PROFESSOR: All right.
Good morning.
And welcome to 8.06.
Let's begin.
Our subject in 8.06 has to do
with applications of quantum
mechanics and using
quantum mechanics
to understand complex
systems, in fact systems
more complex than the ones
you've understood before.
For example, in
previous courses,
you've understood very well
the simple harmonic oscillator.
You've solved for
the Hamiltonian.
You've found all
the eigenstates.
You've found all the energies.
You know about the spectrum.
You can do time evolution
in the harmonic oscillator.
You've discussed
even peculiar states
like coherence states,
squeeze states.
You know a lot about this
very simple Hamiltonian.
You've also studied the
hydrogen atom Hamiltonian.
And you've found the
spectrum of the hydrogen atom
with all the
degeneracies that it
has and understood
some of those wave
functions and the properties.
And those are exact systems.
But it turns out
that in practice,
while those exact systems form
the foundation of what you
learn, many systems,
and most of the systems
you face in real life and
in research, are systems
that are more complicated.
But at least a large fraction
of them have a saving grace.
They can be thought as
that simple Hamiltonian
that you understand very
well plus an extra term
or an extra effect, some sort
of your total Hamiltonian
being essentially the
simple Hamiltonian
but differs from it
by some amount that
makes it a little different.
So your simple
Hamiltonian therefore
can be the harmonic oscillator.
And in that case, for
example, in general,
you may have a potential
for a particle.
And you know, near the
minimum of the potential,
the potential is roughly
quadratic in general.
But then, as a Taylor
expansion around the minimum,
you find a quadratic
term, no linear term,
because it's a minimum,
a quadratic term.
And then you will find maybe
a cubic or a quartic term.
And the oscillations
are a little unharmonic.
But it's dominated by the
simple harmonic oscillator
but some unharmonicity.
This is studied by people that
look at diatomic molecules.
The vibrations have this effect.
It's experimentally detectable.
You can have the hydrogen atom.
And if you want to
study the hydrogen atom,
how does an experimentalist
look at the hydrogen atom?
He puts the hydrogen atom
and inserts a magnetic field
and sees what happens to the
energy levels and then inserts
an electric field and sees what
happens to the energy levels.
And those can be thought
as slight variations
of the original Hamiltonian.
Van der Waals forces are,
you have two neutral atoms,
and they induce on each
other dipole moments
and generate the force,
a very tiny effect
on otherwise simple
hydrogen atom
structure but a very important
force, the Van der Waals force.
So what we're going
to be doing is
trying to understand these
situations in which we have
a Hamiltonian that is equal to
a well known Hamiltonian, this H
0.
0 meaning no perturbation,
no variation.
This is our well known system.
But then there's going
to be an extra piece
to that Hamiltonian.
And we're going to call
it delta H. Delta H is yet
another Hamiltonian.
It may be complicated,
may be simple.
But it's different from H 0.
Now, this will be the
Hamiltonian of the system
that you're really
trying to describe.
And therefore, you
should demand, of course,
that H 0, delta H, and
H are all Hermitian.
All Hamiltonians are
supposed to be Hermitian.
And this is the situation we
want to understand in general.
This is the concrete
mathematical description
of the problem.
But we do a little more here.
We need a tool to help
us deal with this.
And there's a wonderful nice
tool provided by a parameter.
A parameter here makes
all the difference.
What is this parameter?
It's a parameter we
like to put here.
And we'll call it lambda.
You might have said, no.
I don't have such a thing.
This is what I want to do.
But still, it's better
to put a lambda there,
where lambda is unit-free,
no units, and belongs
to the interval 0 to 1.
In that way, you
will have defined
a family of Hamiltonians
that depend on lambda.
And lambda is this quantity
that you can vary from 0 to 1.
So you decide you're going to
solve a more general problem.
Perhaps you knew what is the
extra term in the Hamiltonian.
And you say, why do
I bother with lambda.
The reason you
bother with lambda
is that it's going to help us
solve the equations clearly.
And second, physically,
it's kind of interesting,
because you could think of
lambda as an extra parameter
of the physics in which you
maybe set it equal to 0,
and you recover the
original Hamiltonian.
Or you vary it, and
when it reaches 1,
it is the Hamiltonian
you're trying to solve.
On the other hand,
this parameter
allows you to do
something very nice too.
One of the things
we're going to try
to clarify by the
end of this lecture
is, shouldn't this
thing be rather small
compared to this one.
If we want to, say, deform
the system slightly,
we won the correction
be small compared
to the original Hamiltonian.
So what does it mean
for a Hamiltonian
to be small compared
to another Hamiltonian?
These are operators.
So what does it mean?
Small.
You could say,
well, I don't know
precisely what it means small.
Maybe means that the matrix
elements of this delta H
are small compared to the
matrix elements of that.
And that is true.
Surprisingly, will
not be enough.
On the other hand,
whatever is small--
we could all agree that
if this is not small,
we could put lambda equals 0.01.
Maybe that's small.
But if that's not small,
lambda equal 10 to the minus 9.
If that's not small,
10 to the minus 30.
At some point, this
will be small enough.
And therefore, we could
try to make sense.
This allows you to really think
of this as a perturbation.
For whatever delta H is, for
a sufficiently small lambda,
this is small.
So this is what we're
going to try to solve.
And let's try to imagine
first what can happen.
So I'm going to try to
imagine what's going on.
A plot, that's the best
way to imagine things.
So I'll do a plot.
Here I put lambda.
And here, in the
vertical axis, I
will indicate the
spectrum of H 0.
So this is going
to be an energy.
So it may happen
that, in our systems,
there's a ground state.
And this ground state is
going to be a single state.
I will not put a name to it.
I will just say there
is one state here.
That means the ground
state is non-degenerate.
Degenerate states are
states of the same energy.
And I say there's just one
state, so not degeneracy.
Suppose you go here.
And now you find two states.
So I put two dots here
to indicate that there
are two states there.
Finally, let's go
higher up and assume
that this Hamiltonian
maybe has one state here,
but here it has four states.
And these are the energies.
Those are some numbers.
And the spectrum must
continue to exist.
So this is a spectrum of your
H 0, the Hamiltonian you know.
Certainly, the hydrogen
atom Hamiltonian
has degenerate states.
So that's roughly
what's happening there.
The simple harmonic
oscillator in one dimension
doesn't have degenerate states.
But the isotopic
harmonic oscillator
in two or three dimensions
does have lots of degeneracies.
You've seen those, probably.
So that's typical.
So what are we
aiming to understand?
We're aiming to
understand what happens
to the energy of
those states or what
happens to the
energy eigenstates
as the perturbation
is turned on.
So imagining lambda going
from 0 to 1, the process
of turning on the perturbation.
And eigenstates are
going to change,
because whatever was
an eigenstate of H 0
is not going to be an eigenstate
of the new Hamiltonian.
And the energies
are going to change.
So everything is
going to change.
But presumably, it will
happen continuously
as you change lambda
continuously from 0 to 1.
So this first state, for
example, may do this.
I don't know what it will do.
But it will vary as
a function of lambda.
The energy will do something.
Maybe we can cut it here and say
that lambda is equal to 1 here.
Now we have two states.
So I can analyze this
state with what's
called non-degenerate
perturbation theory, which
means you have a
non-degenerate state.
And there are
techniques that we're
going to do today to understand
how this state varies.
But how about this one?
Here you have two states.
What happens to them?
Well, two states should
remain two states.
And their energies,
what will they do?
Maybe they'll track each other.
But maybe the perturbation
splits the degeneracy.
That's a very
important phenomenon.
Let's assume it does that.
So it may look like this,
like that, for example.
The perturbation makes
one state have more energy
than the other.
Here is another state.
Now, a phenomenon
that might happen--
many things can happen.
This is a very rich subject
because of all the things
that can happen.
It may happen that this thing,
for example, goes like this
and like that.
But there are four states.
It may be that one
state goes here.
And three states go here.
But then after a little
while, they'll depart.
How many did I want?
No, I got too many.
Well, five.
All right, so it may
happen, something like that,
that they split, and
then to a higher order,
they kept splitting.
In fact, they're splitting here
already, but you don't see it.
It's too close, just the
same way as x squared and x
to the fourth and x to
the eighth, at the origin,
they all look the same.
And then they eventually split.
So this is what we're
going to try to understand.
For this, we need non-degenerate
perturbation theory,
for this, degenerate
perturbation theory, for this,
we need sophisticated
degenerate perturbation theory.
This is a very
intricate phenomenon.
But still, it happens and
happens in many applications.
So we're going to
start with a simpler
one, which is non-degenerate
perturbation theory.
And then we say, well, what does
it mean that we understand H 0?
It means that we have found
all the eigenstates with k 1,
maybe up to infinity.
I don't know.
k is not momentum.
These are the energy
eigenstates of H 0
that we're supposed to know.
And they're all orthonormal.
That can always be done.
When you have a
Hermitian Hamiltonian,
you can find an orthonormal
basis of states.
And being orthonormal,
this orthogonality holds.
The states are
eigenstates of the H 0.
So for this, we'll
call this E k 0.
The energy of this
state, k, for the label
of the state, 0,
because we're not doing
anything in perturbation yet.
We're dealing with the
unperturbed 0th order system.
That defines the energies.
And this energy
satisfies a E 0 0
is less than or equal than E 1
0 less than or equal to E 2 0.
So all the energies are ordered.
I need the equals because
of the degeneracy.
They might be degenerate.
So you have this situation.
So let's consider this state
that is non-degenerate.
And let's assume this
is the state n 0.
It's the nth state.
If n 0 is
non-degenerate, it means
that E n 0 it's really
smaller than the next one.
And it's really bigger, the
energy, than the previous one.
No equal signs there.
It really means those
things are separate.
And that's the meaning
of non-degeneracy.
And now, what are
we trying to solve?
Starting from this
n 0, we're trying
to find out how
the energy changes
and how the state changes.
Both things are important.
So we're going to try to solve.
Therefore, for H
of lambda n lambda.
The state n 0 is going to
change when lambda turns on.
And it's going to
become n of lambda.
And this is going to
have an energy E n
lambda instead of E n 0.
This has an energy
E n 0 with respect
to the original Hamiltonian.
Now it's going to have an energy
E n of lambda n of lambda.
So that's the equation
we want to solve.
This is what this state n 0
becomes as you turn on lambda.
And this is what
the energy E n 0
becomes as you turn on lambda.
So we note that, when
n lambda equal to 0,
is what we call
the state n zero.
And the energy E n at lambda
equal 0 is what we call E n 0.
So with this initial
conditions at lambda equals 0,
we're trying to
solve this system
to see what the state becomes.
And now, here comes
a key assumption,
that the way we're
going to solve this
allows us to write a
perturbative serious expansion
for this object.
So in particular, we'll write
n of lambda is equal to n 0.
That's what n of lambda should
be when lambda is equal to 0.
So then there will be a first
order correction, lambda,
times the state n 1 plus lambda
squared times the state n 2.
And it will go on and on.
Moreover, E n lambda,
when lambda is equal to 0,
you're back to the energy E n 0.
But then there will be a
lambda correction times E n 1--
that's a name for the
first order correction--
plus a lambda squared
correction E n 2.
So this is our
hypothesis that there
is a solution in a perturbative
serious expansion of this kind.
And what are our unknowns?
Our unknowns are this object.
This one we know is
the original state.
This object is unknown.
This is unknown.
This is unknown.
All these things are unknown.
And they go on like that.
Most important, all these
objects don't depend on lambda.
The lambda-dependence is
here, lambda, lambda squared,
lambda on.
And these are things that
don't depend on lambda.
These are objects that
have to be calculated.
They're all lambda-independent.
So we are supposed to solve this
equation under this conditions.
And that's what we're
going to do next.
