WEBVTT

00:00:01.960 --> 00:00:03.760
PROFESSOR: So here it is.

00:00:03.760 --> 00:00:10.140
Suppose you have a serious
expansion in lambda.

00:00:10.140 --> 00:00:13.250
So this is the state--
when lambda is equal to 0,

00:00:13.250 --> 00:00:15.320
this should be the state.

00:00:15.320 --> 00:00:17.540
But when lambda is
different from 0,

00:00:17.540 --> 00:00:18.770
that will not be the state.

00:00:18.770 --> 00:00:20.090
We'll have lambda correction.

00:00:20.090 --> 00:00:25.230
So this is the first-order
correction to this state.

00:00:25.230 --> 00:00:27.410
So that's why I put the 1.

00:00:27.410 --> 00:00:30.410
And you should think
of it first order-- oh!

00:00:30.410 --> 00:00:33.110
--because it comes
with a lambda.

00:00:33.110 --> 00:00:35.990
This is the second-order
correction to the state,

00:00:35.990 --> 00:00:38.040
because it comes with
a lambda squared.

00:00:38.040 --> 00:00:39.890
And the same thing here.

00:00:39.890 --> 00:00:45.230
So the superscript is telling
you what order in lambda

00:00:45.230 --> 00:00:46.850
you are working--

00:00:46.850 --> 00:00:48.180
to what accuracy.

00:00:48.180 --> 00:00:50.750
So what is the most
urgent thing to find

00:00:50.750 --> 00:00:52.670
the first order of corrections?

00:00:52.670 --> 00:00:55.720
If you find them, and we
still have time [LAUGH]

00:00:55.720 --> 00:01:00.010
for the second order,
you go more and more.

00:01:00.010 --> 00:01:02.250
OK, let's continue.

00:01:02.250 --> 00:01:03.715
Let's solve some of this.

00:01:06.670 --> 00:01:10.450
So our next task is
to solve this problem.

00:01:10.450 --> 00:01:12.870
And here we go.

00:01:15.720 --> 00:01:18.060
Let's solve that.

00:01:18.060 --> 00:01:19.570
So what am I going to do?

00:01:19.570 --> 00:01:25.840
I'm going to just write this
equation slightly differently.

00:01:25.840 --> 00:01:31.770
I'll write it as h
of lambda, which is--

00:01:31.770 --> 00:01:37.870
OK, I'll write it differently.
h0 plus lambda delta h--

00:01:37.870 --> 00:01:39.930
that's h of lambda--

00:01:39.930 --> 00:01:49.320
minus En of lambda on the state
n of lambda is equal to 0.

00:01:49.320 --> 00:01:52.140
That's your
Schrodinger equation,

00:01:52.140 --> 00:01:54.690
the time-independent
Schrodinger equation,

00:01:54.690 --> 00:01:57.810
we're trying to solve.

00:01:57.810 --> 00:02:01.080
And now I'm going to
just write it out,

00:02:01.080 --> 00:02:03.010
so that you can see what we get.

00:02:03.010 --> 00:02:05.670
So it's going to take a
little bit of writing.

00:02:05.670 --> 00:02:11.370
Let me collect the terms
that have no lambda.

00:02:11.370 --> 00:02:12.300
It's h0.

00:02:12.300 --> 00:02:20.040
This has a lambda, but En begins
with En0 0 that has no lambda.

00:02:20.040 --> 00:02:23.685
So, from this parentheses, this
is a term without a lambda.

00:02:27.020 --> 00:02:30.710
It came from here, En0--

00:02:30.710 --> 00:02:32.150
it's here.

00:02:32.150 --> 00:02:37.340
Now let's look at the
terms with a lambda.

00:02:37.340 --> 00:02:41.750
So I want to see
how I'm writing.

00:02:41.750 --> 00:02:44.750
I want to write it
with a minus sign.

00:02:44.750 --> 00:02:47.000
So, with a lambda,
we have minus--

00:02:50.990 --> 00:03:03.290
from here, we have a
term En1 minus delta h.

00:03:03.290 --> 00:03:06.420
That is all the
terms with a lambda.

00:03:06.420 --> 00:03:09.455
So I should put the
lambda, as well, here.

00:03:09.455 --> 00:03:11.690
Probably I want to
put it in front.

00:03:11.690 --> 00:03:15.920
Minus lambda [INAUDIBLE].

00:03:15.920 --> 00:03:21.020
En1, from there, and
the lambda delta h,

00:03:21.020 --> 00:03:22.385
with a double minus sign.

00:03:25.940 --> 00:03:28.580
Then it goes simple, now.

00:03:28.580 --> 00:03:30.800
I've taken into account
these two terms.

00:03:30.800 --> 00:03:32.210
All the rest come from here.

00:03:32.210 --> 00:03:38.680
So you have a minus
lambda squared En2,

00:03:38.680 --> 00:03:45.560
and, at some point, a
minus lambda to the k Enk.

00:03:45.560 --> 00:03:47.990
And then it goes on.

00:03:47.990 --> 00:03:52.610
And then we write it
like a big bracket, here.

00:03:52.610 --> 00:03:54.870
That's the parentheses.

00:03:54.870 --> 00:03:57.500
And now the state.

00:03:57.500 --> 00:04:11.140
You have n0 lambda n1 plus
lambda squared n2 plus

00:04:11.140 --> 00:04:16.910
lambda to the k, the k-th
correction to the state.

00:04:16.910 --> 00:04:19.880
And it goes on forever.

00:04:19.880 --> 00:04:21.410
And it is here.

00:04:21.410 --> 00:04:25.012
And all that is equal to 0.

00:04:25.012 --> 00:04:30.360
[LAUGH] Looks
daunting, but it's not.

00:04:34.370 --> 00:04:35.730
What should we do?

00:04:35.730 --> 00:04:40.910
Well, here is, again,
lambda helpful for you.

00:04:40.910 --> 00:04:44.250
Lambda is a parameter.

00:04:44.250 --> 00:04:48.090
The left-hand side is
a polynomial on lambda.

00:04:48.090 --> 00:04:51.660
It should vanish for
all values of lambda,

00:04:51.660 --> 00:04:53.460
because the Schrodinger
equation should

00:04:53.460 --> 00:04:55.710
hold for all values of lambda.

00:04:55.710 --> 00:05:00.660
When a polynomial vanishes
for all values of lambda,

00:05:00.660 --> 00:05:04.170
the argument of the polynomial,
all the coefficients

00:05:04.170 --> 00:05:06.330
must vanish, of the polynomial.

00:05:06.330 --> 00:05:13.800
Therefore, we must look at what
is 0-th order in lambda, here,

00:05:13.800 --> 00:05:15.190
and see what we get.

00:05:15.190 --> 00:05:23.490
Well, 0-th order in lambda, we
get this equation, h0 minus En0

00:05:23.490 --> 00:05:30.340
[? on ?] n0 equals 0.

00:05:30.340 --> 00:05:32.790
That's 0-th order in
lambda, and that's

00:05:32.790 --> 00:05:36.495
an equation that is not new.

00:05:36.495 --> 00:05:38.010
[LAUGH] You knew it!

00:05:38.010 --> 00:05:42.300
That's a statement that
n0 was an eigenstate

00:05:42.300 --> 00:05:44.250
of the original Hamiltonian.

00:05:44.250 --> 00:05:45.120
So it's good.

00:05:45.120 --> 00:05:48.030
You know, the 0-th order
things had to work,

00:05:48.030 --> 00:05:53.430
because we said, to 0-th order
you have the known Hamiltonian.

00:05:53.430 --> 00:05:57.045
Let's look at the term
with order lambda.

00:06:02.600 --> 00:06:09.140
Lambda can get from this term
in the Hamiltonian acting on n1.

00:06:09.140 --> 00:06:12.790
That's order lambda,
so let's write it here.

00:06:12.790 --> 00:06:17.190
h0 minus En0 on n1.

00:06:21.260 --> 00:06:25.880
And the other term comes from
a lambda in the first factor

00:06:25.880 --> 00:06:27.470
and no lambda in the second.

00:06:27.470 --> 00:06:32.210
So it's this term, this
acting on that state.

00:06:32.210 --> 00:06:34.760
Look-- there's a lambda,
there's a minus sign,

00:06:34.760 --> 00:06:37.250
so you can put it on
the right-hand side.

00:06:37.250 --> 00:06:45.850
And we get En1 minus
delta h acting on n0.

00:06:51.880 --> 00:06:55.975
Let's be a little daring and
try to get the lambda to the k.

00:07:03.120 --> 00:07:08.600
So h0 minus En0.

00:07:08.600 --> 00:07:11.280
And I want to see what
are the terms that

00:07:11.280 --> 00:07:14.760
have lambda to the k, power k.

00:07:18.050 --> 00:07:23.660
Well, H0 minus En0 acting on
this one has lambda to the k.

00:07:23.660 --> 00:07:27.530
So you have n to
the k, here, nk--

00:07:27.530 --> 00:07:28.610
not "to the k."

00:07:34.110 --> 00:07:41.100
And then, to get lambda to the
k, I could have a lambda here,

00:07:41.100 --> 00:07:46.020
and the term that is before
this, lambda to the k minus 1,

00:07:46.020 --> 00:07:48.180
nk minus 1.

00:07:48.180 --> 00:07:52.930
And it goes with a minus
sign to the right-hand side.

00:07:52.930 --> 00:08:03.060
So you would have En1 minus
delta h on n k minus 1.

00:08:07.730 --> 00:08:21.410
And then you'll have
En2 on Enk minus 2.

00:08:21.410 --> 00:08:24.600
And it will go all
the way until you'll

00:08:24.600 --> 00:08:34.929
have Enk acting on n0,
the original state.

00:08:38.159 --> 00:08:40.873
So let me box this, and, uh--

00:08:46.392 --> 00:08:48.530
those are the
equations that we get.

00:08:54.920 --> 00:08:57.800
And we have to solve them.

00:08:57.800 --> 00:08:59.150
And we can solve them.

00:08:59.150 --> 00:09:02.920
That's the nice
thing about this.

00:09:02.920 --> 00:09:06.250
Well, this one, we argued,
it's simple enough.

00:09:06.250 --> 00:09:08.650
We don't have to
do much about it.

00:09:11.500 --> 00:09:15.100
Then we have to solve for n1.

00:09:15.100 --> 00:09:19.390
Oh, but the second equation
actually has two unknowns.

00:09:19.390 --> 00:09:23.560
We don't know the state
n1, the first correction,

00:09:23.560 --> 00:09:27.510
and we don't know the
energy correction.

00:09:27.510 --> 00:09:31.420
But that's kind of
the useful thing

00:09:31.420 --> 00:09:32.860
that is Schrodinger equation.

00:09:32.860 --> 00:09:35.060
You don't know the
energies, [LAUGH]

00:09:35.060 --> 00:09:36.790
and you don't know
the eigenstate.

00:09:36.790 --> 00:09:39.650
So you couldn't expect this.

00:09:39.650 --> 00:09:41.410
It's kind of interesting.

00:09:41.410 --> 00:09:48.220
If you have solved for n1
and En1 and n2 and En2,

00:09:48.220 --> 00:09:53.320
up to some point, the
next state involves

00:09:53.320 --> 00:09:59.140
nk, the energy of the state
nk, and all the things

00:09:59.140 --> 00:10:03.010
that you already know-- the
lower energies, and the lower

00:10:03.010 --> 00:10:04.120
states.

00:10:04.120 --> 00:10:06.250
So you can solve
this recursively,

00:10:06.250 --> 00:10:08.050
one equation at a time.

00:10:08.050 --> 00:10:10.210
Depending how much
work you want to do,

00:10:10.210 --> 00:10:12.580
you go more and more equations.

00:10:12.580 --> 00:10:17.350
We'll typically go the
first and the second

00:10:17.350 --> 00:10:20.320
and sometimes make some
remarks about these things.

00:10:23.440 --> 00:10:28.040
There's one important
simplifying assumption

00:10:28.040 --> 00:10:33.070
we can make that helps us a lot.

00:10:33.070 --> 00:10:48.240
I can claim you can choose n1
and all the higher ones, n2,

00:10:48.240 --> 00:10:55.890
to be orthogonal to n0.

00:10:55.890 --> 00:10:58.510
Think a little about this.

00:10:58.510 --> 00:11:01.000
What does that tell us?

00:11:01.000 --> 00:11:07.770
It says, oh, this vector should
have no component along n0.

00:11:07.770 --> 00:11:12.990
And these vectors should
have no component along n0.

00:11:12.990 --> 00:11:15.870
The intuitive reason
why this is the case

00:11:15.870 --> 00:11:18.990
that you can choose that
and simplifies your life

00:11:18.990 --> 00:11:23.610
is that, if it had some
component along n0,

00:11:23.610 --> 00:11:27.210
you could just sort
of move it here,

00:11:27.210 --> 00:11:32.130
and now you would have n0 plus
a function of lambda times n0,

00:11:32.130 --> 00:11:36.040
and you can divide
this by this function

00:11:36.040 --> 00:11:41.100
and rescale the state
back, to have an n0 here.

00:11:41.100 --> 00:11:43.830
The normalization
of this state--

00:11:43.830 --> 00:11:46.140
originally, we have
them normalized,

00:11:46.140 --> 00:11:50.820
but it would make our life
extremely more complicated

00:11:50.820 --> 00:11:54.120
if we tried to do this
perturbation series

00:11:54.120 --> 00:11:56.280
and keep the normalization.

00:11:56.280 --> 00:11:58.800
The states are not
going to be normalized,

00:11:58.800 --> 00:12:01.540
but you know that's
not the problem.

00:12:01.540 --> 00:12:05.580
If they are not normalized
but are normalizable,

00:12:05.580 --> 00:12:08.280
you can always work with them.

00:12:08.280 --> 00:12:10.540
So we won't normalize them.

00:12:10.540 --> 00:12:14.970
But the idea is that any piece
that is proportional to and n0,

00:12:14.970 --> 00:12:17.100
you could reabsorb it.

00:12:17.100 --> 00:12:19.450
Now, that's vague.

00:12:19.450 --> 00:12:21.360
If you didn't understand
that argument,

00:12:21.360 --> 00:12:24.930
I commend you, because
it's a vague argument.

00:12:24.930 --> 00:12:28.440
So let me do a more
precise argument.

00:12:28.440 --> 00:12:34.440
Suppose, for example, you're
solving this equation,

00:12:34.440 --> 00:12:37.050
and you solve n0.

00:12:37.050 --> 00:12:40.410
And suppose you've
solved now for n1.

00:12:40.410 --> 00:12:41.940
And you got your n1.

00:12:41.940 --> 00:12:45.390
You're done, you solve
the second equation,

00:12:45.390 --> 00:12:48.150
you're perfectly happy,
but somebody says,

00:12:48.150 --> 00:12:53.070
you know, it has some
component along n0.

00:12:53.070 --> 00:12:55.490
What can you do?

00:12:55.490 --> 00:12:58.020
OK, you say, look--

00:12:58.020 --> 00:13:09.090
if n1 solves this equation,
n1 plus any number c times n0

00:13:09.090 --> 00:13:12.830
still is a solution, I claim.

00:13:12.830 --> 00:13:14.050
Why?

00:13:14.050 --> 00:13:17.950
Because n1, the state n1 that
you're trying to find, only

00:13:17.950 --> 00:13:20.590
appears on the left-hand side.

00:13:20.590 --> 00:13:25.060
And n0 is killed
by this combination

00:13:25.060 --> 00:13:27.340
in the first equation.

00:13:27.340 --> 00:13:32.530
So, if you have a solution n1,
you can replace it by this one.

00:13:32.530 --> 00:13:39.530
And you can choose c to cancel
whatever n0 you had in here.

00:13:39.530 --> 00:13:44.360
So you can always produce a
state that is orthogonal to it.

00:13:44.360 --> 00:13:47.350
And it's easier
to work with that.

00:13:47.350 --> 00:13:49.030
And this goes on forever.

00:13:49.030 --> 00:13:54.100
Suppose you've solved now n1
that has no piece along n0, n2,

00:13:54.100 --> 00:13:54.970
n3, n4--

00:13:54.970 --> 00:13:56.770
all those-- and
you go up to here,

00:13:56.770 --> 00:13:59.380
and nk has a piece along n0.

00:13:59.380 --> 00:14:03.505
You can still add the
constant to nk times n0

00:14:03.505 --> 00:14:04.700
and make it work.

00:14:04.700 --> 00:14:06.860
So you can always do that.

00:14:06.860 --> 00:14:10.820
They're orthogonal to n0.

00:14:10.820 --> 00:14:18.680
So let me say one more
thing that is very amazing.

00:14:18.680 --> 00:14:22.650
Let's look a little a first
look at the equation lambda

00:14:22.650 --> 00:14:27.090
1 of order lambda 1.

00:14:27.090 --> 00:14:32.750
Or-- I'm sorry, these
parentheses are not good.

00:14:32.750 --> 00:14:34.940
This is lambda to the 1.

00:14:34.940 --> 00:14:36.740
This is lambda to the k.

00:14:36.740 --> 00:14:38.000
The parentheses is bad.

00:14:40.580 --> 00:14:41.710
Lambda to the 0.

00:14:45.800 --> 00:14:47.250
Lambda 1.

00:14:47.250 --> 00:14:49.590
So this is our equation.

00:14:49.590 --> 00:15:07.455
We'll have h0 minus En0 n1
equal En1 minus delta h n0.

00:15:11.650 --> 00:15:13.025
I'm going to do one thing.

00:15:16.050 --> 00:15:22.460
I'm going to push a
bra n0 on the left.

00:15:22.460 --> 00:15:24.500
Should I do it on
that same equation?

00:15:24.500 --> 00:15:26.630
Let's save a little time.

00:15:26.630 --> 00:15:29.340
That equation, we
already had it here.

00:15:29.340 --> 00:15:31.430
So let's put an n0 here.

00:15:34.620 --> 00:15:37.810
bra n0 here.

00:15:43.790 --> 00:15:44.860
OK.

00:15:44.860 --> 00:15:47.740
So here is the challenge.

00:15:47.740 --> 00:15:52.090
We've put a lot of
notation on the blackboard.

00:15:52.090 --> 00:15:56.110
And maybe by now all the symbols
are floating in your head

00:15:56.110 --> 00:16:00.040
and not making much sense.

00:16:00.040 --> 00:16:03.580
I want you to figure
out what is the value

00:16:03.580 --> 00:16:06.360
of this left-hand side.