PROFESSOR: So, WKB approximation,
or semiclassical approximation.
So this is work due to three people--
Wentzel, Kramers, and Brillouin--
in that incredible year, 1926, where so much of quantum
mechanics was figured out.
As it turns with many of these discoveries,
once the discoveries were made, people figured out
that somebody did them before.
And that person was a mathematician, Jeffreys,
who did it three years earlier, in 1923.
The work had not become very popular.
So some people write JWKB.
But we will not do that.
We'll note Jeffreys, but we'll follow this more
standard notation.
So these people were dealing with differential equations
with slowly-varying spatial coefficients.
That was the main thing.
So we're continuing our approximation methods.
We've done perturbation theory.
We will add little pieces to the Hamiltonian.
Now we consider things that are slowly varying in space.
You might have a very simple Hamiltonian
where nothing varies in space--
a constant potential.
But as soon as the potential starts varying slowly,
you have approximation methods.
Those are the methods we're considering now.
We will also consider, after we've
finished WKB, time-dependent perturbation theory, where
still things start slowly varying in time.
So we'll have many, many things to do still.
So this is called, also, the semiclassical approximation.
Because classical physics gives you intuition about the quantum
wave function.
So it is a lesson in which you want to learn something
about the quantum wave function, and you learn it
by using classical physics.
A quantity that is relevant here is that the Broglie
wavelength of a particle.
This is the Broglie.
And many people say that semiclassical approximation
has to do with the fact that the quantum mechanic effects are
not that important.
That may happen.
The Broglie wavelength is much smaller
than the physically relevant sizes of your apparatus.
So you have a particle like an electron.
And if the Broglie wavelength is very small
compared to the aperture in the screen,
the letter will go almost like a classical particle.
When the Broglie wavelength is comparable with the size
of the aperture in the screen, you
will get diffraction effects, and the electrons
will do quantum mechanical things.
So the semiclassical approximation
has to do with lambda being smaller
than the length scale, L, of your physical problem.
We will refine this.
In fact, the whole search of understanding
the semiclassical approximation is
all about understanding this better.
Because it's a little subtle.
We will end up deciding that what you need
is that the Broglie wavelength, suitably generalized,
varies very slowly.
We'll have to generalize the concept of the Broglie
wavelength.
We might get to it today.
Mathematically, you can say, OK, I want
this to be sufficiently small.
So semiclassical limit was just take h going to 0.
You should complain, of course.
h is a constant of nature.
I cannot take it equal to 0.
But on the other hand, they could imagine other universes,
maybe, where h has different smaller and smaller values
in which quantum mechanical effects don't set in
until much smaller scales.
But at the end of the day, I will
try to consider h to be small as an idea underlying
a semiclassical approximation.
And the intuition is that for h small--
we cannot tune it, but we can say it--
h small lambda becomes small.
The lambda, the Broglie, dB.
You're taking a quantity with units--
h bar is units.
And saying it's small.
It goes against lots of things.
Things with units are not supposed to be small.
You should compare it with the situation we had before.
We had a very nice and clean situation
with perturbation theory, where we had a unit-free thing
that we consider it to be small.
This time, we're going to try to consider h to be small.
And it's going to be more delicate because it has units.
It's going to be a more complicated story.
The physics is interesting, and that's
why this approximation is harder in some ways
to understand than the ones we've
done in perturbation theory.
So how does this begin?
It begins by thinking of a particle in a potential, V
of x.
And the particle has some energy, E.
And then, if it's classical, as we're imagining now,
it could be a three-dimensional potential.
My sketch of course is just for one-dimensional.
This is E. And you can solve for p squared, 2m, E
minus V. This is a notion of local momentum
because it depends on x.
It's the momentum the particle would have.
When it is at some position, x, you
will have momentum, p, of x.
And now, nobody forbids you from declaring
that you're going to define a position-dependent Broglie
wavelength, which is going to be h over p
of x is your definition, which is equal to 2 pi h bar over p
of x.
And it's going to be local the Broglie.
You see that the Broglie wavelength, when you first
started in 804, was considered for a free particle--
always the [INAUDIBLE].
You have a free particle with some momentum it
has at the Broglie wavelength.
Why was the Broglie wavelength important?
Because when you write the wave function,
it's a wave with wavelength equal to the Broglie
wavelength.
The wave function with the Broglie wavelength
solves the Schrodinger equation.
That's sort of how it all came about.
But it was all defined for a free particle.
The Broglie defined it for a free particle
with some momentum, p.
And it all made sense, because you could write a wave function
using the Broglie momentum.
It was in fact E to the ipx over h bar.
That was your wave function.
But here, you have a particle moving with varying momenta.
And we don't know how to write the solution of the Schrodinger
equation, but there's classically this concept,
and we could define the local, the Broglie wavelength,
and we will have to discover what it means
or how it shows up.
But it probably shows up in some way.
So there's two cases that probably we should consider.
But before that, we looked at the Schrodinger equation--
the time-independent Schrodinger equation.
So it's minus h squared over 2m Laplacian
of psi of x is equal to E minus v of x, psi of x.
All these vectors-- that's why I'm Laplacian, not
second derivatives.
But the right-hand side--
well, if I put the 2m to the other side,
I get minus h squared Laplacian of psi of x,
is equal to 2m E minus vx.
That's p squared of x, psi of x.
That's the local momentum squared.
Maybe it's a curiosity, but it's now nice that the left-hand
side is the momentum operator squared--
so it's p hat squared on the wave function--
is equal to p squared of x times a wave function.
Kind of a nice result. Nice-looking Schrodinger
equation.
I don't know if you've seen it like that before.
It's an operator on the left.
And on the right, almost like an eigenvalue.
It's an illegal eigenvalue.
If it were a real eigenvalue problem,
there should be a number, not a function here.
But it's a function that acts a little like an eigenvalue.
It's a nice way of thinking of the Schrodinger equation
in the semiclassical approximation.
Well, no.
No approximation here so far in the semiclassical language,
in which you call this the local momentum.
And thus defined, it certainly is an exact statement--
no approximation whatsoever.
OK, now I want to know these two circumstances of course.
If you have E greater than v, you're in an allowed region.
And p squared is really 2m E minus v of x.
And it's convenient to define h squared k squared
of x, the wave number.
You remember that p equal hk was good notation.
We use the wave number sometimes.
We'll have here a local wave number as well.
And if you're in the forbidden region,
which energy is less than v--
forbidden region-- then minus p squared
is positive, is 2m v minus E. That is positive
this time, because v is greater than E.
And that, we always used to call the penetration
constant, the kappa, for wave functions
that decay exponentially.
So we call this h squared kappa squared of x.
A local decaying factor wave number.
It can be thought as an imaginary wave number.
But that's notation.
So, so far, so good.
We will need one more piece of intuition
as we work with this semiclassical approximation.
We have not done any approximation yet.
The approximation will come soon,
as we will begin solving the Schrodinger equation
under those circumstances.
And we will take h bar as an expansion parameter--
will be fine and correct, but a little subtle.
So the thing that will help us many times
to understand these things is to write the wave function
as a complex number.
So suppose you have a wave function.
We will write it as a complex number in the polar form--
radius times face.
So the radius is going to be rho of x.
The probability density, I claim, is this.
And there is a phase that I will write as E to the i
over h bar, S of x and t.
It looks a little like an action.
And it's, to some degree, the beginning of the path
into rule formulation.
It has lots of connections with the action principle.
So here, rho and s are going to be real.
So this is truly scale factor in front
that determines the value.
This is a pure phase, because it's an imaginary number
times a real thing.
S has units of h bar, or angular momentum.
And indeed, psi squared is equal to rho of x and t.
If you loop psi squared.
The reason we focus on this wave function
is that our solutions of WKB are going
to have exactly that form.
So we need to have an intuition as to what
the observables of this wave function are.
So the other observable is the current density.
If you remember, it's h bar over m, imaginary part of psi star
gradient psi.
So this must be calculated.
So what is this?
Gradient of psi-- we must take the gradient of this.
Gradient's a derivative, so it acts on one, acts on the other.
When it acts on the first, it first
acts as the relative 1 over 2 squared of rho of x and t,
times the gradient of rho, times the phase factor, plus now
I have to take the gradient on this quantity,
and this will bring down an i, an h bar, the gradient of S,
and then multiplied by the whole wave function,
because this factor remains and the exponential remains.
Now we can multiply by psi star to form what
we need to get for the current.
So psi star [INAUDIBLE] psi.
If I multiply by that, I'm multiplying by the top line.
For the first factor, I get 1/2 gradient of rho.
The exponentials cancel.
And for the second part, we get plus i
over h bar, gradient of S, times psi squared, which is rho.
The imaginary part of this is equal 1
over h bar, rho gradient of s.
So finally, the current, which is
h bar over m, times that imaginary part,
is rho gradient of s, over m.
A very nice formula.
Basically, it says that the phase factor in the wave
function determines the probability current.
And it also says that if you want to think of this,
here are the surfaces of constant phase.
Here is our space.
And S constant.
So here is one valley of the phase,
another valley of the phase.
Those are surfaces in space of constant phase.
The current is orthogonal to that.
So the current is proportional to the gradient.
The gradient of a function is always proportional.
It's a normal vector to the surfaces of constant values.
So the current is orthogonal to the surfaces of constant phase.
If you have a fluid mechanics interpretation,
J is rho v in fluids.
So, so far, everything I've said could have been said in 804.
These are properties of a general wave function.
This is how you compute the current.
The useful thing is that our WKB wave
functions are going to be presented in that language.
So if you think of the analogy with fluid mechanics.
The current is the charge density times the velocity,
and therefore the velocity would be identified
with gradient of S over m.
Or the momentum would be identified with gradient of S.
That's not a quantum mechanical rigorous identification.
Because gradient of S is a function.
And therefore, p there would be a function.
And a function of momentum--
momentum in quantum mechanics is an operator,
and it has eigenvalues, which are numbers.
They're not functions.
But we already have seen the beginning
of some momentum function.
So that analogy is actually quite nice.
Let me give you an example, and conclude with that.
If you have a free particle, you have a wave function psi
of x and t, which is E, to the ipx over h bar,
times minus iEt over h bar.
So in this case, rho is equal to 1, an S--
remember, S is read by having an i over an h bar out.
And that gives you p dot x minus Et.
So for a free particle, the gradient of S
is indeed just the momentum.
You take this gradient, and it's that.
And therefore, that's a rigorous interpretation
when you have a free particle, that the gradient of S
is going to be the momentum.
Interestingly, the derivative of S with respect to time
is minus the energy.
What will happen in the semiclassical approximation
is that this S over there will depend on x,
and this p will depend on x, and it will be
this p that depends on x here.
And we will see how to solve this equation
in an approximation scheme where the changes are a little slow.
And the notation S here is also motivated
because actions in classical mechanics
actually have this property.
The gradient of the action-- if you
think of the action as a function of coordinates,
which is something you don't usually do, but if you do,
in somewhat advanced classical mechanics,
you see that the derivatives of the action-- spatial
derivatives are the momentum, and the time derivatives
are the energy.
It's a nice relation between classical mechanics,
and justifies once more the name of semiclassical approximation,
which we will continue to develop next time.
See you then.