WEBVTT

00:00:00.570 --> 00:00:04.110
PROFESSOR: Let me do a
little exercise using still

00:00:04.110 --> 00:00:05.618
this manipulation.

00:00:11.430 --> 00:00:16.079
And I'll confirm
the way we think

00:00:16.079 --> 00:00:18.860
about expectations values.

00:00:18.860 --> 00:00:23.060
So, suppose exercise.

00:00:23.060 --> 00:00:31.157
Suppose you have indeed that
psi is equal to alpha i psi i.

00:00:34.430 --> 00:00:39.730
Compute the expectation value
of Q in the state of psi.

00:00:39.730 --> 00:00:44.440
Precisely, the expectation
value of this operator

00:00:44.440 --> 00:00:48.490
we've been talking
about on the state.

00:00:48.490 --> 00:00:55.810
So this is equal to the
integral dx psi star Q psi.

00:00:55.810 --> 00:00:59.990
And now I have to
put two sums before.

00:00:59.990 --> 00:01:03.360
And go a little fast here.

00:01:03.360 --> 00:01:15.810
dx sum over i alpha i psi i
star Q sum over j alpha j psi j.

00:01:15.810 --> 00:01:16.750
No star.

00:01:20.520 --> 00:01:29.380
This is equal to sum over i
sum over j alpha i star alpha j

00:01:29.380 --> 00:01:37.054
integral dx psi i star Q psi j.

00:01:37.054 --> 00:01:43.295
But Q psi j is
equal to qj psi j.

00:01:46.790 --> 00:01:57.260
Therefore, this whole
thing is equal to qj

00:01:57.260 --> 00:02:04.060
times the integral dx
of psi i star psi j,

00:02:04.060 --> 00:02:08.280
which is qj delta ij.

00:02:08.280 --> 00:02:09.669
So here we go.

00:02:09.669 --> 00:02:14.750
It's equal to sum
over i, sum over j,

00:02:14.750 --> 00:02:22.200
alpha i star alpha
j, qj delta ij, which

00:02:22.200 --> 00:02:25.866
is equal to the sum over i.

00:02:25.866 --> 00:02:27.760
The j's disappear.

00:02:27.760 --> 00:02:34.610
And this is alpha i squared qi.

00:02:34.610 --> 00:02:36.740
That's it.

00:02:36.740 --> 00:02:37.740
OK.

00:02:37.740 --> 00:02:43.520
Now you're supposed to
look at this and say, yay.

00:02:43.520 --> 00:02:46.516
Now why is that?

00:02:46.516 --> 00:02:48.720
Look.

00:02:48.720 --> 00:02:54.000
How did we define
expectation values?

00:02:54.000 --> 00:03:01.110
We defined it as the sum of
the value times the probability

00:03:01.110 --> 00:03:02.470
that this value have.

00:03:02.470 --> 00:03:04.230
It's for a random variable.

00:03:04.230 --> 00:03:07.560
So here our random variable is
the result of the measurement.

00:03:07.560 --> 00:03:10.570
And what are the
possible values?

00:03:10.570 --> 00:03:11.480
qi's.

00:03:11.480 --> 00:03:16.030
And what are the probabilities
that they have Pi?

00:03:16.030 --> 00:03:17.040
OK.

00:03:17.040 --> 00:03:23.880
So the expectation value
of q should be that,

00:03:23.880 --> 00:03:27.750
should be the sum of
the possible values

00:03:27.750 --> 00:03:32.070
times their probabilities, and
that's what the system gives.

00:03:32.070 --> 00:03:35.170
This is how we defined
expectation value of x.

00:03:35.170 --> 00:03:38.090
Even though it's
expectation value of P.

00:03:38.090 --> 00:03:42.545
And it all comes from
the measurement postulate

00:03:42.545 --> 00:03:44.000
and the definition.

00:03:44.000 --> 00:03:48.690
Now, this definition and
the measurement postulate

00:03:48.690 --> 00:03:52.483
just shows that this
is what we expect.

00:03:52.483 --> 00:03:57.460
This is the result of
the expectation value.

00:03:57.460 --> 00:03:59.610
OK.

00:03:59.610 --> 00:04:01.270
I think I have a nice example.

00:04:01.270 --> 00:04:03.240
I don't know if I
want to go into all

00:04:03.240 --> 00:04:06.060
the detail of these
things, but they illustrate

00:04:06.060 --> 00:04:09.820
things in a nice way.

00:04:09.820 --> 00:04:11.200
So let's try to do it.

00:04:14.720 --> 00:04:19.140
So here it is.

00:04:19.140 --> 00:04:21.350
It's a physical example.

00:04:21.350 --> 00:04:30.560
This is a nice concrete example
because things work out.

00:04:30.560 --> 00:04:35.460
So I think we'll
actually illustrate

00:04:35.460 --> 00:04:37.490
some physical points.

00:04:37.490 --> 00:04:40.120
Example.

00:04:40.120 --> 00:04:43.825
Particle on a circle.

00:04:46.520 --> 00:04:55.560
x 0 to L. Maybe you haven't
seen a circle described by that,

00:04:55.560 --> 00:05:00.660
but you take the x-axis,
and you say yes, the circle

00:05:00.660 --> 00:05:05.850
is 0 to L. L and 0.

00:05:05.850 --> 00:05:09.020
And the way you think
of it is that this point

00:05:09.020 --> 00:05:11.880
is identified with this point.

00:05:11.880 --> 00:05:15.050
If you have a line and you
identify the two endpoints,

00:05:15.050 --> 00:05:18.070
that's called a circle.

00:05:18.070 --> 00:05:19.530
It's in the sense of topology.

00:05:19.530 --> 00:05:25.050
A circle as the set of points
equidistant to a center

00:05:25.050 --> 00:05:30.350
is a geometric description
of a round circle.

00:05:30.350 --> 00:05:33.620
But this, topologically
speaking, anything

00:05:33.620 --> 00:05:37.370
that is closed is
topologically a circle.

00:05:37.370 --> 00:05:42.340
We think of a circle
as this, physically,

00:05:42.340 --> 00:05:46.180
or it could be a curved line
that makes it into a circle.

00:05:46.180 --> 00:05:48.210
But it's not important.

00:05:48.210 --> 00:05:51.620
Let's consider a free
particle on a circle,

00:05:51.620 --> 00:05:57.200
and suppose the circle has
an end L. So x belongs here.

00:05:57.200 --> 00:06:03.020
And here is the wave
function, psi equals 2 over L,

00:06:03.020 --> 00:06:07.000
1 over square root
of 3 sine of 2 pi

00:06:07.000 --> 00:06:17.990
x over L, plus 2 over square
root of 3 cosine 6 pi x over L.

00:06:17.990 --> 00:06:20.930
This is the wave function of
your particle on a circle.

00:06:24.690 --> 00:06:30.750
At some time, time equals
0, it's a free particle.

00:06:30.750 --> 00:06:31.475
No potential.

00:06:31.475 --> 00:06:34.170
And it lives in the
circle, and these functions

00:06:34.170 --> 00:06:35.810
are kind of interesting.

00:06:35.810 --> 00:06:38.200
You see, if you
live on the circle

00:06:38.200 --> 00:06:42.640
you would want to emphasize
the fact that this point 0 is

00:06:42.640 --> 00:06:46.530
the same as the point L,
so you should have that psi

00:06:46.530 --> 00:06:52.130
and L must be equal to psi at 0.

00:06:52.130 --> 00:06:55.410
It's a circle, after
all, it's the same point.

00:06:55.410 --> 00:07:01.810
And therefore for 0 or
for L, the difference here

00:07:01.810 --> 00:07:06.540
is 0 or 2 pi, and the
sine is the same thing.

00:07:06.540 --> 00:07:11.932
And 0, when x equals 0, and 6
pi, so that's also periodic,

00:07:11.932 --> 00:07:14.410
and it's fine.

00:07:14.410 --> 00:07:19.110
It's a good wave
function result.

00:07:19.110 --> 00:07:22.290
The question is,
for this problem,

00:07:22.290 --> 00:07:33.990
what are, if you measure
momentum, measure momentum,

00:07:33.990 --> 00:07:45.960
what are the possible values
and their probabilities?

00:07:45.960 --> 00:07:47.068
Probabilities.

00:07:52.392 --> 00:07:58.970
So you decide to measure
momentum of this particle.

00:07:58.970 --> 00:08:00.070
What can you get?

00:08:03.140 --> 00:08:04.490
OK.

00:08:04.490 --> 00:08:10.240
It looks a little nontrivial,
and it is a little nontrivial.

00:08:10.240 --> 00:08:11.510
Momentum.

00:08:11.510 --> 00:08:19.310
So I must sort of find
the momentum eigenstates.

00:08:19.310 --> 00:08:23.810
Momentum eigenstates, they are
those infinite plane waves,

00:08:23.810 --> 00:08:29.580
e to the ikx, that we
could never normalize.

00:08:29.580 --> 00:08:34.210
Because you square it, it's 1,
and the integral over all space

00:08:34.210 --> 00:08:35.250
is infinite.

00:08:35.250 --> 00:08:39.320
So are we heading
for disaster here?

00:08:39.320 --> 00:08:39.890
No.

00:08:39.890 --> 00:08:43.900
Because it lives
in a finite space.

00:08:43.900 --> 00:08:45.210
Yes, you have a question?

00:08:45.210 --> 00:08:48.623
STUDENT: Should it be a wave
function [INAUDIBLE] complex?

00:08:48.623 --> 00:08:50.956
Because right now, it just
looks like it's a real value.

00:08:50.956 --> 00:08:55.384
And we can't [INAUDIBLE]
real wave functions, can we?

00:08:57.987 --> 00:09:03.390
PROFESSOR: Well, it is the
wave function at time equals 0.

00:09:03.390 --> 00:09:06.980
So the time
derivative would have

00:09:06.980 --> 00:09:10.160
to bring in complex things.

00:09:10.160 --> 00:09:12.900
So you can have a
wave function that

00:09:12.900 --> 00:09:17.590
is 0, that is real at
some particular time.

00:09:17.590 --> 00:09:24.390
Like, any wave function psi of
x e to the minus iEt over h bar

00:09:24.390 --> 00:09:25.835
is a typical wave function.

00:09:25.835 --> 00:09:28.520
And then at time equal
0 it may be real.

00:09:28.520 --> 00:09:30.520
It cannot be real forever.

00:09:30.520 --> 00:09:33.140
So you cannot assume it's real.

00:09:33.140 --> 00:09:37.200
But at some particular
times it could be real.

00:09:37.200 --> 00:09:38.030
Very good question.

00:09:41.252 --> 00:09:42.960
The other thing you
might say, look, this

00:09:42.960 --> 00:09:45.190
is too real to have momentum.

00:09:45.190 --> 00:09:47.580
Momentum has to do with waves.

00:09:47.580 --> 00:09:50.690
That's probably not
a reliable argument.

00:09:50.690 --> 00:09:54.370
OK, so, where do
we go from here?

00:09:54.370 --> 00:09:58.440
Well, let's try to find
the momentum eigenstates.

00:09:58.440 --> 00:10:01.960
They should be things
like that, exponentials.

00:10:01.960 --> 00:10:04.170
So how could they look?

00:10:04.170 --> 00:10:13.750
Well, e to the 2 pi i, maybe.

00:10:16.550 --> 00:10:18.995
What else?

00:10:18.995 --> 00:10:25.680
x, there should be an
x for a momentum thing.

00:10:25.680 --> 00:10:27.760
Now there should
be no units here,

00:10:27.760 --> 00:10:30.080
so there better be an L here.

00:10:33.209 --> 00:10:38.140
And now I could put, maybe,
well the 2 maybe was--

00:10:38.140 --> 00:10:40.350
why did I think of
the 2 or the pi?

00:10:40.350 --> 00:10:42.440
Well, for convenience.

00:10:42.440 --> 00:10:45.302
But let's see what.

00:10:45.302 --> 00:10:48.220
Suppose you have
a number m here.

00:10:52.830 --> 00:11:00.100
Then the good thing about this
is that when x is equal to 0,

00:11:00.100 --> 00:11:04.210
there is some number here,
but when x is equal to L,

00:11:04.210 --> 00:11:09.130
it's a multiple of e to the
2 pi i, so that's periodic.

00:11:09.130 --> 00:11:14.350
So this does satisfy, I
claim, it's the only way

00:11:14.350 --> 00:11:17.960
if m is any integer.

00:11:17.960 --> 00:11:23.560
So it goes from minus
infinity to infinity.

00:11:23.560 --> 00:11:26.650
Those things are periodic.

00:11:26.650 --> 00:11:29.250
They satisfy psi.

00:11:29.250 --> 00:11:34.362
Actually they satisfy psi of
x plus L is equal to psi of x.

00:11:37.210 --> 00:11:37.900
OK.

00:11:37.900 --> 00:11:43.840
That seems to be something that
could be a momentum eigenstate.

00:11:43.840 --> 00:11:46.840
And then I have to normalize it.

00:11:46.840 --> 00:11:51.880
Well, if I square
it and integrate it.

00:11:51.880 --> 00:11:55.110
If I square it then the
phase cancels, so you get 1.

00:11:55.110 --> 00:11:57.990
If you integrate it
you get L. If you put 1

00:11:57.990 --> 00:12:02.290
over the square root of L, when
you square it and integrate,

00:12:02.290 --> 00:12:03.430
you will get 1.

00:12:03.430 --> 00:12:05.970
So here it is.

00:12:05.970 --> 00:12:12.880
Psi m's of x are going to
be defined to be this thing.

00:12:12.880 --> 00:12:17.750
And I claim these things
are momentum eigenstates.

00:12:17.750 --> 00:12:20.845
In fact, what is the
value of the momentum?

00:12:20.845 --> 00:12:29.045
Well, you calculate h
bar over i d dx on psi m.

00:12:29.045 --> 00:12:31.520
And you get what?

00:12:31.520 --> 00:12:44.020
You get 2 pi m over L
times h bar times psi.

00:12:44.020 --> 00:12:47.830
The h bar is there, the i
cancels, and everything then

00:12:47.830 --> 00:12:49.960
multiplies, the x falls down.

00:12:49.960 --> 00:12:53.090
So this is the
state with momentum

00:12:53.090 --> 00:13:09.160
P equals to h bar 2 pi m over L.

00:13:09.160 --> 00:13:09.660
OK.

00:13:09.660 --> 00:13:13.260
Actually, doing that, we've
done the most difficult part

00:13:13.260 --> 00:13:15.140
of the problem.

00:13:15.140 --> 00:13:19.600
You've found the
momentum eigenfunctions.

00:13:19.600 --> 00:13:24.060
So now the rest of the thing
is to rewrite this in terms

00:13:24.060 --> 00:13:25.350
of this kind of objects.

00:13:31.512 --> 00:13:32.525
I'll do it in a second.

00:13:37.680 --> 00:13:39.675
Maybe I'll leave a
little space there

00:13:39.675 --> 00:13:41.610
and you can check
the algebra, and you

00:13:41.610 --> 00:13:44.340
can see it in the notes.

00:13:44.340 --> 00:13:48.420
But you know what
you're supposed to do.

00:13:48.420 --> 00:13:57.650
A sine of x is e to the ix minus
is e to the minus ix over 2i.

00:13:57.650 --> 00:14:02.610
So you'd get these things
converted to exponentials.

00:14:02.610 --> 00:14:06.800
The cosine of x is
equal to e to the ix

00:14:06.800 --> 00:14:12.540
plus e to the minus ix over 2.

00:14:12.540 --> 00:14:15.850
So if you do that with
those things, look.

00:14:15.850 --> 00:14:20.986
What the sine of 2 pi
x going to give you?

00:14:20.986 --> 00:14:25.780
It's going to give you some
exponentials of 2 pi ix over L.

00:14:25.780 --> 00:14:28.640
So suppose that m equals 1.

00:14:28.640 --> 00:14:31.110
And m Equals minus 1.

00:14:31.110 --> 00:14:37.725
And this will give you m
equals 3, 3 times 2 is 6.

00:14:37.725 --> 00:14:39.440
And m equal minus 3.

00:14:39.440 --> 00:14:44.630
So I claim, after some work,
and you could try to do it.

00:14:44.630 --> 00:14:47.410
I think it would
be a nice exercise.

00:14:47.410 --> 00:14:50.560
Psi is equal square
root of 2 over 3,

00:14:50.560 --> 00:14:58.860
1 over 2 i psi 1 minus square
root of 2 over 3, 1 over 2i psi

00:14:58.860 --> 00:15:04.095
minus 1 plus 1 over
square root of 3 psi 3,

00:15:04.095 --> 00:15:08.690
plus 1 over square
root of 3 psi minus 3.

00:15:08.690 --> 00:15:11.080
And it should give
you some satisfaction

00:15:11.080 --> 00:15:13.420
to see something like that.

00:15:13.420 --> 00:15:16.102
You're now seeing
the wave function

00:15:16.102 --> 00:15:20.360
written as a superposition
of momentum eigenstates.

00:15:20.360 --> 00:15:23.060
This theorem came through.

00:15:23.060 --> 00:15:26.200
In this case, as a
particle in the circle,

00:15:26.200 --> 00:15:28.970
the statement is that
the eigenfunctions

00:15:28.970 --> 00:15:33.130
are the exponentials, and
it's Fourier's theorem.

00:15:33.130 --> 00:15:36.070
Again, for a series.

00:15:36.070 --> 00:15:40.420
So finally, here is the answer.

00:15:40.420 --> 00:15:44.510
So psi 1, we can measure psi 1.

00:15:44.510 --> 00:15:47.280
What is the momentum of psi 1?

00:15:47.280 --> 00:15:51.310
So here are p values.

00:15:51.310 --> 00:15:52.650
And probabilities.

00:16:00.870 --> 00:16:11.170
The first value, psi 1, the
momentum is 2 pi h bar over L.

00:16:11.170 --> 00:16:16.750
So 2 pi h bar over L. And
what is its probability?

00:16:16.750 --> 00:16:19.570
It's this whole number squared.

00:16:19.570 --> 00:16:25.060
So square root of 2/3,
1 over 2i squared.

00:16:25.060 --> 00:16:28.510
So how much is that?

00:16:28.510 --> 00:16:34.200
It's 2/3 times 1/4.

00:16:34.200 --> 00:16:39.520
2/3 times 1/4, which is 1/6.

00:16:39.520 --> 00:16:43.930
And the other value that you
can get is minus this one,

00:16:43.930 --> 00:16:49.830
so minus 2 pi h bar over L.
This minus doesn't matter,

00:16:49.830 --> 00:16:53.550
probability also 1/6.

00:16:53.550 --> 00:16:55.570
The next one is with 3.

00:16:55.570 --> 00:17:03.410
So you can get 2, 6
pi, 6 pi h bar over L,

00:17:03.410 --> 00:17:06.405
with probability
square of this, 1/3.

00:17:09.900 --> 00:17:16.800
And minus 6 pi h bar over
L with probability 1/3.

00:17:16.800 --> 00:17:19.329
Happily our
probabilities add up.

00:17:22.310 --> 00:17:23.099
So there you go.

00:17:23.099 --> 00:17:28.230
That's the theorem expressed
in a very clear example.

00:17:28.230 --> 00:17:30.160
We had a wave function.

00:17:30.160 --> 00:17:34.300
You wrote it as a sum of
four momentum eigenstates.

00:17:34.300 --> 00:17:37.240
And now you know, if
you do a measurement,

00:17:37.240 --> 00:17:40.570
what are the possible
values of the momentum.

00:17:40.570 --> 00:17:44.140
This should have
been probably 1/6.

00:17:44.140 --> 00:17:46.820
You can do anything you want.