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PROFESSOR: Now let me
conclude for a few minutes

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by introducing the idea of how
we're going to perturb things.

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So how are we going to set
up our perturbation theory?

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So for a perturbation theory
we will do the following.

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We will take our system
and introduce again--

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so setting up the
perturbation expansion.

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So we want to give you just an
idea of what we're going to do.

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So H of t is going to be H
of 0 plus delta of H of t.

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And now your lambda is going
to be treated as small.

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So again, we're going to use
a power series expansion,

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and everything is going
to depend on lambda.

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And we're going to think
coefficients in that way.

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We're going to work with
the state psi tilde.

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You remember, if
you know psi tilde,

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you put an e to the minus ih 0t
over H bar and you can get H.

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So we'll set the
perturbation for psi tilde.

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And it will have
a zeroth part that

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is time dependent plus a first
part that is time dependent.

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Every term is going to be time
dependent in the perturbation.

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And what was our
Schroedinger equation?

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Our Schroedinger equation was
ih bar d dt of psi tilde of t

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was delta H, like
that, times psi of t.

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But we need to change
things a little bit.

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We replace delta H
by lambda delta H.

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So now it's going to
have a lambda here.

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So I now need to
plug in this thing,

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which is not going
to be too difficult.

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It's easier than what we did in
time independent perturbation

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theory.

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d dt of psi 0--

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let me not put the time
dependence in the case psi 1

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lambda square psi 2.

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And this is equal to lambda
delta H bar psi tilde again.

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So it's psi 0 plus psi
1 plus those terms.

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So for here we'll just
read the first few terms.

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They're pretty easy.

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There's not much of
a lambda thing there.

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So what do we get?

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Terms without lambda.

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ih bar d dt of psi
tilde 0 of t equals 0.

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This is lambda to the 0.

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That's the only term
without the lambda,

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the one that arises here.

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Terms without lambda already.

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Well, there's one
term here, which

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is Schroedinger-like.
d dt of psi 1

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is, in fact, equal
to order of lambda.

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You have delta H
tilde psi 0 of t.

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And the next one, ih
bar d dt of psi 2 of t--

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that's lambda squared-- comes
from the derivative acting

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here.

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We have to look for
lambda squared here.

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And I forgot this lambda.

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And therefore, this time
you get delta H psi 1 of t.

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In general, for lambda n, you
will get ih bar d dt of psi n--

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I'll actually put n
plus 1, n plus 1 here--

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is given by delta H acting
on the previous one.

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So this will be simple.

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Once you know psi 0, which is
a constant, you put it here.

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This will be easily
solved as an integral.

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Once you have psi
1, you put it here.

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You easily solve psi 2 and
start solving one after another.

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The fun thing is that you
can write these equations

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explicitly.

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And just even the first
order result, and sometimes

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the second, give you all
the physics you want,

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which we will explore in
the next few lectures.