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PROFESSOR: Today we're
going to just continue
00:00:26.360 --> 00:00:28.040
what Allan Adams was doing.
00:00:28.040 --> 00:00:32.340
He's away on a trip
in Europe, so he asked
00:00:32.340 --> 00:00:33.900
me to give their lecture today.
00:00:33.900 --> 00:00:39.650
And we'll just follow
what he told me to do.
00:00:39.650 --> 00:00:44.280
He was sort of sad to give me
this lecture, because it's one
00:00:44.280 --> 00:00:46.280
of the most interesting ones.
00:00:46.280 --> 00:00:49.980
This is the one where you get
to see the Schrodinger equation.
00:00:49.980 --> 00:00:56.660
But anyway, had to be a
way, so we'll do it here.
00:00:56.660 --> 00:00:59.980
He also told me to take off
my shoes, but I won't do that.
00:01:02.780 --> 00:01:05.760
So let's go ahead.
00:01:05.760 --> 00:01:07.290
So what do we have so far.
00:01:07.290 --> 00:01:12.910
It's going to be a list
of items that we have.
00:01:12.910 --> 00:01:15.310
And what have we learned?
00:01:15.310 --> 00:01:20.880
We know that
particles, or systems,
00:01:20.880 --> 00:01:29.180
are governed by wave functions,
described by wave functions.
00:01:29.180 --> 00:01:33.710
Wave functions.
00:01:33.710 --> 00:01:38.520
And those are psi
of x at this moment.
00:01:38.520 --> 00:01:44.060
And these are complex numbers,
belong to the complex numbers.
00:01:44.060 --> 00:01:48.500
And they're continuous
and normalizable.
00:01:48.500 --> 00:01:51.030
Their derivatives need
not be continuous,
00:01:51.030 --> 00:01:54.700
but the wave function
has to be continuous.
00:01:54.700 --> 00:01:57.770
It should be also
normalizable, and those
00:01:57.770 --> 00:01:59.900
are two important
properties of it,
00:01:59.900 --> 00:02:02.223
continuous and normalizable.
00:02:07.910 --> 00:02:12.670
Second there's a probability
associated with this thing.
00:02:12.670 --> 00:02:15.360
And this probability
is described
00:02:15.360 --> 00:02:20.200
by this special p of x.
00:02:20.200 --> 00:02:24.700
And given that x is a continuous
variable, you can say well,
00:02:24.700 --> 00:02:28.560
what is the probability that the
particle is just at this point
00:02:28.560 --> 00:02:31.950
would be zero in general.
00:02:31.950 --> 00:02:34.670
You have to ask, typically,
what's the probability that I
00:02:34.670 --> 00:02:37.040
find it in a range.
00:02:37.040 --> 00:02:38.800
It's any continuous
variable that
00:02:38.800 --> 00:02:42.320
postulated to be given
by the square of the wave
00:02:42.320 --> 00:02:45.870
function and the x.
00:02:45.870 --> 00:02:55.276
Third there's superposition
of allowed states.
00:02:58.950 --> 00:03:01.720
So particles can be
in superpositions.
00:03:01.720 --> 00:03:05.330
So a wave function
that depends of x
00:03:05.330 --> 00:03:10.970
may quickly or generally be
given as a superposition of two
00:03:10.970 --> 00:03:11.746
wave functions.
00:03:16.260 --> 00:03:18.590
And this is seen in many ways.
00:03:18.590 --> 00:03:19.760
You have these boxes.
00:03:19.760 --> 00:03:21.420
A particle was a superposition.
00:03:21.420 --> 00:03:24.370
A top, and side,
and hard, and soft.
00:03:24.370 --> 00:03:26.800
And photon superposition
of linearly
00:03:26.800 --> 00:03:29.960
polarized here or that way.
00:03:29.960 --> 00:03:33.300
That's always
possible to explain.
00:03:33.300 --> 00:03:38.270
Now in addition to this,
to motivate the next one
00:03:38.270 --> 00:03:41.606
we talk about relations
between operators.
00:03:44.150 --> 00:03:50.820
There was an abstraction
going on in this course
00:03:50.820 --> 00:03:53.580
in the previous
lectures in which
00:03:53.580 --> 00:03:56.260
the idea of the
momentum of a particle
00:03:56.260 --> 00:04:00.580
became that of an operator
on the wave function.
00:04:03.800 --> 00:04:11.320
So as an aside,
operators momentum,
00:04:11.320 --> 00:04:14.050
we have the momentum
of a particle
00:04:14.050 --> 00:04:17.850
has been associated
with an operator h
00:04:17.850 --> 00:04:21.730
bar over i [INAUDIBLE] x.
00:04:21.730 --> 00:04:25.250
Now two things here.
00:04:25.250 --> 00:04:28.180
My taste, I like to
put h bar over i.
00:04:28.180 --> 00:04:31.000
Allan likes to put I h bar.
00:04:31.000 --> 00:04:35.700
That's exactly the same
thing, but no minus sign
00:04:35.700 --> 00:04:37.640
is a good thing in my opinion.
00:04:37.640 --> 00:04:41.150
So I avoid them when possible.
00:04:41.150 --> 00:04:45.280
Now there's is d dx here
is a partial derivative,
00:04:45.280 --> 00:04:50.130
and there seems to be no need
for partial derivatives here.
00:04:50.130 --> 00:04:51.820
Why partial derivatives?
00:04:51.820 --> 00:04:55.560
I only see functions of x.
00:04:55.560 --> 00:04:56.240
Anybody?
00:04:56.240 --> 00:04:56.930
Why?
00:04:56.930 --> 00:04:57.590
Yes.
00:04:57.590 --> 00:04:59.310
AUDIENCE: The complete
wave function also
00:04:59.310 --> 00:05:01.905
depends on time, doesn't it?
00:05:01.905 --> 00:05:03.280
PROFESSOR: Complete
wave function
00:05:03.280 --> 00:05:04.680
depends on time as well.
00:05:04.680 --> 00:05:06.230
Yes, exactly.
00:05:06.230 --> 00:05:08.590
That's where we're
going to get to today.
00:05:08.590 --> 00:05:12.740
This is the description of
this particle at some instant.
00:05:12.740 --> 00:05:16.480
So within [INAUDIBLE] time,
the time here is implicit.
00:05:16.480 --> 00:05:20.570
It could be at some time,
now, later, some time,
00:05:20.570 --> 00:05:22.090
but that's all we know.
00:05:22.090 --> 00:05:26.360
So in physics you're allowed
to ask the question, well,
00:05:26.360 --> 00:05:28.440
if I know the wave
function of this time
00:05:28.440 --> 00:05:31.290
and that seems to be what I
need to describe the physics,
00:05:31.290 --> 00:05:33.150
what will it be later?
00:05:33.150 --> 00:05:36.190
And later time will come
in, so therefore we'll
00:05:36.190 --> 00:05:40.950
stick to this
partial d dx in here.
00:05:40.950 --> 00:05:45.250
All right, so how
do we use that?
00:05:45.250 --> 00:05:50.980
We think of this operator as
acting on the wave functions
00:05:50.980 --> 00:05:54.430
to give you roughly the
momentum of the particle.
00:05:54.430 --> 00:05:58.540
And we've made it in
such a way that when
00:05:58.540 --> 00:06:02.670
we talk about the expectation
value of the momentum,
00:06:02.670 --> 00:06:06.290
the expected value of the
momentum of the particle,
00:06:06.290 --> 00:06:08.800
we compute the
following quantity.
00:06:08.800 --> 00:06:14.190
We compute the integral from
minus infinity to infinity dx.
00:06:14.190 --> 00:06:17.740
We put the conjugate of
the wave function here.
00:06:17.740 --> 00:06:23.730
And we put the operator,
h bar over i d dx acting
00:06:23.730 --> 00:06:25.640
on the wave function here.
00:06:28.350 --> 00:06:31.260
And that's supposed
to be sort of
00:06:31.260 --> 00:06:37.380
like saying that this evaluates
the momentum of the wave
00:06:37.380 --> 00:06:39.820
function.
00:06:39.820 --> 00:06:41.580
Why is that so?
00:06:41.580 --> 00:06:47.680
It is because if you're trying
to say, oh any wave function,
00:06:47.680 --> 00:06:51.300
a general wave
function need not be
00:06:51.300 --> 00:06:54.410
state in which the particle
has definite momentum.
00:06:54.410 --> 00:06:57.770
So I kind of just say the
momentum of a particle
00:06:57.770 --> 00:07:00.940
is the value of this
operator on the function.
00:07:00.940 --> 00:07:04.930
Because if I act with this
operator on the function,
00:07:04.930 --> 00:07:08.190
on the wave function,
it might not return me,
00:07:08.190 --> 00:07:09.310
the wave function.
00:07:09.310 --> 00:07:14.100
In fact in general, we've
seen that, for special wave
00:07:14.100 --> 00:07:21.220
functions, wave functions
of the form psi, a number,
00:07:21.220 --> 00:07:22.440
e to the ikx.
00:07:26.490 --> 00:07:34.900
Then p, let's think p hat
as the operator on psi.
00:07:34.900 --> 00:07:40.180
Would be h bar
over i d dx on psi.
00:07:42.710 --> 00:07:43.660
Gives you what?
00:07:43.660 --> 00:07:48.760
Well, this h over I,
the h remains there.
00:07:48.760 --> 00:07:52.550
When you differentiate with
respect to x, the ik goes down,
00:07:52.550 --> 00:07:59.040
and the i cancels, so you get
hk times the same wave function.
00:07:59.040 --> 00:08:03.000
And for this, we think that
this wave function is a wave
00:08:03.000 --> 00:08:07.340
function with momentum hk.
00:08:07.340 --> 00:08:10.880
Because if you act with
the operator p on that wave
00:08:10.880 --> 00:08:14.260
function, it returns for you hk.
00:08:14.260 --> 00:08:23.750
So we think of this as
has p equal hk, h bar k.
00:08:23.750 --> 00:08:26.850
So the thing that
we want to do now
00:08:26.850 --> 00:08:29.830
is to make this a
little more general.
00:08:29.830 --> 00:08:31.731
This is just talking
about momentum,
00:08:31.731 --> 00:08:33.230
but in quantum
mechanics we're going
00:08:33.230 --> 00:08:35.150
to have all kinds of operators.
00:08:35.150 --> 00:08:37.860
So we need to be more general.
00:08:37.860 --> 00:08:42.650
So this is going to be, as Allan
calls it, a math interlude.
00:08:50.800 --> 00:08:55.040
Based on the following question,
what this is an operator?
00:08:55.040 --> 00:09:00.050
And then the physics question,
what do measurable things
00:09:00.050 --> 00:09:01.950
have to do with our operators?
00:09:01.950 --> 00:09:12.750
So about operators aren't
measurable things, quantities.
00:09:16.980 --> 00:09:21.950
Now your view of operators
is going to evolve.
00:09:21.950 --> 00:09:23.450
It's going to evolve
in this course,
00:09:23.450 --> 00:09:25.310
It's going to evolve in 805.
00:09:25.310 --> 00:09:27.300
It probably will
continue to evolve.
00:09:27.300 --> 00:09:32.130
So we need to think
of what operators are.
00:09:32.130 --> 00:09:37.630
And there is a simple
way of thinking
00:09:37.630 --> 00:09:41.280
of operators that is going
to be the basis of much
00:09:41.280 --> 00:09:42.480
of the intuition.
00:09:42.480 --> 00:09:46.280
It's a mathematical way
of thinking of operators,
00:09:46.280 --> 00:09:50.180
and what we'll sometimes
use it as a crutch.
00:09:50.180 --> 00:09:55.010
And the idea is that
basically operators
00:09:55.010 --> 00:10:00.390
are things that act on
objects and scramble them.
00:10:00.390 --> 00:10:02.860
So whenever you
have an operator,
00:10:02.860 --> 00:10:05.120
you really have
to figure out what
00:10:05.120 --> 00:10:07.550
are the objects
you're talking about.
00:10:07.550 --> 00:10:10.820
And then the operator
is some instruction
00:10:10.820 --> 00:10:14.290
on how to scramble it,
scramble this object.
00:10:14.290 --> 00:10:16.760
So for example, you
have an operator.
00:10:16.760 --> 00:10:20.560
You must see what it
does to any object here.
00:10:20.560 --> 00:10:24.530
The operator acts on the object
and gives you another object.
00:10:24.530 --> 00:10:28.390
So in a picture
you have all kinds
00:10:28.390 --> 00:10:37.020
of objects sets,
a set of objects.
00:10:37.020 --> 00:10:40.300
And the operators are things.
00:10:40.300 --> 00:10:42.680
You can have a
list of operators.
00:10:42.680 --> 00:10:45.780
And they come here and
move those objects,
00:10:45.780 --> 00:10:48.078
scramble them, do
something to them.
00:10:50.890 --> 00:10:53.960
And we should distinguish them,
because the objects are not
00:10:53.960 --> 00:10:57.610
operators, and the operators
are not the objects.
00:10:57.610 --> 00:11:03.070
So what is the simplest example
in mathematics of this thing
00:11:03.070 --> 00:11:08.850
is vectors and matrices.
00:11:08.850 --> 00:11:09.865
Simplest example.
00:11:15.090 --> 00:11:20.060
Objects are vectors.
00:11:20.060 --> 00:11:24.480
Operators are matrices.
00:11:24.480 --> 00:11:26.920
And how does that work?
00:11:26.920 --> 00:11:31.780
Well, you have a two
by two, the case where
00:11:31.780 --> 00:11:37.440
you have a set in which you have
vectors with two components.
00:11:37.440 --> 00:11:47.250
So example, a vector that
has two components v1 v2.
00:11:47.250 --> 00:11:53.170
And the matrices,
this is the object.
00:11:53.170 --> 00:11:57.060
And the operator is a matrix.
00:11:57.060 --> 00:12:06.390
a 11, a 12, a 21, a
22 as an operator.
00:12:06.390 --> 00:12:11.090
An m on a vector is a vector.
00:12:14.480 --> 00:12:17.870
If you are with a matrix
on a vector, this 2
00:12:17.870 --> 00:12:23.430
by 2 matrix on this common
vector, you get another vector.
00:12:23.430 --> 00:12:29.455
So that's this simplest example
of operators acting on objects.
00:12:31.980 --> 00:12:34.980
In our case, we're going to
talk about a more-- we're
00:12:34.980 --> 00:12:37.170
going to have to begin,
in quantum mechanics
00:12:37.170 --> 00:12:42.240
we're required to begin with a
more sophisticated one in which
00:12:42.240 --> 00:12:48.340
the objects are
going to be-- objects
00:12:48.340 --> 00:12:49.746
are going to be functions.
00:12:53.450 --> 00:13:02.165
In fact, I will write
complex functions of x.
00:13:06.220 --> 00:13:10.090
So let's see it, the
list of operators.
00:13:10.090 --> 00:13:12.680
And what do the operators do?
00:13:12.680 --> 00:13:18.010
The operators act
on the functions.
00:13:25.450 --> 00:13:28.330
So what is an operator?
00:13:28.330 --> 00:13:32.110
It's a rule on how
to take any function,
00:13:32.110 --> 00:13:34.710
and you must give a rule
on how to obtain from that
00:13:34.710 --> 00:13:37.940
function another function.
00:13:37.940 --> 00:13:40.500
So let's start with an examples.
00:13:40.500 --> 00:13:43.200
It's probably the
easiest thing to do.
00:13:43.200 --> 00:13:47.130
So an operator acts
on the functions.
00:13:47.130 --> 00:13:52.340
So an operator for
any function f of x
00:13:52.340 --> 00:13:54.033
will give you another function.
00:13:57.990 --> 00:14:01.760
Function of x.
00:14:04.600 --> 00:14:10.140
And here's operator o, we put
a hat sometimes for operators.
00:14:10.140 --> 00:14:21.750
So the simplest operator,
the operator one.
00:14:21.750 --> 00:14:27.220
We always, mathematicians, love
to begin with trivial examples.
00:14:27.220 --> 00:14:29.910
Illustrate anything
almost, and just kind of
00:14:29.910 --> 00:14:31.780
confuse you many times.
00:14:31.780 --> 00:14:35.170
But actually it's good
to get them of the way.
00:14:35.170 --> 00:14:38.710
So what is the operator one?
00:14:38.710 --> 00:14:41.350
One possibility it
takes any function
00:14:41.350 --> 00:14:44.660
and gives you the number one.
00:14:44.660 --> 00:14:45.850
Do you think that's it?
00:14:45.850 --> 00:14:48.950
Who thinks that's it?
00:14:48.950 --> 00:14:49.710
Nobody?
00:14:49.710 --> 00:14:50.470
Very good.
00:14:50.470 --> 00:14:53.000
that definitely is
not a good thing
00:14:53.000 --> 00:14:54.960
to do to give you
the number one.
00:14:54.960 --> 00:14:57.260
So this operator
does the following.
00:14:57.260 --> 00:14:58.480
I will write it like that.
00:14:58.480 --> 00:15:05.517
The operator one takes f
of x and gives you what?
00:15:05.517 --> 00:15:06.225
AUDIENCE: f of x.
00:15:06.225 --> 00:15:06.975
PROFESSOR: f of x.
00:15:06.975 --> 00:15:08.070
Correct.
00:15:08.070 --> 00:15:08.570
Good.
00:15:08.570 --> 00:15:12.440
So it's a very simple
operator, but it's an operator.
00:15:12.440 --> 00:15:15.640
It's like what matrix?
00:15:15.640 --> 00:15:17.980
The identity matrix.
00:15:17.980 --> 00:15:18.770
Very good.
00:15:18.770 --> 00:15:22.370
There could be a zero operator
that gives you nothing
00:15:22.370 --> 00:15:23.980
and would be the zero matrix.
00:15:23.980 --> 00:15:27.330
So let's write the more
interesting operator.
00:15:27.330 --> 00:15:30.164
The operator would d dx.
00:15:30.164 --> 00:15:30.955
That's interesting.
00:15:30.955 --> 00:15:35.240
The derivative can be
thought of as an operator
00:15:35.240 --> 00:15:38.670
because if you
start with f of x,
00:15:38.670 --> 00:15:43.390
it gives you another
function, d dx of f of x.
00:15:46.040 --> 00:15:48.830
And that's a rule to get
from one function to another.
00:15:48.830 --> 00:15:53.890
Therefore that's an operator,
qualifies as an operator.
00:15:53.890 --> 00:15:57.010
Another operator that
typically can confuse you
00:15:57.010 --> 00:16:01.560
is the operator x.
00:16:01.560 --> 00:16:02.850
x an operator?
00:16:02.850 --> 00:16:04.820
What does that mean?
00:16:04.820 --> 00:16:07.040
Well, you just
have to define it.
00:16:07.040 --> 00:16:10.500
At this moment, it
could mean many things.
00:16:10.500 --> 00:16:13.220
But you will see that
[INAUDIBLE] is the only thing
00:16:13.220 --> 00:16:15.820
that probably makes some sense.
00:16:15.820 --> 00:16:18.450
So what is the operator x?
00:16:18.450 --> 00:16:21.310
Well, it's the operator
that, given f of x,
00:16:21.310 --> 00:16:27.090
gives you the function
x times f of x.
00:16:27.090 --> 00:16:28.930
That's a reasonable thing to do.
00:16:28.930 --> 00:16:30.350
It's multiplying by x.
00:16:30.350 --> 00:16:32.710
It changes the function.
00:16:32.710 --> 00:16:35.590
You could define the operator
x squared that multiplies
00:16:35.590 --> 00:16:36.610
by x squared.
00:16:36.610 --> 00:16:41.090
And the only reasonable
thing is to multiply it by.
00:16:41.090 --> 00:16:43.330
You could divide
by it, and you may
00:16:43.330 --> 00:16:45.630
need to divide by it as well.
00:16:45.630 --> 00:16:47.460
And you could define
the operator 1
00:16:47.460 --> 00:16:51.650
over x gives you the
function times 1 over x.
00:16:51.650 --> 00:16:54.940
We will need that
sometime, but not now.
00:16:54.940 --> 00:16:59.410
Let's see another set of
operators where we give a name.
00:16:59.410 --> 00:17:01.380
It doesn't have a
name because it's not
00:17:01.380 --> 00:17:03.800
all that's useful in fact.
00:17:03.800 --> 00:17:05.640
But it's good to
illustrate things.
00:17:05.640 --> 00:17:11.810
They operator s
sub q for squared.
00:17:11.810 --> 00:17:15.710
S q for the first two
letters of the word square.
00:17:15.710 --> 00:17:19.910
That takes f of x
into f of x squared.
00:17:19.910 --> 00:17:21.069
That's another function.
00:17:21.069 --> 00:17:23.960
You could define more
functions like that.
00:17:23.960 --> 00:17:30.020
The operator p 42.
00:17:30.020 --> 00:17:33.295
That's another silly operator.
00:17:33.295 --> 00:17:35.960
Well certainly a lot
more silly than this one.
00:17:35.960 --> 00:17:37.330
That's not too bad.
00:17:37.330 --> 00:17:47.880
But the p 42 takes f of x And
gives you the number 42 times
00:17:47.880 --> 00:17:50.760
the constant function.
00:17:50.760 --> 00:17:52.300
So that's a function of x.
00:17:52.300 --> 00:17:54.045
It's trivial function of x.
00:17:57.860 --> 00:18:00.360
Now enough examples.
00:18:00.360 --> 00:18:01.660
So you get the idea.
00:18:01.660 --> 00:18:05.400
Operators act on functions
and give you functions.
00:18:05.400 --> 00:18:08.260
And we just need to
define them, and then we
00:18:08.260 --> 00:18:09.760
know what we're talking about.
00:18:09.760 --> 00:18:11.645
Yes?
00:18:11.645 --> 00:18:14.400
AUDIENCE: Is the Dirac
delta and operator?
00:18:14.400 --> 00:18:18.430
PROFESSOR: The Dirac
delta, well, you
00:18:18.430 --> 00:18:20.226
can think of it as an operator.
00:18:23.600 --> 00:18:26.650
So it all depends how
you define things.
00:18:26.650 --> 00:18:30.110
So how could I do find
the Dirac delta function
00:18:30.110 --> 00:18:32.140
to be an operator?
00:18:32.140 --> 00:18:34.590
So delta of x minus a.
00:18:34.590 --> 00:18:39.410
Can I call it the
operator delta hat of a?
00:18:39.410 --> 00:18:42.500
Well, I would have to tell
you what it does in functions.
00:18:42.500 --> 00:18:45.110
And probably I
would say delta had
00:18:45.110 --> 00:18:50.670
on a on a function of x is equal
to delta of x minus a times
00:18:50.670 --> 00:18:53.920
the function of x.
00:18:53.920 --> 00:18:55.820
And I'd say it's an operator.
00:18:55.820 --> 00:18:59.910
Now the question is, really,
is it a useful operator?
00:18:59.910 --> 00:19:04.350
And sometimes it will
be useful in fact.
00:19:04.350 --> 00:19:07.080
This is a more general
case of another operator
00:19:07.080 --> 00:19:08.330
that maybe I could define.
00:19:11.340 --> 00:19:17.960
o sub h of x is
the operator that
00:19:17.960 --> 00:19:23.825
takes f of x into h
of x times f of x.
00:19:27.330 --> 00:19:30.870
So that would be
another operator.
00:19:30.870 --> 00:19:34.900
Now there are operators
that are particularly nice,
00:19:34.900 --> 00:19:38.040
and there are the
so-called linear operators.
00:19:40.700 --> 00:19:42.685
So what is a linear operator?
00:19:45.610 --> 00:19:47.970
It's one that respects
superposition.
00:19:47.970 --> 00:19:54.590
So linear operator
respects superposition.
00:19:59.420 --> 00:20:03.780
So o hat is linear.
00:20:03.780 --> 00:20:08.980
o hat is a linear operator.
00:20:08.980 --> 00:20:16.420
If o hat on a times f
of x plus b times g of x
00:20:16.420 --> 00:20:19.310
does what you would
imagine it should do,
00:20:19.310 --> 00:20:22.790
it that's on the first, and
then it acts on the second.
00:20:22.790 --> 00:20:25.800
Acting on the first,
the number goes out
00:20:25.800 --> 00:20:28.620
and doesn't do anything,
say, on the number.
00:20:28.620 --> 00:20:29.510
It's linear.
00:20:29.510 --> 00:20:31.720
It's part of that idea.
00:20:31.720 --> 00:20:42.690
And it gives you o on f of
x plus b times o on g of x.
00:20:42.690 --> 00:20:48.005
So your operator may be linear,
or it may not be linear.
00:20:54.670 --> 00:20:56.890
And we have to just guess them.
00:20:56.890 --> 00:21:00.680
And you would imagine
that we can decide that,
00:21:00.680 --> 00:21:02.930
of the list of
operators that we have,
00:21:02.930 --> 00:21:12.910
let's see, one d dx-- how much?
00:21:12.910 --> 00:21:13.470
Which one?
00:21:13.470 --> 00:21:21.750
Sq, p 42, and o sub h of x.
00:21:25.770 --> 00:21:27.070
Let's see.
00:21:27.070 --> 00:21:31.130
Let's vote on each one
whether it's linear or not.
00:21:31.130 --> 00:21:35.190
A shouting match whether I
hear a stronger yes or no.
00:21:35.190 --> 00:21:35.750
OK?
00:21:35.750 --> 00:21:38.700
One is a linear operator, yes?
00:21:38.700 --> 00:21:39.340
AUDIENCE: Yes.
00:21:39.340 --> 00:21:40.860
PROFESSOR: No?
00:21:40.860 --> 00:21:42.350
Yes.
00:21:42.350 --> 00:21:43.680
All right.
00:21:43.680 --> 00:21:45.050
d dx linear.
00:21:45.050 --> 00:21:45.782
Yes?
00:21:45.782 --> 00:21:46.554
AUDIENCE: Yes.
00:21:46.554 --> 00:21:47.220
PROFESSOR: Good.
00:21:47.220 --> 00:21:48.660
That's strong enough.
00:21:48.660 --> 00:21:50.800
Don't need to hear
the other one.
00:21:50.800 --> 00:21:51.800
x hat.
00:21:51.800 --> 00:21:52.580
Linear operator?
00:21:52.580 --> 00:21:53.690
Yes or no?
00:21:53.690 --> 00:21:54.410
AUDIENCE: Yes.
00:21:54.410 --> 00:21:56.870
PROFESSOR: Yes, good.
00:21:56.870 --> 00:21:58.850
Squaring, linear operator?
00:21:58.850 --> 00:21:59.550
AUDIENCE: No.
00:21:59.550 --> 00:22:00.390
PROFESSOR: No.
00:22:00.390 --> 00:22:03.540
No way it could be
a linear operator.
00:22:03.540 --> 00:22:06.520
It just doesn't happen.
00:22:06.520 --> 00:22:14.330
If you have sq on f plus g,
it would be f plus g squared,
00:22:14.330 --> 00:22:18.470
which is f squared plus g
squared plus, unfortunately
00:22:18.470 --> 00:22:20.200
too, fg.
00:22:20.200 --> 00:22:28.600
And this thing ruins it, because
this is sq of f plus sq of g.
00:22:28.600 --> 00:22:30.210
It's even worse than that.
00:22:30.210 --> 00:22:37.680
You put sq on af,
by linearity it
00:22:37.680 --> 00:22:39.880
should be a times the operator.
00:22:39.880 --> 00:22:44.060
But when you square a times f,
you get a squared f squared.
00:22:44.060 --> 00:22:46.260
So you don't even
need two functions
00:22:46.260 --> 00:22:48.110
to see that it's not real.
00:22:48.110 --> 00:22:52.600
So definitely no.
00:22:52.600 --> 00:22:55.070
How about p 42?
00:22:55.070 --> 00:22:55.685
AUDIENCE: No.
00:22:55.685 --> 00:22:57.100
PROFESSOR: No, of course not.
00:22:57.100 --> 00:22:59.930
Because if you
add two functions,
00:22:59.930 --> 00:23:01.460
it still gives you 42.
00:23:01.460 --> 00:23:05.130
It doesn't get you 84, so no.
00:23:05.130 --> 00:23:07.840
How about oh of x?
00:23:07.840 --> 00:23:08.810
AUDIENCE: Yes.
00:23:08.810 --> 00:23:10.080
PROFESSOR: Yes, it does that.
00:23:10.080 --> 00:23:15.100
If you act with this operator on
a sum of functions distributive
00:23:15.100 --> 00:23:16.070
law, it works.
00:23:16.070 --> 00:23:18.210
So this is linear.
00:23:18.210 --> 00:23:18.710
Good.
00:23:18.710 --> 00:23:21.080
Linear operators
are important to us
00:23:21.080 --> 00:23:24.250
because we have
some superposition
00:23:24.250 --> 00:23:25.690
of allowed states.
00:23:25.690 --> 00:23:28.390
So if this is a state
and this is a state,
00:23:28.390 --> 00:23:29.910
this is also good state.
00:23:29.910 --> 00:23:35.120
So if we want superposition
to work well with our theory,
00:23:35.120 --> 00:23:36.860
we want linear operator.
00:23:36.860 --> 00:23:38.470
So that's good.
00:23:38.470 --> 00:23:41.730
So we have those
linear operators.
00:23:41.730 --> 00:23:45.420
And now operators have
another thing that
00:23:45.420 --> 00:23:48.980
makes them something special.
00:23:48.980 --> 00:23:56.380
It is the idea that there's
simpler object they can act on.
00:23:56.380 --> 00:23:59.920
We don't assume you've studied
linear algebra in this course,
00:23:59.920 --> 00:24:02.770
so whatever I'm
going to say, take it
00:24:02.770 --> 00:24:07.630
as motivation to learn some
linear algebra at some stage.
00:24:07.630 --> 00:24:10.830
You will be a little more
linear algebra in 805.
00:24:10.830 --> 00:24:13.970
But at this moment,
just basic idea.
00:24:13.970 --> 00:24:19.660
So whenever you have matrices,
one thing that people do
00:24:19.660 --> 00:24:23.450
is to see if there
are special vectors.
00:24:23.450 --> 00:24:27.170
Any arbitrary vector, when
you act within the matrix,
00:24:27.170 --> 00:24:30.060
is going to just jump and
go somewhere else, point
00:24:30.060 --> 00:24:31.920
in another direction.
00:24:31.920 --> 00:24:33.970
But there are some
special vectors
00:24:33.970 --> 00:24:38.040
that do act-- if you
have a given matrix m,
00:24:38.040 --> 00:24:43.560
there are some funny vectors
sometimes that acted by n
00:24:43.560 --> 00:24:45.390
remain the same direction.
00:24:45.390 --> 00:24:47.950
They may grow a little
or become smaller,
00:24:47.950 --> 00:24:50.020
but they remain
the same direction.
00:24:50.020 --> 00:24:52.340
These are called eigenvectors.
00:24:52.340 --> 00:24:55.000
And that constant
of proportionality,
00:24:55.000 --> 00:24:58.530
proportional to the action of
the operator on the vector,
00:24:58.530 --> 00:25:00.650
is called the eigenvalue.
00:25:00.650 --> 00:25:04.930
So these things
have generalizations
00:25:04.930 --> 00:25:06.290
for our operators.
00:25:06.290 --> 00:25:16.410
So operators can have special
functions, eigenfunctions.
00:25:23.630 --> 00:25:25.110
What are these eigenfunctions?
00:25:25.110 --> 00:25:28.640
So let's consider
that operator a hat.
00:25:28.640 --> 00:25:29.700
It's some operator.
00:25:29.700 --> 00:25:31.950
I don't know which
one of these, be we're
00:25:31.950 --> 00:25:33.710
going to talk about
linear operator.
00:25:33.710 --> 00:25:37.020
So linear operators
have eigenfunctions.
00:25:37.020 --> 00:25:38.560
A hat.
00:25:38.560 --> 00:25:41.040
So a hat.
00:25:41.040 --> 00:25:45.790
There may be functions that,
when you act with the operator,
00:25:45.790 --> 00:25:48.330
you sort of get the
same function up
00:25:48.330 --> 00:25:51.770
to possibly a constant a.
00:25:51.770 --> 00:25:55.250
So you may get a
times the function.
00:25:55.250 --> 00:25:58.690
And that's a pretty
unusual function,
00:25:58.690 --> 00:26:01.680
because, on most functions,
any given operator
00:26:01.680 --> 00:26:03.880
is going to make a mess
out of the function.
00:26:03.880 --> 00:26:05.560
But sometimes it does that.
00:26:05.560 --> 00:26:10.100
So to label them better
with respect to operator,
00:26:10.100 --> 00:26:13.280
I would put a subscript a,
which means that there's
00:26:13.280 --> 00:26:17.050
some special function
that has a parameter a,
00:26:17.050 --> 00:26:19.450
for which this
operator gives you
00:26:19.450 --> 00:26:21.870
a times that special function.
00:26:21.870 --> 00:26:24.670
And that special
function is called--
00:26:24.670 --> 00:26:29.526
this is the eigenfunction
and this is the eigenvalue.
00:26:36.830 --> 00:26:40.040
And that the
eigenvalue is a number.
00:26:40.040 --> 00:26:46.120
It's a complex number
c there over there.
00:26:46.120 --> 00:26:48.270
So these are special things.
00:26:48.270 --> 00:26:51.180
They don't necessarily
happen all the time to exist,
00:26:51.180 --> 00:26:54.500
but sometimes they do, and
then they're pretty useful.
00:26:54.500 --> 00:26:59.130
And we have one example of
them that is quite nice.
00:26:59.130 --> 00:27:12.650
For the operator a equal
p, we have eigenfunctions
00:27:12.650 --> 00:27:23.710
e to the ikx with eigenvalue hk.
00:27:23.710 --> 00:27:26.750
So this is the connection
to this whole thing.
00:27:26.750 --> 00:27:30.660
We wanted to make clear for
you that what you saw here,
00:27:30.660 --> 00:27:33.170
that this operator
acting on this function
00:27:33.170 --> 00:27:37.400
gives you something times this
function is a general fact
00:27:37.400 --> 00:27:39.580
about the operators.
00:27:39.580 --> 00:27:41.920
Operators have eigenfunctions.
00:27:41.920 --> 00:27:48.000
So eigenfunction e of x with
eigenvalue hk, because indeed
00:27:48.000 --> 00:27:54.000
p hat on this e to ikx,
as you see this h bar
00:27:54.000 --> 00:27:58.060
k times e to the ikx.
00:27:58.060 --> 00:28:00.160
So here you have p hat is the a.
00:28:00.160 --> 00:28:05.220
This is the function
labeled a would be like k.
00:28:05.220 --> 00:28:08.540
Here is something like k again.
00:28:08.540 --> 00:28:09.760
And here is this thing.
00:28:09.760 --> 00:28:14.140
But the main thing operator
on the function number
00:28:14.140 --> 00:28:15.990
times the function
is an eigenfunction.
00:28:15.990 --> 00:28:17.230
Yes?
00:28:17.230 --> 00:28:22.250
AUDIENCE: For a given operator,
is the eigenvalue [INAUDIBLE]?
00:28:22.250 --> 00:28:25.940
PROFESSOR: Well, for a given
operator good question.
00:28:25.940 --> 00:28:29.390
a is a list of values.
00:28:29.390 --> 00:28:32.580
So there may be many,
many, many eigenfunctions.
00:28:32.580 --> 00:28:35.410
Many cases infinitely
many eigenfunctions.
00:28:35.410 --> 00:28:40.730
In fact, here I can put
for k any number I want,
00:28:40.730 --> 00:28:42.500
and I get a different function.
00:28:42.500 --> 00:28:55.390
So a belongs to c and may take
many, or even infinite, values.
00:28:55.390 --> 00:29:01.650
If you remember for nice
matrices, n by n matrix
00:29:01.650 --> 00:29:05.950
may be a nice n by n matrix
because n eigenvectors
00:29:05.950 --> 00:29:09.730
and eigenvalues are
sometimes hard to generate,
00:29:09.730 --> 00:29:12.690
sometimes eigenvalues have
the same numbers and things
00:29:12.690 --> 00:29:13.330
like that.
00:29:17.510 --> 00:29:18.080
OK.
00:29:18.080 --> 00:29:24.680
Linearity is this some of two
eigenvectors and eigenvector.
00:29:24.680 --> 00:29:26.960
Yes?
00:29:26.960 --> 00:29:28.169
No?
00:29:28.169 --> 00:29:28.710
AUDIENCE: No.
00:29:28.710 --> 00:29:29.980
PROFESSOR: No, no.
00:29:29.980 --> 00:29:30.650
Correct.
00:29:30.650 --> 00:29:33.380
That's not necessarily true.
00:29:33.380 --> 00:29:39.400
If you have two
eigenvectors, they
00:29:39.400 --> 00:29:41.080
have different eigenvalues.
00:29:41.080 --> 00:29:44.580
So things don't work
out well necessarily.
00:29:44.580 --> 00:29:47.680
So an eigenvector plus
another eigenvector
00:29:47.680 --> 00:29:49.020
is not an eigenvector.
00:29:49.020 --> 00:29:56.920
So you have here, for
example, A f1 equals a1f1.
00:29:56.920 --> 00:30:07.340
And A f2 equal a2f2,
then a on f1 plus f2
00:30:07.340 --> 00:30:13.970
would be a1f1 plus
a2f2, and that's
00:30:13.970 --> 00:30:20.940
not equal to something
times f1 plus f2.
00:30:20.940 --> 00:30:24.000
It would have to be
something times f1 plus f2
00:30:24.000 --> 00:30:25.650
to be an eigenvector.
00:30:25.650 --> 00:30:29.420
So this is not necessarily
an eigenvector.
00:30:29.420 --> 00:30:33.290
And it doesn't help to put
a constant in front of here.
00:30:33.290 --> 00:30:34.320
Nothing helps.
00:30:34.320 --> 00:30:37.980
There's no way to
construct an eigenvector
00:30:37.980 --> 00:30:41.690
from two eigenvectors by
adding or subtracting.
00:30:41.690 --> 00:30:45.720
The size of the eigenvector
is not fixed either.
00:30:45.720 --> 00:30:56.516
If f is an eigenvector, then 3
times f is also an eigenvector.
00:30:59.700 --> 00:31:02.270
And we call it the
same eigenvector.
00:31:02.270 --> 00:31:05.700
Nobody would call it a
different eigenvector.
00:31:05.700 --> 00:31:07.860
It's really the same.
00:31:07.860 --> 00:31:10.710
OK, so how does that
relate to physics?
00:31:10.710 --> 00:31:12.700
Well, we've seen
it here already.
00:31:12.700 --> 00:31:16.640
that one operator that
we've learned to work with
00:31:16.640 --> 00:31:17.870
is the momentum operator.
00:31:17.870 --> 00:31:20.460
It has those eigenfunctions.
00:31:20.460 --> 00:31:25.370
So back to physics.
00:31:25.370 --> 00:31:27.190
We have other operators.
00:31:27.190 --> 00:31:29.370
Therefore we have
the P operator.
00:31:29.370 --> 00:31:30.660
That's good.
00:31:30.660 --> 00:31:32.970
We have the X operator.
00:31:32.970 --> 00:31:34.730
That's nice.
00:31:34.730 --> 00:31:36.850
It's multiplication by x.
00:31:36.850 --> 00:31:39.000
And why do we use it?
00:31:39.000 --> 00:31:41.560
Because sometimes you
have the energy operator.
00:31:46.430 --> 00:31:48.560
And what is the energy operator?
00:31:48.560 --> 00:31:51.640
The energy operator
is just the energy
00:31:51.640 --> 00:31:55.790
that you've always known, but
think of it as an operator.
00:31:55.790 --> 00:31:58.040
So how do we do that?
00:31:58.040 --> 00:32:01.750
Well, what is the energy
of a particle we've written
00:32:01.750 --> 00:32:07.300
p squared over 2m plus v of x.
00:32:07.300 --> 00:32:09.920
Well, that was the
energy of a particle,
00:32:09.920 --> 00:32:13.200
the momentum squared
over 2m plus v of x.
00:32:13.200 --> 00:32:17.470
So the energy operator
is hat here, hat there.
00:32:24.250 --> 00:32:28.460
And now it becomes an
interesting object.
00:32:28.460 --> 00:32:32.300
This energy operator
will be called E hat.
00:32:32.300 --> 00:32:34.190
It acts and functions.
00:32:34.190 --> 00:32:35.810
It is not a number.
00:32:35.810 --> 00:32:39.020
The energy is a number,
but the energy operator
00:32:39.020 --> 00:32:40.060
is not a number.
00:32:40.060 --> 00:32:43.220
Far from a number in fact.
00:32:43.220 --> 00:32:49.340
The energy operator is
minus h squared over 2m.
00:32:49.340 --> 00:32:52.340
d second the x squared.
00:32:52.340 --> 00:32:53.440
Why that?
00:32:53.440 --> 00:32:59.270
Well, because p was h bar
over i d dx as an operator.
00:32:59.270 --> 00:33:04.270
So this sort of arrow here,
it sort of the introduction.
00:33:04.270 --> 00:33:14.090
But after a while you just say
P hat is h bar over a i d dx.
00:33:14.090 --> 00:33:15.160
End of story.
00:33:15.160 --> 00:33:18.250
It's not like double arrow.
00:33:18.250 --> 00:33:19.670
It's just what it is.
00:33:19.670 --> 00:33:20.740
That operator.
00:33:20.740 --> 00:33:21.710
That's what we call it.
00:33:21.710 --> 00:33:26.390
So when we square it, the i
squares, the minus h squares,
00:33:26.390 --> 00:33:31.400
and d dx and d dx applied
twice is the second derivative.
00:33:31.400 --> 00:33:34.850
And here we get
v of X hat, which
00:33:34.850 --> 00:33:38.820
is your good potential, whatever
potential you're interested in,
00:33:38.820 --> 00:33:43.380
in which, whenever you see
an x, you put an X hat.
00:33:43.380 --> 00:33:45.310
And now this is an operator.
00:33:45.310 --> 00:33:50.000
So you see this is not a
number, not the function.
00:33:50.000 --> 00:33:52.020
It's just an operator.
00:33:52.020 --> 00:33:55.900
The operator has this
sort of operator v of x.
00:33:55.900 --> 00:34:00.590
Now what is this v of
x here as an operator?
00:34:00.590 --> 00:34:04.130
This is v of x as an operator
is just multiplication
00:34:04.130 --> 00:34:07.120
by the function v of x.
00:34:07.120 --> 00:34:12.120
You see, you have here that the
operator x is f of x like that.
00:34:12.120 --> 00:34:16.630
I could have written the
operator X hat to the n.
00:34:16.630 --> 00:34:18.380
What would it be?
00:34:18.380 --> 00:34:23.239
Well, if I add to
the function, this
00:34:23.239 --> 00:34:29.750
is a lot of X hats
acting on the function.
00:34:29.750 --> 00:34:31.440
Well, let the first one out.
00:34:31.440 --> 00:34:33.510
You let x times f of x.
00:34:33.510 --> 00:34:36.090
The second, that's
another x, another x.
00:34:36.090 --> 00:34:42.590
So this is just x to
the n times f of x.
00:34:42.590 --> 00:34:44.550
So lots of X hats.
00:34:44.550 --> 00:34:47.469
X hats To the
100th on a function
00:34:47.469 --> 00:34:49.889
is just X to the 100th
times a function.
00:34:49.889 --> 00:34:54.650
So v of x on a
function is just v
00:34:54.650 --> 00:34:56.520
of the number x on a function.
00:34:56.520 --> 00:35:00.240
It's just like this
operator, the O in which you
00:35:00.240 --> 00:35:03.550
multiply by a function.
00:35:03.550 --> 00:35:07.780
So please I hope this
is completely clear what
00:35:07.780 --> 00:35:10.500
this means as an operator.
00:35:10.500 --> 00:35:13.220
You take the wave function,
take two derivatives,
00:35:13.220 --> 00:35:18.790
and add the product of the wave
function times v of plane x.
00:35:18.790 --> 00:35:20.970
So I'll write it here maybe.
00:35:20.970 --> 00:35:22.040
So important.
00:35:22.040 --> 00:35:28.160
E hat and psi of x
is therefore minus h
00:35:28.160 --> 00:35:33.920
squared over 2m the [INAUDIBLE]
the x squared of psi of x
00:35:33.920 --> 00:35:38.122
plus just plain v
of x times psi of x.
00:35:42.157 --> 00:35:42.990
That's what it does.
00:35:42.990 --> 00:35:45.610
That's an operator.
00:35:45.610 --> 00:35:52.100
And for these
operators in general.
00:35:52.100 --> 00:35:53.830
Math interlude, is it over?
00:35:53.830 --> 00:35:55.560
Not quite.
00:35:55.560 --> 00:35:56.470
Wow.
00:35:56.470 --> 00:35:57.860
No, yes.
00:35:57.860 --> 00:36:00.196
Allan said at this
moment it's over,
00:36:00.196 --> 00:36:01.320
when you introduce it here.
00:36:04.760 --> 00:36:09.810
I'll say something more here,
but it's going to be over now.
00:36:09.810 --> 00:36:12.970
Our three continues
here then with four.
00:36:19.910 --> 00:36:37.563
Four, to each observable we
have an associated operator.
00:36:43.810 --> 00:36:53.420
So for momentum, we
have the operator P hat.
00:36:53.420 --> 00:37:00.460
And for position we
have the operator X hat.
00:37:00.460 --> 00:37:04.780
And for energy we have
the operator E hat.
00:37:08.150 --> 00:37:12.240
And these are
examples of operators.
00:37:12.240 --> 00:37:17.840
Example operator A hat
could be any of those.
00:37:17.840 --> 00:37:21.340
And there may be more
observables depending
00:37:21.340 --> 00:37:23.460
on the system you're working.
00:37:23.460 --> 00:37:25.790
If you have particles
in a line, there's
00:37:25.790 --> 00:37:28.719
not too many more
observables at this moment.
00:37:28.719 --> 00:37:30.135
If you have a
particle in general,
00:37:30.135 --> 00:37:32.840
you can have angular momentum.
00:37:32.840 --> 00:37:35.050
That's an interesting
observable.
00:37:35.050 --> 00:37:36.110
It can be others.
00:37:36.110 --> 00:37:41.240
So for any of those,
our definition
00:37:41.240 --> 00:37:44.360
is just like with
it for momentum.
00:37:44.360 --> 00:37:48.490
The expectation
value of the operator
00:37:48.490 --> 00:37:53.400
is computed by doing what
you did for momentum.
00:37:53.400 --> 00:37:56.210
You act with the operator
on the wave function
00:37:56.210 --> 00:38:02.950
here and multiply by the
compass conjugate function.
00:38:02.950 --> 00:38:06.150
And integrate just like
you did for momentum.
00:38:06.150 --> 00:38:11.080
This is going to be the
value that you expect.
00:38:11.080 --> 00:38:14.680
After many trials on
this wave function,
00:38:14.680 --> 00:38:19.040
you would expect the measured
value of this exhibit
00:38:19.040 --> 00:38:23.270
a distribution which its
expectation value, the mean,
00:38:23.270 --> 00:38:25.640
is given by this.
00:38:25.640 --> 00:38:29.430
Now there are other definitions.
00:38:29.430 --> 00:38:32.650
One definition that
already has been mentioned
00:38:32.650 --> 00:38:41.010
is that the uncertainty of
the operator on the state
00:38:41.010 --> 00:38:45.810
psi, the uncertainty,
is computed
00:38:45.810 --> 00:38:49.880
by taking the square
root of the expectation
00:38:49.880 --> 00:39:04.030
value of A squared minus
the expectation value of A,
00:39:04.030 --> 00:39:07.230
as a number, squared.
00:39:07.230 --> 00:39:10.810
Now the expectation value
of A squared, just simply
00:39:10.810 --> 00:39:12.830
here instead of A
you put A squared,
00:39:12.830 --> 00:39:14.830
so you've got A squared here.
00:39:14.830 --> 00:39:19.430
That unless the function is very
special, it's very different
00:39:19.430 --> 00:39:24.400
whole is bigger than the
expectation value of A squared.
00:39:24.400 --> 00:39:28.160
So this is a number, and
it's called the uncertainty.
00:39:28.160 --> 00:39:32.270
And that's the uncertainty
of the uncertainty principle.
00:39:32.270 --> 00:39:37.530
So for operators, we need to
have another observation that
00:39:37.530 --> 00:39:39.930
comes from matrices that
is going to be crucial
00:39:39.930 --> 00:39:45.580
for us is the observation that
operators don't necessarily
00:39:45.580 --> 00:39:47.510
commute.
00:39:47.510 --> 00:39:52.120
And we'll do the most
important example of that.
00:39:52.120 --> 00:39:56.470
So we'll try to see in
the operators associated
00:39:56.470 --> 00:40:00.530
with momentum and
position commute.
00:40:00.530 --> 00:40:04.050
And what we mean by commute
or don't communicate?
00:40:04.050 --> 00:40:07.420
Whether the order of
multiplication matters.
00:40:07.420 --> 00:40:10.820
Now we talked about
matrices at the beginning,
00:40:10.820 --> 00:40:14.800
and we said matrices act on
vectors to give you vectors.
00:40:14.800 --> 00:40:16.560
So do they commute?
00:40:16.560 --> 00:40:18.145
Well, matrices don't commute.
00:40:18.145 --> 00:40:21.260
The order matters for
matrices multiplication.
00:40:21.260 --> 00:40:25.500
So these operators we're
inventing here for physics,
00:40:25.500 --> 00:40:27.900
the order does matter as well.
00:40:27.900 --> 00:40:29.526
So commutation.
00:40:36.480 --> 00:40:42.210
So let's try to see if
we compute the operator p
00:40:42.210 --> 00:40:43.230
and x hat.
00:40:43.230 --> 00:40:49.360
Is it equal to the
operator x hat times p?
00:40:49.360 --> 00:40:51.220
This is a very good question.
00:40:51.220 --> 00:40:54.890
These are two operators
that we've defined.
00:40:54.890 --> 00:40:59.160
And we just want to know
if the order matters
00:40:59.160 --> 00:41:01.560
or if it doesn't matter.
00:41:01.560 --> 00:41:02.900
So how can I check it?
00:41:02.900 --> 00:41:06.260
I cannot just
check it like this,
00:41:06.260 --> 00:41:11.110
because operators are only clear
what they do is when they act
00:41:11.110 --> 00:41:12.540
on functions.
00:41:12.540 --> 00:41:15.220
So the only thing
that I can do is test
00:41:15.220 --> 00:41:19.430
if this thing acting on
functions give the same.
00:41:19.430 --> 00:41:23.710
So I'm going act with this
on the function f of x.
00:41:23.710 --> 00:41:26.960
And I'm going to have act with
this on the function f of x.
00:41:29.620 --> 00:41:33.040
Now what do I mean by
acting with p times
00:41:33.040 --> 00:41:35.010
x hat on the function f of x.
00:41:35.010 --> 00:41:39.060
This is by definition
you act first
00:41:39.060 --> 00:41:41.925
with the operator that
is next to the f and then
00:41:41.925 --> 00:41:42.550
with the other.
00:41:42.550 --> 00:41:48.285
So this is p hat on the
function x hat times f of x.
00:41:51.440 --> 00:41:57.050
So here I would have, this is
x hat on the function p hat
00:41:57.050 --> 00:41:58.700
f of x.
00:41:58.700 --> 00:42:04.160
See, if you have a series of
matrices, m1, m2, m3 acting
00:42:04.160 --> 00:42:06.330
on a vector, what do you mean?
00:42:06.330 --> 00:42:09.070
Act with this on the vector,
then with this on the vector,
00:42:09.070 --> 00:42:10.590
then with this.
00:42:10.590 --> 00:42:12.090
That's multiplication.
00:42:12.090 --> 00:42:14.370
So we're doing that.
00:42:14.370 --> 00:42:15.500
So let's evaluate.
00:42:15.500 --> 00:42:17.990
What is x operator on f of x?
00:42:17.990 --> 00:42:22.300
This is p hat on x times f of x.
00:42:25.510 --> 00:42:28.140
That's what the x operator
in the function is.
00:42:28.140 --> 00:42:30.570
Here, what this x hat?
00:42:30.570 --> 00:42:39.100
And now I have this, so I
have here h over i d dx of f.
00:42:39.100 --> 00:42:41.530
Let's go one more step here.
00:42:41.530 --> 00:42:50.460
This is h over i d ddx now
of this function, x f of x.
00:42:50.460 --> 00:42:55.440
And here I have just the x
function times this function.
00:42:55.440 --> 00:43:00.080
So h over i x df dx.
00:43:03.630 --> 00:43:05.840
Well, are these the same?
00:43:05.840 --> 00:43:11.750
No, because this d dx here is
not only acting on f like here.
00:43:11.750 --> 00:43:12.940
It's acting on the x.
00:43:12.940 --> 00:43:14.860
So this gives you two terms.
00:43:14.860 --> 00:43:18.100
One extra term on the
d dx acts on the x.
00:43:18.100 --> 00:43:20.390
And then one term
that is equal to this.
00:43:20.390 --> 00:43:22.060
So you don't get the same.
00:43:22.060 --> 00:43:28.340
So you get from here h over i
f of x, when you [INAUDIBLE]
00:43:28.340 --> 00:43:34.980
the x plus h over i x the df dx.
00:43:38.470 --> 00:43:39.790
So you don't get the same.
00:43:39.790 --> 00:43:43.050
So when I subtract
them, so when I
00:43:43.050 --> 00:43:54.280
do xp minus px acting on the
function f of x, what do I get?
00:43:54.280 --> 00:43:57.670
Well, I put them in this
order, x before the p.
00:43:57.670 --> 00:43:59.540
Doesn't matter
which one you take,
00:43:59.540 --> 00:44:01.280
but many people like this.
00:44:01.280 --> 00:44:05.150
Well, these terms cancel
and I get minus this thing.
00:44:05.150 --> 00:44:16.600
So I get minus h over i f
of x, or i h bar f of x.
00:44:16.600 --> 00:44:18.460
Wow.
00:44:18.460 --> 00:44:21.470
You got something very strange.
00:44:21.470 --> 00:44:26.740
The x times p minus p times
x gives you a number--
00:44:26.740 --> 00:44:30.900
an imaginary number, even
worse-- times f of x.
00:44:30.900 --> 00:44:34.630
So from this we
write the following.
00:44:34.630 --> 00:44:38.830
We say look, operators
are defined by the action
00:44:38.830 --> 00:44:42.970
and function, but for any
function, the only effect of xp
00:44:42.970 --> 00:44:51.600
minus px, which we call
the commutator of x with p.
00:44:51.600 --> 00:44:56.010
This definition, the bracket
of two things, of A B.
00:44:56.010 --> 00:45:00.830
Is defined to be A B minus B
A. It's called the commutator.
00:45:00.830 --> 00:45:06.280
x p is an operator acting
on f of x, gives you
00:45:06.280 --> 00:45:09.240
i h bar times f of x.
00:45:09.240 --> 00:45:13.530
So our kind of silly
operator that does nothing
00:45:13.530 --> 00:45:14.670
has appeared here.
00:45:14.670 --> 00:45:20.140
Because I could now
say that x hat with p
00:45:20.140 --> 00:45:24.070
is equal to i h bar
times the unit operator.
00:45:28.700 --> 00:45:30.790
Apart from the
Schrodinger equation,
00:45:30.790 --> 00:45:33.160
this is probably the
most important equation
00:45:33.160 --> 00:45:34.190
in quantum mechanics.
00:45:37.260 --> 00:45:41.960
It's the fact that x and b
are incompatible operators
00:45:41.960 --> 00:45:43.010
as you will see later.
00:45:43.010 --> 00:45:45.270
They don't commute.
00:45:45.270 --> 00:45:47.270
Their order matters.
00:45:47.270 --> 00:45:50.210
What's going to mean is
that when you measure one,
00:45:50.210 --> 00:45:52.280
you have difficulties
measuring the other.
00:45:52.280 --> 00:45:53.310
They interfere.
00:45:53.310 --> 00:45:55.560
They cannot be measured
simultaneously.
00:45:55.560 --> 00:45:58.950
All those things
are encapsulated
00:45:58.950 --> 00:46:02.530
in a very lovely
mathematical formula,
00:46:02.530 --> 00:46:06.355
which says that this is the
way these operators work.
00:46:09.560 --> 00:46:12.460
Any questions?
00:46:12.460 --> 00:46:14.410
Yes?
00:46:14.410 --> 00:46:17.140
AUDIENCE: When x-- the
commutator of x and p
00:46:17.140 --> 00:46:18.592
is itself an operator, right?
00:46:18.592 --> 00:46:19.300
PROFESSOR: RIght.
00:46:19.300 --> 00:46:22.190
AUDIENCE: So is that
what we're saying?
00:46:22.190 --> 00:46:27.370
When we had operators
before, we can't simply
00:46:27.370 --> 00:46:29.190
just cancel the f of x.
00:46:29.190 --> 00:46:31.520
I mean we're not really
canceling it, but it just
00:46:31.520 --> 00:46:35.820
because I h bar is the only
eigenvalue of the operator?
00:46:35.820 --> 00:46:40.240
PROFESSOR: Well, basically what
we've shown by this calculation
00:46:40.240 --> 00:46:43.910
is that this operator,
this combination
00:46:43.910 --> 00:46:46.890
is really the same as
the identity operator.
00:46:46.890 --> 00:46:50.480
That's all we've shown, that
some particular combination is
00:46:50.480 --> 00:46:52.530
the identity operator.
00:46:52.530 --> 00:46:56.200
Now this is very
deep, this equation.
00:46:56.200 --> 00:46:59.750
In fact, that's
the way Heisenberg
00:46:59.750 --> 00:47:02.800
invented quantum mechanics.
00:47:02.800 --> 00:47:05.610
He called it the
matrix mechanics,
00:47:05.610 --> 00:47:11.370
because he knew that operators
were related to matrices.
00:47:11.370 --> 00:47:14.420
It's a beautiful story how
he came up with this idea.
00:47:14.420 --> 00:47:16.660
It's very different from
what we're doing today
00:47:16.660 --> 00:47:19.340
that we're going to
follow Schrodinger today.
00:47:19.340 --> 00:47:25.850
But basically his analysis
led very quickly to this idea.
00:47:25.850 --> 00:47:28.030
And this is deep.
00:47:34.530 --> 00:47:35.590
Why is it deep?
00:47:38.350 --> 00:47:40.440
Depends who you ask.
00:47:40.440 --> 00:47:44.370
If you ask a mathematician,
they would probably tell you
00:47:44.370 --> 00:47:45.560
this equation is not deep.
00:47:45.560 --> 00:47:47.970
This is scary equation.
00:47:47.970 --> 00:47:50.730
And why is it scary?
00:47:50.730 --> 00:47:54.890
Because whenever a
mathematician see operators,
00:47:54.890 --> 00:47:58.630
they want to write matrices.
00:47:58.630 --> 00:48:02.120
So the mathematician, you
show him this equation,
00:48:02.120 --> 00:48:05.470
will say OK, Let me
try to figure out which
00:48:05.470 --> 00:48:08.010
matrices you're talking about.
00:48:08.010 --> 00:48:10.580
And this mathematician will
start doing calculations
00:48:10.580 --> 00:48:13.660
with two by two matrices,
and will say, no,
00:48:13.660 --> 00:48:15.850
I can't find two by
two matrices that
00:48:15.850 --> 00:48:18.550
behave like these operators.
00:48:18.550 --> 00:48:21.140
I can't find three by
three matrices either.
00:48:21.140 --> 00:48:22.570
And four by four.
00:48:22.570 --> 00:48:23.990
And five by five.
00:48:23.990 --> 00:48:26.400
And finds no matrix
really can do
00:48:26.400 --> 00:48:32.300
that, except if the matrix
is infinite dimensional.
00:48:32.300 --> 00:48:34.130
Infinite by infinite matrices.
00:48:34.130 --> 00:48:38.070
So that's why it's very
hard for a mathematician.
00:48:38.070 --> 00:48:40.070
This is the beginning
of quantum mechanics.
00:48:40.070 --> 00:48:43.960
This looks like a
trivial equation,
00:48:43.960 --> 00:48:47.090
and mathematicians
get scared by it.
00:48:47.090 --> 00:48:51.930
You show them for physicists
there will be angular momentum.
00:48:51.930 --> 00:48:56.010
The operators are like this,
and there's complicated
00:48:56.010 --> 00:48:59.720
into the [INAUDIBLE].
00:48:59.720 --> 00:49:02.170
The three components
of angular momentum
00:49:02.170 --> 00:49:04.340
have this commutation relation.
00:49:04.340 --> 00:49:07.100
And h bar here as well.
00:49:07.100 --> 00:49:08.000
Complicated.
00:49:08.000 --> 00:49:09.540
Three operators.
00:49:09.540 --> 00:49:10.930
They mix with each other.
00:49:10.930 --> 00:49:14.000
Show it to a mathematician,
he starts laughing at you.
00:49:14.000 --> 00:49:18.380
He says that best, the
simplest case, this is easy.
00:49:18.380 --> 00:49:19.390
This is complicated.
00:49:22.010 --> 00:49:23.550
It's very strange.
00:49:23.550 --> 00:49:26.420
But the reason this is
easier, the mathematician
00:49:26.420 --> 00:49:29.530
goes and, after five minutes,
comes to you with three
00:49:29.530 --> 00:49:33.450
by three matrices that
satisfies this relation.
00:49:33.450 --> 00:49:35.150
And here there weren't.
00:49:35.150 --> 00:49:38.250
And four by four that
satisfy, and five by five,
00:49:38.250 --> 00:49:40.310
and two by two, and all of them.
00:49:40.310 --> 00:49:43.010
We can calculate
all of them for you.
00:49:43.010 --> 00:49:45.770
But this one it's infinite
dimensional matrices,
00:49:45.770 --> 00:49:50.570
and it's very surprising, very
interesting, and very deep.
00:49:50.570 --> 00:49:55.300
All right, so we move
on a little bit more
00:49:55.300 --> 00:49:56.820
to the other observable.
00:49:56.820 --> 00:50:08.930
So after this, we have
more general observable.
00:50:08.930 --> 00:50:11.460
So let's talk a
little about them.
00:50:11.460 --> 00:50:15.290
That's another postulate
of quantum mechanics
00:50:15.290 --> 00:50:18.640
that continues with this
one, postulate number five.
00:50:18.640 --> 00:50:27.650
So once you measure,
upon measuring
00:50:27.650 --> 00:50:42.640
an observable A associated
with the operator A hat,
00:50:42.640 --> 00:50:43.605
two things happen.
00:50:46.130 --> 00:50:49.340
You measure this quantity
that could be momentum,
00:50:49.340 --> 00:50:55.360
could be energy, could
be position, you name it.
00:50:55.360 --> 00:51:02.560
The measured value
must be a number.
00:51:02.560 --> 00:51:13.460
It's one of the
eigenvalues of A hat.
00:51:13.460 --> 00:51:17.450
So actually those eigenvalues,
remember the definition
00:51:17.450 --> 00:51:18.920
of the eigenvalues.
00:51:18.920 --> 00:51:19.990
It's there.
00:51:19.990 --> 00:51:23.940
I said many, but
whenever you measure,
00:51:23.940 --> 00:51:27.750
the only possibilities
that you get this number.
00:51:27.750 --> 00:51:32.600
So you measure the momentum, you
must get this hk, for example.
00:51:32.600 --> 00:51:37.870
So observables, we have
an associated operator,
00:51:37.870 --> 00:51:40.480
and the measured values
are the eigenvalues.
00:51:43.630 --> 00:51:47.120
Now these eigenvalues, in
order to be observable,
00:51:47.120 --> 00:51:48.940
they should be a real numbers.
00:51:48.940 --> 00:51:51.160
And we said oh,
they can be complex.
00:51:51.160 --> 00:51:56.190
Well, we will limit
the kind of observables
00:51:56.190 --> 00:51:59.380
to things that have
real eigenvalues,
00:51:59.380 --> 00:52:03.830
and these are going to be called
later on permission operators.
00:52:03.830 --> 00:52:07.770
At this moment, the notes,
they don't mention them.
00:52:07.770 --> 00:52:11.240
You're going to
get them confused.
00:52:11.240 --> 00:52:17.650
So anyway, special operators
that have real eigenvalues.
00:52:17.650 --> 00:52:21.350
So we mentioned here they
will have to be a real.
00:52:21.350 --> 00:52:26.050
Have to be real.
00:52:26.050 --> 00:52:30.330
And then the second one, which
is an even stranger thing that
00:52:30.330 --> 00:52:34.000
happens is something you've
already seen in examples.
00:52:34.000 --> 00:52:37.360
After you measure, the
whole wave function
00:52:37.360 --> 00:52:42.970
goes into the state which is the
eigenfunction of the operator.
00:52:42.970 --> 00:53:00.950
So after measurement system
collapses into psi a.
00:53:04.060 --> 00:53:10.080
The measure value is one
over the eigenvalues a of A.
00:53:10.080 --> 00:53:14.020
And the system
collapses into psi a.
00:53:14.020 --> 00:53:21.780
So psi a is such that
A hat psi a is a psi a.
00:53:21.780 --> 00:53:26.960
So this is the eigenvector with
eigenvalue a that you measured.
00:53:26.960 --> 00:53:29.570
So after you
measure the momentum
00:53:29.570 --> 00:53:34.160
and you found that its h
bar k, the wave function
00:53:34.160 --> 00:53:37.140
is the wave function
of momentum h bar k.
00:53:37.140 --> 00:53:40.450
If at the beginning, it was
a superposition of many,
00:53:40.450 --> 00:53:44.140
as Fourier told you,
then after measuring,
00:53:44.140 --> 00:53:46.420
if you get one
component of momentum,
00:53:46.420 --> 00:53:48.670
that's all that is left
of the wave function.
00:53:48.670 --> 00:53:50.560
It collapses.
00:53:50.560 --> 00:53:53.950
This collapse is a
very strange thing,
00:53:53.950 --> 00:53:57.370
and is something about
quantum mechanics
00:53:57.370 --> 00:54:02.740
that people are a little
uncomfortable with,
00:54:02.740 --> 00:54:07.230
and try to understand better,
but surprisingly nobody
00:54:07.230 --> 00:54:14.170
has understood it better after
60 years of thinking about it.
00:54:14.170 --> 00:54:15.670
And it works very well.
00:54:15.670 --> 00:54:18.250
It's a very strange thing.
00:54:18.250 --> 00:54:22.320
Because for example, if you
have a wave function that
00:54:22.320 --> 00:54:28.240
says your particle can be
anywhere, after you measure it
00:54:28.240 --> 00:54:30.710
where it is, the
whole wave function
00:54:30.710 --> 00:54:35.240
becomes a delta function at
the position that you measure.
00:54:35.240 --> 00:54:38.640
So everything on
the wave function,
00:54:38.640 --> 00:54:41.540
when you do a
measurement, basically
00:54:41.540 --> 00:54:45.110
collapses as we'll see.
00:54:45.110 --> 00:54:49.650
Now for example,
let's do an example.
00:54:49.650 --> 00:54:50.150
Position.
00:54:54.420 --> 00:54:59.130
So you have a wave
function psi of x.
00:54:59.130 --> 00:55:09.810
You find measure and
find the particle at x0.
00:55:13.350 --> 00:55:16.830
Measure and you find
the particle at x0.
00:55:16.830 --> 00:55:21.930
So measure what?
00:55:21.930 --> 00:55:22.900
I should be clear.
00:55:22.900 --> 00:55:23.630
Measure position.
00:55:27.000 --> 00:55:28.580
So we said two things.
00:55:28.580 --> 00:55:32.860
The measured value is one
of the eigenvalues of a,
00:55:32.860 --> 00:55:34.690
and after measurement,
the system
00:55:34.690 --> 00:55:36.790
collapses to eigenfunctions.
00:55:36.790 --> 00:55:42.310
Now here we really need a
little of your intuition.
00:55:42.310 --> 00:55:47.010
Our position eigenstate is a
particle a localized at one
00:55:47.010 --> 00:55:47.510
place.
00:55:47.510 --> 00:55:49.850
What is the best
function associated
00:55:49.850 --> 00:55:51.430
to a position eigenstate?
00:55:51.430 --> 00:55:52.990
It's a delta function.
00:55:52.990 --> 00:55:57.440
The function that says it's at
some point and nowhere else.
00:55:57.440 --> 00:56:04.040
So eigenfunctions delta
of x minus x0, it's
00:56:04.040 --> 00:56:08.510
a function as a function of x.
00:56:08.510 --> 00:56:14.140
It peaks at x0, and
it's 0 everywhere else.
00:56:14.140 --> 00:56:18.330
And this is, when you
find a particle at x0,
00:56:18.330 --> 00:56:19.810
this is the wave function.
00:56:19.810 --> 00:56:24.830
The wave function must be
proportional to this quantity.
00:56:24.830 --> 00:56:27.595
Now you can't normalize
this wave function.
00:56:27.595 --> 00:56:30.665
It's a small complication, but
we shouldn't worry about it
00:56:30.665 --> 00:56:31.165
too much.
00:56:33.680 --> 00:56:37.760
Basically you really can't
localize a particle perfectly,
00:56:37.760 --> 00:56:40.890
so that's the little problem
with this wave function.
00:56:40.890 --> 00:56:43.540
You've studied how you can
represent delta functions
00:56:43.540 --> 00:56:47.130
as limits and probably
intuitively those limits
00:56:47.130 --> 00:56:48.740
are the best things.
00:56:48.740 --> 00:56:52.430
But this is the wave
function, so after you
00:56:52.430 --> 00:56:56.400
measure the system, you go into
an eigenstate of the operator.
00:56:56.400 --> 00:56:59.420
Is this an eigenstate
of the x operator?
00:56:59.420 --> 00:57:01.740
What a strange question.
00:57:01.740 --> 00:57:02.850
But it is.
00:57:02.850 --> 00:57:10.310
Look, if you put the x operator
on delta of x minus x zero,
00:57:10.310 --> 00:57:11.870
what is it supposed to do?
00:57:11.870 --> 00:57:14.620
It's supposed to multiply by x.
00:57:14.620 --> 00:57:21.160
So it's x times delta
of x minus x zero.
00:57:21.160 --> 00:57:23.950
If you had a little experience
with delta functions,
00:57:23.950 --> 00:57:27.270
you'd know that this
function is 0 everywhere,
00:57:27.270 --> 00:57:33.230
except when x is equal to x0,
so this x can be turned into x0.
00:57:33.230 --> 00:57:37.220
It just never at any other
place it contributes.
00:57:37.220 --> 00:57:44.490
This x really can be turned into
x0 times delta of x minus x0.
00:57:44.490 --> 00:57:47.680
Because delta functions are
really used to do integrals.
00:57:47.680 --> 00:57:51.160
And if you do the
integral of this function,
00:57:51.160 --> 00:57:53.860
you will see that it gives you
the same value as the integral
00:57:53.860 --> 00:57:55.770
of this function.
00:57:55.770 --> 00:57:57.230
So there you have it.
00:57:57.230 --> 00:58:00.290
The operator acting
on the eigenfunction
00:58:00.290 --> 00:58:01.790
is a number times this.
00:58:01.790 --> 00:58:04.970
So these are indeed
eigenfunctions
00:58:04.970 --> 00:58:07.390
of the x operator.
00:58:07.390 --> 00:58:11.610
And what you measured was an
eigenvalue of the x operator.
00:58:11.610 --> 00:58:16.780
Eigenvalue of x.
00:58:16.780 --> 00:58:20.970
And this is an
eigenfunction of x.
00:58:26.980 --> 00:58:34.570
So we can do the same
with the momentum.
00:58:34.570 --> 00:58:39.255
Eigenvalues and eigenfunctions,
we've seen them more properly.
00:58:42.000 --> 00:58:46.270
Now we'll go to the sixth
postulate, the last postulate
00:58:46.270 --> 00:58:55.220
that we'll want to talk about
is the one of general operators,
00:58:55.220 --> 00:58:59.970
and general eigenfunctions,
and what happens with them.
00:58:59.970 --> 00:59:10.590
So let's take now our operator
A and its functions that
00:59:10.590 --> 00:59:12.310
can be found.
00:59:12.310 --> 00:59:32.300
So six, given an observable A
hat and its eigenfunctions phi
00:59:32.300 --> 00:59:34.620
a of x.
00:59:34.620 --> 00:59:40.285
So an a runs over many
values, many values.
00:59:43.910 --> 00:59:47.730
OK so let's consider this case.
00:59:47.730 --> 00:59:51.310
Now eigenfunctions
of an operator
00:59:51.310 --> 00:59:54.610
are very interesting objects.
00:59:54.610 --> 01:00:00.290
You see the eigenfunctions of
momentum were of this form.
01:00:00.290 --> 01:00:04.610
And they allow you to
expand via the Fourier
01:00:04.610 --> 01:00:08.750
any wave function as super
positions of these things.
01:00:08.750 --> 01:00:12.140
Fourier told you you
can't expand any function
01:00:12.140 --> 01:00:14.940
in eigenfunctions
of the momentum,
01:00:14.940 --> 01:00:17.900
or the result is more general.
01:00:17.900 --> 01:00:21.790
For observables
in general you can
01:00:21.790 --> 01:00:24.660
expand functions,
arbitrary functions,
01:00:24.660 --> 01:00:27.630
in terms of the eigenfunctions.
01:00:27.630 --> 01:00:31.940
Now for that, remember,
an eigenfunction
01:00:31.940 --> 01:00:35.070
is not determined up to scale.
01:00:35.070 --> 01:00:39.030
You change multiplied by three,
it's an eigenfunction still.
01:00:39.030 --> 01:00:43.470
So people like to normalize
them nicely, the eigenfunctions.
01:00:43.470 --> 01:00:47.580
You construct them like this,
and you normalize them nicely.
01:00:47.580 --> 01:00:50.300
So how do you normalize them?
01:00:50.300 --> 01:00:59.150
Normalize them by saying
that the integral over x
01:00:59.150 --> 01:01:06.930
of psi a star of x psi b
of x is going to be what?
01:01:06.930 --> 01:01:12.560
OK, basically what you want
is that these eigenfunctions
01:01:12.560 --> 01:01:14.190
be basically orthogonal.
01:01:14.190 --> 01:01:17.020
Each one orthogonal to the next.
01:01:17.020 --> 01:01:20.360
So you want this to
be 0 unless these two
01:01:20.360 --> 01:01:22.260
different eigenfunctions
are different.
01:01:22.260 --> 01:01:24.840
And when they are
the same, you want
01:01:24.840 --> 01:01:26.820
them to be just
like wave functions,
01:01:26.820 --> 01:01:30.640
that their total integral of
psi squared is equal to 1.
01:01:30.640 --> 01:01:32.695
So what you put
here is delta ab.
01:01:35.380 --> 01:01:38.050
Now this is something
that you can always
01:01:38.050 --> 01:01:40.200
do with eigenfunctions.
01:01:40.200 --> 01:01:42.680
It's proven in
mathematics books.
01:01:42.680 --> 01:01:44.760
It's not all that
simple to prove,
01:01:44.760 --> 01:01:47.980
but this can always be done.
01:01:47.980 --> 01:01:53.740
And when we need examples,
we'll do it ourselves.
01:01:53.740 --> 01:01:58.370
So given an operator that
you have its eigenfunctions
01:01:58.370 --> 01:02:00.230
like that, two things happen.
01:02:00.230 --> 01:02:12.800
One can expand psi as psi of x.
01:02:12.800 --> 01:02:17.160
Any arbitrary wave
function as the sum.
01:02:17.160 --> 01:02:18.770
Or sometimes an integral.
01:02:18.770 --> 01:02:22.340
So some people like to write
this and put them an integral
01:02:22.340 --> 01:02:24.910
on top of that.
01:02:24.910 --> 01:02:26.990
You can write it
whichever way you want.
01:02:26.990 --> 01:02:28.150
It doesn't matter.
01:02:28.150 --> 01:02:33.850
Of coefficients times
the eigenfunctions.
01:02:33.850 --> 01:02:38.330
So just like any wave could be
a written, a Fourier coefficient
01:02:38.330 --> 01:02:40.330
[INAUDIBLE] Fourier function.
01:02:40.330 --> 01:02:47.180
Any state can be written a
superposition of these things.
01:02:47.180 --> 01:02:49.580
So that's one.
01:02:49.580 --> 01:03:10.940
And two, the probability of
measuring A hat and getting a.
01:03:10.940 --> 01:03:16.150
So a one of the particular
values that you can get.
01:03:16.150 --> 01:03:22.390
That probability is given by
the square of this coefficient.
01:03:24.930 --> 01:03:29.010
Ca squared.
01:03:29.010 --> 01:03:36.870
So this is P, the probability,
to measure in psi and to get a.
01:03:36.870 --> 01:03:40.150
I think, actually,
let's put an a0 there.
01:03:44.430 --> 01:03:49.360
So here we go.
01:03:49.360 --> 01:03:52.600
So it's a very
interesting thing.
01:03:52.600 --> 01:03:55.190
Basically you expand
the wave function
01:03:55.190 --> 01:03:57.950
in terms of these
eigenfunctions,
01:03:57.950 --> 01:04:02.070
and these coefficients
give you the probabilities
01:04:02.070 --> 01:04:05.130
of measuring these numbers.
01:04:05.130 --> 01:04:10.380
So we can illustrate that
again with the delta functions,
01:04:10.380 --> 01:04:12.450
and we'll do it
quick, because we
01:04:12.450 --> 01:04:17.000
get to the punchline of this
lecture with the Schrodinger
01:04:17.000 --> 01:04:18.720
equation.
01:04:18.720 --> 01:04:24.540
So what do we have?
01:04:24.540 --> 01:04:29.920
Well, let me think of
the operator X example.
01:04:29.920 --> 01:04:33.220
Operator X, the
eigenfunctions are
01:04:33.220 --> 01:04:38.812
delta of x minus x0 for all x0.
01:04:38.812 --> 01:04:40.020
These are the eigenfunctions.
01:04:44.290 --> 01:04:47.270
And we'll write sort
of a trivial equation,
01:04:47.270 --> 01:04:50.730
but it sort of illustrates
what's going on.
01:04:50.730 --> 01:04:59.410
Psi of x as a superposition
over an integral over x0
01:04:59.410 --> 01:05:02.930
of delta of x minus x0.
01:05:06.260 --> 01:05:07.520
Psi of x0.
01:05:10.520 --> 01:05:13.550
Delta of x minus x zero.
01:05:20.280 --> 01:05:23.725
OK, first let's check
that this make sense.
01:05:23.725 --> 01:05:26.500
Here we're integrating over x0.
01:05:26.500 --> 01:05:28.290
x0 is the variable.
01:05:28.290 --> 01:05:32.430
This thing shoots
and fires whenever
01:05:32.430 --> 01:05:37.030
x is equal to x--
whenever x0 is equal to x.
01:05:37.030 --> 01:05:38.890
Therefore the whole
result of that integral
01:05:38.890 --> 01:05:42.590
is psi of x is a little
funny how it's written,
01:05:42.590 --> 01:05:46.250
because you have x minus
0, which is the same
01:05:46.250 --> 01:05:50.660
as delta of x0 minus x
is just the same thing.
01:05:50.660 --> 01:05:54.690
And you integrate over x0,
and you get just psi of x.
01:05:54.690 --> 01:05:57.490
But what have you achieved here?
01:05:57.490 --> 01:06:03.410
You've achieved the analogue
of this equation in which these
01:06:03.410 --> 01:06:08.660
are the psi a.
01:06:08.660 --> 01:06:10.945
These are the coefficients Ca.
01:06:14.700 --> 01:06:19.980
And this is the
sum, this integral.
01:06:19.980 --> 01:06:21.420
So there you go.
01:06:21.420 --> 01:06:26.000
Any wave function can be written
as the sum of coefficients
01:06:26.000 --> 01:06:30.010
times the eigenfunctions
of the operator.
01:06:30.010 --> 01:06:33.920
And what is the probability
to find the particle at x0?
01:06:33.920 --> 01:06:35.340
Well, it's from here.
01:06:35.340 --> 01:06:37.040
The coefficients, the a squared.
01:06:37.040 --> 01:06:41.910
That's exactly
what we had before.
01:06:41.910 --> 01:06:46.080
So this is getting
basically what we want.
01:06:46.080 --> 01:06:48.440
So this brings us
to the final stage
01:06:48.440 --> 01:06:53.430
of this lecture in which
we have to get the time
01:06:53.430 --> 01:06:55.580
evolution finally.
01:06:55.580 --> 01:07:01.180
So how does it happen?
01:07:01.180 --> 01:07:04.270
Well it happens in a
very interesting way.
01:07:04.270 --> 01:07:07.175
So maybe I'll call it
seven, Schrodinger equation.
01:07:13.950 --> 01:07:21.910
So as with any fundamental
equation in physics,
01:07:21.910 --> 01:07:27.530
there's experimental evidence
and suddenly, however, you
01:07:27.530 --> 01:07:31.100
have to do a conceptual leap.
01:07:31.100 --> 01:07:34.440
Experimental evidence doesn't
tell you the equation.
01:07:34.440 --> 01:07:36.790
It suggests the equation.
01:07:36.790 --> 01:07:40.500
And it tells you probably
what you're doing is right.
01:07:40.500 --> 01:07:44.110
So what we're going to do now
is collect some of the evidence
01:07:44.110 --> 01:07:49.220
we had and look at an
equation, and then just have
01:07:49.220 --> 01:07:53.670
a flash of inspiration,
change something very little,
01:07:53.670 --> 01:07:55.545
and suddenly that's the
Schrodinger equation.
01:07:59.260 --> 01:08:02.240
Allan told me, in
fact, still sometimes
01:08:02.240 --> 01:08:05.540
are disappointed that we
don't derive the Schrodinger
01:08:05.540 --> 01:08:06.160
equation.
01:08:06.160 --> 01:08:08.690
Now let's derive
it mathematically.
01:08:08.690 --> 01:08:11.700
But you also don't derive
Newton's equations.
01:08:11.700 --> 01:08:14.520
F equal ma.
01:08:14.520 --> 01:08:16.800
You have an
inspiration, you get it.
01:08:16.800 --> 01:08:17.979
Newton got it.
01:08:17.979 --> 01:08:21.319
And then you use it and
you see it makes sense.
01:08:21.319 --> 01:08:23.580
It has to be a
sensible equation,
01:08:23.580 --> 01:08:26.069
and you can test very quickly
whether your equation is
01:08:26.069 --> 01:08:27.020
sensible.
01:08:27.020 --> 01:08:29.300
But you can't quite derive it.
01:08:29.300 --> 01:08:33.020
In 805 we come a little closer
to deriving the Schrodinger
01:08:33.020 --> 01:08:37.390
equation, which we say unitary
time evolution, something
01:08:37.390 --> 01:08:39.819
that I haven't
explained what it is,
01:08:39.819 --> 01:08:41.910
implies the
Schrodinger equation.
01:08:41.910 --> 01:08:43.740
And that's a mathematical fact.
01:08:43.740 --> 01:08:47.720
And you can begin unitary
time evolution, define it,
01:08:47.720 --> 01:08:50.279
and you derive the
Schrodinger equation.
01:08:50.279 --> 01:08:53.180
But that's just
saying that you've
01:08:53.180 --> 01:08:55.840
substituted the Schrodinger
equation by saying there
01:08:55.840 --> 01:08:57.950
is unitary time evolution.
01:08:57.950 --> 01:09:00.689
The Schrodinger question
really comes from something
01:09:00.689 --> 01:09:03.660
a little deeper than that.
01:09:03.660 --> 01:09:06.279
Experimentally it comes
from something else.
01:09:06.279 --> 01:09:08.790
So how does it come?
01:09:08.790 --> 01:09:12.220
Well, you've studied some of
the history of this subject,
01:09:12.220 --> 01:09:18.979
and you've seen that Planck
postulated quantized energies
01:09:18.979 --> 01:09:24.660
in multiples of h bar omega.
01:09:24.660 --> 01:09:26.770
And then came Einstein
and said look,
01:09:26.770 --> 01:09:32.000
in fact, the energy of
a photon is h bar omega.
01:09:32.000 --> 01:09:36.960
And the momentum of
the photon was h bar k.
01:09:36.960 --> 01:09:44.510
So all these people, starting
with Planck and then Einstein,
01:09:44.510 --> 01:09:47.290
understood what the photon is.
01:09:47.290 --> 01:09:53.399
The quantum of
photons for photons,
01:09:53.399 --> 01:09:58.890
you have E is equal h bar
omega, and the momentum
01:09:58.890 --> 01:10:02.090
is equal to h bar k.
01:10:02.090 --> 01:10:04.610
I write them as a vector,
because the momentum
01:10:04.610 --> 01:10:09.160
is a vector, but we also write
them in this because p equal h
01:10:09.160 --> 01:10:13.530
bar k, assuming you move
just in one direction.
01:10:13.530 --> 01:10:15.280
And that's the way
it's been written.
01:10:15.280 --> 01:10:22.710
So this is the result of much
work beginning by Planck,
01:10:22.710 --> 01:10:26.110
Einstein, and Compton.
01:10:29.230 --> 01:10:32.480
So you may recall
Einstein said in 1905
01:10:32.480 --> 01:10:36.050
for that there seemed to be
this quantum of light that
01:10:36.050 --> 01:10:38.350
carry energy h omega.
01:10:38.350 --> 01:10:41.130
Planck didn't quite like that.
01:10:41.130 --> 01:10:43.080
And people were not
all that convinced.
01:10:43.080 --> 01:10:47.620
Experiments were done
by Millikan in 1915,
01:10:47.620 --> 01:10:51.350
and people were still
not quite convinced.
01:10:51.350 --> 01:10:54.640
And then came Compton and
did Compton scattering.
01:10:54.640 --> 01:10:57.250
And then people said, yeah,
they seem to be particles.
01:10:57.250 --> 01:10:58.790
No way out of that.
01:10:58.790 --> 01:11:03.960
And they satisfy
such a relation.
01:11:03.960 --> 01:11:10.200
Now there was
something about this
01:11:10.200 --> 01:11:15.460
that was quite nice, that
these photons are associated
01:11:15.460 --> 01:11:21.140
with waves, and that
was not too surprising,
01:11:21.140 --> 01:11:28.960
because people understood that
electromagnetic waves are waves
01:11:28.960 --> 01:11:30.700
that correspond to photons.
01:11:30.700 --> 01:11:33.550
So you can also
see that this says
01:11:33.550 --> 01:11:40.220
that E p is equal
to h bar omega k
01:11:40.220 --> 01:11:42.950
as an equation between vectors.
01:11:42.950 --> 01:11:48.000
You see the E is the first, and
the p is the second equation.
01:11:48.000 --> 01:11:52.540
And this is actually a
relativistic equation.
01:11:52.540 --> 01:11:55.310
It's a wonderful
relativistic equation,
01:11:55.310 --> 01:11:59.910
because energy and momentum
form what is called
01:11:59.910 --> 01:12:03.340
a relativity of four vector.
01:12:03.340 --> 01:12:08.550
It's the four vector-- this is
a little aside on relativity--
01:12:08.550 --> 01:12:11.310
four vector.
01:12:11.310 --> 01:12:16.080
The index mew runs
from 0, 1, 2, 3.
01:12:16.080 --> 01:12:20.760
Just like the x mews,
which are t and x.
01:12:20.760 --> 01:12:25.790
Run from x0, which is t, x1,
which is x, x2 which is y,
01:12:25.790 --> 01:12:28.290
three-- these are four vectors.
01:12:28.290 --> 01:12:30.500
And this is a four vector.
01:12:30.500 --> 01:12:31.760
This is a four vector.
01:12:31.760 --> 01:12:34.720
This all seemed quite
pretty, and this
01:12:34.720 --> 01:12:38.200
was associated to photons.
01:12:38.200 --> 01:12:42.080
But then came De Broglie.
01:12:42.080 --> 01:12:46.020
And De Broglie had
a very daring idea
01:12:46.020 --> 01:12:49.370
that even though this
was written for photons,
01:12:49.370 --> 01:12:53.150
it was true for particles
as well, for any particle.
01:12:53.150 --> 01:13:01.805
De Broglie says good for
particles, all particles.
01:13:04.770 --> 01:13:08.810
And these particles
are really waves.
01:13:08.810 --> 01:13:13.810
So what if he write--
he wrote psi of x and t
01:13:13.810 --> 01:13:17.530
is equal a wave associated
to a matter particle.
01:13:17.530 --> 01:13:25.790
And it would be an e to
the i kx minus omega t.
01:13:25.790 --> 01:13:26.505
That's a wave.
01:13:29.220 --> 01:13:37.970
And you know that this wave
has momentum p equal h bar k.
01:13:37.970 --> 01:13:42.690
If k is positive,
look at this sign.
01:13:42.690 --> 01:13:46.080
If this sign is like
that, then k is positive.
01:13:46.080 --> 01:13:49.640
This is a wave that is
moving to the right.
01:13:49.640 --> 01:13:51.885
So p being hk.
01:13:51.885 --> 01:13:55.080
If k is positive, p is positive,
is moving to the right,
01:13:55.080 --> 01:14:03.890
this is a wave moving to the
right, and has this momentum.
01:14:03.890 --> 01:14:07.160
So it should also
have an energy.
01:14:07.160 --> 01:14:11.520
Compton said that this is
relativistic because this all
01:14:11.520 --> 01:14:12.820
comes from photons.
01:14:12.820 --> 01:14:19.920
So if the momentum is given
by that, and the energy
01:14:19.920 --> 01:14:23.160
must also be given by
a similar relation.
01:14:23.160 --> 01:14:26.520
In fact, he mostly
said, look, you
01:14:26.520 --> 01:14:32.120
must have the energy being
equal to h bar omega.
01:14:32.120 --> 01:14:37.960
The momentum, therefore,
would be equal to h bar k.
01:14:37.960 --> 01:14:42.560
And I will sometimes
erase these things.
01:14:42.560 --> 01:14:46.310
So what happens with this thing?
01:14:46.310 --> 01:14:49.570
Well, momentum equal to hk.
01:14:49.570 --> 01:14:56.120
We've already understood
this as momentum operator
01:14:56.120 --> 01:14:59.655
being h bar over i d dx.
01:15:04.510 --> 01:15:09.290
So this fact that these
two must go together and be
01:15:09.290 --> 01:15:13.030
true for particles was
De Broglie's insight,
01:15:13.030 --> 01:15:15.850
and the connection
to relativity.
01:15:15.850 --> 01:15:17.790
Now here we have this.
01:15:17.790 --> 01:15:22.820
So now we just have to try
to figure out what could we
01:15:22.820 --> 01:15:25.060
do for the energy.
01:15:25.060 --> 01:15:27.500
Could we have an
energy operator?
01:15:27.500 --> 01:15:31.820
What would the energy
operator have to do?
01:15:31.820 --> 01:15:34.730
Well, if the energy
operator is supposed
01:15:34.730 --> 01:15:38.995
to give us h bar omega,
the only thing it could be
01:15:38.995 --> 01:15:45.400
is that the energy
is i h bar d dt.
01:15:45.400 --> 01:15:45.940
Why?
01:15:45.940 --> 01:15:50.400
Because you go again
at the wave function.
01:15:50.400 --> 01:15:55.740
And you think i h bar d
dt, and what do you get?
01:15:55.740 --> 01:16:08.675
i h bar d dt on the wave
function is equal to i h bar.
01:16:08.675 --> 01:16:14.420
You take the d dt,
you get minus i omega
01:16:14.420 --> 01:16:16.450
times the whole wave function.
01:16:16.450 --> 01:16:22.984
So this is equal h bar omega
times the wave function,
01:16:22.984 --> 01:16:25.240
times the wave
function like that.
01:16:25.240 --> 01:16:26.490
So here it is.
01:16:26.490 --> 01:16:30.380
This is the operator
that realizes the energy,
01:16:30.380 --> 01:16:34.870
just like this is the operator
that realizes the momentum.
01:16:34.870 --> 01:16:40.760
You could say these are the
main relations that we have.
01:16:40.760 --> 01:16:46.110
So if you have
this wave function,
01:16:46.110 --> 01:16:55.480
it corresponds to a particle
with momentum hk and energy h
01:16:55.480 --> 01:16:56.680
omega.
01:16:56.680 --> 01:17:01.680
So now we write this.
01:17:01.680 --> 01:17:15.670
So for this psi
that we have here,
01:17:15.670 --> 01:17:22.030
h bar over i d dx
of psi of x and t
01:17:22.030 --> 01:17:27.890
is equal the value of the
momentum times psi of x and t.
01:17:27.890 --> 01:17:30.880
That is something we've seen.
01:17:30.880 --> 01:17:32.810
But then there's a second one.
01:17:32.810 --> 01:17:42.940
For this psi, we also that i
h bar d dt of psi of x and t
01:17:42.940 --> 01:17:48.270
is equal to the energy of
that particle times x and t,
01:17:48.270 --> 01:17:52.010
because the energy of that
particle is h bar omega.
01:17:56.430 --> 01:18:02.770
And look, this is familiar.
01:18:02.770 --> 01:18:08.820
And here the t plays no role,
but here the t plays a role.
01:18:08.820 --> 01:18:14.670
And this is prescribing you
how a wave function of energy E
01:18:14.670 --> 01:18:17.830
evolves in time.
01:18:17.830 --> 01:18:19.260
So you're almost there.
01:18:19.260 --> 01:18:22.930
You have something
very deep in here.
01:18:22.930 --> 01:18:25.290
It's telling you if you
know the wave function
01:18:25.290 --> 01:18:30.240
and it has energy E, this
is how it looks later.
01:18:30.240 --> 01:18:33.620
You can take this derivative
and solve this differential
01:18:33.620 --> 01:18:35.570
equation.
01:18:35.570 --> 01:18:38.200
Now this differential
equation is kind of trivial
01:18:38.200 --> 01:18:40.260
because E is a number here.
01:18:45.330 --> 01:18:51.180
But if you know that you have
a particle with energy E,
01:18:51.180 --> 01:18:54.800
that's how it evolves in time.
01:18:54.800 --> 01:18:59.190
So came Schrodinger and
looked at this equation.
01:19:02.300 --> 01:19:09.970
Psi of x and t equal
E psi of x and t.
01:19:09.970 --> 01:19:14.240
This is true for any
particle that has energy E.
01:19:14.240 --> 01:19:18.580
How can I make out of
this a full equation?
01:19:18.580 --> 01:19:22.390
Because maybe I don't
know what is the energy E.
01:19:22.390 --> 01:19:26.410
The energy E might be
anything in general.
01:19:26.410 --> 01:19:27.490
What can I do?
01:19:30.260 --> 01:19:32.090
Very simple.
01:19:32.090 --> 01:19:34.710
One single replacement
in that equation.
01:19:38.670 --> 01:19:40.790
Done.
01:19:40.790 --> 01:19:41.490
It's over.
01:19:41.490 --> 01:19:43.000
That's the Schrodinger equation.
01:19:43.000 --> 01:19:48.308
It's the energy operator
that we introduced before.
01:19:48.308 --> 01:19:48.807
Inspiration.
01:19:53.150 --> 01:19:57.110
Change E to E hat.
01:19:57.110 --> 01:19:58.920
This is the
Schrodinger equation.
01:19:58.920 --> 01:20:02.570
Now what has really
happened here,
01:20:02.570 --> 01:20:07.520
this equation that was trivial
for a wave function that
01:20:07.520 --> 01:20:10.700
represented a particle
with energy E,
01:20:10.700 --> 01:20:15.360
if this is the energy operator,
this is not so easy anymore.
01:20:15.360 --> 01:20:18.970
Because remember, the energy
operator, for example,
01:20:18.970 --> 01:20:26.590
was p squared over
2m plus v of x.
01:20:26.590 --> 01:20:33.590
And this was minus h squared
over 2m d second dx squared
01:20:33.590 --> 01:20:39.640
plus v of x acting
on wave functions.
01:20:39.640 --> 01:20:44.040
So now you've got a really
interesting equation,
01:20:44.040 --> 01:20:49.070
because you don't assume
that the energy is a number,
01:20:49.070 --> 01:20:50.740
because you don't know it.
01:20:50.740 --> 01:20:52.740
In general, if the
particle is moving
01:20:52.740 --> 01:20:55.830
in a complicated
potential, you don't know
01:20:55.830 --> 01:20:58.180
what are the possible energies.
01:20:58.180 --> 01:21:02.830
But this is symbolically
what must be happening,
01:21:02.830 --> 01:21:06.630
because if this particle
has a definite energy,
01:21:06.630 --> 01:21:11.040
then this energy operator
gives you the energy acting
01:21:11.040 --> 01:21:13.560
on the function,
and then you recover
01:21:13.560 --> 01:21:18.160
what you know is true for a
particle of a given energy.
01:21:18.160 --> 01:21:20.820
So in general, the
Schrodinger equation
01:21:20.820 --> 01:21:22.810
is a complicated equation.
01:21:22.810 --> 01:21:24.810
Let's write it now completely.
01:21:24.810 --> 01:21:26.540
So this is the
Schrodinger equation.
01:21:30.390 --> 01:21:32.590
And if we write
it completely, it
01:21:32.590 --> 01:21:41.030
will read i h bar d psi dt is
equal to minus h bar squared
01:21:41.030 --> 01:21:49.880
over 2m d second dx squared
of psi plus v of x times psi--
01:21:49.880 --> 01:21:53.900
psi of x and t, psi of x and t.
01:21:59.650 --> 01:22:02.090
So it's an equation, a
differential equation.
01:22:02.090 --> 01:22:07.920
It's first order in time,
and second order in space.
01:22:07.920 --> 01:22:13.210
So let me say three things
about this equation and finish.
01:22:13.210 --> 01:22:17.790
First, it requires
complex numbers.
01:22:17.790 --> 01:22:21.330
If psi would be real, everything
on the right hand side
01:22:21.330 --> 01:22:22.150
would be real.
01:22:22.150 --> 01:22:25.980
But with an i it would
spoil it, so complex numbers
01:22:25.980 --> 01:22:28.300
have to be there.
01:22:28.300 --> 01:22:31.290
Second, it's a linear equation.
01:22:31.290 --> 01:22:33.310
It satisfies the proposition.
01:22:33.310 --> 01:22:36.270
So if one wave function
satisfies the Schrodinger
01:22:36.270 --> 01:22:38.860
equation, the sum
of wave functions,
01:22:38.860 --> 01:22:42.380
and another wave function
does, the sum does.
01:22:42.380 --> 01:22:44.490
Third, it's deterministic.
01:22:44.490 --> 01:22:49.430
If you know psi at
x and time equals 0,
01:22:49.430 --> 01:22:53.320
you can calculate psi
at any later time,
01:22:53.320 --> 01:22:57.890
because this is a first order
differential equation in time.
01:22:57.890 --> 01:23:00.030
This equation will
be the subject of all
01:23:00.030 --> 01:23:02.300
what we'll do in this course.
01:23:02.300 --> 01:23:03.940
So that's it for today.
01:23:03.940 --> 01:23:05.620
Thank you.
01:23:05.620 --> 01:23:08.370
[APPLAUSE]