Tip:
Highlight text to annotate it
X
[MUSIC PLAYING]
Here we're going to introduce one of the most widely used
approximation theories in quantum mechanics--
perturbation theory.
Perturbation theory is useful to us, because there are many
situations in quantum mechanics where important and
useful phenomenon occur, even when we're making relatively
small changes.
As I said one scientific example is the effect of
electric fields on atoms.
The fields that we can practically apply are very
small compared to the fields inside the atoms themselves.
So our applied perturbation is small.
Despite that, we can quite easily see the consequences.
For example, in the splitting of spectral lines, the
so-called Stark splitting.
Perturbation theory is, at its core, a theory of successive
approximations.
We essentially can calculate the first small changes we
make, based on the properties of the unperturbed system.
This kind of approach is also common in classical systems.
Perturbation theory is by no means
restricted to quantum mechanics.
Indeed, most situations in classical physics where we
pretend the problem is approximately linear for small
displacements have much in common mathematically with
perturbation theories.
If we want to calculate how easy it would be to stretch a
spring by small amount, we do that calculation based on the
shape and properties of the spring before we stretch it.
That is a situation where we can calculate the change we
get based on the mechanical properties of
the unperturbed system.
This would be a first order calculation.
Of course, if we pull the spring by some amount, we may
in fact change how difficult it is to make further changes,
just like trying to stretch elastic band too far.
So when we start to stretch the elastic band, it's
relatively easy to stretch it.
And the amount it stretches is probably roughly proportional
to how hard we pull.
But if we pull too much, the band gets very
much harder to stretch.
So we have to take account of that in our theory, that as we
stretch it it gets harder to stretch it.
And that is not something we can do in a simple first order
linear theory.
And we might have to think about quadratic or second
order terms in our theory.
We have to use a theory of successive approximations.
We have to put in the fact that by stretching the elastic
band, we've somewhat changed its properties.
And we have to put that in to calculating what would happen
if we try to stretch it some more.
Another good example of a spring like system is a
loudspeaker cone.
So a first approximation, they amount that the cone moves in
an out is proportional to how hard you drive it by passing
electrical current through the magnetic coil.
But the movement of the loudspeaker cone is not quite
linearly proportional to that drive.
The plot of how much a loudspeaker cone moves in and
out in response to the current we feed into it might not be
exactly a straight line.
Most likely at high drives, for example if I push in too
far here, the loudspeaker cone can't go any further.
And that line, therefore, of the output versus input is
something that would have to bend over.
And we might have to describe that line not a straight line,
but there's some higher order polynomial.
And these particular saturations of the behavior
would introduce, for example, cubic components, representing
the fact that there's only so far you could
go in either direction.
Such higher order behaviors can give rise to quite
specific effects, such as harmonic distortion in load
speakers, generating new frequencies in the sound that
were not there in the driving current.
And they can be calculated by theories that have the same
basic mathematical flavor as perturbation theories.
Perturbation theories are ways of formalizing these ideas of
successive approximations, or polynomial behavior, of the
response to some drive.
And we'll now start to set up a perturbation theory for
quantum mechanics.
In quantum mechanics, it's particularly common to want to
know the change in the eigenenergies as we apply some
perturbation or a small change to a
quantum mechanical system.
So we will discuss our perturbation theory for energy
eigenvalues and eigenfunctions.
This means that we will be considering the Hamiltonian of
the system, and imagining we're going to be changing the
system slightly by slightly changing the Hamiltonian.
Or, as we will be describing it, by adding a perturbing
Hamiltonian.
For the moment, we will be doing this for problems that
are not changing in time.
Hence, this theory will be called time independent, or
sometimes stationary perturbation.
We will only be considering here the case where our energy
states are non-degenerate.
That is, there's only one eigenfunction for one
eigenvalue.
We can construct perturbation theories for degenerate cases.
And these are actually quite important.
But for the moment here, we're going to avoid them.
Because they turn out to be rather different in character.
Now at this point, I should warn you that the first time
most students go through perturbation theory they find
it very confusing.
There are a lot of subscripts and superscripts involved.
So the algebra is necessarily somewhat complicated.
Also, for many modern students, it's the first time
they've looked at a theory like this, one based on what
are, essentially, [INAUDIBLE]
approximations to quite complicated problems.
Before modern computers become available, workers in fields
such as classical and even quantum mechanics were very
used to making such analytic approximations.
Because fully numerical calculations
were hard to perform.
But you might wonder again why, therefore, we are
bothering to look at such an analytic approximation theory
now that we have very sophisticated
abilities to calculate.
And why, therefore, we're taking this time to go through
what can initially be very confusing subject.
Again, perhaps the most important reason here is for
conceptual understanding of
processes in quantum mechanics.
It's also true that in many cases, these approaches enable
us to come up with quite simple and generally useful
formulas, which will save us much time and calculations.
So please bear with me as we construct what initially can
look like quite formidable and forbidding formulas.
So formally then, we're going to construct time independent
perturbation theory.
We start out by presuming we have some so-called
unperturbed Hamiltonian, H0 And that has some presumably
known and normalized eigensolutions.
So here's H0 operating on its eigensolution, some particular
psi n with an eigenenergy Em.
Now we imagine that our perturbation could somehow be
progressively turned on.
This isn't in a time dependent sense.
It's really in a mathematical sense.
We could progressively maker perturbation stronger and
stronger, mathematically.
And, for example, we might be imagining that we are
progressively increasing some applied field or voltage on
some structure.
And we would imagine increasing it from 0 to some
specific value.
So we could look successively for changes in the solutions
that we're going to see.
For example, in the nth energy eigenvalue, Em.
So here's what it is to start out with.
We've called it Em with a superscript 0 here.
And at 0 field we presume it starts out at some value here.
And then perhaps first of all as we turn on our field here
the eigenenergy changes linearly with field.
So this is our eigenenergy here on this vertical axis.
And as we turn the field on, perhaps with some
proportionality constant, a, the energy, for
example, maybe increases.
So that would be a change in energy that was, at least for
small fields here, proportional to the applied
electric field, for example.
We would call those first order
corrections to the energy.
So we're saying the energy is equal to some constant--
we could call that the zeroth order result--
plus some change in the eigenenergy that's somehow
proportional to the amount of perturbation that we're
putting onto this system.
Of course, that might not be quite enough.
The change of energy E with field might not be linear.
Perhaps there's some part of it that's proportional also to
the square of the field, that would give some curvature, for
example, to this line we have here.
And we would call the additional changes that were
proportional to the square of the field, we would call those
second order corrections.
And we could keep on going with this to
higher and higher orders.
Now it's more convenient and general to imagine that we
actually physically have a fixed perturbation.
For example, a fixed applied field, E. But instead we
mathematically increase what is typically called a
house-keeping parameter--
gamma here.
So gamma is going from 0 up to 1.
And does it does so, that means that mathematically
we're increasing our field from 0 up to
this specific value.
That's one way to think of it, anyway.
So our perturbation at any particular point on this graph
here is really gamma times E. And E, we said, we imagine is
actually fixed.
And this is just a mathematical parameter that
we're looking at here.
So now we can express changes as orders of gamma--
gamma to the 1, gamma to the 2.
And this one, incidentally, would correspond to gamma to
the power of 0.
Rather than changes that are in the field itself.
So this is merely a mathematical change, compared
to what we were thinking about previously.
But this is the way that perturbation theory is
typically written down, even if at the moment it seems
somewhat unnecessary to make this mathematical change.
So instead of writing that the eigenenergy is the initial
value plus something proportional to the
perturbation, plus something proportional to its squared,
we write instead the initial value plus gamma times some
constant here, plus gamma squared times some other
constant here.
This constant is really taking the role
that the a took there.
And this one is really taking the role
that the b took there.
But instead of working out a and b, we're actually going to
work out these parameters, Em1 and Em2, and so on.
These terms here have dimensions of energy.
Gamma is a dimensionless quantity.
And they reflect the first order and second order
corrections to the energy that result from the specific
perturbation that we're putting on here, for example,
a specific value of field.
In general, then, we imagine that our perturbed system has
some additional term in the Hamiltonian.
20 And we call that the perturbing Hamiltonian.
So so far I've talked about applying some electric field
that would therefore correspond to some perturbing
Hamiltonian.
In this theory, we're a bit more general than that.
But we imagine that we've somehow made of perturbation
to the system that corresponds to adding in some perturbing
Hamiltonian to the Hamiltonian of the system.
So for example, in the case of our infinitely deep potential
well with an applied field, that perturbing Hamiltonian
would be this expression here--
the electronic charge times the field times basically the
position, although we've referred the position to the
center of the well.
That's not particularly important.
But that's the formula we chose to write down here.
So that would be our perturbing Hamiltonian.
You see it has the dimensions of energy in it.
And in this theory, we write the perturbing Hamiltonian as
gamma times the actual perturbation we're finally
going to be putting on here.
That's the notion, anyway, mathematically.
And we use gamma to keep track of the order of the
corrections that we're talking about.
And we do that by tracking the different powers of gamma in
the various expressions we come up with.
And of course, once we've done that, conceptually at least,
we can set gamma equal to 1 at the end of all of this, if
we'd like to.
So we could have set up the theory this way.
And it could still be a perturbation theory.
There would be nothing wrong with that.
In which case, we would work a and b in some way or other,
and possibly some other parameters.
But to make it more general, and to correspond to the
notation that's typically used in discussing quantum
mechanical perturbation theory, we're going to write
in this form instead.
And we're going to work out the parameters Em1 and Em2,
and possibly some others.
If this is confusing at first, then just think of gamma as
the strength of the electric field in our specific problem.
With this way of thinking about the problem
mathematically, we can write our perturbed Schrodinger
equation in the following way.
So here's our original unperturbed Hamiltonian.
And here's the perturbation we're adding.
And formally we put in this gamma house-keeping parameter
in front of it.
And that is the total Hamiltonian of the system.
And it's acting on the wave function.
And the idea is we're going to think about the eigenenergies,
E over the system corresponding to some solution
wave function here.
So we now presume that we can express the resulting
perturbed eigenfunction and eigenvalue as power series in
this parameter gamma, that is in the following kind of form.
So the function we are talking about, we think of it as some
zeroth order function plus some function proportional to
gamma plus some other function proportional to gamma squared
plus some other function proportional
to gamma cubed here.
And similarly for the energy, we imagine it starts out as
some constant plus something that's proportional to gamma
plus something that's proportional to gamma squared
plus something that's proportional to gamma cubed.
And as I said, if the house-keeping parameter seems
a little abstract, just think of gamma as a dimensionless
measure of the strength of the field that we're applying in
our specific problem, for example.
Now what we do mathematically is we substitute these power
series into our perturbed Schrodinger equation.
That is, in this equation here we're going to substitute this
power series for phi into this equation here and here.
And we're going to substitute this power series for E for
the E here.
That gives us something that looks like this.
So here's our Hamiltonian.
And here's our wave function phi.
We get it on the left, and also over on the right written
out as a power series.
And here's our E, also written out as a power series.
Now at any specific point in space, each of these functions
with the superscript n's in here, corresponding to what we
call the orders of these functions, and each function
of this form-- remember that the Hamiltonian acting on some
function is just a function.
If you think of this as a matrix and think of this as a
vector, multiplying the two of them
together gives us a vector.
So this also is a function.
So if you think that at any specific point in space this
function and also this other function here
is just some number.
It has a value at a particular point in space.
So at any specific point in space, the left-hand side of
this equation it's just a power series in gamma.
For example, something that looks like this.
And so is the right-hand side.
It's also just a power series in gamma.
So that is, we choose some point in space.
Each of these functions is a specific value.
But we're operating on them with this operator here.
So it's this whole function that has some value
at a point in space.
But on the right hand side, we simply have the values of
these functions at the specific point in space.
But therefore, at any point in space we have a number from
the left-hand side.
And we have a number from the right-hand side.
So this is a simple equation between two numbers.
And we could write those numbers out as power series.
So it's an equation between two power series.
Because a power series expansion is unique, the only
way that two power series can be equal to one another, like
these ones here--
and that can work for every value of gamma, at least
within some convergence range, for example, between 0 and 1--
is if the series are equal, term by term.
That is, if a0 is equal to b naught, a1 is equal to b1, a2
is equal to b2, and so on.
That's the only way this can work.
Because power series have to be unique.
Hence in this expression here that we've just written down
we can equate each term with a specific power of gamma.
That is, we find terms with a specific power of gamma on the
left, and they have to be equal to the terms with the
same power of gamma on the right.
And hence we can obtain a progressive set of equations,
which we can solve to evaluate corrections.
So we're going to look for equations equating gamma to
the power of 0 on the left with gamma to the power of 0
on the right, gamma to the power 1 on the left with gamma
to the power 1 on the right, and so on.
And we can do this to whatever order we wish.
We can keep on going terms in gamma cubed, gamma to the
fourth, and so on.
So therefore in this equation here, for example starting
with gamma to the power of 0, that is terms without any
gammas in them, we equate the ones with no gammas in them on
the left to the ones with no gammas in them on the right.
So that would give us our zeroth order equation.
So if we have to have no gammas on the left, we have H0
phi 0 is equal to E0 phi 0.
Any other terms on the left would have a gamma in them,
and any other terms on the right would
have a gamma in them.
And this, of course, is just the unperturbed equation.
We already know the solutions to this equation.
And we know, therefore, that phi 0 is actually just one of
these original eigenfunctions.
And E0 is just one of these original corresponding
eigenvalues.
So if we now presume that we're starting in the specific
eigenstate psi m--
and we'll take that presumption from this point
onwards in this derivation--
then we're going to write psi m and Em instead
of phi 0 and E0.
These are the same things.
So we're going to change over to this notation now.
So with our equation now rewritten with our new
notation, we've got psi m here instead of phi 0.
And we've got Em here instead of E0, and also psi m instead
of phi 0, we can set up a progressive set of equations,
each equating a different power of gamma up on the left
and on the right.
So our first one, the one we already have, is H0 times psi
m is equal to Em times psi m.
Any other terms on the left would have a gamma in them.
And any other terms on the right would
have a gamma in them.
The next one we get by equating gamma to the power of
1 on both sides.
So we could have H0 times gamma phi 1.
And we can have gamma Hp times psi m.
So here are those two terms.
And then on the right-hand side we can have Em times
gamma phi 1.
So that's this term.
Plus gamma E1 times psi m.
That's this term.
And of course, we can drop the gammas through all of these
terms, because they all have a gamma in them.
And we can keep going.
Second order, we have H0 operating on
gamma squared phi 2.
We have gamma Hp operating on gamma phi 1.
And on the right-hand side we have Em times
gamma squared phi 2.
And E1 times gamma phi 1, and gamma squared E2 times psi m.
And again, we can drop the gamma squareds on both sides.
And we could keep on going like this.
So we've separated out three-- and more if we want them--
different equations by equating powers of gamma on
both sides.
We can rewrite these equations slightly.
It's a simple manipulation.
So this equation we will rewrite as this one here, H0
minus Em times psi m equals 0.
Now incidentally, you see we're using the same loose
notation here as we used before.
This is an operator.
And this, on the face of it, is just a number.
If you like, you can put in an identity operator here.
But typically in quantum mechanics it's presumed that
we understand what is going on here, and we're
not confused by it.
The first order equation we can rewrite rather simply as
one in this form.
So again, we've taken the same kind of operator on the left,
but now operating on phi 1.
And on the right we'll have E1 minus the perturbing
Hamiltonian in terms of psi m, with the same understanding
about implicit identity operators, if you like.
And for the second order one, rearrangement can give us,
again, the same kind of form on the left.
And now two terms on the right-hand side here.
And we can keep going with that.
[MUSIC PLAYING]