WEBVTT
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PROFESSOR: Last time, we talked
about the Broglie wavelength.
00:00:03.820 --> 00:00:07.090
And our conclusion was,
at the end of the day,
00:00:07.090 --> 00:00:12.730
that we could write the
plane wave that corresponded
00:00:12.730 --> 00:00:17.840
to a matter particle, with some
momentum, p, and some energy,
00:00:17.840 --> 00:00:21.030
E. So that was our
main result last time,
00:00:21.030 --> 00:00:23.890
the final form for the wave.
00:00:23.890 --> 00:00:30.250
So we had psi of
x and t that was
00:00:30.250 --> 00:00:37.150
e to the i k x minus i omega t.
00:00:37.150 --> 00:00:41.680
And that was the matter
wave with the relations
00:00:41.680 --> 00:00:46.150
that p is equal to h bar k.
00:00:46.150 --> 00:00:50.020
So this represents a
particle with momentum,
00:00:50.020 --> 00:00:54.200
p, where p is h bar times
this number that appears here,
00:00:54.200 --> 00:00:56.810
the wave number,
and with energy,
00:00:56.810 --> 00:01:03.310
E, equal to h bar
omega, where omega
00:01:03.310 --> 00:01:08.720
is that number that appears
in the [? term ?] exponential.
00:01:08.720 --> 00:01:11.410
Nevertheless, we
were talking, or we
00:01:11.410 --> 00:01:16.700
could talk, about
non-relativistic particles.
00:01:21.270 --> 00:01:23.970
And this is our
focus of attention.
00:01:23.970 --> 00:01:30.190
And in this case, E is
equal to p squared over 2m.
00:01:30.190 --> 00:01:33.400
That formula that expresses
the kinetic energy
00:01:33.400 --> 00:01:37.060
in terms of the momentum, mv.
00:01:37.060 --> 00:01:42.880
So this is the wave function
for a free particle.
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And the task that
we have today is
00:01:54.730 --> 00:01:59.920
to try to use this insight,
this wave function,
00:01:59.920 --> 00:02:02.620
to figure out what
is the equation that
00:02:02.620 --> 00:02:05.380
governs general wave functions.
00:02:05.380 --> 00:02:11.740
So, you see, we've been
led to this wave function
00:02:11.740 --> 00:02:16.120
by postulates of the Broglie
and experiments of Davisson,
00:02:16.120 --> 00:02:18.850
and Germer, and
others, that prove
00:02:18.850 --> 00:02:24.310
that particles like electrons
have wave properties.
00:02:24.310 --> 00:02:27.490
But to put this
on a solid footing
00:02:27.490 --> 00:02:31.630
you need to obtain this
from some equation, that
00:02:31.630 --> 00:02:34.720
will say, OK, if you have
a free particle, what
00:02:34.720 --> 00:02:35.770
are the solutions.
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And you should
find this solution.
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Perhaps you will
find more solutions.
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And you will understand
the problem better.
00:02:44.390 --> 00:02:48.340
And finally, if you understand
the problem of free particle,
00:02:48.340 --> 00:02:53.330
there is a good chance you
can generalize this and write
00:02:53.330 --> 00:02:56.500
the equation for a
particle that moves
00:02:56.500 --> 00:02:59.300
under the influence
of potentials.
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So basically, what
I'm going to do
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by trying to figure out how this
wave emerges from an equation,
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is motivate and
eventually give you,
00:03:13.510 --> 00:03:17.540
by the middle of this lecture,
the Schrodinger equation.
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So that's what we're
going to try to do.
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And the first thing is
to try to understand
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what kind of equation this
wave function satisfies.
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So you want to think of
differential equations
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like wave equations.
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Maybe it's some kind
of wave equation.
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We'll see it's kind
of a variant of that.
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But one thing we
could say, is that you
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have this wave function here.
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And you wish to know, for
example, what is the momentum.
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Well you should look at k, the
number that multiplies the x
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here, and multiply by h bar.
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And that would give
you the momentum.
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But another way of doing it
would be to do the following.
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To say, well, h bar over
i d dx of psi of x and t,
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calculate this thing.
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Now, if I differentiate
with respect to x,
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I get here, i
times k going down.
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The i cancels this
i, and I get h bar k.
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So, I get h bar k
times the exponential.
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And that is equal to the value
of the momentum times the wave.
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So here is this wave actually
satisfies a funny equation,
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not quite the differential
equation we're looking for yet,
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but you can act with a
differential operator.
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A derivative is something
of a differential operator.
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It operates in functions,
and takes the derivative.
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And when it acts on
this wave function,
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it gives you the momentum
times the wave function.
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And this momentum
here is a number.
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Here you have an operator.
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An operator just means something
that acts on functions,
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and gives you functions.
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So taking a derivative of a
function is still a function.
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So that's an operator.
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So we are left here to
think of this operator
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as the operator that
reveals for you the momentum
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of the free particle, because
acting on the wave function,
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it gives you the momentum
times the wave function.
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Now it couldn't be that
acting on the wave function
00:05:58.570 --> 00:06:02.620
just gives you the momentum,
because the exponential doesn't
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disappear after the
differential operator acts.
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So it's actually
the operator acting
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on the wave function gives you a
number times the wave function.
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And that number is the momentum.
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So we will call
this operator, given
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that it gives us the momentum,
the momentum operator, so
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momentum operator.
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And to distinguish it
from p, we'll put a hat,
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is defined to be
h bar over i d dx.
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And therefore, for
our free particle,
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you can write what we've
just derived in a brief way,
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writing p hat
acting on psi, where
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this means the
operator acting on psi,
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gives you the momentum of this
state times psi of x and t.
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And that's a number.
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So this is an operator
state, number state.
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So we say a few things,
this language that we're
00:07:22.680 --> 00:07:25.780
going to be using all the time.
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We call this wave function,
this psi, if this is true,
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this holds, then we
say the psi of x and t
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is an eigenstate of
the momentum operator.
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And that language comes from
matrix algebra, linear algebra,
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in which you have a
matrix and a vector.
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And when the matrix
on a vector gives you
00:08:12.520 --> 00:08:15.250
a number times the
same vector, we
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say that that vector is an
eigenvector of the matrix.
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Here, we call it an eigenstate.
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Probably, nobody
would complain if you
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called it an eigenvector,
but eigenstate
00:08:28.910 --> 00:08:30.450
would be more appropriate.
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So it's an eigenstate of p.
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So, in general, if you have an
operator, A, under a function,
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phi, such that A acting
on phi is alpha phi,
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we say that phi is an
eigenstate of the operator,
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and in fact eigenvalue alpha.
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So, here is an eigenstate
of p with eigenvalue
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of p, the number p, because
acting on the wave function
00:09:18.375 --> 00:09:21.150
gives you the number p
times that wave function.
00:09:21.150 --> 00:09:24.900
Not every wave function
will be an eigenstate.
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Just like, when you have a
matrix acting on most vectors,
00:09:28.930 --> 00:09:31.800
a matrix will rotate
the vector and move it
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into something else.
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But sometimes, a matrix
acting in a vector
00:09:35.910 --> 00:09:39.510
will give you the same
vector up to a constant,
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and then you've
got an eigenvector.
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And here, we have an eigenstate.
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So another way of
expressing this,
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is we say that psi of x
and t, this psi of x and t,
00:09:56.610 --> 00:10:04.950
is a state of definite momentum.
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It's important terminology,
definite momentum means
00:10:11.010 --> 00:10:16.230
that if you measured it, you
would find the momentum p.
00:10:16.230 --> 00:10:20.010
And the momentum-- there
would be no uncertainty
00:10:20.010 --> 00:10:20.970
on this measurement.
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You measure, and
you always get p.
00:10:25.040 --> 00:10:27.300
And that's what,
intuitively, we have,
00:10:27.300 --> 00:10:30.090
because we decided
that this was the wave
00:10:30.090 --> 00:10:34.170
function for a free
particle with momentum, p.
00:10:34.170 --> 00:10:38.160
So as long as we
just have that, we
00:10:38.160 --> 00:10:42.600
have that psi is a state
of definite momentum.
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This is an interesting statement
that will apply for many things
00:10:48.512 --> 00:10:49.470
as we go in the course.
00:10:49.470 --> 00:10:54.410
But now let's consider another
aspect of this equation.
00:10:54.410 --> 00:10:56.040
So we succeeded with that.
00:10:56.040 --> 00:10:58.710
And we can ask if there
is a similar thing
00:10:58.710 --> 00:11:04.720
that we can do to figure out
the energy of the particle.
00:11:04.720 --> 00:11:07.400
And indeed we can
do the following.
00:11:07.400 --> 00:11:12.555
We can do i h bar d dt of psi.
00:11:16.610 --> 00:11:21.730
And if we have that,
we'll take the derivative.
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Now, this time,
we'll have i h bar.
00:11:24.810 --> 00:11:27.770
And when we differentiate that
wave function with respect
00:11:27.770 --> 00:11:33.740
to time, we get minus i omega
times the wave function.
00:11:33.740 --> 00:11:36.230
So i times minus i is 1.
00:11:36.230 --> 00:11:39.110
And you get h bar omega psi.
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Success, that was the energy
of the particle times psi.
00:11:55.490 --> 00:11:57.990
And this looks quite
interesting already.
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This is a number, again.
00:12:00.460 --> 00:12:06.710
And this is a time derivative
of the wave function.
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But we can put more physics into
this, because in a sense, well,
00:12:15.580 --> 00:12:20.700
this differential
equation tells you
00:12:20.700 --> 00:12:24.390
how a wave function
with energy, E,
00:12:24.390 --> 00:12:28.540
what the time dependence
of that wave function is.
00:12:28.540 --> 00:12:31.690
But that wave function
already, in our case,
00:12:31.690 --> 00:12:35.110
is a wave function
of definite momentum.
00:12:35.110 --> 00:12:39.190
So somehow, the information
that is missing there,
00:12:39.190 --> 00:12:44.320
is that the energy
is p squared over 2m.
00:12:44.320 --> 00:12:52.080
So we have that the energy
is p squared over 2m.
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So let's try to think of
the energy as an operator.
00:13:00.990 --> 00:13:04.830
And look, you could
say the energy,
00:13:04.830 --> 00:13:08.790
well, this is the energy
operator acting on the function
00:13:08.790 --> 00:13:10.350
gives you the energy.
00:13:10.350 --> 00:13:15.630
That this true, but it's too
general, not interesting enough
00:13:15.630 --> 00:13:16.680
at this point.
00:13:16.680 --> 00:13:21.780
What is really interesting is
that the energy has a formula.
00:13:21.780 --> 00:13:24.430
And that's the physics
of the particle,
00:13:24.430 --> 00:13:27.090
the formula for the energy
depends on the momentum.
00:13:27.090 --> 00:13:29.970
So we want to capture that.
00:13:29.970 --> 00:13:32.340
So let's look what
we're going to do.
00:13:32.340 --> 00:13:36.210
We're going to do a
relatively simple thing, which
00:13:36.210 --> 00:13:38.070
we are going to walk back this.
00:13:38.070 --> 00:13:42.490
So I'm going to
start with E psi.
00:13:42.490 --> 00:13:47.940
And I'm going to invent
an operator acting on psi
00:13:47.940 --> 00:13:49.065
that gives you this energy.
00:13:51.590 --> 00:14:01.517
So I'm going to invent an O.
00:14:01.517 --> 00:14:07.050
So how do we do that?
00:14:07.050 --> 00:14:15.140
Well, E is equal to p
squared over 2m times psi.
00:14:15.140 --> 00:14:17.630
It's a number times psi.
00:14:17.630 --> 00:14:21.500
But then you say, oh,
p, but I remember p.
00:14:21.500 --> 00:14:25.420
I could write it as an operator.
00:14:25.420 --> 00:14:30.750
So if I have p times
psi, I could write it
00:14:30.750 --> 00:14:44.792
as p over 2m h bar
over i d dx of psi.
00:14:44.792 --> 00:14:50.380
Now please, listen
with lots of attention.
00:14:50.380 --> 00:14:52.930
I'm going to do a simple
thing, but it's very easy
00:14:52.930 --> 00:14:54.820
to get confused
with the notation.
00:14:54.820 --> 00:14:57.540
If I make a little typo
in what I'm writing
00:14:57.540 --> 00:15:00.480
it can confuse you
for a long time.
00:15:00.480 --> 00:15:05.990
So, so far these are numbers.
00:15:05.990 --> 00:15:09.100
Number, this is a
number times psi.
00:15:09.100 --> 00:15:13.780
But this p times
psi is p hat psi
00:15:13.780 --> 00:15:16.150
which is that operator, there.
00:15:16.150 --> 00:15:17.720
So I wrote it this way.
00:15:20.590 --> 00:15:22.660
I want to make one more-- yes?
00:15:22.660 --> 00:15:25.610
AUDIENCE: Should that say E psi?
00:15:25.610 --> 00:15:27.550
PROFESSOR: Oh yes,
thank you very much.
00:15:31.520 --> 00:15:32.020
Thank you.
00:15:34.530 --> 00:15:43.985
Now, the question is, can I
move this p close to the psi.
00:15:50.860 --> 00:15:51.360
Opinions?
00:15:53.960 --> 00:15:54.900
Yes?
00:15:54.900 --> 00:15:57.260
AUDIENCE: Are you asking
if it's just a constant?
00:15:57.260 --> 00:15:59.710
PROFESSOR: Correct,
p is a constant.
00:15:59.710 --> 00:16:01.460
p hat is not a constant.
00:16:01.460 --> 00:16:02.330
Derivatives are not.
00:16:02.330 --> 00:16:04.490
But p at this
moment is a number.
00:16:04.490 --> 00:16:06.750
So it doesn't care
about the derivatives.
00:16:06.750 --> 00:16:07.610
And it goes in.
00:16:07.610 --> 00:16:16.180
So I'll write it as
1 over 2m h/i d dx,
00:16:16.180 --> 00:16:22.480
and here, output p psi,
where is that number.
00:16:22.480 --> 00:16:28.750
But now, p psi, I can
write it as whatever it is,
00:16:28.750 --> 00:16:41.550
which is h/i d dx, and p
psi is again, h/i d dx psi.
00:16:41.550 --> 00:16:44.340
So here we go.
00:16:44.340 --> 00:16:48.180
We have obtained, and
let me write the equation
00:16:48.180 --> 00:16:51.240
in slightly reversed form.
00:16:51.240 --> 00:16:57.780
Minus, because of the two
i's, 1 over 2m, two partials
00:16:57.780 --> 00:17:03.660
derivatives is a second order
partial derivative on psi,
00:17:03.660 --> 00:17:09.329
h bar squared over
2m d second dx psi.
00:17:09.329 --> 00:17:13.904
That's the whole right-hand
side, is equal to E psi.
00:17:21.200 --> 00:17:26.890
So the number E
times psi is this.
00:17:26.890 --> 00:17:33.732
So we could call this
thing the energy operator.
00:17:45.040 --> 00:17:48.090
And this is the energy operator.
00:17:54.190 --> 00:18:03.800
And it has the property that
the energy operator acting
00:18:03.800 --> 00:18:06.620
on this wave function
is, in fact, equal
00:18:06.620 --> 00:18:09.530
to the energy times
the wave function.
00:18:12.780 --> 00:18:19.970
So this state again is
an energy eigenstate.
00:18:19.970 --> 00:18:22.710
Energy operator on the
state is the energy
00:18:22.710 --> 00:18:23.920
times the same state.
00:18:23.920 --> 00:18:46.110
So psi is an energy eigenstate,
or a state of definite energy,
00:18:46.110 --> 00:18:48.970
or an energy
eigenstate with energy,
00:18:48.970 --> 00:18:56.100
E. I can make it clear for
you that, in fact, this energy
00:18:56.100 --> 00:18:58.890
operator, as you've
noticed, the only thing
00:18:58.890 --> 00:19:07.870
that it is is minus h squared
over 2m d second dx squared.
00:19:07.870 --> 00:19:09.850
But where it came
from, it's clear
00:19:09.850 --> 00:19:17.420
that it's nothing else but
1 over 2m p hat squared,
00:19:17.420 --> 00:19:23.210
because p hat is
indeed h/i d dx.
00:19:23.210 --> 00:19:25.430
So if you do this computation.
00:19:25.430 --> 00:19:26.510
How much is this?
00:19:26.510 --> 00:19:32.000
This is A p hat times p
hat, that's p hat squared.
00:19:32.000 --> 00:19:38.440
And that's h/i d dx h/i d dx.
00:19:38.440 --> 00:19:42.470
X And that gives you the answer.
00:19:42.470 --> 00:19:49.650
So the energy operator
is p hat squared over 2m.
00:19:52.540 --> 00:19:56.110
All right, so actually,
at this moment,
00:19:56.110 --> 00:20:00.970
we do have a Schrodinger
equation, for the first time.
00:20:00.970 --> 00:20:06.050
If we combine the
top line over there.
00:20:06.050 --> 00:20:17.050
I h bar d dt of psi
is equal to E psi,
00:20:17.050 --> 00:20:23.900
but E psi I will write it
as minus h squared over 2m d
00:20:23.900 --> 00:20:28.530
second dx squared psi.