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PROFESSOR: Good morning.

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We're going to continue doing
perturbation theory today.

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We've finished doing
non-degenerate perturbation

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theory, and then turned to the
degenerate perturbation theory

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as well.

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We had computed already
the first order corrections

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to the state, first order
corrections to the energy,

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and second order
correction to the energy.

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So here in this formulas
that I've written

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are all the results
we have so far.

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And a useful thing
is to ask yourself

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whether you know what
every symbol here means.

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This is the state who
change the deformation

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of the original state n0
of the Hamiltonian H0.

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We have the Hamiltonian
H of lambda,

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which was equal to H0
plus lambda delta H.

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And n0 was an eigenstate
of H0 with energy En0.

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And we found that as
you turn on lambda,

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lambda becomes non-zero.

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The energies change.

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The eigenstate changes.

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And here is lambda, first
order in lambda, second order

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in lambda.

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We didn't compute the state
to second order in lambda.

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Moreover, we have
the symbol delta Hkn

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that was defined to
be the matrix element

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on the unperturbed eigenstate
of the operator delta H. Well,

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that's another symbol in here.

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We are summing over all
states in the spectrum,

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except for the state n here
because the denominator here

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would give you a 0.

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And this would be
problematic in general.

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So that was what we derived.

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And these are the
formulas we have.

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It's worth making a
few remarks as to what

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we see from these
formulas because they're

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a little complicated.

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And you may not have
immediate intuition about it.

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So a few remarks.

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And the first one--

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in fact, both remarks
that I'm going

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to make concern the behavior
of the energy as represented

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by this formula.

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First remark is that if you
were to look at the ground state

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energy of the system, the
first two terms here, so for n

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equals 0, if we call the
ground state by the label n

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equals 0 for this
state n equals 0,

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these first two terms on
the energy overestimate

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the ground state energy.

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They always give you more than
what the true ground state

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energy is for any
value of lambda.

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That is kind of plausible,
given that you see here

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that for the ground state
energy, when n is equal to 0,

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and it's the lowest
energy state,

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these differences
are all positive.

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All these states
have more energy.

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Therefore, this
term is negative.

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And the next correction
to order lambda square

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tends to lower the energy
of the ground state.

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So the claim is that the
first v lambda ground state

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energy overestimate the
true ground state energy.

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And so how do we see that?

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So we consider E0, because
we're doing n equals 0.

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This is the ground state.

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We start labeling with
n equals 0 sometimes.

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Plus lambda H00.

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That is the order lambda
estimate for the ground state

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energy.

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The first term can be
viewed as the expectation

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value of the original
Hamiltonian on the ground

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state.

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That is E0 because
this H0 and 0 is E00.

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So this is E00.

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Plus lambda.

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I'm sorry here.

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I made a little mistake.

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That should be
delta H. Delta H00

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plus lambda, the
expectation value of delta H

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on the same state.

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So these two things
together are nothing

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but the expectation value on the
ground state of H0 plus lambda

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delta H.

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That's kind of nice.

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Isn't it?

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That these two terms
really are nothing else

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but the expectation value on
the ground state of this thing

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that we called H of lambda.

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And now comes the
variational principal

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that says that if you evaluate
the Hamiltonian on an arbitrary

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state, the expectation value
of the Hamilton in an arbitrary

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state, you get more energy
than the ground state

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energy of the system.

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You always get more.

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And when you hit
the ground state,

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you get the lowest value.

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So look at that.

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This is the Hamiltonian.

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And this is the
unperturbed ground state.

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This is not the
real ground state.

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The real ground state is
what we're trying to find.

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So this is like
saying, OK, you're

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evaluating the exact Hamiltonian
on some arbitrary state.

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Therefore, this is greater than
or equal to the true ground

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state energy that we
would call E0 lambda.

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True ground state energy.

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Which in that notation,
indeed, is E0 lambda.

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So that's a nice result.
And it matches with the idea

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that for the ground state
the order lambda squared

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correction is minus lambda
squared, the sum over k

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different from 0 delta Hk0
squared over Ek minus E00.

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And this thing, as we mentioned,
the numerator is positive.

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The denominator is
positive because this

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was the ground state.

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And this is a
non-degenerate ground state.

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And K, therefore, has
more energy here, Ek0.

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So everything is positive.

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So this is negative.

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So actually here, there
is a generalization

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of this observation.

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And you can imagine that you
have now a particular state,

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n here.

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Then you have all these
states that are above

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and all these states
that are below.

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So let's look at the
second order correction

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to the energy of the state En,
the second order correction.

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The first order
correction shifts.

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The energy is
proportional to delta Hmn.

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But the second
order correction is

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minus lambda squared, the
sum over k different from n.

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So let's split it.

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k greater than n.

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And we'll write the same thing
Hkn squared over Ek0 minus En0.

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And then I want to write the
other states, the states where

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k is less than n.

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But let me do one thing here.

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On those states, I'll
switch the order of the sum.

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I'll put En0, the order of the
sign in the denominator, Ek0,

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and change the sign here.

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So these two changes
of sign are correlated.

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I changed the sign
in the denominator.

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And I changed the sign in front.

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And now, we see the following.

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All these states that are
above n, k is greater than n.

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This difference is positive.

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This is manifestly positive.

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This is negative.

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So these states are kind
of pushing your state down.

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The fact of this states on
top is a negative correction

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are pushing the states down.

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On the other hand,
the lower states,

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the states with k less
than n, again, I've

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ordered now convenient.

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The other states k
now have less energy.

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So this is positive.

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This is positive.

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This is positive.

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So these states over here are
actually pushing that state up.

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So the upper states
are pushing it down.

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The lower states
are pushing it up.

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This is usually referred
to as level repulsion.

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The levels repel each other.

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The upper states don't want the
state n to go close to them.

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The lower states don't want
them to go close to them either.

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So it's a nice dynamic
that helps you understand

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what's going on here.

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Another point we
want to make has

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to do with the validity
of this expansion.

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So in general, when you
have series expansions

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the issues of
convergence are delicate.

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So we can get a lot of insight
just by doing an example.

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So let me talk about the
validity of the perturbation

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series.

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This is supposed to
give us some insight.

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One thing we've said is
that we think delta H is

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supposed to be smaller than h0.

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Now, these are operators.

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So the statement
that they're smaller

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has to be made more precise.

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What is the size of an operator?

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And you could think,
well, we could

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say the entries of
the matrix elements

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should be much smaller.

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And that is true,
but it's not enough.

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For that, let's
consider an example.

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So this will be a two by two
matrix Hamiltonian, in which

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H of lambda will be given
by some H0 plus lambda v

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hat, which will be E1
0 E2 0 plus lambda.

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That v hat matrix will
be of the form v v star.

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And must be her mission.

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So v is a number.

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v star is the complex conjugate.

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These are the two energies,
because the Hamiltonian

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is known.

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Each time we talk about the
Hamiltonian that is known,

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we write that there's
a diagonal matrix.

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We're saying, yes, that
is the matrix element

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of the Hamiltonian in the basis
of eigenstates, which we know.

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So a known Hamiltonian you can
represent by a diagonal matrix.

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So here is our whole
matrix H of lambda.

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And you could say, all right,
our perturbation theory,

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practically speaking,
is these formulas that

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allow us to compute the
eigenvalues of this matrices,

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which are of this
matrix, which is

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the energies of the
system of the eigenstate

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and compute the eigenstates.

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For H0, these are
the eigenvalues.

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And the eigenstates
are 1 0 and 0 1.

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So perturbation
theory is really,

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you can think in
terms of Hamiltonians,

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but also you can think of
it in terms of matrices.

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It's allowing you to find the
eigenvalues and eigenvectors

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of this matrices.

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So this is a two by two
matrix with numbers.

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You know how to compute
the eigenvalues.

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00:16:06,360 --> 00:16:10,930
So I'll give you the answer.

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The eigenvalues E plus
minus are E1 plus E2/2

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plus minus E1 minus E2/2
times the square root of 1

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plus lambda squared absolute
value of V2 squared E1

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minus E2/2 squared.

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Long formula.

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That's it.

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Those are the exact things.

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If you wanted to see this as a
result in perturbation theory,

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you would say, well,
lambda squared v squared

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or lambda v is small.

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And I'm going to think
of this term as small.

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And I'm going to
expand the square root.

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And if I expand
this square root,

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I'm going to get
all kinds of terms

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with different powers of lambda.

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You can also see there that
there's no energy correction

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linear in lambda.

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Because when you expand 1
plus epsilon square root,

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it's roughly 1 plus epsilon
over 2 plus dot, dot.

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And therefore, when you
span the square root,

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the first correction
is going to be

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proportional to lambda squared.

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And that actually
conforms to that

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because the matrix elements
of the Hamiltonian delta

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Hnn along the diagonals
for the perturbation are 0.

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So there is no order
lambda correction.

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00:18:12,460 --> 00:18:15,580
So what do you do here then?

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You must do this expansion.

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And here, I write
the relevant series.

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f of z, if you define
it, as 1 plus z squared

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00:18:27,400 --> 00:18:34,660
is equal 1 plus z squared over
2 minus z 4th over 8 plus z6/6.

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And it's not that
simple after that.

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00:18:39,370 --> 00:18:47,080
Minus 5/128z to the 8
plus order z to the 10.

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00:18:47,080 --> 00:18:50,770
So that's this perturbative
series expansion

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00:18:50,770 --> 00:18:53,770
that you would use here.

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And how good is the
convergence of this series?

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Well, it's OK.

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It's something that when
you study complex analysis

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00:19:07,690 --> 00:19:13,090
you see this function 1 plus
square root of z squared

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has branch cuts at i and at
minus i in the complex plane,

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in the z complex plane.

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Those are the branch cuts.

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These are the places where
this square root becomes 0.

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And you have to deal with them.

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And the function is
convergent only up to here.

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There is a radius
of convergence.

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A radius of convergence is 1.

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You can try it with
Mathematica or with the program

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00:19:51,140 --> 00:19:54,710
and take 30 terms, 40 terms.

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And you will see that as long
as you take a point z here,

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it converges.

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00:20:00,380 --> 00:20:03,920
You take a point a little bit
out, it blows up the series.

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00:20:03,920 --> 00:20:06,650
So it has a radius
of convergence.

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00:20:06,650 --> 00:20:08,900
That's not so bad.

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00:20:08,900 --> 00:20:11,330
Radius of convergence is OK.

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00:20:11,330 --> 00:20:18,020
But basically, we need z to be
small, which corresponds here

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to the statement that lambda v
z small for a fast convergence

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00:20:27,320 --> 00:20:32,150
corresponds to lambda
v absolute value

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00:20:32,150 --> 00:20:43,840
being smaller than E1 0 minus
E2 0 absolute value over 2.

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Roughly smaller.

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00:20:47,040 --> 00:20:54,910
And now, we see that the thing
that matters for a perturbation

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00:20:54,910 --> 00:21:01,200
series to be good is also
that the perturbation be

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00:21:01,200 --> 00:21:05,720
small compared with
the energy differences.

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00:21:05,720 --> 00:21:08,700
Not just it should be
small, the perturbation.

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00:21:08,700 --> 00:21:11,350
It should be small compared
with the energy difference.

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00:21:11,350 --> 00:21:15,480
So if you have a Hamiltonian
with a state of energy 100

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00:21:15,480 --> 00:21:20,150
and a state of energy
101, you might say, well,

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00:21:20,150 --> 00:21:22,510
if I take a
perturbation of size two

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00:21:22,510 --> 00:21:25,550
or three, that's very small
compared to the energies.

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00:21:25,550 --> 00:21:29,650
But it's not been small compared
to the difference of energies.

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00:21:29,650 --> 00:21:32,920
And that can cause the
perturbation expansion

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00:21:32,920 --> 00:21:34,280
to go wrong.

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00:21:34,280 --> 00:21:38,320
So this gives you extra
insight that, in fact,

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00:21:38,320 --> 00:21:42,160
being a small perturbation
not only means small compared

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00:21:42,160 --> 00:21:45,040
to the energies, but
also small compared

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00:21:45,040 --> 00:21:47,020
with the energy differences.

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Something that you
see here as well.

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Energy differences are
controlling things.

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00:21:52,480 --> 00:21:57,750
And if the energy
differences are--

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00:21:57,750 --> 00:22:02,110
if the perturbation is not small
compared to energy differences,

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00:22:02,110 --> 00:22:06,520
then these ratios
can be rather large.

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00:22:06,520 --> 00:22:09,610
And the perturbation
terms are very large.

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00:22:09,610 --> 00:22:14,300
And nothing is very simple.

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00:22:14,300 --> 00:22:20,380
So that's what I wanted to
say about the convergence

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00:22:20,380 --> 00:22:22,270
of a perturbation expansion.

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00:22:22,270 --> 00:22:24,310
More rigorous
statements can be made,

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00:22:24,310 --> 00:22:27,540
but we're not going to
try to make them here.