WEBVTT
00:00:00.500 --> 00:00:03.385
PROFESSOR: That brings us to
claim number four, which is
00:00:03.385 --> 00:00:07.480
perhaps the most important one.
00:00:07.480 --> 00:00:18.960
I may have said it already.b
The eigenfunctions of Q
00:00:18.960 --> 00:00:41.870
form a set of basis functions,
and then any reasonable psi
00:00:41.870 --> 00:00:53.010
can be written as
a superposition
00:00:53.010 --> 00:00:56.030
of Q eigenfunctions.
00:01:01.670 --> 00:01:05.523
OK, so let's just
make sense of this.
00:01:05.523 --> 00:01:09.170
Because not only, I
think we understand
00:01:09.170 --> 00:01:14.660
what this means, but let's
write it out mathematically.
00:01:14.660 --> 00:01:26.250
So the statement is any psi
of x, or this physical state,
00:01:26.250 --> 00:01:29.470
can be written as a
superposition of all
00:01:29.470 --> 00:01:34.000
these eigenfunctions So there
are numbers, alpha 1 psi
00:01:34.000 --> 00:01:40.160
1 of x plus alpha 2 psi 2 of x.
00:01:40.160 --> 00:01:43.780
Those are the expansion
coefficients with alphas.
00:01:43.780 --> 00:01:52.610
And in summary, we say from
sum over i, alpha i psi i of x.
00:01:52.610 --> 00:01:56.750
So the idea is that
those alpha i's exist
00:01:56.750 --> 00:01:59.990
and you can write them.
00:01:59.990 --> 00:02:02.780
So any wave function
that you have,
00:02:02.780 --> 00:02:05.810
you can write it
in a superposition
00:02:05.810 --> 00:02:11.050
of those eigenfunctions
of the Hermitian operator.
00:02:11.050 --> 00:02:16.460
And there are two
things to say here.
00:02:16.460 --> 00:02:20.080
One is that, how would you
calculate those alpha i's?
00:02:23.300 --> 00:02:26.190
Well, actually, if you
assume this equation,
00:02:26.190 --> 00:02:28.990
the calculation of
alpha i's is simple,
00:02:28.990 --> 00:02:32.650
because of this property.
00:02:32.650 --> 00:02:35.370
You're supposed to know
the eigenfunctions.
00:02:35.370 --> 00:02:39.180
You must have done the work to
calculate the eigenfunctions.
00:02:39.180 --> 00:02:41.920
So here is what you can do.
00:02:41.920 --> 00:02:44.950
You can do the
following integral.
00:02:44.950 --> 00:02:52.980
You can do this one, psi i psi.
00:02:52.980 --> 00:02:55.290
Let's calculate this thing.
00:02:55.290 --> 00:02:57.330
Remember what this is.
00:02:57.330 --> 00:03:04.255
This is an integral,
dx, of psi i star.
00:03:04.255 --> 00:03:06.560
That's psi.
00:03:06.560 --> 00:03:14.420
And psi is the sum over
j of alpha j psi j.
00:03:14.420 --> 00:03:15.870
You can use any letter.
00:03:15.870 --> 00:03:20.670
I used i for the sum, but
since I put that psi i,
00:03:20.670 --> 00:03:25.230
I would make a great
confusion if I used another i.
00:03:25.230 --> 00:03:30.010
So I should use j there.
00:03:30.010 --> 00:03:31.970
And what is this?
00:03:31.970 --> 00:03:33.915
Well, you're integrating
the part of this.
00:03:33.915 --> 00:03:34.840
That's a sum.
00:03:34.840 --> 00:03:36.700
So the sum can go out.
00:03:36.700 --> 00:03:46.940
It's the sum over j alpha j
integral of psi i star psi j d.
00:03:46.940 --> 00:03:49.610
And what is this delta ij?
00:03:49.610 --> 00:03:52.030
That is our nice orthonormality.
00:03:52.030 --> 00:04:00.210
So this is sum over
j alpha j, delta i j.
00:04:00.210 --> 00:04:03.110
Now, this is kind
of a simple sum.
00:04:03.110 --> 00:04:04.590
You can always be done.
00:04:04.590 --> 00:04:07.980
You should just think a second.
00:04:07.980 --> 00:04:12.180
You're summing over
j, and i is fixed.
00:04:12.180 --> 00:04:16.620
The only case when this
gives something is when j,
00:04:16.620 --> 00:04:19.089
and you're summing
over, is equal to i,
00:04:19.089 --> 00:04:21.779
which is a fixed number.
00:04:21.779 --> 00:04:24.540
Therefore, the only
thing that survives
00:04:24.540 --> 00:04:27.200
is j equals to i, so this is 1.
00:04:27.200 --> 00:04:30.450
And therefore, this is alpha i.
00:04:30.450 --> 00:04:33.910
So we did succeed
in calculating this,
00:04:33.910 --> 00:04:42.760
and in fact, alpha i is equal
to this integral of psi i
00:04:42.760 --> 00:04:44.310
with psi.
00:04:44.310 --> 00:04:47.030
So how do you
compute it now for i?
00:04:47.030 --> 00:04:48.390
You must do an integral.
00:04:48.390 --> 00:04:49.340
Of what?
00:04:49.340 --> 00:04:52.820
Of psi i star times
your wave function.
00:04:52.820 --> 00:04:55.830
So in this common interval.
00:04:55.830 --> 00:05:00.795
So the alpha i's are
given by these numbers.
00:05:00.795 --> 00:05:04.707
This would prove.
00:05:04.707 --> 00:05:09.710
The other thing
that you can check
00:05:09.710 --> 00:05:18.890
is if the wave function
squared dx is equal to 1.
00:05:22.940 --> 00:05:30.090
What does it imply
for the alpha i's?
00:05:30.090 --> 00:05:32.680
You see, the wave
function is normalized,
00:05:32.680 --> 00:05:36.902
but it's not a function of alpha
1, alpha 2, alpha 3, alpha 4,
00:05:36.902 --> 00:05:37.610
all these things.
00:05:37.610 --> 00:05:39.980
So I must calculate this.
00:05:39.980 --> 00:05:42.970
And now let's do it,
quickly, but do it.
00:05:42.970 --> 00:05:53.170
Sum over i, alpha i, psi i star,
sum over j, alpha j, psi j.
00:05:53.170 --> 00:05:56.662
See, that's the integral
of these things squared dx.
00:06:00.160 --> 00:06:01.612
I'm sorry.
00:06:01.612 --> 00:06:06.202
I went wrong here.
00:06:06.202 --> 00:06:08.760
The star is there.
00:06:08.760 --> 00:06:13.390
The first psi, starred,
the second psi.
00:06:13.390 --> 00:06:15.030
Now I got it right.
00:06:15.030 --> 00:06:24.070
Now, I take out the sums i, sum
over j, alpha i star alpha j,
00:06:24.070 --> 00:06:30.350
integral dx psi i star psi j.
00:06:30.350 --> 00:06:35.640
This is delta i j, therefore
j becomes equal to i,
00:06:35.640 --> 00:06:41.850
and you get sum over i
of alpha i star alpha
00:06:41.850 --> 00:06:48.100
i, which is the sum over i
of, then alpha i squared.
00:06:48.100 --> 00:06:49.810
OK.
00:06:49.810 --> 00:06:51.630
So that's what it says.
00:06:51.630 --> 00:06:52.140
Look.
00:06:52.140 --> 00:06:56.170
This is something that should
be internalized as well.
00:06:56.170 --> 00:07:01.690
The sum over i of the alpha
i squared is equal to 1.
00:07:01.690 --> 00:07:05.255
Whenever you have a
superposition of wave
00:07:05.255 --> 00:07:09.960
functions, and the whole
thing is normalized,
00:07:09.960 --> 00:07:15.640
and your wave functions
are orthonormal,
00:07:15.640 --> 00:07:16.875
then it's very simple.
00:07:16.875 --> 00:07:22.050
The normalization is computed
by doing the sums of squares
00:07:22.050 --> 00:07:23.760
of each coefficient.
00:07:23.760 --> 00:07:31.570
The mixings don't exist
because there's no mixes here.
00:07:31.570 --> 00:07:33.280
So everything is separate.
00:07:33.280 --> 00:07:34.700
Everything is unmixed.
00:07:34.700 --> 00:07:36.070
Everything is nice.
00:07:38.960 --> 00:07:40.030
So there you go.
00:07:40.030 --> 00:07:48.090
This is how you expand any
state in the collection
00:07:48.090 --> 00:07:51.970
of eigenfunctions of
any Hermitian operator
00:07:51.970 --> 00:07:55.071
that you are looking at.
00:07:55.071 --> 00:07:55.570
OK.
00:07:55.570 --> 00:07:57.960
So finally, we get it.
00:07:57.960 --> 00:08:02.790
We've done all the
work necessary to state
00:08:02.790 --> 00:08:04.840
the measurement possibility.
00:08:04.840 --> 00:08:09.710
How do we find what we measure?
00:08:09.710 --> 00:08:10.910
So here it is.
00:08:18.260 --> 00:08:19.730
Measurement Postulate.
00:08:32.510 --> 00:08:33.980
So here's the issue.
00:08:33.980 --> 00:08:35.715
We want to measure.
00:08:35.715 --> 00:08:39.890
I'm going to say
these things in words.
00:08:39.890 --> 00:08:43.590
You want to measure the
operator, q, of your state.
00:08:43.590 --> 00:08:47.120
The operator might be the
momentum, might be the energy,
00:08:47.120 --> 00:08:50.280
might be the angular momentum,
could be kinetic energy,
00:08:50.280 --> 00:08:51.830
could be potential energy.
00:08:51.830 --> 00:08:54.620
Any Hermitian operator.
00:08:54.620 --> 00:08:58.700
You want to measure
it in your state.
00:08:58.700 --> 00:09:02.450
The first thing that
the postulate will say
00:09:02.450 --> 00:09:07.310
is that you will, in general,
obtain just one number
00:09:07.310 --> 00:09:10.220
each time you do a
measurement, but that number
00:09:10.220 --> 00:09:15.020
is one of the eigenvalues
of this operator.
00:09:15.020 --> 00:09:18.800
So the set of
possible measurements,
00:09:18.800 --> 00:09:23.030
possible outcomes,
better say, is the set
00:09:23.030 --> 00:09:25.440
of eigenvalues of the operator.
00:09:25.440 --> 00:09:28.380
Those are the only
numbers you can get.
00:09:28.380 --> 00:09:31.430
But you can get them with
different probabilities.
00:09:31.430 --> 00:09:35.610
And for that, you
must use this plane.
00:09:35.610 --> 00:09:41.090
And you must, in a
sense, rewrite your state
00:09:41.090 --> 00:09:45.894
as a superposition of the
eigenfunctions, those alphas.
00:09:45.894 --> 00:09:51.090
And the probability
to measure q1
00:09:51.090 --> 00:09:53.617
is the probability
that you end up
00:09:53.617 --> 00:09:55.840
of this part of
the superposition,
00:09:55.840 --> 00:10:00.646
and it will be given by
alpha 1 squared, [INAUDIBLE].
00:10:00.646 --> 00:10:03.500
The probability
to measure q will
00:10:03.500 --> 00:10:08.390
be given by alpha 2 squared
and all of these numbers.
00:10:08.390 --> 00:10:15.490
So, and finally, that
after the measurement,
00:10:15.490 --> 00:10:17.060
another funny thing happens.
00:10:17.060 --> 00:10:23.630
The state that was this whole
sum collapses to that state
00:10:23.630 --> 00:10:25.290
that you obtained.
00:10:25.290 --> 00:10:31.280
So if you obtained q1, well, the
whole thing collapses to psi 1.
00:10:31.280 --> 00:10:32.840
After you've done
the measurement,
00:10:32.840 --> 00:10:36.260
the state of the
system becomes psi 1.
00:10:36.260 --> 00:10:38.630
So this is the spirit
of what happens.
00:10:38.630 --> 00:10:42.590
Let me write it out.
00:10:42.590 --> 00:10:57.390
If we measure Q
in the state psi,
00:10:57.390 --> 00:11:09.770
the possible values
obtained are q1, q2.
00:11:12.922 --> 00:11:26.070
The probability, p
i, to measure q i
00:11:26.070 --> 00:11:31.980
is p i equals alpha i squared.
00:11:31.980 --> 00:11:37.310
And remember what this
alpha i we calculated it.
00:11:37.310 --> 00:11:47.030
This overlap of psi
i with psi squared.
00:11:52.880 --> 00:11:58.370
And finally, after finding--
00:12:02.125 --> 00:12:09.970
after, let's write
it, the outcome, q i,
00:12:09.970 --> 00:12:23.020
the state of the
system becomes psi
00:12:23.020 --> 00:12:28.070
of x is equal to psi i of x.
00:12:28.070 --> 00:12:34.541
And this is a collapse
of the wave function.
00:12:42.720 --> 00:12:46.370
And it also means that after
you've done the measurement
00:12:46.370 --> 00:12:51.720
and you did obtain the value
of q i, you stay with psi i,
00:12:51.720 --> 00:12:54.872
if you measure it again, you
would keep obtaining q i.
00:13:02.420 --> 00:13:06.190
Why did it all become possible?
00:13:06.190 --> 00:13:10.190
It all became possible
because Hermitian operators
00:13:10.190 --> 00:13:14.060
are rich enough to
allow you to write
00:13:14.060 --> 00:13:17.030
any state as a superposition.
00:13:17.030 --> 00:13:21.200
And therefore, if you
want to measure momentum,
00:13:21.200 --> 00:13:23.510
you must find all the
eigenfunctions of momentum
00:13:23.510 --> 00:13:27.830
and rewrite your state as a
superposition of momentum.
00:13:27.830 --> 00:13:29.880
You want to do energy?
00:13:29.880 --> 00:13:31.900
Well, you must
rewrite your state
00:13:31.900 --> 00:13:34.370
as a superposition of
energy eigenstates,
00:13:34.370 --> 00:13:35.700
and then you can measure.
00:13:35.700 --> 00:13:37.970
Want to measure
angular momentum?
00:13:37.970 --> 00:13:41.000
Find the eigenstates
of angular momentum,
00:13:41.000 --> 00:13:45.440
use the theorem to rewrite your
whole state in different ways.
00:13:45.440 --> 00:13:48.820
And this is something we
said in the first lecture
00:13:48.820 --> 00:13:53.600
of this course, that any
vector in a vector space
00:13:53.600 --> 00:13:56.510
can be written in
infinitely many ways
00:13:56.510 --> 00:13:59.660
as different
superpositions of vectors.
00:13:59.660 --> 00:14:02.530
We wrote the arrow
and said, this vector
00:14:02.530 --> 00:14:06.772
is the sum of this and this,
and this plus this plus this,
00:14:06.772 --> 00:14:09.830
and this plus this plus this.
00:14:09.830 --> 00:14:13.040
And yes, you need
all that flexibility.
00:14:13.040 --> 00:14:17.750
For any measurement,
you rewrite the vector
00:14:17.750 --> 00:14:21.410
as the sum of the
eigenvectors, and then you
00:14:21.410 --> 00:14:24.910
can tell what are
your predictions.
00:14:24.910 --> 00:14:30.050
You need that flexibility that
any vector in a vector space
00:14:30.050 --> 00:14:32.660
can be written in
infinitely many ways
00:14:32.660 --> 00:14:37.320
as different linear
superpositions.
00:14:37.320 --> 00:14:41.060
So there's a couple
of things we can
00:14:41.060 --> 00:14:45.970
do to add intuition to this.
00:14:45.970 --> 00:14:48.690
I'll do, first, a
consistency check,
00:14:48.690 --> 00:14:51.230
and maybe I'll do
an example as well.
00:14:53.850 --> 00:14:57.934
And then we have to
define uncertainties,
00:14:57.934 --> 00:15:00.970
those of that phase.
00:15:00.970 --> 00:15:04.940
So any question about this
measurement postulate?
00:15:04.940 --> 00:15:07.530
Is there something
unclear about it?
00:15:10.590 --> 00:15:13.040
It's a very strange postulate.
00:15:13.040 --> 00:15:18.080
You see, it divides quantum
mechanics into two realms.
00:15:18.080 --> 00:15:20.840
There's the realm of the
Schrodinger equation,
00:15:20.840 --> 00:15:23.970
your wave function
evolves in time.
00:15:23.970 --> 00:15:27.190
And then there's a
realm of measurement.
00:15:27.190 --> 00:15:29.600
The Schroedinger
equation doesn't tell you
00:15:29.600 --> 00:15:31.760
what you're supposed
to do with measurement.
00:15:31.760 --> 00:15:35.540
But consistency with a
Schroedinger equations
00:15:35.540 --> 00:15:37.130
doesn't allow you many things.
00:15:37.130 --> 00:15:39.980
And this is apparently
the only thing we can do.
00:15:39.980 --> 00:15:45.380
And then we do a measurement,
but somehow, this psi of x
00:15:45.380 --> 00:15:49.472
collapses and becomes one of
the results of your measurement.
00:15:52.310 --> 00:15:55.160
People have wondered, if
the Schroedinger equation is
00:15:55.160 --> 00:15:58.400
all there is in the world,
why doesn't the result
00:15:58.400 --> 00:16:01.550
of the measurement come out
of the Schroedinger equation?
00:16:01.550 --> 00:16:05.150
Well, people think
very hard about it,
00:16:05.150 --> 00:16:10.460
and they come up with all
kinds of interesting things.
00:16:10.460 --> 00:16:13.590
Nevertheless, nothing
that comes out
00:16:13.590 --> 00:16:17.120
is sufficiently clear
and sufficiently useful
00:16:17.120 --> 00:16:19.970
to merit a discussion
at this moment.
00:16:19.970 --> 00:16:23.600
It's very interesting, and
it's subject of research,
00:16:23.600 --> 00:16:28.340
but nobody has found a flaw
with this way of stating things.
00:16:28.340 --> 00:16:31.190
And it's the simplest
way of stating things.
00:16:31.190 --> 00:16:35.415
And therefore, the measurement
is an extra assumption,
00:16:35.415 --> 00:16:36.940
an extra postulate.
00:16:36.940 --> 00:16:39.210
That's how a measurement works.
00:16:39.210 --> 00:16:42.290
And after you measure,
you leave the system,
00:16:42.290 --> 00:16:45.740
the Schroedinger equation
takes over and keeps evolving.
00:16:45.740 --> 00:16:48.410
You measure again,
something happens,
00:16:48.410 --> 00:16:51.360
there's some answer
that gets realized.
00:16:51.360 --> 00:16:56.230
Some answers are not
realized, and it so continues.