WEBVTT
00:00:00.300 --> 00:00:01.674
PROFESSOR: Let's
do a work check.
00:00:04.200 --> 00:00:05.845
So main check.
00:00:11.970 --> 00:00:26.270
If integral psi star x t0,
psi x t0 dx is equal to 1
00:00:26.270 --> 00:00:32.680
at t equal to t0,
as we say there,
00:00:32.680 --> 00:00:45.160
then it must hold for later
times, t greater than t0.
00:00:45.160 --> 00:00:49.240
This is what we want to
check, or verify, or prove.
00:00:49.240 --> 00:00:53.860
Now, to do it, we're
going to take our time.
00:00:53.860 --> 00:00:57.970
So it's not going to happen in
five minutes, not 10 minutes,
00:00:57.970 --> 00:01:01.450
maybe not even half an hour.
00:01:01.450 --> 00:01:04.209
Not because it's
so difficult. It's
00:01:04.209 --> 00:01:05.980
because there's so
many things that one
00:01:05.980 --> 00:01:08.620
can say in between
that teach you
00:01:08.620 --> 00:01:10.270
a lot about quantum mechanics.
00:01:10.270 --> 00:01:13.700
So we're going to
take our time here.
00:01:13.700 --> 00:01:21.840
So we're going to first rewrite
it with better notation.
00:01:21.840 --> 00:01:30.090
So we'll define rho
of x and t, which
00:01:30.090 --> 00:01:35.155
is going to be called
the probability density.
00:01:38.590 --> 00:01:42.280
And it's nothing else than
what you would expect, psi star
00:01:42.280 --> 00:01:46.960
of x and t, psi of x and t.
00:01:52.660 --> 00:01:56.520
It's a probability density.
00:01:56.520 --> 00:01:59.690
You know that has the
right interpretation,
00:01:59.690 --> 00:02:02.120
it's psi squared.
00:02:02.120 --> 00:02:05.820
And that's the kind of thing
that integrated over space
00:02:05.820 --> 00:02:08.250
gives you the total probability.
00:02:08.250 --> 00:02:14.150
So this is a
positive number given
00:02:14.150 --> 00:02:17.490
by this quantity is called
the probability density.
00:02:17.490 --> 00:02:18.680
Fine.
00:02:18.680 --> 00:02:21.500
What do we know about
this probability density
00:02:21.500 --> 00:02:26.180
that we're trying to
find about its integral?
00:02:26.180 --> 00:02:36.500
So define next N of t to be the
integral of rho of x and t dx.
00:02:39.800 --> 00:02:42.830
Integrate this
probability density
00:02:42.830 --> 00:02:50.130
throughout space, and that's
going to give you N of t.
00:02:50.130 --> 00:02:51.730
Now, what do we know?
00:02:51.730 --> 00:02:57.980
We know that N of t, or
let's assume that N of t0
00:02:57.980 --> 00:03:00.660
is equal to 1.
00:03:00.660 --> 00:03:02.650
N is that normalization.
00:03:02.650 --> 00:03:05.160
It's that total integral
of the probability
00:03:05.160 --> 00:03:07.620
what had to be equal to 1.
00:03:07.620 --> 00:03:13.290
Well, let's assume N
at t0 is equal to 1.
00:03:13.290 --> 00:03:15.930
That's good.
00:03:15.930 --> 00:03:22.270
The question is, will
the Schrodinger equation
00:03:22.270 --> 00:03:28.400
guarantee that--
00:03:28.400 --> 00:03:29.670
and here's the claim--
00:03:29.670 --> 00:03:34.880
dN dt is equal to 0?
00:03:39.060 --> 00:03:41.930
Will the Schrodinger
equation guarantee this?
00:03:46.470 --> 00:03:49.730
If the Schrodinger
equation guarantees
00:03:49.730 --> 00:03:52.980
that this derivative
is, indeed, zero,
00:03:52.980 --> 00:03:54.960
then we're in good business.
00:03:54.960 --> 00:03:59.580
Because the derivative
is zero, the value's 1,
00:03:59.580 --> 00:04:01.600
will remain 1 forever.
00:04:01.600 --> 00:04:02.344
Yes?
00:04:02.344 --> 00:04:03.885
AUDIENCE: May I ask
why you specified
00:04:03.885 --> 00:04:05.256
for t greater than t0?
00:04:08.160 --> 00:04:13.380
Well, I don't have to specify
for t greater than t naught.
00:04:13.380 --> 00:04:19.029
I could do it for all t
different than t naught.
00:04:19.029 --> 00:04:29.160
But if I say this way, as
imagining that somebody
00:04:29.160 --> 00:04:31.500
prepares the system
at some time,
00:04:31.500 --> 00:04:35.310
t naught, and maybe the system
didn't exist for other times
00:04:35.310 --> 00:04:36.180
below.
00:04:36.180 --> 00:04:39.750
Now, if a system
existed for long time
00:04:39.750 --> 00:04:44.340
and you look at it at t naught,
then certainly the Schrodinger
00:04:44.340 --> 00:04:49.740
equation should imply
that it works later
00:04:49.740 --> 00:04:52.050
and it works before.
00:04:52.050 --> 00:04:57.172
So it's not really necessary,
but no loss of generality.
00:05:01.240 --> 00:05:02.750
OK, so that's it.
00:05:02.750 --> 00:05:04.510
Will it guarantee that?
00:05:04.510 --> 00:05:06.380
Well, that's our thing to do.
00:05:06.380 --> 00:05:15.020
So let's begin the work by doing
a little bit of a calculation.
00:05:15.020 --> 00:05:18.190
And so what do we need to do?
00:05:18.190 --> 00:05:22.610
We need to find the
derivative of this quantity.
00:05:22.610 --> 00:05:34.690
So what is this derivative of N
dN dt will be the integral d dt
00:05:34.690 --> 00:05:43.160
of rho of x and t dx.
00:05:43.160 --> 00:05:49.040
So I went here and
brought in the d dt, which
00:05:49.040 --> 00:05:50.990
became a partial derivative.
00:05:50.990 --> 00:05:55.910
Because this is just a
function of t, but inside here,
00:05:55.910 --> 00:05:58.160
there's a function of
t and a function of x.
00:05:58.160 --> 00:06:03.740
So I must make clear that
I'm just differentiating t.
00:06:03.740 --> 00:06:07.490
So is d dt of rho.
00:06:07.490 --> 00:06:12.680
And now we can write
it as integral dx.
00:06:12.680 --> 00:06:14.120
What this rho?
00:06:14.120 --> 00:06:15.860
Psi star psi.
00:06:15.860 --> 00:06:28.240
So we would have d dt of psi
star times psi plus psi star
00:06:28.240 --> 00:06:30.620
d dt of psi.
00:06:40.240 --> 00:06:40.740
OK.
00:06:44.480 --> 00:06:49.120
And here you see, if you
were waiting for that,
00:06:49.120 --> 00:06:52.220
that the Schrodinger
equation has to be necessary.
00:06:52.220 --> 00:06:54.950
Because we have the psi dt.
00:06:54.950 --> 00:06:59.550
And that information is there
with Schrodinger's equation.
00:06:59.550 --> 00:07:03.540
So let's do that.
00:07:03.540 --> 00:07:04.730
So what do we have?
00:07:04.730 --> 00:07:12.780
ih bar d psi dt equal h psi.
00:07:12.780 --> 00:07:15.590
We'll write it like
that for the time being
00:07:15.590 --> 00:07:18.500
without copying all what h is.
00:07:18.500 --> 00:07:20.090
That would take a lot of time.
00:07:23.490 --> 00:07:28.200
And from this equation,
you can find immediately
00:07:28.200 --> 00:07:39.440
that d psi dt is minus
i over h bar h hat psi.
00:07:44.220 --> 00:07:47.270
Now we need to complex
conjugate this equation,
00:07:47.270 --> 00:07:49.620
and that is always
a little more scary.
00:07:59.650 --> 00:08:02.140
Actually, the way
to do this in a way
00:08:02.140 --> 00:08:05.530
that you never get into
scary or strange things.
00:08:05.530 --> 00:08:09.280
So let me take the complex
conjugate of this equation.
00:08:09.280 --> 00:08:14.810
Here I would have i
goes to minus i h bar,
00:08:14.810 --> 00:08:16.880
and now I would have--
00:08:16.880 --> 00:08:18.860
we can go very slow--
00:08:18.860 --> 00:08:29.480
d psi dt star equals, and then
I'll be simple minded here.
00:08:29.480 --> 00:08:32.030
I think it's the best.
00:08:32.030 --> 00:08:36.350
I'll just start the
right hand side.
00:08:36.350 --> 00:08:41.340
I start the left hand side
and start the right hand side.
00:08:41.340 --> 00:08:48.310
Now here, the complex conjugate
of a derivative, in this case
00:08:48.310 --> 00:08:50.290
I want to clarify what it is.
00:08:50.290 --> 00:08:53.780
It's just the derivative
of the complex conjugate.
00:08:53.780 --> 00:09:04.140
So this is minus ih
bar d/dt of psi star
00:09:04.140 --> 00:09:12.540
equals h hat psi
star, that's fine.
00:09:12.540 --> 00:09:21.450
And from here, if I multiply
again by i divided by h bar,
00:09:21.450 --> 00:09:36.042
we get d psi star dt is equal
to i over h star h hat psi star.
00:09:41.560 --> 00:09:47.040
We obtain this useful formula
and this useful formula,
00:09:47.040 --> 00:09:53.440
and both go into our
calculation of dN dt.
00:09:53.440 --> 00:09:56.620
So what do we have here?
00:09:56.620 --> 00:10:13.000
dN dt equals integral dx, and
I will put an i over h bar,
00:10:13.000 --> 00:10:15.356
I think, here.
00:10:15.356 --> 00:10:15.855
Yes.
00:10:18.990 --> 00:10:20.400
i over h bar.
00:10:26.400 --> 00:10:28.055
Look at this term first.
00:10:30.630 --> 00:10:40.840
We have i over h
bar, h psi star psi.
00:10:40.840 --> 00:10:44.410
And the second term
involves a d psi dt that
00:10:44.410 --> 00:10:47.320
comes with an opposite sign.
00:10:47.320 --> 00:10:56.690
Same factor of i over h bar,
so minus psi star h psi.
00:11:06.610 --> 00:11:10.680
So the virtue of what
we've done so far
00:11:10.680 --> 00:11:13.530
is that it doesn't
look so bad yet.
00:11:13.530 --> 00:11:21.900
And looks relatively clean, and
it's very suggestive, actually.
00:11:21.900 --> 00:11:23.350
So what's happening?
00:11:23.350 --> 00:11:27.295
We want to show that
dN dt is equal to 0.
00:11:31.220 --> 00:11:35.210
Now, are we going to be
able to show that simply
00:11:35.210 --> 00:11:39.080
that to do a lot of algebra
and say, oh, it's 0?
00:11:39.080 --> 00:11:41.480
Well, it's kind of
going to work that way,
00:11:41.480 --> 00:11:43.880
but we're going to
do the work and we're
00:11:43.880 --> 00:11:50.870
going to get to dN dt being
an integral of something.
00:11:50.870 --> 00:11:53.410
And it's just not
going to look like 0,
00:11:53.410 --> 00:11:56.900
but it will be
manipulated in such a way
00:11:56.900 --> 00:12:01.550
that you can argue it's 0
using the boundary condition.
00:12:01.550 --> 00:12:05.210
So it's kind of interesting
how it's going to work.
00:12:05.210 --> 00:12:08.930
But here structurally,
you see what
00:12:08.930 --> 00:12:13.470
must happen for this
calculation to succeed.
00:12:13.470 --> 00:12:15.670
So we need for this to be 0.
00:12:22.630 --> 00:12:27.250
We need the following
thing to happen.
00:12:27.250 --> 00:12:36.580
The integral of h
hat psi star psi
00:12:36.580 --> 00:12:42.776
be equal to the integral
of psi star h psi.
00:12:45.692 --> 00:12:48.510
And I should write the dx's.
00:12:48.510 --> 00:12:49.260
They are there.
00:12:53.030 --> 00:12:59.310
So this would guarantee
that dN dt is equal to 0.
00:12:59.310 --> 00:13:05.500
So that's a very nice
statement, and it's kind of nice
00:13:05.500 --> 00:13:09.890
is that you have one
function starred,
00:13:09.890 --> 00:13:12.540
one function non-starred.
00:13:12.540 --> 00:13:16.110
The h is where the function
needs to be starred,
00:13:16.110 --> 00:13:18.520
but on the other
side of the equation,
00:13:18.520 --> 00:13:21.520
the h is on the other side.
00:13:21.520 --> 00:13:27.300
So you've kind of moved the
h from the complex conjugated
00:13:27.300 --> 00:13:30.570
function to the non-complex
conjugated function.
00:13:30.570 --> 00:13:34.720
From the first function
to this second function.
00:13:34.720 --> 00:13:40.320
And that's a very nice thing
to demand of the Hamiltonian.
00:13:40.320 --> 00:13:43.290
So actually what
seems to be happening
00:13:43.290 --> 00:13:47.190
is that this conservation
of probability
00:13:47.190 --> 00:13:50.490
will work if your
Hamiltonian is good enough
00:13:50.490 --> 00:13:52.500
to do something like this.
00:13:55.530 --> 00:13:59.295
And this is a nice formula,
it's a famous formula.
00:14:05.670 --> 00:14:15.600
This is true if H is
a Hermitian operator.
00:14:22.990 --> 00:14:25.990
It's a very interesting
new name that
00:14:25.990 --> 00:14:29.750
shows up that an
operator being Hermitian.
00:14:29.750 --> 00:14:33.650
So this is what I
was promising you,
00:14:33.650 --> 00:14:35.420
that we're going to
do this, and we're
00:14:35.420 --> 00:14:40.500
going to be learning all kinds
of funny things as it happens.
00:14:40.500 --> 00:14:45.200
So what is it for a
Hermitian operator?
00:14:45.200 --> 00:14:56.650
Well, a Hermitian
operator, H, would actually
00:14:56.650 --> 00:14:58.310
satisfy the following.
00:15:06.810 --> 00:15:17.590
That the integral,
H psi 1 star psi 2
00:15:17.590 --> 00:15:29.100
is equal to the integral
of psi 1 star H psi 2.
00:15:29.100 --> 00:15:37.230
So an operator is said to be
Hermitian if you can move it
00:15:37.230 --> 00:15:44.310
from the first part to the
second part in this sense,
00:15:44.310 --> 00:15:47.950
and with two
different functions.
00:15:47.950 --> 00:15:51.720
So this should be possible
to do if an operator is
00:15:51.720 --> 00:15:53.370
to be called Hermitian.
00:15:55.940 --> 00:16:00.640
Now, of course, if it holds
for two arbitrary functions,
00:16:00.640 --> 00:16:05.400
it holds when the two functions
are the same, in this case.
00:16:05.400 --> 00:16:09.110
So what we need is
a particular case
00:16:09.110 --> 00:16:12.350
of the condition of hermiticity.
00:16:12.350 --> 00:16:16.700
Hermiticity simply means that
the operator does this thing.
00:16:20.410 --> 00:16:27.080
Any two functions that you put
here, this equality is true.
00:16:27.080 --> 00:16:31.300
Now if you ask yourself, how
do I even understand that?
00:16:31.300 --> 00:16:35.550
What allows me to move the H
from one side to the other?
00:16:35.550 --> 00:16:37.130
We'll see it very soon.
00:16:37.130 --> 00:16:41.380
But it's the fact that H
has second derivatives,
00:16:41.380 --> 00:16:44.240
and maybe you can
integrate them by parts
00:16:44.240 --> 00:16:47.980
and move the derivatives
from the psi 1 to the psi 2,
00:16:47.980 --> 00:16:50.210
and do all kinds of things.
00:16:50.210 --> 00:16:55.220
But you should try to think at
this moment structurally, what
00:16:55.220 --> 00:16:58.610
kind of objects you have, what
kind of properties you have.
00:16:58.610 --> 00:17:02.840
And the objects
are this operator
00:17:02.840 --> 00:17:05.540
that controls the
time evolution, called
00:17:05.540 --> 00:17:07.170
the Hamiltonian.
00:17:07.170 --> 00:17:12.420
And if I want probability
interpretation to make sense,
00:17:12.420 --> 00:17:17.420
we need this equality, which is
a consequence of hermiticity.
00:17:17.420 --> 00:17:23.170
Now, I'll maybe use a
little of this blackboard.
00:17:23.170 --> 00:17:28.280
I haven't used it much before.
00:17:28.280 --> 00:17:31.460
In terms of Hermitian
operators, I'm
00:17:31.460 --> 00:17:35.540
almost there with a definition
of a Hermitian operator.
00:17:35.540 --> 00:17:40.550
I haven't quite given it to
you, but let's let state it,
00:17:40.550 --> 00:17:46.750
given that we're already in
this discussion of hermiticity.
00:17:46.750 --> 00:17:53.690
So this is what is called the
Hermitian operator, does that.
00:17:53.690 --> 00:18:11.580
But in general, rho,
given an operator T,
00:18:11.580 --> 00:18:29.980
one defines its hermitian
conjugate P dagger as follows.
00:18:29.980 --> 00:18:37.135
So you have the
integral of psi 1 star T
00:18:37.135 --> 00:18:44.990
psi 2, and that must be
rearranged until it looks
00:18:44.990 --> 00:18:56.000
like T dagger psi 1 star psi 2.
00:18:56.000 --> 00:18:59.270
Now, these things
are the beginning
00:18:59.270 --> 00:19:04.970
of a whole set of ideas that are
terribly important in quantum
00:19:04.970 --> 00:19:05.910
mechanics.
00:19:05.910 --> 00:19:09.960
Hermitian operators, or
eigenvalues and eigenvectors.
00:19:09.960 --> 00:19:12.170
So it's going to take
a little time for you
00:19:12.170 --> 00:19:14.040
to get accustomed to them.
00:19:14.040 --> 00:19:15.770
But this is the beginning.
00:19:15.770 --> 00:19:17.990
You will explore a little
bit of these things
00:19:17.990 --> 00:19:20.690
in future homework, and
start getting familiar.
00:19:20.690 --> 00:19:24.950
For now, it looks very
strange and unmotivated.
00:19:24.950 --> 00:19:28.550
Maybe you will see that
that will change soon, even
00:19:28.550 --> 00:19:32.190
throughout today's lecture.
00:19:32.190 --> 00:19:35.900
So this is the
Hermitian conjugate.
00:19:35.900 --> 00:19:39.320
So if you want to calculate
the Hermitian conjugate,
00:19:39.320 --> 00:19:42.710
you must start with this thing,
and start doing manipulations
00:19:42.710 --> 00:19:48.410
to clean up the psi 2,
have nothing at the psi 2,
00:19:48.410 --> 00:19:51.680
everything acting on
psi 1, and that thing
00:19:51.680 --> 00:19:53.780
is called the dagger.
00:19:53.780 --> 00:20:06.830
And then finally, T is Hermitian
if T dagger is equal to T.
00:20:06.830 --> 00:20:10.880
So its Hermitian
conjugate is itself.
00:20:10.880 --> 00:20:13.730
It's almost like people
say a real number is
00:20:13.730 --> 00:20:17.300
a number whose complex
conjugate is equal to itself.
00:20:17.300 --> 00:20:23.240
So a Hermitian operator is
one whose Hermitian conjugate
00:20:23.240 --> 00:20:28.400
is equal to itself, and
you see if T is Hermitian,
00:20:28.400 --> 00:20:34.370
well then it's back to T
and T in both places, which
00:20:34.370 --> 00:20:36.800
is what we've been saying here.
00:20:36.800 --> 00:20:39.910
This is a Hermitian operator.