need to arbitrarily
high momentum modes?
00:23:22.290 --> 00:23:24.170
Well, it's because of this.
00:23:24.170 --> 00:23:24.950
We have a kink.
00:23:27.610 --> 00:23:32.330
And this feature, what's the
length scale of that feature?
00:23:32.330 --> 00:23:34.170
It's infinitesimally
small, which
00:23:34.170 --> 00:23:35.840
means I'm going to have to--
in order to reproduce that,
00:23:35.840 --> 00:23:37.465
in order to probe
it, I'm going to need
00:23:37.465 --> 00:23:40.130
a momentum that's
arbitrarily large.
00:23:40.130 --> 00:23:42.940
So it's really about the
range, not just the number.
00:23:42.940 --> 00:23:45.930
But you need arbitrarily
large momentum.
00:23:45.930 --> 00:23:50.890
To construct or detect an
arbitrarily small feature
00:23:50.890 --> 00:23:52.640
you need arbitrarily
large momentum modes.
00:23:52.640 --> 00:23:53.140
Yeah?
00:23:53.140 --> 00:23:56.390
AUDIENCE: Why do
you [INAUDIBLE]?
00:23:56.390 --> 00:23:58.330
Why don't you just
say, oh you need
00:23:58.330 --> 00:24:00.020
an arbitrary small wavelength?
00:24:00.020 --> 00:24:02.754
Why wouldn't you just
phrase that [INAUDIBLE]?
00:24:02.754 --> 00:24:04.420
PROFESSOR: I chose
to phrase it that way
00:24:04.420 --> 00:24:06.711
because I want an emphasize
and encourage-- I emphasize
00:24:06.711 --> 00:24:10.740
you to think and
encourage you to conflate
00:24:10.740 --> 00:24:13.140
short distance and
large momentum.
00:24:13.140 --> 00:24:16.764
I want the connection between
momentum and the length scale
00:24:16.764 --> 00:24:18.680
to be something that
becomes intuitive to you.
00:24:18.680 --> 00:24:20.440
So when I talk about
something with short features,
00:24:20.440 --> 00:24:22.940
I'm going to talk about it as
something with large momentum.
00:24:22.940 --> 00:24:25.960
And that's because in a
quantum mechanical system,
00:24:25.960 --> 00:24:27.990
something with
short wavelength is
00:24:27.990 --> 00:24:31.236
something that carries
large momentum.
00:24:31.236 --> 00:24:32.351
That cool?
00:24:32.351 --> 00:24:32.850
Great.
00:24:32.850 --> 00:24:33.930
Good question.
00:24:33.930 --> 00:24:36.270
AUDIENCE: So earlier you
said that any reasonable wave
00:24:36.270 --> 00:24:40.122
function, a possible
wave function,
00:24:40.122 --> 00:24:41.580
does that mean
they're not supposed
00:24:41.580 --> 00:24:43.100
to be Fourier transformable?
00:24:43.100 --> 00:24:45.640
PROFESSOR: That's
usually a condition.
00:24:45.640 --> 00:24:46.360
Yeah, exactly.
00:24:46.360 --> 00:24:47.600
We don't quite
phrase it that way.
00:24:47.600 --> 00:24:49.683
And in fact, there's a
problem on your problem set
00:24:49.683 --> 00:24:52.060
that will walk you
through what we will mean.
00:24:52.060 --> 00:24:53.700
What should be
true of the Fourier
00:24:53.700 --> 00:24:56.410
transform in order for this
to reasonably function.
00:24:56.410 --> 00:24:58.450
And among other things--
and your intuition
00:24:58.450 --> 00:25:00.710
here is exactly right--
among other things,
00:25:00.710 --> 00:25:03.190
being able to have a Fourier
transform where you don't have
00:25:03.190 --> 00:25:04.822
arbitrarily high
momentum modes is
00:25:04.822 --> 00:25:06.280
going to be an
important condition.
00:25:06.280 --> 00:25:09.460
That's going to turn to be
related to the derivative
00:25:09.460 --> 00:25:11.014
being continuous.
00:25:11.014 --> 00:25:12.180
That's a very good question.
00:25:12.180 --> 00:25:17.144
So that's the optional
problem 8 on problem set 2.
00:25:17.144 --> 00:25:17.810
Other questions?
00:25:22.340 --> 00:25:25.130
PROFESSOR: Cool, so that's
it for the clicker questions.
00:25:25.130 --> 00:25:26.490
Sorry for the technology fail.
00:25:29.400 --> 00:25:32.505
So I'm just going to
turn this off in disgust.
00:25:37.384 --> 00:25:38.425
That's really irritating.
00:25:41.580 --> 00:25:44.880
So today what I want
to start on is pick up
00:25:44.880 --> 00:25:47.070
on the discussion of the
uncertainty principle
00:25:47.070 --> 00:25:49.440
that we sort of
outlined previously.
00:25:49.440 --> 00:25:51.950
The fact that when we have a
wave function with reasonably
00:25:51.950 --> 00:25:53.460
well defined position
corresponding
00:25:53.460 --> 00:25:55.744
to a particle with reasonably
well defined position,
00:25:55.744 --> 00:25:58.160
it didn't have a reasonably
well defined momentum and vice
00:25:58.160 --> 00:25:58.850
versa.
00:25:58.850 --> 00:26:00.630
The certainty of
the momentum seems
00:26:00.630 --> 00:26:04.570
to imply lack of knowledge about
the position and vice versa.
00:26:04.570 --> 00:26:09.520
So in order to do that, we
need to define uncertainty.
00:26:09.520 --> 00:26:11.930
So I need to define for
you delta x and delta p.
00:26:14.470 --> 00:26:16.740
So first I just want
to run through what
00:26:16.740 --> 00:26:19.290
should be totally
remedial probability,
00:26:19.290 --> 00:26:23.570
but it's always useful
to just remember
00:26:23.570 --> 00:26:25.310
how these basic things work.
00:26:25.310 --> 00:26:28.620
So consider a set
of people in a room,
00:26:28.620 --> 00:26:32.370
and I want to plot the number
of people with a particular age
00:26:32.370 --> 00:26:36.240
as a function of the
age of possible ages.
00:26:36.240 --> 00:26:41.540
So let's say we have 16
people, and at 14 we have one,
00:26:41.540 --> 00:26:46.720
and at 15 we have 1,
and at 16 we have 3.
00:26:46.720 --> 00:26:50.170
And that's 16.
00:26:50.170 --> 00:26:53.705
And at 20 we have 2.
00:26:56.680 --> 00:26:58.715
And at 21 we have 4.
00:27:02.270 --> 00:27:03.935
And at 22 we have 5.
00:27:09.040 --> 00:27:10.070
And that's it.
00:27:10.070 --> 00:27:12.180
OK.
00:27:12.180 --> 00:27:23.020
So 1, 1, 3, 2, 4, 5.
00:27:23.020 --> 00:27:26.950
OK, so what's the
probability that any given
00:27:26.950 --> 00:27:29.750
person in this group of
16 has a particular age?
00:27:29.750 --> 00:27:30.647
I'll call it a.
00:27:33.340 --> 00:27:34.840
So how do we compute
the probability
00:27:34.840 --> 00:27:36.630
that they have age a?
00:27:36.630 --> 00:27:37.380
Well this is easy.
00:27:37.380 --> 00:27:40.460
It's the number that have
age a over the total number.
00:27:44.660 --> 00:27:47.140
So note an important thing,
an important side note,
00:27:47.140 --> 00:27:49.860
which is that the sum
over all possible ages
00:27:49.860 --> 00:27:52.955
of the probability that you
have age a is equal to 1,
00:27:52.955 --> 00:27:55.080
because it's just going to
be the sum of the number
00:27:55.080 --> 00:27:57.340
with a particular age over the
total number, which is just
00:27:57.340 --> 00:27:59.048
the sum of the number
with any given age.
00:28:04.540 --> 00:28:05.750
So here's some questions.
00:28:05.750 --> 00:28:09.264
So what's the most likely age?
00:28:09.264 --> 00:28:10.680
If you grabbed one
of these people
00:28:10.680 --> 00:28:12.495
from the room with
a giant Erector set,
00:28:12.495 --> 00:28:14.030
and pull out a person,
and let them dangle,
00:28:14.030 --> 00:28:15.613
and ask them what
their age is, what's
00:28:15.613 --> 00:28:17.288
the most likely they'll have?
00:28:17.288 --> 00:28:18.382
AUDIENCE: 22.
00:28:18.382 --> 00:28:18.965
PROFESSOR: 22.
00:28:21.211 --> 00:28:22.960
On the other hand,
what's the average age?
00:28:31.700 --> 00:28:35.420
Well, just by eyeball roughly
what do you think it is?
00:28:35.420 --> 00:28:37.030
So around 19 or 20.
00:28:37.030 --> 00:28:40.220
It turns out to
be 19.2 for this.
00:28:40.220 --> 00:28:40.840
OK.
00:28:40.840 --> 00:28:43.490
But if everyone had a little
sticker on their lapel
00:28:43.490 --> 00:28:47.770
that says I'm 14, 15, 16, 20,
21 or 22, how many people have
00:28:47.770 --> 00:28:49.910
the age 19.2?
00:28:49.910 --> 00:28:51.010
None, right?
00:28:51.010 --> 00:28:54.020
So a useful thing is
that the average need not
00:28:54.020 --> 00:28:55.940
be an observable value.
00:28:57.985 --> 00:28:59.610
This is going to come
back to haunt us.
00:28:59.610 --> 00:29:02.210
Oops, 19.4.
00:29:02.210 --> 00:29:03.550
That's what I got.
00:29:03.550 --> 00:29:11.300
So in particular how
did I get the average?
00:29:11.300 --> 00:29:13.050
I'm going to define
some notation.
00:29:13.050 --> 00:29:14.320
This notation is
going to stick with us
00:29:14.320 --> 00:29:15.736
for the rest of
quantum mechanics.
00:29:15.736 --> 00:29:18.800
The average age,
how do I compute it?
00:29:18.800 --> 00:29:21.330
So we all know this, but let
me just be explicit about it.
00:29:21.330 --> 00:29:25.240
It's the sum over
all possible ages
00:29:25.240 --> 00:29:29.610
of the number of
the number of people
00:29:29.610 --> 00:29:32.950
with that age times
the age divided
00:29:32.950 --> 00:29:34.800
by the total number of people.
00:29:37.100 --> 00:29:37.600
OK?
00:29:41.000 --> 00:29:46.892
So in this case, I'd go 14,14,
16, 16, 16, 20, 20, 21, 21,
00:29:46.892 --> 00:29:49.620
21 21, 22, 22, 22, 22, 22.
00:29:49.620 --> 00:29:51.662
And so that's all
I've written here.
00:29:51.662 --> 00:29:53.620
But notice that I can
write this in a nice way.
00:29:53.620 --> 00:29:56.462
This is equal to the sum
over all possible ages
00:29:56.462 --> 00:30:01.759
of a times the ratio of Na to N
with a ratio of Na to n total.
00:30:01.759 --> 00:30:03.800
That's just the probability
that any given person
00:30:03.800 --> 00:30:05.490
has a probability a.
00:30:05.490 --> 00:30:07.610
a times probability of a.
00:30:07.610 --> 00:30:11.680
So the expected value is the
sum over all possible values
00:30:11.680 --> 00:30:14.460
of the value times the
probability to get that value.
00:30:14.460 --> 00:30:15.840
Yeah?
00:30:15.840 --> 00:30:18.520
This is the same equation,
but I'm going to box it.
00:30:18.520 --> 00:30:21.820
It's a very useful relation.
00:30:21.820 --> 00:30:24.290
And so, again, does the
average have to be measurable?
00:30:24.290 --> 00:30:26.225
No, it certainly doesn't.
00:30:26.225 --> 00:30:29.310
And it usually isn't.
00:30:29.310 --> 00:30:33.590
So let's ask the same thing
for the square of ages.
00:30:33.590 --> 00:30:39.190
What is the average
of a squared?
00:30:39.190 --> 00:30:39.920
Square the ages.
00:30:39.920 --> 00:30:42.500
You might say, well, why
would I ever care about that?
00:30:42.500 --> 00:30:43.999
But let's just be
explicit about it.
00:30:43.999 --> 00:30:45.630
So following the
same logic here,
00:30:45.630 --> 00:30:48.460
the average of a squared,
the average value
00:30:48.460 --> 00:30:50.092
of the square of the
ages is, well, I'm
00:30:50.092 --> 00:30:51.550
going to do exactly
the same thing.
00:30:51.550 --> 00:30:53.060
It's just a squared, right?
00:30:53.060 --> 00:30:56.317
14 squared, 15 squared, 16
square, 16 squared, 16 squared.
00:30:56.317 --> 00:30:58.650
So this is going to give me
exactly the same expression.
00:30:58.650 --> 00:31:03.330
So over a of a squared
probability of measuring a.
00:31:07.140 --> 00:31:12.230
And more generally, the expected
value, or the average value
00:31:12.230 --> 00:31:14.802
of some function of
a is equal-- and this
00:31:14.802 --> 00:31:16.260
is something you
don't usually do--
00:31:16.260 --> 00:31:21.610
is equal to the sum over a of
f of a, the value of f given
00:31:21.610 --> 00:31:24.309
a particular value of a,
times the probability that you
00:31:24.309 --> 00:31:26.100
measure that value of
a in the first place.
00:31:29.320 --> 00:31:33.855
It's exactly the same
logic as averages.
00:31:37.340 --> 00:31:39.330
Right, cool.
00:31:39.330 --> 00:31:41.940
So here's a quick question.
00:31:41.940 --> 00:31:47.700
Is a squared equal to the
expected value of a squared?
00:31:47.700 --> 00:31:48.844
AUDIENCE: No.
00:31:48.844 --> 00:31:50.885
PROFESSOR: Right, in
general no, not necessarily.
00:31:56.810 --> 00:32:00.480
So for example,
the average value--
00:32:00.480 --> 00:32:02.780
suppose we have a Gaussian
centered at the origin.
00:32:02.780 --> 00:32:04.760
So here's a.
00:32:04.760 --> 00:32:07.076
Now a isn't age, but it's
something-- I don't know.
00:32:07.076 --> 00:32:09.570
You include infants or whatever.
00:32:09.570 --> 00:32:10.320
It's not age.
00:32:10.320 --> 00:32:13.340
Its happiness on a given day.
00:32:13.340 --> 00:32:17.920
So what's the average value?
00:32:17.920 --> 00:32:19.000
Meh.
00:32:19.000 --> 00:32:19.580
Right?
00:32:19.580 --> 00:32:21.910
Sort of vaguely neutral, right?
00:32:21.910 --> 00:32:24.080
But on the other hand,
if you take a squared,
00:32:24.080 --> 00:32:26.080
very few people have
a squared as zero.
00:32:26.080 --> 00:32:28.340
Most people have a
squared as not a 0 value.
00:32:28.340 --> 00:32:30.120
And most people are
sort of in the middle.
00:32:30.120 --> 00:32:34.130
Most people are sort of
hazy on what the day is.
00:32:34.130 --> 00:32:37.430
So in this case,
the expected value
00:32:37.430 --> 00:32:39.170
of a, or the average
value of a is 0.
00:32:39.170 --> 00:32:43.620
The average value of a
squared is not equal to 0.
00:32:43.620 --> 00:32:44.500
Yeah?
00:32:44.500 --> 00:32:46.791
And that's because the squared
has everything positive.
00:32:48.810 --> 00:32:51.560
So how do we characterize--
this gives us
00:32:51.560 --> 00:32:54.390
a useful tool for characterizing
the width of a distribution.
00:32:54.390 --> 00:32:56.890
So here we have a distribution
where its average value is 0,
00:32:56.890 --> 00:32:58.000
but its width is non-zero.
00:32:58.000 --> 00:33:00.810
And then the expectation
value of a squared,
00:33:00.810 --> 00:33:03.310
the expected value of
a squared, is non-zero.
00:33:03.310 --> 00:33:07.700
So how do we define the
width of a distribution?
00:33:07.700 --> 00:33:10.020
This is going to be
like our uncertainty.
00:33:10.020 --> 00:33:11.100
How happy are you today?
00:33:11.100 --> 00:33:11.910
Well, I'm not sure.
00:33:11.910 --> 00:33:12.880
How unsure are you?
00:33:12.880 --> 00:33:14.980
Well, that should give
us a precise measure.
00:33:14.980 --> 00:33:17.170
So let me define three things.
00:33:17.170 --> 00:33:19.060
First the deviation.
00:33:19.060 --> 00:33:22.770
So the deviation is going to be
a minus the average value of a.
00:33:22.770 --> 00:33:24.520
So this is just take
the actual value of a
00:33:24.520 --> 00:33:26.341
and subtract off the
average value of a.
00:33:26.341 --> 00:33:28.340
So we always get something
that's centered at 0.
00:33:32.087 --> 00:33:33.420
I'm going to write it like this.
00:33:36.240 --> 00:33:38.730
Note, by the way, just a
convenient thing to note.
00:33:38.730 --> 00:33:43.920
The average value of a
minus it's average value.
00:33:43.920 --> 00:33:46.564
Well, what's the
average value of 7?
00:33:46.564 --> 00:33:47.447
AUDIENCE: 7.
00:33:47.447 --> 00:33:48.280
PROFESSOR: OK, good.
00:33:48.280 --> 00:33:52.832
So that first term is
the average value of a.
00:33:52.832 --> 00:33:54.540
And that second term
is the average value
00:33:54.540 --> 00:33:58.090
of this number, which is
just this number minus a.
00:33:58.090 --> 00:33:59.931
So this is 0.
00:33:59.931 --> 00:34:00.430
Yeah?
00:34:04.092 --> 00:34:05.550
The average value
of a number is 0.
00:34:05.550 --> 00:34:06.966
The average value
of this variable
00:34:06.966 --> 00:34:10.469
is the average value of
that variable, but that's 0.
00:34:10.469 --> 00:34:12.750
So deviation is not a terribly
good thing on average,
00:34:12.750 --> 00:34:14.625
because on average the
deviation is always 0.
00:34:14.625 --> 00:34:17.602
That's what it means to
say this is the average.
00:34:17.602 --> 00:34:19.060
So the derivation
is saying how far
00:34:19.060 --> 00:34:21.560
is any particular
instance from the average.
00:34:21.560 --> 00:34:23.060
And if you average
those deviations,
00:34:23.060 --> 00:34:24.260
they always give you 0.
00:34:24.260 --> 00:34:25.909
So this is not a
very good measure
00:34:25.909 --> 00:34:28.630
of the actual width
of the system.
00:34:28.630 --> 00:34:31.559
But we can get a nice measure by
getting the deviation squared.
00:34:34.600 --> 00:34:36.880
And let's take the mean
of the derivation squared.
00:34:36.880 --> 00:34:40.829
So the mean of the derivation
squared, mean of a minus
00:34:40.829 --> 00:34:42.120
the average value of a squared.
00:34:45.179 --> 00:34:48.669
This is what I'm going to
call the standard deviation.
00:34:48.669 --> 00:34:50.460
Which is a little odd,
because really you'd
00:34:50.460 --> 00:34:52.418
want to call it the
standard deviation squared.
00:34:52.418 --> 00:34:53.886
But whatever.
00:34:53.886 --> 00:34:55.219
We use funny words.
00:34:59.450 --> 00:35:03.225
So now what does it mean if
the average value of a is 0?
00:35:03.225 --> 00:35:05.100
It means it's centered
at 0, but what does it
00:35:05.100 --> 00:35:07.830
mean if the standard
deviation of a is 0?
00:35:10.880 --> 00:35:15.610
So if the standard
deviation is 0,
00:35:15.610 --> 00:35:21.280
one then the distribution
has no width, right?
00:35:21.280 --> 00:35:23.450
Because if there was
any amplitude away
00:35:23.450 --> 00:35:25.600
from the average
value, then that
00:35:25.600 --> 00:35:27.960
would give a non-zero
strictly positive contribution
00:35:27.960 --> 00:35:31.280
to this average expectation,
and this wouldn't be 0 anymore.
00:35:31.280 --> 00:35:33.120
So standard deviation
is 0, as long
00:35:33.120 --> 00:35:35.578
as there's no width, which is
why the standard deviation is
00:35:35.578 --> 00:35:40.970
a good useful measure
of width or uncertainty.
00:35:40.970 --> 00:35:45.660
And just as a note, taking
this seriously and taking
00:35:45.660 --> 00:35:49.200
the square, so standard
deviation squared,
00:35:49.200 --> 00:35:52.320
this is equal to the
average value of a squared
00:35:52.320 --> 00:35:57.150
minus twice a times the average
value of a plus average value
00:35:57.150 --> 00:35:59.500
of a quantity squared.
00:35:59.500 --> 00:36:01.110
But if you do this
out, this is going
00:36:01.110 --> 00:36:05.770
to be equal to a squared
minus 2 average value
00:36:05.770 --> 00:36:07.260
of a average value of a.
00:36:07.260 --> 00:36:11.060
That's just minus
twice the average value
00:36:11.060 --> 00:36:14.180
of a quantity squared.
00:36:14.180 --> 00:36:16.360
And then plus average
value of a squared.
00:36:16.360 --> 00:36:19.230
So this is an alternate way of
writing the standard deviation.
00:36:19.230 --> 00:36:19.730
OK?
00:36:19.730 --> 00:36:23.170
So we can either write it in
this fashion or this fashion.
00:36:23.170 --> 00:36:29.075
And the notation for
this is delta a squared.
00:36:32.750 --> 00:36:33.580
OK?
00:36:33.580 --> 00:36:35.840
So when I talk about
an uncertainty, what
00:36:35.840 --> 00:36:38.520
I mean is, given
my distribution,
00:36:38.520 --> 00:36:40.094
I compute the
standard deviation.
00:36:40.094 --> 00:36:41.510
And the uncertainty
is going to be
00:36:41.510 --> 00:36:44.302
the square root of the
standard deviations squared.
00:36:44.302 --> 00:36:45.990
OK?
00:36:45.990 --> 00:36:48.880
So delta a, the words
I'm going to use for this
00:36:48.880 --> 00:36:59.430
is the uncertainty in a given
some probability distribution.
00:37:01.612 --> 00:37:03.820
Different probability
distributions are going to give
00:37:03.820 --> 00:37:05.020
me different delta a's.
00:37:09.240 --> 00:37:10.880
So one thing that's
sort of annoying
00:37:10.880 --> 00:37:12.706
is that when you
write delta a, there's
00:37:12.706 --> 00:37:14.330
nothing in the notation
that says which
00:37:14.330 --> 00:37:16.360
distribution you
were talking about.
00:37:16.360 --> 00:37:18.030
When you have multiple
distributions,
00:37:18.030 --> 00:37:21.582
or multiple possible probability
distributions, sometimes it's
00:37:21.582 --> 00:37:23.790
useful to just put given
the probability distribution
00:37:23.790 --> 00:37:25.670
p of a.
00:37:25.670 --> 00:37:28.170
This is not very often used,
but sometimes it's very helpful
00:37:28.170 --> 00:37:30.984
when you're doing calculations
just to keep track.
00:37:30.984 --> 00:37:34.010
Everyone cool with that?
00:37:34.010 --> 00:37:35.702
Yeah, questions?
00:37:35.702 --> 00:37:37.590
AUDIENCE: [INAUDIBLE]
delta a squared, right?
00:37:37.590 --> 00:37:38.980
PROFESSOR: Yeah, exactly.
00:37:38.980 --> 00:37:40.145
Of delta a squared.
00:37:40.145 --> 00:37:41.450
Yeah.
00:37:41.450 --> 00:37:43.550
Other questions?
00:37:43.550 --> 00:37:44.100
Yeah?
00:37:44.100 --> 00:37:45.890
AUDIENCE: So really it should
be parentheses [INAUDIBLE].
00:37:45.890 --> 00:37:47.620
PROFESSOR: Yeah, it's just
this is notation that's
00:37:47.620 --> 00:37:50.010
used typically, so I didn't
put the parentheses around
00:37:50.010 --> 00:37:53.140
precisely to alert you to the
stupidities of this notation.
00:37:59.580 --> 00:38:02.810
So any other questions?
00:38:02.810 --> 00:38:03.310
Good.
00:38:03.310 --> 00:38:06.970
OK, so let's just do the same
thing for continuous variables.
00:38:06.970 --> 00:38:08.610
Now for continuous variables.
00:38:14.516 --> 00:38:16.140
I'm just going to
write the expressions
00:38:16.140 --> 00:38:17.790
and just get them
out of the way.
00:38:17.790 --> 00:38:20.900
So the average value of
some x, given a probability
00:38:20.900 --> 00:38:23.220
distribution on x where x
is a continuous variable,
00:38:23.220 --> 00:38:24.952
is going to be equal
to the integral.
00:38:24.952 --> 00:38:26.910
Let's just say x is
defined from minus infinity
00:38:26.910 --> 00:38:31.500
to infinity, which is pretty
useful, or pretty typical.
00:38:31.500 --> 00:38:37.644
dx probability
distribution of x times x.
00:38:37.644 --> 00:38:38.560
I shouldn't use curvy.
00:38:38.560 --> 00:38:39.620
I should just use x.
00:38:42.150 --> 00:38:44.960
And similarly for
x squared, or more
00:38:44.960 --> 00:38:47.920
generally, for f of x,
the average value of f
00:38:47.920 --> 00:38:52.050
of x, or the expected value of
f of x given this probability
00:38:52.050 --> 00:38:54.900
distribution, is going to
be equal to the integral dx
00:38:54.900 --> 00:38:56.960
minus infinity to infinity.
00:38:56.960 --> 00:39:01.124
The probability distribution
of x times f of x.
00:39:01.124 --> 00:39:02.790
In direct analogy to
what we had before.
00:39:05.890 --> 00:39:08.850
So this is all just mathematics.
00:39:08.850 --> 00:39:11.620
And we define the
uncertainty in x
00:39:11.620 --> 00:39:15.940
is equal to the expectation
value of x squared
00:39:15.940 --> 00:39:20.155
minus the expected value
of x quantity squared.
00:39:23.560 --> 00:39:24.760
And this is delta x squared.
00:39:27.430 --> 00:39:30.210
If you see me dropping an
exponent or a factor of 2,
00:39:30.210 --> 00:39:33.110
please, please, please tell me.
00:39:33.110 --> 00:39:36.060
So thank you for that.
00:39:36.060 --> 00:39:39.720
All of that is just straight up
classical probability theory.
00:39:39.720 --> 00:39:42.750
And I just want to write this
in the notation of quantum
00:39:42.750 --> 00:39:43.660
mechanics.
00:39:43.660 --> 00:39:45.850
Given that the
system is in a state
00:39:45.850 --> 00:39:50.210
described by the wave function
psi of x, the average value,
00:39:50.210 --> 00:39:53.560
the expected value of x, the
typical value if you just
00:39:53.560 --> 00:39:56.160
observe the particle
at some moment,
00:39:56.160 --> 00:40:01.200
is equal to the integral over
all possible values of x.
00:40:01.200 --> 00:40:05.790
The probability distribution,
psi of x norm squared x.
00:40:08.480 --> 00:40:12.480
And similarly, for
any function of x,
00:40:12.480 --> 00:40:15.250
the expected value is going to
be equal to the integral dx.
00:40:15.250 --> 00:40:17.583
The probability distribution,
which is given by the norm
00:40:17.583 --> 00:40:23.710
squared of the wave function
times f of x minus infinity
00:40:23.710 --> 00:40:26.140
to infinity.
00:40:26.140 --> 00:40:30.010
And same definition
for uncertainty.
00:40:30.010 --> 00:40:33.070
And again, this notation
is really dangerous,
00:40:33.070 --> 00:40:37.010
because the expected value of
x depends on the probability
00:40:37.010 --> 00:40:37.790
distribution.
00:40:37.790 --> 00:40:39.985
In a physical system,
the expected value of x
00:40:39.985 --> 00:40:41.610
depends on what the
state of the system
00:40:41.610 --> 00:40:43.610
is, what the wave function
is, and this notation
00:40:43.610 --> 00:40:44.970
doesn't indicate that.
00:40:44.970 --> 00:40:47.450
So there are a couple of ways
to improve this notation.
00:40:47.450 --> 00:40:52.540
One of which is-- so this is,
again, a sort of side note.
00:40:52.540 --> 00:40:54.810
One way to improve
this notation x
00:40:54.810 --> 00:40:59.450
is to write the expected
value of x in the state psi,
00:40:59.450 --> 00:41:01.036
so you write psi as a subscript.
00:41:01.036 --> 00:41:02.910
Another notation that
will come back-- you'll
00:41:02.910 --> 00:41:05.040
see why this is a
useful notation later
00:41:05.040 --> 00:41:10.511
in the semester-- is
this notation, psi.
00:41:10.511 --> 00:41:12.510
And we will give meaning
to this notation later,
00:41:12.510 --> 00:41:13.610
but I just want to
alert you that it's
00:41:13.610 --> 00:41:15.690
used throughout books, and
it means the same thing
00:41:15.690 --> 00:41:17.814
as what we're talking about
the expected value of x
00:41:17.814 --> 00:41:19.880
given a particular state psi.
00:41:19.880 --> 00:41:20.640
OK?
00:41:20.640 --> 00:41:21.140
Yeah?
00:41:21.140 --> 00:41:23.480
AUDIENCE: To calculate the
expected value of momentum
00:41:23.480 --> 00:41:25.630
do you need to transform the--
00:41:25.630 --> 00:41:26.880
PROFESSOR: Excellent question.
00:41:26.880 --> 00:41:27.840
Excellent, excellent question.
00:41:27.840 --> 00:41:29.290
OK, so the question
is, how do we
00:41:29.290 --> 00:41:30.961
do the same thing for momentum?
00:41:30.961 --> 00:41:33.210
If you want to compute the
expected value of momentum,
00:41:33.210 --> 00:41:34.170
what do you have to do?
00:41:34.170 --> 00:41:36.790
Do you have to do some Fourier
transform to the wave function?
00:41:36.790 --> 00:41:39.830
So this is a
question that you're
00:41:39.830 --> 00:41:41.360
going to answer
on the problem set
00:41:41.360 --> 00:41:43.334
and that we made a
guess for last time.
00:41:43.334 --> 00:41:45.000
But quickly, let's
just think about what
00:41:45.000 --> 00:41:46.820
it's going to be
purely formally.
00:41:46.820 --> 00:41:50.540
Formally, if we want to know the
likely value of the momentum,
00:41:50.540 --> 00:41:53.210
the likely value the momentum,
it's a continuous variable.
00:41:53.210 --> 00:41:55.400
Just like any other
observable variable,
00:41:55.400 --> 00:41:58.700
we can write as the integral
over all possible values
00:41:58.700 --> 00:42:01.270
of momentum from,
let's say, it could
00:42:01.270 --> 00:42:03.980
be minus infinity to infinity.
00:42:03.980 --> 00:42:09.141
The probability of having that
momentum times momentum, right?
00:42:09.141 --> 00:42:10.140
Everyone cool with that?
00:42:10.140 --> 00:42:11.590
This is a tautology, right?
00:42:11.590 --> 00:42:14.930
This is what you
mean by probability.
00:42:14.930 --> 00:42:17.600
But we need to know if we have
a quantum mechanical system
00:42:17.600 --> 00:42:19.830
described by state
psi of x, how do
00:42:19.830 --> 00:42:23.066
we can get the probability
that you measure p?
00:42:23.066 --> 00:42:25.096
Do I want to do this now?
00:42:25.096 --> 00:42:27.430
Yeah, OK I do.
00:42:27.430 --> 00:42:30.780
And we need a guess.
00:42:30.780 --> 00:42:31.391
Question mark.
00:42:31.391 --> 00:42:33.140
We made a guess at the
end of last lecture
00:42:33.140 --> 00:42:34.765
that, in quantum
mechanics, this should
00:42:34.765 --> 00:42:41.820
be dp minus infinity to infinity
of the Fourier transform.
00:42:41.820 --> 00:42:46.700
Psi tilde of p up
to an h bar factor.
00:42:46.700 --> 00:42:53.160
Psi tilde of p, the Fourier
transform p norm squared.
00:42:53.160 --> 00:42:56.730
OK, so we're guessing that the
Fourier transform norm squared
00:42:56.730 --> 00:42:59.190
is equal to the probability
of measuring the associated
00:42:59.190 --> 00:43:00.850
momentum.
00:43:00.850 --> 00:43:03.370
So that's a guess.
00:43:03.370 --> 00:43:04.090
That's a guess.
00:43:04.090 --> 00:43:06.400
And so on your problem set
you're going to prove it.
00:43:06.400 --> 00:43:07.220
OK?
00:43:07.220 --> 00:43:08.940
So exactly the same
logic goes through.
00:43:08.940 --> 00:43:10.050
It's a very good
question, thanks.
00:43:10.050 --> 00:43:10.850
Other questions?
00:43:10.850 --> 00:43:12.698
Yeah?
00:43:12.698 --> 00:43:14.674
AUDIENCE: Is that p
the momentum itself?
00:43:14.674 --> 00:43:17.150
Or is that the probability?
00:43:17.150 --> 00:43:19.100
PROFESSOR: So this
is the probability
00:43:19.100 --> 00:43:20.990
of measuring momentum p.
00:43:20.990 --> 00:43:22.770
And that's the value p.
00:43:22.770 --> 00:43:23.980
We're summing over all p's.
00:43:23.980 --> 00:43:27.660
This is the probability,
and that's actually p.
00:43:27.660 --> 00:43:29.351
So the Fourier
transform is a function
00:43:29.351 --> 00:43:31.600
of the momentum in the same
way that the wave function
00:43:31.600 --> 00:43:34.380
is a function of
the position, right?
00:43:34.380 --> 00:43:36.080
So this is a function
of the momentum.
00:43:36.080 --> 00:43:39.917
It's norm squared
defines the probability.
00:43:39.917 --> 00:43:41.500
And then the p on
the right is this p,
00:43:41.500 --> 00:43:43.500
because we're computing
the expected value of p,
00:43:43.500 --> 00:43:44.640
or the average value of p.
00:43:44.640 --> 00:43:46.100
That make sense?
00:43:46.100 --> 00:43:47.530
Cool.
00:43:47.530 --> 00:43:48.426
Yeah?
00:43:48.426 --> 00:43:50.590
AUDIENCE: Are we then
multiplying by p squared
00:43:50.590 --> 00:43:52.027
if we're doing all p's?
00:43:52.027 --> 00:43:56.340
Because we have the dp times
p for each [INAUDIBLE].
00:43:56.340 --> 00:43:56.960
PROFESSOR: No.
00:43:56.960 --> 00:43:58.100
So that's a very good question.
00:43:58.100 --> 00:43:58.920
So let's go back.
00:43:58.920 --> 00:44:00.490
Very good question.
00:44:00.490 --> 00:44:02.470
Let me phrase it in
terms of position,
00:44:02.470 --> 00:44:03.430
because the same
question comes up.
00:44:03.430 --> 00:44:04.580
Thank you for asking that.
00:44:04.580 --> 00:44:05.250
Look at this.
00:44:05.250 --> 00:44:06.070
This is weird.
00:44:06.070 --> 00:44:08.570
I'm going to phrase this as a
dimensional analysis question.
00:44:08.570 --> 00:44:10.520
Tell me if this is the same
question as you're asking.
00:44:10.520 --> 00:44:12.310
This is a thing with
dimensions of what?
00:44:12.310 --> 00:44:13.542
Length, right?
00:44:13.542 --> 00:44:15.000
But over on the
right hand side, we
00:44:15.000 --> 00:44:19.305
have a length and a probability,
which is a number, and then
00:44:19.305 --> 00:44:19.930
another length.
00:44:19.930 --> 00:44:21.940
That looks like
x squared, right?
00:44:21.940 --> 00:44:23.570
So why are we getting something
with dimensions of length,
00:44:23.570 --> 00:44:25.610
not something with
dimensions of length squared?
00:44:25.610 --> 00:44:27.443
And the answer is this
is not a probability.
00:44:27.443 --> 00:44:29.970
It is a probability density.
00:44:29.970 --> 00:44:34.180
So it's got units of
probability per unit length.
00:44:34.180 --> 00:44:36.240
So this has dimensions
of one over length.
00:44:36.240 --> 00:44:39.380
So this quantity,
p of x dx, tells me
00:44:39.380 --> 00:44:41.930
the probability, which is a
pure number, no dimensions.
00:44:41.930 --> 00:44:44.690
The probability to find the
particle between x and x
00:44:44.690 --> 00:44:46.550
plus dx.
00:44:46.550 --> 00:44:47.110
Cool?
00:44:47.110 --> 00:44:50.100
So that was our
second postulate.
00:44:50.100 --> 00:44:52.455
Psi of x dx squared
is the probability
00:44:52.455 --> 00:44:54.640
of finding it in this domain.
00:44:54.640 --> 00:44:59.040
And so what we're doing is we're
summing over all such domains
00:44:59.040 --> 00:45:02.610
the probability times the value.
00:45:02.610 --> 00:45:03.490
Cool?
00:45:03.490 --> 00:45:05.730
So this is the difference
between discrete,
00:45:05.730 --> 00:45:09.115
where we didn't have these
probability densities,
00:45:09.115 --> 00:45:11.490
we just had numbers, pure
numbers and pure probabilities.
00:45:11.490 --> 00:45:14.776
Now we have probability
densities per unit whatever.
00:45:14.776 --> 00:45:15.405
Yeah?
00:45:15.405 --> 00:45:17.405
AUDIENCE: How do you
pronounce the last notation
00:45:17.405 --> 00:45:18.707
that you wrote?
00:45:18.707 --> 00:45:20.040
PROFESSOR: How do you pronounce?
00:45:20.040 --> 00:45:21.248
Good, that's a good question.
00:45:21.248 --> 00:45:23.500
The question is, how do
we pronounce these things.
00:45:23.500 --> 00:45:25.160
So this is called
the expected value
00:45:25.160 --> 00:45:28.190
of x, or the average value of
x, or most typically in quantum
00:45:28.190 --> 00:45:31.130
mechanics, the
expectation value of x.
00:45:31.130 --> 00:45:33.160
So you can call it
anything you want.
00:45:33.160 --> 00:45:34.830
This is the same thing.
00:45:34.830 --> 00:45:38.029
The psi is just to denote
that this is in the state psi.
00:45:38.029 --> 00:45:39.570
And it can be
pronounced in two ways.
00:45:39.570 --> 00:45:41.236
You can either say
the expectation value
00:45:41.236 --> 00:45:45.600
of x, or the expectation
of x in the state psi.
00:45:45.600 --> 00:45:48.610
And this would be
pronounced one of two ways.
00:45:48.610 --> 00:45:55.451
The expectation value of x in
the state psi, or psi x psi.
00:45:55.451 --> 00:45:55.950
Yeah.
00:45:55.950 --> 00:45:58.030
That's a very good question.
00:45:58.030 --> 00:45:59.240
But they mean the same thing.
00:45:59.240 --> 00:46:02.812
Now, I should emphasize that you
can have two ways of describing
00:46:02.812 --> 00:46:04.270
something that mean
the same thing,
00:46:04.270 --> 00:46:06.144
but they carry different
connotations, right?
00:46:08.500 --> 00:46:12.727
Like have a friend
who's a really nice guy.
00:46:12.727 --> 00:46:13.310
He's a mensch.
00:46:13.310 --> 00:46:14.310
He's a good guy.
00:46:14.310 --> 00:46:16.020
And so I could see
he's a nice guy,
00:46:16.020 --> 00:46:18.200
I could say he's
[? carinoso ?], and they
00:46:18.200 --> 00:46:21.490
mean different things
in different languages.
00:46:21.490 --> 00:46:24.500
It's the same idea, but they
have different flavors, right?
00:46:24.500 --> 00:46:27.310
So whatever your
native language is,
00:46:27.310 --> 00:46:29.082
you've got some analog of this.
00:46:29.082 --> 00:46:32.045
This means something in
a particular mathematical
00:46:32.045 --> 00:46:33.920
language for talking
about quantum mechanics.
00:46:33.920 --> 00:46:35.450
And this has a different flavor.
00:46:35.450 --> 00:46:36.896
It carries different
implications,
00:46:36.896 --> 00:46:38.270
and we'll see what
that is later.
00:46:38.270 --> 00:46:39.120
We haven't got there yet.
00:46:39.120 --> 00:46:39.620
Yeah?
00:46:39.620 --> 00:46:41.890
AUDIENCE: Why is there a
double notation of psi?
00:46:41.890 --> 00:46:44.460
PROFESSOR: Why is there
a double notation of psi?
00:46:44.460 --> 00:46:47.830
Yeah, we'll see later.
00:46:47.830 --> 00:46:50.530
Roughly speaking, it's because
in computing this expectation
00:46:50.530 --> 00:46:52.480
value, there's a psi squared.
00:46:52.480 --> 00:46:55.940
And so this is to
remind you of that.
00:46:55.940 --> 00:46:58.010
Other questions?
00:46:58.010 --> 00:47:00.060
Terminology is one of the
most annoying features
00:47:00.060 --> 00:47:00.640
of quantum mechanics.
00:47:00.640 --> 00:47:01.139
Yeah?
00:47:01.139 --> 00:47:04.435
AUDIENCE: So it seems like
this [INAUDIBLE] variance
00:47:04.435 --> 00:47:07.301
is a really convenient
way of doing it.
00:47:07.301 --> 00:47:08.800
How is it the
Heisenberg uncertainty
00:47:08.800 --> 00:47:13.480
works exactly as it does for
this definition of variance.
00:47:13.480 --> 00:47:15.105
PROFESSOR: That's a
very good question.
00:47:17.285 --> 00:47:18.660
In order to answer
that question,
00:47:18.660 --> 00:47:20.440
we need to actually work out
the Heisenberg uncertainty
00:47:20.440 --> 00:47:20.940
relation.
00:47:20.940 --> 00:47:25.182
So the question is, look, this
is some choice of uncertainty.
00:47:25.182 --> 00:47:27.640
You could have chosen some
other definition of uncertainly.
00:47:27.640 --> 00:47:29.590
We could have considered
the expectation
00:47:29.590 --> 00:47:32.230
value of x to the fourth
minus x to the fourth
00:47:32.230 --> 00:47:33.920
and taken the
fourth root of that.
00:47:33.920 --> 00:47:36.100
So why this one?
00:47:36.100 --> 00:47:38.460
And one answer is, indeed,
the uncertainty relation
00:47:38.460 --> 00:47:40.050
works out quite nicely.
00:47:40.050 --> 00:47:43.170
But then I think
important to say
00:47:43.170 --> 00:47:45.837
here is that there are many ways
you could construct quantities.
00:47:45.837 --> 00:47:47.378
This is a convenient
one, and we will
00:47:47.378 --> 00:47:49.530
discover that it has nice
properties that we like.
00:47:49.530 --> 00:47:53.662
There is no God given reason why
this had to be the right thing.
00:47:53.662 --> 00:47:56.120
I can say more, but I don't
want to take the time to do it,
00:47:56.120 --> 00:47:58.560
so ask in office hours.
00:47:58.560 --> 00:48:01.180
OK, good.
00:48:01.180 --> 00:48:02.680
The second part of
your question was
00:48:02.680 --> 00:48:04.690
why does the Heisenberg
relation work out
00:48:04.690 --> 00:48:05.750
nicely in terms of
these guys, and we
00:48:05.750 --> 00:48:07.400
will study that in
extraordinary detail.
00:48:07.400 --> 00:48:08.025
We'll see that.
00:48:08.025 --> 00:48:11.240
So we're going to derive it
twice soon and then later.
00:48:11.240 --> 00:48:12.690
The later version is better.
00:48:12.690 --> 00:48:14.115
So let me work
out some examples.
00:48:17.330 --> 00:48:21.380
Or actually, I'm going
to skip the examples
00:48:21.380 --> 00:48:22.740
in the interest of time.
00:48:22.740 --> 00:48:23.850
They're in the
notes, and so they'll
00:48:23.850 --> 00:48:24.933
be posted on the web page.
00:48:24.933 --> 00:48:28.341
By the way, the first 18
lectures of notes are posted.
00:48:28.341 --> 00:48:29.590
I had a busy night last night.
00:48:32.600 --> 00:48:34.547
So let's come back to
computing expectation
00:48:34.547 --> 00:48:35.380
values for momentum.
00:48:46.020 --> 00:48:48.000
So I want to go
back to this and ask
00:48:48.000 --> 00:48:51.230
a silly-- I want to make some
progress towards deriving
00:48:51.230 --> 00:48:51.910
this relation.
00:48:51.910 --> 00:48:55.070
So I want to start over on
the definition of the expected
00:48:55.070 --> 00:48:55.820
value of momentum.
00:48:58.119 --> 00:49:00.660
And I'd like to do it directly
in terms of the wave function.
00:49:00.660 --> 00:49:01.990
So how would we do this?
00:49:01.990 --> 00:49:04.900
So one way of saying this is
what's the average value of p.
00:49:04.900 --> 00:49:06.910
Well, I can phrase this
in terms of the wave
00:49:06.910 --> 00:49:08.035
function the following way.
00:49:08.035 --> 00:49:10.630
I'm going to sum over
all positions dx.
00:49:10.630 --> 00:49:15.140
Expectation value of x
squared from minus infinity
00:49:15.140 --> 00:49:16.620
to infinity.
00:49:16.620 --> 00:49:19.065
And then the momentum
associated to the value x.
00:49:23.287 --> 00:49:25.370
So it's tempting to write
something like this down
00:49:25.370 --> 00:49:26.950
to think maybe
there's some p of x.
00:49:29.476 --> 00:49:31.100
This is a tempting
thing to write down.
00:49:33.700 --> 00:49:34.200
Can we?
00:49:40.490 --> 00:49:46.490
Are we ever in a position
to say intelligently
00:49:46.490 --> 00:49:50.010
that a particle--
that an electron
00:49:50.010 --> 00:49:55.094
is both hard and white?
00:49:55.094 --> 00:49:56.010
AUDIENCE: No.
00:49:56.010 --> 00:49:58.060
PROFESSOR: No,
because being hard
00:49:58.060 --> 00:50:01.676
is a superposition of being
black and white, right?
00:50:01.676 --> 00:50:06.210
Are we ever in a position
to say that our particle has
00:50:06.210 --> 00:50:08.850
a definite position
x and correspondingly
00:50:08.850 --> 00:50:11.450
a definite momentum p.
00:50:11.450 --> 00:50:12.930
It's not that we don't get too.
00:50:12.930 --> 00:50:15.664
It's that it doesn't
make sense to do so.
00:50:15.664 --> 00:50:17.330
In general, being in
a definite position
00:50:17.330 --> 00:50:21.230
means being in a
superposition of having
00:50:21.230 --> 00:50:24.520
different values for momentum.
00:50:24.520 --> 00:50:27.100
And if you want a sharp
way of saying this,
00:50:27.100 --> 00:50:29.065
look at these relations.
00:50:29.065 --> 00:50:32.730
They claim that any
function can be expressed
00:50:32.730 --> 00:50:37.040
as a superposition of states
with definite momentum, right?
00:50:37.040 --> 00:50:41.880
Well, among other things a
state with definite position,
00:50:41.880 --> 00:50:46.990
x0, can be written
as a superposition, 1
00:50:46.990 --> 00:50:52.100
over 2pi integral dk.
00:50:52.100 --> 00:50:56.659
I'll call this delta tilde of k.
00:50:56.659 --> 00:50:57.200
e to the ikx.
00:51:05.009 --> 00:51:07.050
If you haven't played with
delta functions before
00:51:07.050 --> 00:51:09.655
and you haven't seen this,
then you will on the problem
00:51:09.655 --> 00:51:11.230
set, because we
have a problem that
00:51:11.230 --> 00:51:13.020
works through a
great many details.
00:51:13.020 --> 00:51:17.840
But in particular, it's
clear that this is not--
00:51:17.840 --> 00:51:20.672
this quantity can't be
a delta function of k,
00:51:20.672 --> 00:51:22.880
because, if it were, this
would be just e to the ikx.
00:51:22.880 --> 00:51:25.060
And that's definitely
not a delta function.
00:51:25.060 --> 00:51:29.910
Meanwhile, what can you say
about the continuity structure
00:51:29.910 --> 00:51:31.060
of a delta function.
00:51:31.060 --> 00:51:33.300
Is it continuous?
00:51:33.300 --> 00:51:33.960
No.
00:51:33.960 --> 00:51:35.364
Its derivative isn't continuous.
00:51:35.364 --> 00:51:36.280
Its second derivative.
00:51:36.280 --> 00:51:38.363
None of its derivatives
are in any way continuous.
00:51:38.363 --> 00:51:40.879
They're all absolutely
horrible, OK?
00:51:40.879 --> 00:51:42.420
So how many momentum
modes am I going
00:51:42.420 --> 00:51:44.919
to need to superimpose in order
to reproduce a function that
00:51:44.919 --> 00:51:47.900
has this sort of structure?
00:51:47.900 --> 00:51:48.790
An infinite number.
00:51:48.790 --> 00:51:49.650
And it turns out
it's going to be
00:51:49.650 --> 00:51:51.970
an infinite number with the
same amplitude, slightly
00:51:51.970 --> 00:51:54.420
different phase, OK?
00:51:54.420 --> 00:51:57.885
So you can never say
that you're in a state
00:51:57.885 --> 00:51:59.760
with definite position
and definite momentum.
00:51:59.760 --> 00:52:01.400
Being in a state with
definite position
00:52:01.400 --> 00:52:07.210
means being in a superposition
of being in a superposition.
00:52:07.210 --> 00:52:11.570
In fact, I'm just going
right down the answer here.
00:52:11.570 --> 00:52:12.702
e to the ikx0.
00:52:17.030 --> 00:52:21.060
Being in a state with
definite position
00:52:21.060 --> 00:52:22.850
means being in a
superposition of states
00:52:22.850 --> 00:52:26.420
with arbitrary momentum
and vice versa.
00:52:26.420 --> 00:52:28.651
You cannot be in a state
with definite position,
00:52:28.651 --> 00:52:29.400
definite momentum.
00:52:29.400 --> 00:52:30.274
So this doesn't work.
00:52:30.274 --> 00:52:33.280
So what we want is we
want some good definition.
00:52:33.280 --> 00:52:34.780
So this does not work.
00:52:34.780 --> 00:52:37.500
We want some good
definition of p
00:52:37.500 --> 00:52:40.950
given that we're working
with a wave function which
00:52:40.950 --> 00:52:43.500
is a function of x.
00:52:43.500 --> 00:52:45.415
What is that good
definition of the momentum?
00:52:51.310 --> 00:52:54.560
We have a couple of hints.
00:52:54.560 --> 00:52:55.930
So hint the first.
00:53:02.860 --> 00:53:04.810
So this is what we're after.
00:53:04.810 --> 00:53:06.520
Hint the first is
that a wave-- we
00:53:06.520 --> 00:53:09.950
know that given a wave with wave
number k, which is equal 2pi
00:53:09.950 --> 00:53:15.470
over lambda, is associated,
according to de Broglie
00:53:15.470 --> 00:53:19.460
and according to
Davisson-Germer experiments,
00:53:19.460 --> 00:53:22.070
to a particle-- so
having a particle--
00:53:22.070 --> 00:53:25.280
a wave, with wave number k or
wavelength lambda associated
00:53:25.280 --> 00:53:29.590
particle with momentum
p is equal to h bar k.
00:53:29.590 --> 00:53:31.670
Yeah?
00:53:31.670 --> 00:53:34.540
But in particular, what is a
plane with wavelength lambda
00:53:34.540 --> 00:53:36.570
or wave number k look like?
00:53:36.570 --> 00:53:39.500
That's e to the iks.
00:53:39.500 --> 00:53:41.890
And if I have a
wave, a plane wave
00:53:41.890 --> 00:53:45.190
e to the iks, how do I
get h bar k out of it?
00:53:52.650 --> 00:53:58.504
Note the following, the
derivative with respect to x.
00:53:58.504 --> 00:53:59.920
Actually let me
do this down here.
00:54:08.060 --> 00:54:12.910
Note that the derivative with
respect to x of e to the ikx
00:54:12.910 --> 00:54:17.230
is equal to ik e to the ikx.
00:54:24.030 --> 00:54:27.050
There's nothing up my sleeves.
00:54:27.050 --> 00:54:30.140
So in particular, if
I want to get h bar k,
00:54:30.140 --> 00:54:32.820
I can multiply by h
bar and divide by i.
00:54:32.820 --> 00:54:35.510
Multiply by h bar, divide by
i, derivative with respect
00:54:35.510 --> 00:54:39.140
to x e to the ikx.
00:54:39.140 --> 00:54:42.490
And this is equal to
h bar k e to the ikx.
00:54:45.160 --> 00:54:45.960
That's suggestive.
00:54:49.200 --> 00:54:53.330
And I can write this
as p e to the ikx.
00:55:00.620 --> 00:55:02.196
So let's quickly
check the units.
00:55:05.170 --> 00:55:08.740
So first off, what are
the units of h bar?
00:55:08.740 --> 00:55:10.130
Here's the super
easy to remember
00:55:10.130 --> 00:55:13.450
the units of-- or
dimensions of h bar are.
00:55:13.450 --> 00:55:18.550
Delta x delta p is h bar.
00:55:18.550 --> 00:55:19.050
OK?
00:55:19.050 --> 00:55:21.300
If you're ever in doubt,
if you just remember,
00:55:21.300 --> 00:55:24.852
h bar has units of
momentum times length.
00:55:24.852 --> 00:55:26.560
It's just the easiest
way to remember it.
00:55:26.560 --> 00:55:27.893
You'll never forget it that way.
00:55:27.893 --> 00:55:31.070
So if h bar has units of
momentum times length,
00:55:31.070 --> 00:55:33.130
what are the units of k?
00:55:33.130 --> 00:55:33.850
1 over length.
00:55:33.850 --> 00:55:35.433
So does this
dimensionally make sense?
00:55:35.433 --> 00:55:36.170
Yeah.
00:55:36.170 --> 00:55:40.630
Momentum times length divided
by length number momentum.
00:55:40.630 --> 00:55:41.130
Good.
00:55:41.130 --> 00:55:42.781
So dimensionally we
haven't lied yet.
00:55:46.000 --> 00:55:48.250
So this makes it tempting
to say something like, well,
00:55:48.250 --> 00:55:51.530
hell h bar upon i
derivative with respect
00:55:51.530 --> 00:55:57.425
to x is equal in some-- question
mark, quotation mark-- p.
00:56:01.110 --> 00:56:03.510
Right?
00:56:03.510 --> 00:56:06.250
So at this point it's just
tempting to say, look, trust
00:56:06.250 --> 00:56:07.915
me, p is h bar upon idx.
00:56:07.915 --> 00:56:09.290
But I don't know
about you, but I
00:56:09.290 --> 00:56:11.910
find that deeply,
deeply unsatisfying.
00:56:11.910 --> 00:56:14.540
So let me ask the question
slightly differently.
00:56:14.540 --> 00:56:16.580
We've followed the
de Broglie relations,
00:56:16.580 --> 00:56:18.586
and we've been led
to the idea that
00:56:18.586 --> 00:56:19.960
using wave functions
that there's
00:56:19.960 --> 00:56:22.400
some relationship
between the momentum,
00:56:22.400 --> 00:56:24.850
the observable quantity that
you measure with sticks,
00:56:24.850 --> 00:56:28.450
and meters, and stuff, and this
operator, this differential
00:56:28.450 --> 00:56:32.170
operator, h bar upon on i
derivative with respect to x.
00:56:32.170 --> 00:56:36.560
By the way, my notation for
dx is the partial derivative
00:56:36.560 --> 00:56:38.652
with respect to x.
00:56:38.652 --> 00:56:39.235
Just notation.
00:56:42.350 --> 00:56:45.670
So if this is supposed
to be true in some sense,
00:56:45.670 --> 00:56:49.590
what is momentum have
to do with a derivative?
00:56:49.590 --> 00:56:51.090
Momentum is about
velocities, which
00:56:51.090 --> 00:56:53.090
is like derivatives with
respect to time, right?
00:56:53.090 --> 00:56:53.860
Times mass.
00:56:53.860 --> 00:56:56.180
Mass times derivative with
respect to time, velocity.
00:56:56.180 --> 00:56:57.960
So what does it have to do with
the derivative with respect
00:56:57.960 --> 00:56:58.735
to position?
00:57:02.480 --> 00:57:07.550
And this ties into the
most beautiful theorem
00:57:07.550 --> 00:57:13.627
in classical mechanics, which
is the Noether's theorem, named
00:57:13.627 --> 00:57:15.960
after the mathematician who
discovered it, Emmy Noether.
00:57:19.070 --> 00:57:22.870
And just out of
curiosity, how many people
00:57:22.870 --> 00:57:25.626
have seen Noether's
theorem in class.
00:57:25.626 --> 00:57:26.970
Oh that's so sad.
00:57:26.970 --> 00:57:27.640
That's a sin.
00:57:27.640 --> 00:57:31.212
OK, so here's a statement
of Noether's theorem,
00:57:31.212 --> 00:57:33.670
and it underlies an enormous
amount of classical mechanics,
00:57:33.670 --> 00:57:34.920
but also of quantum mechanics.
00:57:37.839 --> 00:57:39.630
Noether, incidentally,
was a mathematician.
00:57:39.630 --> 00:57:41.899
There's a whole wonderful
story about Emmy Noether.
00:57:41.899 --> 00:57:43.440
Ville went to her
and was like, look,
00:57:43.440 --> 00:57:45.690
I'm trying to understand
the notion of energy.
00:57:45.690 --> 00:57:48.130
And this guy down the
hall, Einstein, he
00:57:48.130 --> 00:57:50.660
has a theory called general
relativity about curved space
00:57:50.660 --> 00:57:52.650
times and how that has
something to do with gravity.
00:57:52.650 --> 00:57:53.246
But it doesn't
make a lot of sense
00:57:53.246 --> 00:57:54.870
to me, because I don't even
know how to define the energy.
00:57:54.870 --> 00:57:56.620
So how do you define
momentum and energy
00:57:56.620 --> 00:57:59.860
in this guy's crazy theory?
00:57:59.860 --> 00:58:02.390
And so Noether, who
was a mathematician,
00:58:02.390 --> 00:58:05.050
did all sorts of beautiful
stuff in algebra,
00:58:05.050 --> 00:58:06.080
looked at the problem
and was like I don't even
00:58:06.080 --> 00:58:07.450
know what it means in
classical mechanics.
00:58:07.450 --> 00:58:08.835
So what is a mean in
classical mechanics?
00:58:08.835 --> 00:58:09.970
So she went back to
classical mechanics
00:58:09.970 --> 00:58:11.520
and, from first
principles, came up
00:58:11.520 --> 00:58:13.680
with a good definition of
momentum, which turns out
00:58:13.680 --> 00:58:16.190
to underlie the modern idea
of conserved quantities
00:58:16.190 --> 00:58:17.250
and symmetries.
00:58:17.250 --> 00:58:21.210
And it's had enormous
far reaching impact,
00:58:21.210 --> 00:58:24.430
and say her name would praise.
00:58:24.430 --> 00:58:28.780
So Noether tells us the
following statement,
00:58:28.780 --> 00:58:39.230
to every symmetry-- and I
should say continuous symmetry--
00:58:39.230 --> 00:58:46.060
to every symmetry is associated
a conserved quantity.
00:58:52.590 --> 00:58:53.310
OK?
00:58:53.310 --> 00:58:54.920
So in particular, what
do I mean by symmetry?
00:58:54.920 --> 00:58:56.253
Well, for example, translations.
00:58:56.253 --> 00:58:59.530
x goes to x plus some length l.
00:58:59.530 --> 00:59:01.280
This could be done for
arbitrary length l.
00:59:01.280 --> 00:59:03.571
So for example, translation
by this much or translation
00:59:03.571 --> 00:59:04.150
by that much.
00:59:04.150 --> 00:59:05.440
These are translations.
00:59:05.440 --> 00:59:08.870
To every symmetry is associated
a conserved quantity.
00:59:08.870 --> 00:59:11.165
What symmetry is
associated to translations?
00:59:16.270 --> 00:59:18.142
Conservation of momentum, p dot.
00:59:20.980 --> 00:59:27.910
Time translations, t
goes to t plus capital
00:59:27.910 --> 00:59:30.790
T. What's a conserved
quantity associated
00:59:30.790 --> 00:59:32.660
with time
translational symmetry?
00:59:32.660 --> 00:59:35.230
Energy, which is
time independent.
00:59:38.270 --> 00:59:40.290
And rotations.
00:59:40.290 --> 00:59:41.280
Rotational symmetries.
00:59:45.350 --> 00:59:51.680
x, as a vector, goes to
some rotation times x.
00:59:54.540 --> 00:59:56.875
What's conserved by virtue
of rotational symmetry?
00:59:56.875 --> 00:59:58.000
AUDIENCE: Angular momentum.
00:59:58.000 --> 00:59:58.970
PROFESSOR: Angular momentum.
00:59:58.970 --> 00:59:59.500
Rock on.
01:00:03.760 --> 01:00:08.910
OK So quickly, I'm not going to
prove to you Noether's theorem.
01:00:08.910 --> 01:00:12.800
It's one of the most beautiful
and important theorems
01:00:12.800 --> 01:00:14.700
in physics, and you
should all study it.
01:00:14.700 --> 01:00:16.200
But let me just
convince you quickly
01:00:16.200 --> 01:00:18.002
that it's true in
classical mechanics.
01:00:18.002 --> 01:00:20.210
And this was observed long
before Noether pointed out
01:00:20.210 --> 01:00:23.080
why it was true in general.
01:00:23.080 --> 01:00:25.930
What does it mean to have
transitional symmetry?
01:00:25.930 --> 01:00:28.570
It means that, if I
do an experiment here
01:00:28.570 --> 01:00:31.650
and I do it here, I get
exactly the same results.
01:00:31.650 --> 01:00:33.820
I translate the system
and nothing changes.
01:00:33.820 --> 01:00:34.320
Cool?
01:00:34.320 --> 01:00:36.278
That's what I mean by
saying I have a symmetry.
01:00:36.278 --> 01:00:38.540
You do this thing,
and nothing changes.
01:00:38.540 --> 01:00:42.324
OK, so imagine I have a
particle, a classical particle,
01:00:42.324 --> 01:00:43.740
and it's moving
in some potential.
01:00:46.890 --> 01:00:49.564
This is u of x, right?
01:00:49.564 --> 01:00:51.230
And we know what the
equations of motion
01:00:51.230 --> 01:00:53.550
are in classical
mechanics from f
01:00:53.550 --> 01:00:58.560
equals ma p dot is equal to
the force, which is minus
01:00:58.560 --> 01:00:59.440
the gradient of u.
01:00:59.440 --> 01:01:01.510
Minus the gradient of u.
01:01:01.510 --> 01:01:02.010
Right?
01:01:02.010 --> 01:01:05.680
That's f equals ma in
terms of the potential.
01:01:05.680 --> 01:01:08.710
Now is the gradient of u 0?
01:01:08.710 --> 01:01:10.175
No.
01:01:10.175 --> 01:01:11.425
In this case, there's a force.
01:01:13.671 --> 01:01:15.920
So if I do an experiment
here, do I get the same thing
01:01:15.920 --> 01:01:17.583
as doing my experiment here?
01:01:17.583 --> 01:01:18.430
AUDIENCE: No.
01:01:18.430 --> 01:01:19.080
PROFESSOR: Certainly not.
01:01:19.080 --> 01:01:21.163
The [? system ?] is not
translationally invariant.
01:01:21.163 --> 01:01:23.210
The potential breaks that
translational symmetry.
01:01:23.210 --> 01:01:26.490
What potential has
translational symmetry?
01:01:26.490 --> 01:01:27.490
AUDIENCE: [INAUDIBLE].
01:01:27.490 --> 01:01:28.870
PROFESSOR: Yeah, constant.
01:01:28.870 --> 01:01:32.100
The only potential that has full
translational symmetry in one
01:01:32.100 --> 01:01:35.970
dimension is translation
invariant, i.e.
01:01:35.970 --> 01:01:37.500
constant.
01:01:37.500 --> 01:01:38.650
OK?
01:01:38.650 --> 01:01:40.180
What's the force?
01:01:40.180 --> 01:01:40.680
AUDIENCE: 0.
01:01:40.680 --> 01:01:41.221
PROFESSOR: 0.
01:01:41.221 --> 01:01:41.890
0 gradient.
01:01:41.890 --> 01:01:43.810
So what's p dot?
01:01:43.810 --> 01:01:44.910
Yep.
01:01:44.910 --> 01:01:45.720
Noether's theorem.
01:01:45.720 --> 01:01:48.040
Solid.
01:01:48.040 --> 01:01:49.270
OK.
01:01:49.270 --> 01:01:51.360
Less trivial is
conservation of energy.
01:01:51.360 --> 01:01:53.670
I claim and she claims--
and she's right--
01:01:53.670 --> 01:01:56.770
that if the system has
the same dynamics at one
01:01:56.770 --> 01:01:58.562
moment and a few moments
later and, indeed,
01:01:58.562 --> 01:02:00.561
any amount of time later,
if the laws of physics
01:02:00.561 --> 01:02:02.550
don't change in
time, then there must
01:02:02.550 --> 01:02:05.260
be a conserved
quantity called energy.
01:02:05.260 --> 01:02:08.740
There must be a
conserved quantity.
01:02:08.740 --> 01:02:09.960
And that's Noether's theorem.
01:02:09.960 --> 01:02:11.870
So this is the first
step, but this still
01:02:11.870 --> 01:02:14.190
doesn't tell us what
momentum exactly
01:02:14.190 --> 01:02:16.874
has to do with a derivative
with respect to space.
01:02:16.874 --> 01:02:18.290
We see that there's
a relationship
01:02:18.290 --> 01:02:22.380
between translations and
momentum conservation,
01:02:22.380 --> 01:02:24.890
but what's the relationship?
01:02:24.890 --> 01:02:26.270
So let's do this.
01:02:26.270 --> 01:02:29.060
I'm going to define an
operation called translate by L.
01:02:29.060 --> 01:02:30.520
And what translate
by L does is it
01:02:30.520 --> 01:02:39.211
takes f of x and it maps
it to f of x minus L.
01:02:39.211 --> 01:02:41.210
So this is a thing that
affects the translation.
01:02:41.210 --> 01:02:43.880
And why do I say that's
a translation by L rather
01:02:43.880 --> 01:02:47.490
than minus L. Well,
the point-- if you have
01:02:47.490 --> 01:02:50.420
some function like this,
and it has a peak at 0,
01:02:50.420 --> 01:02:57.630
then after the translation, the
peak is when x is equal to L.
01:02:57.630 --> 01:02:58.130
OK?
01:02:58.130 --> 01:02:59.940
So just to get the
signs straight.
01:02:59.940 --> 01:03:02.660
So define this operation,
which takes a function of x
01:03:02.660 --> 01:03:05.235
and translates it by L, but
leaves it otherwise identical.
01:03:09.084 --> 01:03:10.500
So let's consider
how translations
01:03:10.500 --> 01:03:13.160
behave on functions.
01:03:13.160 --> 01:03:15.160
And this is really cute.
01:03:15.160 --> 01:03:20.180
f of x minus L can be
written as a Taylor expansion
01:03:20.180 --> 01:03:23.820
around the point x-- around
the point L equals 0.
01:03:23.820 --> 01:03:26.250
So let's do Taylor
expansion for small L.
01:03:26.250 --> 01:03:32.210
So this is equal to f of x
minus L derivative with respect
01:03:32.210 --> 01:03:39.050
to x of f of x plus L squared
over 2 derivative squared,
01:03:39.050 --> 01:03:42.780
two derivatives of x, f
of x plus dot, dot, dot.
01:03:42.780 --> 01:03:43.280
Right?
01:03:43.280 --> 01:03:46.306
I'm just Taylor expanding.
01:03:46.306 --> 01:03:46.930
Nothing sneaky.
01:03:49.510 --> 01:03:52.550
Let's add the next
term, actually.
01:03:52.550 --> 01:03:54.820
Let me do this on
a whole new board.
01:04:06.210 --> 01:04:08.865
All right, so we
have translate by L
01:04:08.865 --> 01:04:15.130
on f of x is equal to f of x
minus L is equal to f of x.
01:04:15.130 --> 01:04:17.530
Now Taylor expanding
minus L derivative
01:04:17.530 --> 01:04:26.376
with respect to x of f
plus L squared over 2--
01:04:26.376 --> 01:04:27.900
I'm not giving
myself enough space.
01:04:27.900 --> 01:04:28.400
I'm sorry.
01:04:30.990 --> 01:04:37.600
f of x minus L is equal to
f of x minus L with respect
01:04:37.600 --> 01:04:46.150
to x of f of x plus L
squared over 2 to derivatives
01:04:46.150 --> 01:04:51.430
of x f of x minus
L cubed over 6--
01:04:51.430 --> 01:04:55.030
we're just Taylor expanding--
cubed with respect
01:04:55.030 --> 01:05:00.800
to x of f of x and so on.
01:05:00.800 --> 01:05:01.871
Yeah?
01:05:01.871 --> 01:05:04.370
But I'm going to write this in
the following suggestive way.
01:05:04.370 --> 01:05:13.610
This is equal to 1 times
f of x minus L derivative
01:05:13.610 --> 01:05:17.730
with respect to x f
of x plus L squared
01:05:17.730 --> 01:05:22.740
over 2 derivative with
respect to x squared times f
01:05:22.740 --> 01:05:28.000
of x minus L cubed over 6
derivative cubed with respect
01:05:28.000 --> 01:05:31.470
to x plus dot, dot, dot.
01:05:31.470 --> 01:05:32.580
Everybody good with that?
01:05:35.750 --> 01:05:38.740
But this is a series that
you should recognize,
01:05:38.740 --> 01:05:41.700
a particular Taylor series
for a particular function.
01:05:41.700 --> 01:05:44.754
It's a Taylor expansion for the
01:05:44.754 --> 01:05:45.670
AUDIENCE: Exponential.
01:05:45.670 --> 01:05:46.880
PROFESSOR: Exponential.
01:05:46.880 --> 01:05:51.230
e to the minus L derivative
with respect to x f of x.
01:05:59.689 --> 01:06:00.730
Which is kind of awesome.
01:06:00.730 --> 01:06:02.220
So let's just check to make
sure that this makes sense
01:06:02.220 --> 01:06:03.460
from dimensional grounds.
01:06:03.460 --> 01:06:05.710
So that's a derivative with
respect to x as units of 1
01:06:05.710 --> 01:06:06.120
over length.
01:06:06.120 --> 01:06:07.870
That's a length, so
this is dimensionless,
01:06:07.870 --> 01:06:09.370
so we can exponentiate it.
01:06:09.370 --> 01:06:12.450
Now you might look at me and
say, look, this is silly.
01:06:12.450 --> 01:06:14.680
You've taken an
operation like derivative
01:06:14.680 --> 01:06:15.760
and exponentiated it.
01:06:15.760 --> 01:06:17.631
What does that mean?
01:06:17.631 --> 01:06:19.290
And that is what it means?
01:06:19.290 --> 01:06:21.000
[LAUGHTER]
01:06:21.000 --> 01:06:21.610
OK?
01:06:21.610 --> 01:06:24.180
So we're going to do this all
the time in quantum mechanics.
01:06:24.180 --> 01:06:26.452
We're going to do things
like exponentiate operations.
01:06:26.452 --> 01:06:27.910
We'll talk about
it in more detail,
01:06:27.910 --> 01:06:29.368
but we're always
going to define it
01:06:29.368 --> 01:06:31.925
in this fashion as a
formal power series.
01:06:31.925 --> 01:06:32.425
Questions?
01:06:35.019 --> 01:06:36.560
AUDIENCE: Can you
transform operators
01:06:36.560 --> 01:06:38.170
from one space to another?
01:06:38.170 --> 01:06:39.680
PROFESSOR: Oh, you totally can.
01:06:39.680 --> 01:06:40.930
But we'll come back to that.
01:06:40.930 --> 01:06:43.400
We're going to talk about
operators next time.
01:06:43.400 --> 01:06:48.750
OK, so here's where we are.
01:06:48.750 --> 01:06:55.880
So from this what
is a derivative
01:06:55.880 --> 01:06:56.950
with respect to x mean?
01:06:56.950 --> 01:06:58.820
What does a derivative
with respect to x do?
01:06:58.820 --> 01:07:00.280
Well a derivative
with respect to x
01:07:00.280 --> 01:07:03.850
is something that generates
translations with respect
01:07:03.850 --> 01:07:06.285
to x through a Taylor expansion.
01:07:09.550 --> 01:07:12.240
If we have L be
arbitrarily small, right?
01:07:12.240 --> 01:07:14.951
L is arbitrarily small.
01:07:14.951 --> 01:07:17.200
What is the translation by
an arbitrarily small amount
01:07:17.200 --> 01:07:18.140
of f of x?
01:07:18.140 --> 01:07:19.610
Well, if L is
arbitrarily small, we
01:07:19.610 --> 01:07:21.110
can drop all the
higher order terms,
01:07:21.110 --> 01:07:24.542
and the change is just Ldx.
01:07:24.542 --> 01:07:26.000
So the derivative
with respect to x
01:07:26.000 --> 01:07:30.330
is telling us about
infinitesimal translations.
01:07:30.330 --> 01:07:31.360
Cool?
01:07:31.360 --> 01:07:33.490
The derivative with
respect to a position
01:07:33.490 --> 01:07:35.520
is something that
tells you, or controls,
01:07:35.520 --> 01:07:39.429
or generates infinitesimal
translations.
01:07:39.429 --> 01:07:41.220
And if you exponentiate
it, you do it many,
01:07:41.220 --> 01:07:43.350
many, many times in
a particular way,
01:07:43.350 --> 01:07:46.410
you get a macroscopic
finite translation.
01:07:46.410 --> 01:07:48.410
Cool?
01:07:48.410 --> 01:07:50.820
So this gives us three things.
01:07:50.820 --> 01:08:02.730
Translations in x are generated
by derivative with respect
01:08:02.730 --> 01:08:03.230
to x.
01:08:05.920 --> 01:08:14.410
But through Noether's
theorem translations,
01:08:14.410 --> 01:08:20.390
in x are associated to
conservation of momentum.
01:08:27.270 --> 01:08:30.720
So you shouldn't be so
shocked-- it's really not
01:08:30.720 --> 01:08:44.050
totally shocking-- that in
quantum mechanics, where we're
01:08:44.050 --> 01:08:48.500
very interested in the action
of things on functions,
01:08:48.500 --> 01:08:51.319
not just in positions, but
on functions of position,
01:08:51.319 --> 01:08:56.415
it shouldn't be totally shocking
that in quantum mechanics,
01:08:56.415 --> 01:08:58.970
the derivative with
respect to x is related
01:08:58.970 --> 01:09:00.810
to the momentum in
some particular way.
01:09:03.950 --> 01:09:07.968
Similarly, translations
in t are going
01:09:07.968 --> 01:09:09.384
to be generated
by what operation?
01:09:12.380 --> 01:09:15.140
Derivative with respect to time.
01:09:15.140 --> 01:09:18.520
So derivative with respect to
time from Noether's theorem
01:09:18.520 --> 01:09:20.779
is associated with
conservation of energy.
01:09:23.640 --> 01:09:25.810
That seems plausible.
01:09:25.810 --> 01:09:27.689
Derivative with respect
to, I don't know,
01:09:27.689 --> 01:09:31.779
an angle, a rotation.
01:09:31.779 --> 01:09:34.260
That's going to be
associated with what?
01:09:34.260 --> 01:09:36.390
Angular momentum?
01:09:36.390 --> 01:09:38.080
But angular momentum
around the axis
01:09:38.080 --> 01:09:39.240
for whom this is
the angle, so I'll
01:09:39.240 --> 01:09:40.364
call that z for the moment.
01:09:43.859 --> 01:09:48.140
And we're going to see these
pop up over and over again.
01:09:48.140 --> 01:09:49.520
But here's the thing.
01:09:52.819 --> 01:09:58.880
We started out with these
three principles today,
01:09:58.880 --> 01:10:02.140
and we've let ourselves to
some sort of association
01:10:02.140 --> 01:10:08.350
between the momentum and
the derivative like this.
01:10:08.350 --> 01:10:08.850
OK?
01:10:08.850 --> 01:10:10.558
And I've given you
some reason to believe
01:10:10.558 --> 01:10:11.950
that this isn't totally insane.
01:10:11.950 --> 01:10:13.440
Translations are
deeply connected
01:10:13.440 --> 01:10:14.500
with conservation of momentum.
01:10:14.500 --> 01:10:15.950
Transitional symmetry
is deeply connected
01:10:15.950 --> 01:10:17.310
with conservation momentum.
01:10:17.310 --> 01:10:18.770
And an infinitesimal
translation is
01:10:18.770 --> 01:10:22.070
nothing but a derivative
with respect to position.
01:10:22.070 --> 01:10:24.330
Those are deeply
linked concepts.
01:10:24.330 --> 01:10:27.900
But I didn't derive anything.
01:10:27.900 --> 01:10:29.570
I gave you no
derivation whatsoever
01:10:29.570 --> 01:10:33.000
of the relationship between
d dx and the momentum.
01:10:33.000 --> 01:10:36.170
Instead, I'm simply
going to declare it.
01:10:36.170 --> 01:10:39.620
I'm going to declare that,
in quantum mechanics--
01:10:39.620 --> 01:10:42.150
you cannot stop me--
in quantum mechanics,
01:10:42.150 --> 01:10:48.300
p is represented by an
operator, it's represented
01:10:48.300 --> 01:10:51.720
by the specific operator h bar
upon I derivative with respect
01:10:51.720 --> 01:10:53.840
to x.
01:10:53.840 --> 01:10:54.990
And this is a declaration.
01:10:58.680 --> 01:11:00.890
OK?
01:11:00.890 --> 01:11:02.700
It is simply a fact.
01:11:02.700 --> 01:11:05.730
And when they say it's a fact,
I mean two things by that.
01:11:05.730 --> 01:11:08.037
The first is it is a fact
that, in quantum mechanics,
01:11:08.037 --> 01:11:10.120
momentum is represented
by derivative with respect
01:11:10.120 --> 01:11:12.220
to x times h bar upon i.
01:11:12.220 --> 01:11:15.639
Secondly, it is a fact that,
if you take this expression
01:11:15.639 --> 01:11:17.930
and you work with the rest
of the postulates of quantum
01:11:17.930 --> 01:11:19.346
mechanics, including
what's coming
01:11:19.346 --> 01:11:22.730
next lecture about operators
and time evolution,
01:11:22.730 --> 01:11:24.580
you reproduce the physics
of the real world.
01:11:24.580 --> 01:11:26.030
You reproduce it beautifully.
01:11:26.030 --> 01:11:28.540
You reproduce it so well that
no other models have even
01:11:28.540 --> 01:11:32.000
ever vaguely come close to the
explanatory power of quantum
01:11:32.000 --> 01:11:32.971
mechanics.
01:11:32.971 --> 01:11:33.470
OK?
01:11:33.470 --> 01:11:34.210
It is a fact.
01:11:34.210 --> 01:11:36.710
It is not true in
some epistemic sense.
01:11:36.710 --> 01:11:38.910
You can't sit back
and say, ah a priori
01:11:38.910 --> 01:11:41.990
starting with the integers we
derive that p is equal to-- no,
01:11:41.990 --> 01:11:43.050
it's a model.
01:11:43.050 --> 01:11:44.509
But that's what physics does.
01:11:44.509 --> 01:11:46.050
Physics doesn't tell
you what's true.
01:11:46.050 --> 01:11:48.390
Physics doesn't tell
you what a priori
01:11:48.390 --> 01:11:49.820
did the world have to look like.
01:11:49.820 --> 01:11:52.412
Physics tells you
this is a good model,
01:11:52.412 --> 01:11:54.370
and it works really well,
and it fits the data.
01:11:54.370 --> 01:11:56.328
And to the degree that
it doesn't fit the data,
01:11:56.328 --> 01:11:57.850
it's wrong.
01:11:57.850 --> 01:11:58.414
OK?
01:11:58.414 --> 01:11:59.705
This isn't something we derive.
01:11:59.705 --> 01:12:01.250
This is something we declare.
01:12:01.250 --> 01:12:03.940
We call it our model, and then
we use it to calculate stuff,
01:12:03.940 --> 01:12:06.100
and we see if it
fits the real world.
01:12:09.600 --> 01:12:12.041
Out, please, please leave.
01:12:12.041 --> 01:12:12.540
Thank you.
01:12:15.473 --> 01:12:24.730
[LAUGHTER]
01:12:24.730 --> 01:12:25.750
I love MIT.
01:12:25.750 --> 01:12:26.390
I really do.
01:12:43.340 --> 01:12:45.280
So let me close
off at this point
01:12:45.280 --> 01:12:48.450
with the following observation.
01:12:48.450 --> 01:12:51.180
[LAUGHTER]
01:12:51.180 --> 01:12:55.540
We live in a world
governed by probabilities.
01:12:55.540 --> 01:12:57.910
There's a finite probability
that, at any given moment,
01:12:57.910 --> 01:13:00.976
that two pirates might
walk into a room, OK?
01:13:00.976 --> 01:13:01.475
[LAUGHTER]
01:13:01.475 --> 01:13:03.340
You just never know.
01:13:03.340 --> 01:13:08.350
[APPLAUSE]
01:13:08.350 --> 01:13:13.240
But those probabilities can be
computed in quantum mechanics.
01:13:13.240 --> 01:13:15.302
And they're computed
in the following ways.
01:13:15.302 --> 01:13:16.760
They're computed
the following ways
01:13:16.760 --> 01:13:18.420
as we'll study in great detail.
01:13:18.420 --> 01:13:24.840
If I take a state, psi of x,
which is equal to e to the ikx,
01:13:24.840 --> 01:13:28.000
this is a state that has
definite momentum h bar k.
01:13:28.000 --> 01:13:28.990
Right?
01:13:28.990 --> 01:13:29.720
We claimed this.
01:13:29.720 --> 01:13:33.090
This was de Broglie
and Davisson-Germer.
01:13:33.090 --> 01:13:35.450
Note the following,
take this operator
01:13:35.450 --> 01:13:37.770
and act on this wave
function with this operator.
01:13:37.770 --> 01:13:38.925
What do you get?
01:13:38.925 --> 01:13:40.300
Well, we already
know, because we
01:13:40.300 --> 01:13:42.760
constructed it to
have this property.
01:13:42.760 --> 01:13:44.605
P hat on psi of
x-- and I'm going
01:13:44.605 --> 01:13:46.230
to call this psi sub
k of x, because it
01:13:46.230 --> 01:13:51.070
has a definite k-- is equal
to h bar k psi k of x.
01:13:55.110 --> 01:13:58.455
A state with a definite
momentum has the property
01:13:58.455 --> 01:14:00.580
that, when you hit it with
the operation associated
01:14:00.580 --> 01:14:03.204
with momentum, you get back the
same function times a constant,
01:14:03.204 --> 01:14:06.120
and that constant is
exactly the momentum we
01:14:06.120 --> 01:14:09.486
ascribe to that plane wave.
01:14:09.486 --> 01:14:10.730
Is that cool?
01:14:10.730 --> 01:14:11.280
Yeah?
01:14:11.280 --> 01:14:12.071
AUDIENCE: Question.
01:14:12.071 --> 01:14:13.735
Just with notation,
what does the hat
01:14:13.735 --> 01:14:14.744
above the p [INAUDIBLE]?
01:14:14.744 --> 01:14:15.410
PROFESSOR: Good.
01:14:15.410 --> 01:14:15.610
Excellent.
01:14:15.610 --> 01:14:17.360
So the hat above the
P is to remind you
01:14:17.360 --> 01:14:18.740
that P is on a number.
01:14:18.740 --> 01:14:20.250
It's an operation.
01:14:20.250 --> 01:14:23.370
It's a rule for
acting on functions.
01:14:23.370 --> 01:14:25.670
We'll talk about that in
great detail next time.
01:14:25.670 --> 01:14:27.560
But here's what I
want to emphasize.
01:14:27.560 --> 01:14:30.792
This is a state which is equal
to all others in the sense
01:14:30.792 --> 01:14:32.750
that it's a perfectly
reasonable wave function,
01:14:32.750 --> 01:14:35.980
but it's more equal because it
has a simple interpretation.
01:14:35.980 --> 01:14:37.350
Right?
01:14:37.350 --> 01:14:39.904
The probability that I measure
the momentum to be h bar k
01:14:39.904 --> 01:14:41.320
is one, and the
probability that I
01:14:41.320 --> 01:14:45.010
measure it to be anything
else is 0, correct?
01:14:45.010 --> 01:14:47.870
But I can always consider a
state which is a superposition.
01:14:47.870 --> 01:14:54.426
Psi is equal to alpha, let's
just do 1 over 2 e to the ikx.
01:14:54.426 --> 01:14:59.180
k1 x plus 1 over root
2 e to the minus ikx.
01:15:09.070 --> 01:15:12.930
Is this state a state
with definite momentum?
01:15:12.930 --> 01:15:15.170
If I act on this
state-- I'll call this i
01:15:15.170 --> 01:15:18.590
sub s-- if I act on this state
with the momentum operator,
01:15:18.590 --> 01:15:21.771
do I get back this
state times a constant?
01:15:21.771 --> 01:15:22.270
No.
01:15:22.270 --> 01:15:23.080
That's interesting.
01:15:23.080 --> 01:15:24.659
And so it seems to
be that if we have
01:15:24.659 --> 01:15:27.200
a state with definite momentum
and we act on it with momentum
01:15:27.200 --> 01:15:28.810
operator, we get
back its momentum.
01:15:28.810 --> 01:15:30.340
If we have a state
that's a superposition
01:15:30.340 --> 01:15:33.006
of different momentum and we act
on it with a momentum operator,
01:15:33.006 --> 01:15:35.330
this gives us h bar k 1,
this gives us h bar k2.
01:15:35.330 --> 01:15:36.957
So it changes
which superposition
01:15:36.957 --> 01:15:37.790
we're talking about.
01:15:37.790 --> 01:15:41.330
We don't get back
our same state.
01:15:41.330 --> 01:15:43.065
So the action of this
operator on a state
01:15:43.065 --> 01:15:45.440
is going to tell us something
about whether the state has
01:15:45.440 --> 01:15:48.570
definite value of the momentum.
01:15:48.570 --> 01:15:50.490
And these coefficients
are going to turn out
01:15:50.490 --> 01:15:52.910
to contain all the information
about the probability
01:15:52.910 --> 01:15:53.660
of the system.
01:15:53.660 --> 01:15:55.560
This is the
probability when norm
01:15:55.560 --> 01:15:59.000
squared that will measure the
system to have momentum k1.
01:15:59.000 --> 01:16:00.530
And this coefficient
norm squared
01:16:00.530 --> 01:16:02.450
is going to tell us
the probability that we
01:16:02.450 --> 01:16:05.750
have momentum k2.
01:16:05.750 --> 01:16:10.730
So I think the
current wave function
01:16:10.730 --> 01:16:21.020
is something like a
superposition of 1/10 psi
01:16:21.020 --> 01:16:32.890
pirates plus 1 minus
is 1/100 square root.
01:16:32.890 --> 01:16:35.760
To normalize it
properly psi no pirates.
01:16:41.340 --> 01:16:43.670
And I'll leave you with
pondering this probability.
01:16:43.670 --> 01:16:47.330
See you guys next time.
01:16:47.330 --> 01:17:11.525
[APPLAUSE]
01:17:11.525 --> 01:17:13.150
CHRISTOPHER SMITH:
We've come for Prof.
01:17:13.150 --> 01:17:15.140
Allan Adams.
01:17:15.140 --> 01:17:16.880
PROFESSOR: It is I.
01:17:16.880 --> 01:17:21.110
CHRISTOPHER SMITH: When in
the chronicles of wasted time,
01:17:21.110 --> 01:17:24.900
I see descriptions
of fairest rights,
01:17:24.900 --> 01:17:29.650
and I see lovely
shows of lovely dames.
01:17:29.650 --> 01:17:34.440
And descriptions of ladies
dead and lovely nights.
01:17:34.440 --> 01:17:37.860
Then in the bosom of
fair loves depths.
01:17:37.860 --> 01:17:43.730
Of eyes, of foot,
of eye, of brow.
01:17:43.730 --> 01:17:48.090
I see the antique pens
do but express the beauty
01:17:48.090 --> 01:17:50.770
that you master now.
01:17:50.770 --> 01:17:55.550
So are all their praises but
prophecies of this, our time.
01:17:55.550 --> 01:17:59.120
All you prefiguring.
01:17:59.120 --> 01:18:02.942
But though they had
but diving eyes--
01:18:02.942 --> 01:18:04.900
PROFESSOR: I was wrong
about the probabilities.
01:18:04.900 --> 01:18:05.376
[LAUGHTER]
01:18:05.376 --> 01:18:06.792
CHRISTOPHER SMITH:
But though they
01:18:06.792 --> 01:18:09.120
had but diving eyes,
they had not skill
01:18:09.120 --> 01:18:11.710
enough you're worth to sing.
01:18:11.710 --> 01:18:15.020
For we which now behold
these present days
01:18:15.020 --> 01:18:17.600
have eyes to behold.
01:18:17.600 --> 01:18:21.440
[LAUGHTER]
01:18:21.440 --> 01:18:23.180
But not tongues to praise.
01:18:25.840 --> 01:18:28.200
[APPLAUSE]
01:18:28.200 --> 01:18:29.144
It's not over.
01:18:29.144 --> 01:18:31.990
You wait.
01:18:31.990 --> 01:18:35.020
ARSHIA SURTI: Not marbled with
gilded monuments of princes
01:18:35.020 --> 01:18:37.160
shall outlive this
powerful rhyme.
01:18:37.160 --> 01:18:40.690
But you shall shine more
bright in these contents
01:18:40.690 --> 01:18:43.370
that unswept stone
besmear its sluttish tide.
01:18:43.370 --> 01:18:48.970
When wasteful war shall
statues overturn and broils
01:18:48.970 --> 01:18:50.230
root out the work of masonry.
01:18:50.230 --> 01:18:53.060
Nor Mars his sword.
01:18:53.060 --> 01:18:57.390
Nor war's quick fire shall
burn the living record
01:18:57.390 --> 01:18:59.640
of your memory.
01:18:59.640 --> 01:19:04.810
Gainst death and all oblivious
enmity shall you pace forth.
01:19:04.810 --> 01:19:06.570
Your praise shall
still find room,
01:19:06.570 --> 01:19:08.233
even in the eyes
of all posterity.
01:19:11.220 --> 01:19:15.910
So no judgment arise till
you yourself judgment arise.
01:19:15.910 --> 01:19:18.060
You live in this and
dwell in lover's eyes.
01:19:20.862 --> 01:19:23.200
[APPLAUSE]
01:19:23.200 --> 01:19:26.220
CHRISTOPHER SMITH: Verily
happy Valentine's day upon you.
01:19:26.220 --> 01:19:28.170
May your day be filled
with love and poetry.
01:19:28.170 --> 01:19:31.448
Whatever state you're in,
we will always love you.
01:19:31.448 --> 01:19:33.918
[LAUGHTER]
01:19:33.918 --> 01:19:37.880
[APPLAUSE]
01:19:37.880 --> 01:19:41.690
Signed, Jack Florian,
James [INAUDIBLE].
01:19:41.690 --> 01:19:42.590
[LAUGHTER]
01:19:42.590 --> 01:19:44.390
PROFESSOR: Thank you, sir.
01:19:44.390 --> 01:19:45.582
Thank you.
01:19:45.582 --> 01:19:46.790
CHRISTOPHER SMITH: Now we go.
01:19:50.390 --> 01:19:51.940
[APPLAUSE]