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PROFESSOR: All right.

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Good morning.

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And welcome to 8.06.

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Let's begin.

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Our subject in 8.06 has to do
with applications of quantum

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mechanics and using
quantum mechanics

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to understand complex
systems, in fact systems

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more complex than the ones
you've understood before.

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For example, in
previous courses,

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you've understood very well
the simple harmonic oscillator.

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You've solved for
the Hamiltonian.

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You've found all
the eigenstates.

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You've found all the energies.

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You know about the spectrum.

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You can do time evolution
in the harmonic oscillator.

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You've discussed
even peculiar states

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like coherence states,
squeeze states.

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You know a lot about this
very simple Hamiltonian.

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You've also studied the
hydrogen atom Hamiltonian.

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And you've found the
spectrum of the hydrogen atom

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with all the
degeneracies that it

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has and understood
some of those wave

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functions and the properties.

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And those are exact systems.

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But it turns out
that in practice,

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while those exact systems form
the foundation of what you

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learn, many systems,
and most of the systems

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you face in real life and
in research, are systems

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that are more complicated.

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But at least a large fraction
of them have a saving grace.

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They can be thought as
that simple Hamiltonian

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that you understand very
well plus an extra term

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or an extra effect, some sort
of your total Hamiltonian

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being essentially the
simple Hamiltonian

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but differs from it
by some amount that

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makes it a little different.

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So your simple
Hamiltonian therefore

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can be the harmonic oscillator.

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And in that case, for
example, in general,

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you may have a potential
for a particle.

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And you know, near the
minimum of the potential,

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the potential is roughly
quadratic in general.

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But then, as a Taylor
expansion around the minimum,

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you find a quadratic
term, no linear term,

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because it's a minimum,
a quadratic term.

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And then you will find maybe
a cubic or a quartic term.

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And the oscillations
are a little unharmonic.

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But it's dominated by the
simple harmonic oscillator

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but some unharmonicity.

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This is studied by people that
look at diatomic molecules.

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The vibrations have this effect.

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It's experimentally detectable.

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You can have the hydrogen atom.

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And if you want to
study the hydrogen atom,

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how does an experimentalist
look at the hydrogen atom?

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He puts the hydrogen atom
and inserts a magnetic field

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and sees what happens to the
energy levels and then inserts

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an electric field and sees what
happens to the energy levels.

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And those can be thought
as slight variations

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of the original Hamiltonian.

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Van der Waals forces are,
you have two neutral atoms,

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and they induce on each
other dipole moments

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and generate the force,
a very tiny effect

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on otherwise simple
hydrogen atom

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structure but a very important
force, the Van der Waals force.

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So what we're going
to be doing is

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trying to understand these
situations in which we have

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a Hamiltonian that is equal to
a well known Hamiltonian, this H

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0.

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0 meaning no perturbation,
no variation.

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This is our well known system.

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But then there's going
to be an extra piece

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to that Hamiltonian.

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And we're going to call
it delta H. Delta H is yet

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another Hamiltonian.

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It may be complicated,
may be simple.

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But it's different from H 0.

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Now, this will be the
Hamiltonian of the system

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that you're really
trying to describe.

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And therefore, you
should demand, of course,

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that H 0, delta H, and
H are all Hermitian.

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All Hamiltonians are
supposed to be Hermitian.

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And this is the situation we
want to understand in general.

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This is the concrete
mathematical description

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of the problem.

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But we do a little more here.

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We need a tool to help
us deal with this.

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And there's a wonderful nice
tool provided by a parameter.

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A parameter here makes
all the difference.

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What is this parameter?

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It's a parameter we
like to put here.

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And we'll call it lambda.

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You might have said, no.

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I don't have such a thing.

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This is what I want to do.

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But still, it's better
to put a lambda there,

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where lambda is unit-free,
no units, and belongs

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to the interval 0 to 1.

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In that way, you
will have defined

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a family of Hamiltonians
that depend on lambda.

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And lambda is this quantity
that you can vary from 0 to 1.

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So you decide you're going to
solve a more general problem.

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Perhaps you knew what is the
extra term in the Hamiltonian.

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And you say, why do
I bother with lambda.

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The reason you
bother with lambda

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is that it's going to help us
solve the equations clearly.

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And second, physically,
it's kind of interesting,

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because you could think of
lambda as an extra parameter

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of the physics in which you
maybe set it equal to 0,

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and you recover the
original Hamiltonian.

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Or you vary it, and
when it reaches 1,

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it is the Hamiltonian
you're trying to solve.

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On the other hand,
this parameter

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allows you to do
something very nice too.

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One of the things
we're going to try

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to clarify by the
end of this lecture

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is, shouldn't this
thing be rather small

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compared to this one.

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If we want to, say, deform
the system slightly,

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we won the correction
be small compared

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to the original Hamiltonian.

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So what does it mean
for a Hamiltonian

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to be small compared
to another Hamiltonian?

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These are operators.

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So what does it mean?

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Small.

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You could say,
well, I don't know

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precisely what it means small.

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Maybe means that the matrix
elements of this delta H

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are small compared to the
matrix elements of that.

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And that is true.

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Surprisingly, will
not be enough.

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On the other hand,
whatever is small--

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we could all agree that
if this is not small,

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we could put lambda equals 0.01.

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Maybe that's small.

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But if that's not small,
lambda equal 10 to the minus 9.

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If that's not small,
10 to the minus 30.

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At some point, this
will be small enough.

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And therefore, we could
try to make sense.

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This allows you to really think
of this as a perturbation.

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For whatever delta H is, for
a sufficiently small lambda,

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this is small.

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So this is what we're
going to try to solve.

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And let's try to imagine
first what can happen.

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So I'm going to try to
imagine what's going on.

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A plot, that's the best
way to imagine things.

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So I'll do a plot.

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Here I put lambda.

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And here, in the
vertical axis, I

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will indicate the
spectrum of H 0.

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So this is going
to be an energy.

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So it may happen
that, in our systems,

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there's a ground state.

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And this ground state is
going to be a single state.

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I will not put a name to it.

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I will just say there
is one state here.

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That means the ground
state is non-degenerate.

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Degenerate states are
states of the same energy.

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And I say there's just one
state, so not degeneracy.

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Suppose you go here.

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And now you find two states.

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So I put two dots here
to indicate that there

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are two states there.

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Finally, let's go
higher up and assume

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that this Hamiltonian
maybe has one state here,

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but here it has four states.

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And these are the energies.

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Those are some numbers.

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And the spectrum must
continue to exist.

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So this is a spectrum of your
H 0, the Hamiltonian you know.

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Certainly, the hydrogen
atom Hamiltonian

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has degenerate states.

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So that's roughly
what's happening there.

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The simple harmonic
oscillator in one dimension

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doesn't have degenerate states.

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But the isotopic
harmonic oscillator

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in two or three dimensions
does have lots of degeneracies.

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You've seen those, probably.

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So that's typical.

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So what are we
aiming to understand?

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We're aiming to
understand what happens

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to the energy of
those states or what

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happens to the
energy eigenstates

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as the perturbation
is turned on.

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So imagining lambda going
from 0 to 1, the process

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of turning on the perturbation.

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And eigenstates are
going to change,

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because whatever was
an eigenstate of H 0

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is not going to be an eigenstate
of the new Hamiltonian.

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And the energies
are going to change.

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So everything is
going to change.

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But presumably, it will
happen continuously

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as you change lambda
continuously from 0 to 1.

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So this first state, for
example, may do this.

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I don't know what it will do.

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But it will vary as
a function of lambda.

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The energy will do something.

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Maybe we can cut it here and say
that lambda is equal to 1 here.

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Now we have two states.

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So I can analyze this
state with what's

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called non-degenerate
perturbation theory, which

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means you have a
non-degenerate state.

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And there are
techniques that we're

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going to do today to understand
how this state varies.

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But how about this one?

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Here you have two states.

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What happens to them?

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Well, two states should
remain two states.

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And their energies,
what will they do?

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Maybe they'll track each other.

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But maybe the perturbation
splits the degeneracy.

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That's a very
important phenomenon.

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Let's assume it does that.

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So it may look like this,
like that, for example.

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The perturbation makes
one state have more energy

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than the other.

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Here is another state.

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Now, a phenomenon
that might happen--

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many things can happen.

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This is a very rich subject
because of all the things

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that can happen.

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It may happen that this thing,
for example, goes like this

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and like that.

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But there are four states.

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It may be that one
state goes here.

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And three states go here.

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But then after a little
while, they'll depart.

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How many did I want?

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No, I got too many.

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Well, five.

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All right, so it may
happen, something like that,

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that they split, and
then to a higher order,

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they kept splitting.

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In fact, they're splitting here
already, but you don't see it.

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It's too close, just the
same way as x squared and x

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to the fourth and x to
the eighth, at the origin,

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they all look the same.

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And then they eventually split.

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So this is what we're
going to try to understand.

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For this, we need non-degenerate
perturbation theory,

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for this, degenerate
perturbation theory, for this,

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we need sophisticated
degenerate perturbation theory.

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This is a very
intricate phenomenon.

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But still, it happens and
happens in many applications.

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So we're going to
start with a simpler

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one, which is non-degenerate
perturbation theory.

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And then we say, well, what does
it mean that we understand H 0?

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It means that we have found
all the eigenstates with k 1,

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maybe up to infinity.

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I don't know.

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k is not momentum.

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These are the energy
eigenstates of H 0

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that we're supposed to know.

255
00:15:58,370 --> 00:16:01,370
And they're all orthonormal.

256
00:16:01,370 --> 00:16:03,540
That can always be done.

257
00:16:03,540 --> 00:16:06,110
When you have a
Hermitian Hamiltonian,

258
00:16:06,110 --> 00:16:09,350
you can find an orthonormal
basis of states.

259
00:16:09,350 --> 00:16:20,090
And being orthonormal,
this orthogonality holds.

260
00:16:20,090 --> 00:16:25,040
The states are
eigenstates of the H 0.

261
00:16:25,040 --> 00:16:34,040
So for this, we'll
call this E k 0.

262
00:16:34,040 --> 00:16:37,340
The energy of this
state, k, for the label

263
00:16:37,340 --> 00:16:41,630
of the state, 0,
because we're not doing

264
00:16:41,630 --> 00:16:44,390
anything in perturbation yet.

265
00:16:44,390 --> 00:16:48,860
We're dealing with the
unperturbed 0th order system.

266
00:16:54,160 --> 00:16:56,150
That defines the energies.

267
00:16:56,150 --> 00:17:01,900
And this energy
satisfies a E 0 0

268
00:17:01,900 --> 00:17:13,150
is less than or equal than E 1
0 less than or equal to E 2 0.

269
00:17:13,150 --> 00:17:15,244
So all the energies are ordered.

270
00:17:17,800 --> 00:17:22,660
I need the equals because
of the degeneracy.

271
00:17:22,660 --> 00:17:24,310
They might be degenerate.

272
00:17:24,310 --> 00:17:28,600
So you have this situation.

273
00:17:28,600 --> 00:17:40,000
So let's consider this state
that is non-degenerate.

274
00:17:40,000 --> 00:17:45,940
And let's assume this
is the state n 0.

275
00:17:45,940 --> 00:17:46,960
It's the nth state.

276
00:17:49,640 --> 00:17:58,870
If n 0 is
non-degenerate, it means

277
00:17:58,870 --> 00:18:07,700
that E n 0 it's really
smaller than the next one.

278
00:18:12,390 --> 00:18:17,250
And it's really bigger, the
energy, than the previous one.

279
00:18:20,150 --> 00:18:22,220
No equal signs there.

280
00:18:22,220 --> 00:18:27,170
It really means those
things are separate.

281
00:18:27,170 --> 00:18:30,710
And that's the meaning
of non-degeneracy.

282
00:18:34,560 --> 00:18:37,680
And now, what are
we trying to solve?

283
00:18:41,430 --> 00:18:46,590
Starting from this
n 0, we're trying

284
00:18:46,590 --> 00:18:49,620
to find out how
the energy changes

285
00:18:49,620 --> 00:18:51,780
and how the state changes.

286
00:18:51,780 --> 00:18:53,950
Both things are important.

287
00:18:53,950 --> 00:18:58,020
So we're going to try to solve.

288
00:18:58,020 --> 00:19:04,080
Therefore, for H
of lambda n lambda.

289
00:19:04,080 --> 00:19:08,980
The state n 0 is going to
change when lambda turns on.

290
00:19:08,980 --> 00:19:11,570
And it's going to
become n of lambda.

291
00:19:11,570 --> 00:19:15,880
And this is going to
have an energy E n

292
00:19:15,880 --> 00:19:22,920
lambda instead of E n 0.

293
00:19:22,920 --> 00:19:28,530
This has an energy
E n 0 with respect

294
00:19:28,530 --> 00:19:30,480
to the original Hamiltonian.

295
00:19:30,480 --> 00:19:36,090
Now it's going to have an energy
E n of lambda n of lambda.

296
00:19:36,090 --> 00:19:41,040
So that's the equation
we want to solve.

297
00:19:41,040 --> 00:19:48,160
This is what this state n 0
becomes as you turn on lambda.

298
00:19:48,160 --> 00:19:51,630
And this is what
the energy E n 0

299
00:19:51,630 --> 00:19:53,770
becomes as you turn on lambda.

300
00:19:53,770 --> 00:20:00,840
So we note that, when
n lambda equal to 0,

301
00:20:00,840 --> 00:20:05,010
is what we call
the state n zero.

302
00:20:05,010 --> 00:20:16,790
And the energy E n at lambda
equal 0 is what we call E n 0.

303
00:20:16,790 --> 00:20:21,410
So with this initial
conditions at lambda equals 0,

304
00:20:21,410 --> 00:20:23,600
we're trying to
solve this system

305
00:20:23,600 --> 00:20:27,660
to see what the state becomes.

306
00:20:27,660 --> 00:20:32,280
And now, here comes
a key assumption,

307
00:20:32,280 --> 00:20:34,470
that the way we're
going to solve this

308
00:20:34,470 --> 00:20:39,870
allows us to write a
perturbative serious expansion

309
00:20:39,870 --> 00:20:41,890
for this object.

310
00:20:41,890 --> 00:20:49,890
So in particular, we'll write
n of lambda is equal to n 0.

311
00:20:49,890 --> 00:20:54,390
That's what n of lambda should
be when lambda is equal to 0.

312
00:20:54,390 --> 00:20:58,680
So then there will be a first
order correction, lambda,

313
00:20:58,680 --> 00:21:06,990
times the state n 1 plus lambda
squared times the state n 2.

314
00:21:06,990 --> 00:21:10,890
And it will go on and on.

315
00:21:10,890 --> 00:21:18,450
Moreover, E n lambda,
when lambda is equal to 0,

316
00:21:18,450 --> 00:21:21,870
you're back to the energy E n 0.

317
00:21:25,610 --> 00:21:35,450
But then there will be a
lambda correction times E n 1--

318
00:21:35,450 --> 00:21:39,200
that's a name for the
first order correction--

319
00:21:39,200 --> 00:21:43,940
plus a lambda squared
correction E n 2.

320
00:21:49,270 --> 00:21:53,710
So this is our
hypothesis that there

321
00:21:53,710 --> 00:22:01,170
is a solution in a perturbative
serious expansion of this kind.

322
00:22:01,170 --> 00:22:03,060
And what are our unknowns?

323
00:22:03,060 --> 00:22:06,190
Our unknowns are this object.

324
00:22:06,190 --> 00:22:08,790
This one we know is
the original state.

325
00:22:08,790 --> 00:22:11,050
This object is unknown.

326
00:22:11,050 --> 00:22:13,300
This is unknown.

327
00:22:13,300 --> 00:22:15,510
This is unknown.

328
00:22:15,510 --> 00:22:17,310
All these things are unknown.

329
00:22:17,310 --> 00:22:19,560
And they go on like that.

330
00:22:19,560 --> 00:22:25,935
Most important, all these
objects don't depend on lambda.

331
00:22:28,590 --> 00:22:33,120
The lambda-dependence is
here, lambda, lambda squared,

332
00:22:33,120 --> 00:22:33,990
lambda on.

333
00:22:33,990 --> 00:22:37,320
And these are things that
don't depend on lambda.

334
00:22:37,320 --> 00:22:42,530
These are objects that
have to be calculated.

335
00:22:42,530 --> 00:22:46,070
They're all lambda-independent.

336
00:22:46,070 --> 00:22:52,190
So we are supposed to solve this
equation under this conditions.

337
00:22:52,190 --> 00:22:55,900
And that's what we're
going to do next.