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PROFESSOR: So,
WKB approximation,

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or semiclassical approximation.

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So this is work due
to three people--

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Wentzel, Kramers,
and Brillouin--

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in that incredible year,
1926, where so much of quantum

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mechanics was figured out.

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As it turns with many
of these discoveries,

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once the discoveries were
made, people figured out

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that somebody did them before.

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And that person was a
mathematician, Jeffreys,

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who did it three years
earlier, in 1923.

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The work had not
become very popular.

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So some people write JWKB.

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But we will not do that.

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We'll note Jeffreys, but
we'll follow this more

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standard notation.

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So these people were dealing
with differential equations

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with slowly-varying
spatial coefficients.

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That was the main thing.

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So we're continuing our
approximation methods.

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We've done perturbation theory.

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We will add little pieces
to the Hamiltonian.

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Now we consider things that
are slowly varying in space.

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You might have a very
simple Hamiltonian

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where nothing varies in space--

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a constant potential.

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But as soon as the potential
starts varying slowly,

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you have approximation methods.

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Those are the methods
we're considering now.

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We will also
consider, after we've

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finished WKB, time-dependent
perturbation theory, where

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still things start
slowly varying in time.

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So we'll have many,
many things to do still.

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So this is called, also, the
semiclassical approximation.

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Because classical physics gives
you intuition about the quantum

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wave function.

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So it is a lesson in which
you want to learn something

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about the quantum wave
function, and you learn it

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by using classical physics.

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A quantity that is relevant
here is that the Broglie

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wavelength of a particle.

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This is the Broglie.

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And many people say that
semiclassical approximation

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has to do with the fact that
the quantum mechanic effects are

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not that important.

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That may happen.

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The Broglie wavelength
is much smaller

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than the physically relevant
sizes of your apparatus.

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So you have a particle
like an electron.

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And if the Broglie
wavelength is very small

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compared to the
aperture in the screen,

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the letter will go almost
like a classical particle.

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When the Broglie wavelength
is comparable with the size

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of the aperture
in the screen, you

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will get diffraction
effects, and the electrons

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will do quantum
mechanical things.

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So the semiclassical
approximation

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has to do with
lambda being smaller

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than the length scale, L,
of your physical problem.

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We will refine this.

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In fact, the whole
search of understanding

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the semiclassical
approximation is

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all about understanding
this better.

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Because it's a little subtle.

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We will end up deciding
that what you need

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is that the Broglie wavelength,
suitably generalized,

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varies very slowly.

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We'll have to generalize
the concept of the Broglie

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wavelength.

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We might get to it today.

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Mathematically, you
can say, OK, I want

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this to be sufficiently small.

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So semiclassical limit was
just take h going to 0.

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You should complain, of course.

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h is a constant of nature.

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I cannot take it equal to 0.

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But on the other hand, they
could imagine other universes,

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maybe, where h has different
smaller and smaller values

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in which quantum mechanical
effects don't set in

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until much smaller scales.

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But at the end of
the day, I will

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try to consider h to be
small as an idea underlying

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a semiclassical approximation.

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And the intuition is
that for h small--

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we cannot tune it,
but we can say it--

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h small lambda becomes small.

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The lambda, the Broglie, dB.

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You're taking a
quantity with units--

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h bar is units.

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And saying it's small.

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It goes against lots of things.

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Things with units are
not supposed to be small.

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You should compare it with
the situation we had before.

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We had a very nice
and clean situation

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with perturbation theory,
where we had a unit-free thing

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that we consider it to be small.

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This time, we're going to try
to consider h to be small.

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And it's going to be more
delicate because it has units.

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It's going to be a
more complicated story.

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The physics is
interesting, and that's

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why this approximation
is harder in some ways

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to understand than
the ones we've

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done in perturbation theory.

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So how does this begin?

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It begins by thinking of a
particle in a potential, V

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of x.

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And the particle
has some energy, E.

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And then, if it's classical,
as we're imagining now,

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it could be a
three-dimensional potential.

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My sketch of course is
just for one-dimensional.

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This is E. And you can
solve for p squared, 2m, E

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minus V. This is a
notion of local momentum

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because it depends on x.

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It's the momentum the
particle would have.

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When it is at some
position, x, you

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will have momentum, p, of x.

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And now, nobody forbids
you from declaring

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that you're going to define
a position-dependent Broglie

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wavelength, which is
going to be h over p

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of x is your definition, which
is equal to 2 pi h bar over p

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of x.

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And it's going to be
local the Broglie.

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You see that the Broglie
wavelength, when you first

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started in 804, was considered
for a free particle--

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always the [INAUDIBLE].

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You have a free particle
with some momentum it

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has at the Broglie wavelength.

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Why was the Broglie
wavelength important?

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Because when you write
the wave function,

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it's a wave with wavelength
equal to the Broglie

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wavelength.

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The wave function with
the Broglie wavelength

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solves the Schrodinger equation.

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That's sort of how
it all came about.

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But it was all defined
for a free particle.

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The Broglie defined
it for a free particle

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with some momentum, p.

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And it all made sense, because
you could write a wave function

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using the Broglie momentum.

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It was in fact E to
the ipx over h bar.

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That was your wave function.

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But here, you have a particle
moving with varying momenta.

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And we don't know how to write
the solution of the Schrodinger

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equation, but there's
classically this concept,

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and we could define the
local, the Broglie wavelength,

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and we will have to
discover what it means

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or how it shows up.

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But it probably
shows up in some way.

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So there's two cases that
probably we should consider.

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But before that, we looked
at the Schrodinger equation--

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the time-independent
Schrodinger equation.

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So it's minus h squared
over 2m Laplacian

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of psi of x is equal to
E minus v of x, psi of x.

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All these vectors-- that's
why I'm Laplacian, not

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second derivatives.

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But the right-hand side--

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well, if I put the
2m to the other side,

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I get minus h squared
Laplacian of psi of x,

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is equal to 2m E minus vx.

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That's p squared of x, psi of x.

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That's the local
momentum squared.

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Maybe it's a curiosity, but
it's now nice that the left-hand

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side is the momentum
operator squared--

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so it's p hat squared
on the wave function--

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is equal to p squared of
x times a wave function.

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Kind of a nice result.
Nice-looking Schrodinger

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equation.

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I don't know if you've
seen it like that before.

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It's an operator on the left.

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And on the right, almost
like an eigenvalue.

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It's an illegal eigenvalue.

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If it were a real
eigenvalue problem,

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there should be a number,
not a function here.

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But it's a function that acts
a little like an eigenvalue.

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It's a nice way of thinking
of the Schrodinger equation

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in the semiclassical
approximation.

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Well, no.

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No approximation here so far
in the semiclassical language,

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in which you call this
the local momentum.

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And thus defined, it certainly
is an exact statement--

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no approximation whatsoever.

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OK, now I want to know these
two circumstances of course.

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If you have E greater than v,
you're in an allowed region.

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And p squared is really
2m E minus v of x.

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And it's convenient to
define h squared k squared

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of x, the wave number.

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You remember that p equal
hk was good notation.

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We use the wave
number sometimes.

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We'll have here a local
wave number as well.

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And if you're in the
forbidden region,

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which energy is less than v--

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forbidden region--
then minus p squared

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is positive, is 2m v
minus E. That is positive

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this time, because
v is greater than E.

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And that, we always used
to call the penetration

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constant, the kappa,
for wave functions

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that decay exponentially.

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So we call this h squared
kappa squared of x.

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A local decaying
factor wave number.

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00:13:30,060 --> 00:13:34,050
It can be thought as an
imaginary wave number.

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But that's notation.

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So, so far, so good.

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00:13:41,170 --> 00:13:43,890
We will need one more
piece of intuition

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00:13:43,890 --> 00:13:48,930
as we work with this
semiclassical approximation.

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00:13:48,930 --> 00:13:52,360
We have not done any
approximation yet.

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00:13:52,360 --> 00:13:54,390
The approximation
will come soon,

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00:13:54,390 --> 00:13:57,930
as we will begin solving
the Schrodinger equation

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00:13:57,930 --> 00:14:01,900
under those circumstances.

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00:14:01,900 --> 00:14:06,270
And we will take h bar as
an expansion parameter--

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00:14:09,495 --> 00:14:15,400
will be fine and correct,
but a little subtle.

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00:14:15,400 --> 00:14:18,130
So the thing that will
help us many times

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00:14:18,130 --> 00:14:22,680
to understand these things
is to write the wave function

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00:14:22,680 --> 00:14:24,210
as a complex number.

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00:14:24,210 --> 00:14:26,245
So suppose you have
a wave function.

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00:14:28,900 --> 00:14:32,710
We will write it as a complex
number in the polar form--

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00:14:32,710 --> 00:14:34,900
radius times face.

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00:14:34,900 --> 00:14:41,960
So the radius is
going to be rho of x.

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00:14:41,960 --> 00:14:45,440
The probability density,
I claim, is this.

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00:14:45,440 --> 00:14:50,530
And there is a phase that
I will write as E to the i

220
00:14:50,530 --> 00:14:55,660
over h bar, S of x and t.

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00:14:55,660 --> 00:14:57,680
It looks a little
like an action.

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00:14:57,680 --> 00:15:00,370
And it's, to some degree,
the beginning of the path

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00:15:00,370 --> 00:15:03,690
into rule formulation.

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00:15:03,690 --> 00:15:07,810
It has lots of connections
with the action principle.

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00:15:07,810 --> 00:15:14,020
So here, rho and s
are going to be real.

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00:15:14,020 --> 00:15:18,840
So this is truly
scale factor in front

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00:15:18,840 --> 00:15:20,010
that determines the value.

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00:15:20,010 --> 00:15:23,230
This is a pure phase, because
it's an imaginary number

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00:15:23,230 --> 00:15:25,560
times a real thing.

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00:15:25,560 --> 00:15:31,350
S has units of h bar,
or angular momentum.

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00:15:31,350 --> 00:15:40,620
And indeed, psi squared is
equal to rho of x and t.

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00:15:40,620 --> 00:15:44,210
If you loop psi squared.

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00:15:44,210 --> 00:15:46,970
The reason we focus
on this wave function

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00:15:46,970 --> 00:15:50,060
is that our solutions
of WKB are going

235
00:15:50,060 --> 00:15:52,190
to have exactly that form.

236
00:15:52,190 --> 00:15:56,330
So we need to have an
intuition as to what

237
00:15:56,330 --> 00:16:00,660
the observables of
this wave function are.

238
00:16:00,660 --> 00:16:04,640
So the other observable
is the current density.

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00:16:04,640 --> 00:16:15,270
If you remember, it's h bar over
m, imaginary part of psi star

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00:16:15,270 --> 00:16:18,510
gradient psi.

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00:16:18,510 --> 00:16:20,190
So this must be calculated.

242
00:16:26,080 --> 00:16:27,090
So what is this?

243
00:16:27,090 --> 00:16:33,270
Gradient of psi-- we must
take the gradient of this.

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00:16:33,270 --> 00:16:37,680
Gradient's a derivative, so it
acts on one, acts on the other.

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00:16:37,680 --> 00:16:39,510
When it acts on
the first, it first

246
00:16:39,510 --> 00:16:45,690
acts as the relative 1 over
2 squared of rho of x and t,

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00:16:45,690 --> 00:16:59,840
times the gradient of rho,
times the phase factor, plus now

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00:16:59,840 --> 00:17:03,890
I have to take the
gradient on this quantity,

249
00:17:03,890 --> 00:17:12,740
and this will bring down an i,
an h bar, the gradient of S,

250
00:17:12,740 --> 00:17:15,630
and then multiplied by
the whole wave function,

251
00:17:15,630 --> 00:17:18,449
because this factor remains
and the exponential remains.

252
00:17:26,720 --> 00:17:31,560
Now we can multiply by
psi star to form what

253
00:17:31,560 --> 00:17:33,140
we need to get for the current.

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00:17:38,990 --> 00:17:45,040
So psi star [INAUDIBLE] psi.

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00:17:45,040 --> 00:17:50,570
If I multiply by that, I'm
multiplying by the top line.

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00:17:50,570 --> 00:17:54,995
For the first factor, I
get 1/2 gradient of rho.

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00:17:54,995 --> 00:17:58,820
The exponentials cancel.

258
00:17:58,820 --> 00:18:03,140
And for the second
part, we get plus i

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00:18:03,140 --> 00:18:10,620
over h bar, gradient of S,
times psi squared, which is rho.

260
00:18:10,620 --> 00:18:17,920
The imaginary part
of this is equal 1

261
00:18:17,920 --> 00:18:21,430
over h bar, rho gradient of s.

262
00:18:25,420 --> 00:18:27,880
So finally, the
current, which is

263
00:18:27,880 --> 00:18:32,020
h bar over m, times
that imaginary part,

264
00:18:32,020 --> 00:18:36,420
is rho gradient of s, over m.

265
00:18:41,270 --> 00:18:43,770
A very nice formula.

266
00:18:43,770 --> 00:18:48,810
Basically, it says that the
phase factor in the wave

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00:18:48,810 --> 00:18:52,230
function determines the
probability current.

268
00:18:52,230 --> 00:18:58,020
And it also says that if
you want to think of this,

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00:18:58,020 --> 00:19:01,340
here are the surfaces
of constant phase.

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00:19:01,340 --> 00:19:02,880
Here is our space.

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00:19:02,880 --> 00:19:07,210
And S constant.

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00:19:07,210 --> 00:19:09,104
So here is one
valley of the phase,

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00:19:09,104 --> 00:19:10,270
another valley of the phase.

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00:19:10,270 --> 00:19:13,650
Those are surfaces in
space of constant phase.

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00:19:13,650 --> 00:19:16,510
The current is
orthogonal to that.

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00:19:16,510 --> 00:19:20,880
So the current is
proportional to the gradient.

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00:19:20,880 --> 00:19:24,260
The gradient of a function
is always proportional.

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00:19:24,260 --> 00:19:28,360
It's a normal vector to the
surfaces of constant values.

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00:19:28,360 --> 00:19:33,570
So the current is orthogonal to
the surfaces of constant phase.

280
00:19:36,270 --> 00:19:40,140
If you have a fluid
mechanics interpretation,

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00:19:40,140 --> 00:19:44,130
J is rho v in fluids.

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00:19:47,050 --> 00:19:53,680
So, so far, everything I've said
could have been said in 804.

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00:19:53,680 --> 00:19:57,370
These are properties of
a general wave function.

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00:19:57,370 --> 00:19:58,870
This is how you
compute the current.

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00:19:58,870 --> 00:20:01,930
The useful thing is
that our WKB wave

286
00:20:01,930 --> 00:20:06,200
functions are going to be
presented in that language.

287
00:20:06,200 --> 00:20:09,680
So if you think of the
analogy with fluid mechanics.

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00:20:09,680 --> 00:20:14,140
The current is the charge
density times the velocity,

289
00:20:14,140 --> 00:20:19,060
and therefore the velocity
would be identified

290
00:20:19,060 --> 00:20:23,990
with gradient of S over m.

291
00:20:23,990 --> 00:20:31,110
Or the momentum would be
identified with gradient of S.

292
00:20:31,110 --> 00:20:35,990
That's not a quantum mechanical
rigorous identification.

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00:20:35,990 --> 00:20:39,440
Because gradient
of S is a function.

294
00:20:39,440 --> 00:20:42,830
And therefore, p there
would be a function.

295
00:20:42,830 --> 00:20:45,920
And a function of momentum--

296
00:20:45,920 --> 00:20:49,280
momentum in quantum
mechanics is an operator,

297
00:20:49,280 --> 00:20:51,950
and it has eigenvalues,
which are numbers.

298
00:20:51,950 --> 00:20:53,000
They're not functions.

299
00:20:53,000 --> 00:20:56,480
But we already have
seen the beginning

300
00:20:56,480 --> 00:20:59,360
of some momentum function.

301
00:20:59,360 --> 00:21:03,890
So that analogy is
actually quite nice.

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00:21:03,890 --> 00:21:07,880
Let me give you an example,
and conclude with that.

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00:21:07,880 --> 00:21:14,920
If you have a free particle,
you have a wave function psi

304
00:21:14,920 --> 00:21:23,020
of x and t, which is E,
to the ipx over h bar,

305
00:21:23,020 --> 00:21:27,800
times minus iEt over h bar.

306
00:21:30,640 --> 00:21:37,230
So in this case, rho
is equal to 1, an S--

307
00:21:37,230 --> 00:21:43,350
remember, S is read by having
an i over an h bar out.

308
00:21:43,350 --> 00:21:48,080
And that gives you
p dot x minus Et.

309
00:21:51,750 --> 00:21:56,223
So for a free particle,
the gradient of S

310
00:21:56,223 --> 00:22:00,460
is indeed just the momentum.

311
00:22:00,460 --> 00:22:02,400
You take this gradient,
and it's that.

312
00:22:02,400 --> 00:22:05,620
And therefore, that's a
rigorous interpretation

313
00:22:05,620 --> 00:22:08,880
when you have a free particle,
that the gradient of S

314
00:22:08,880 --> 00:22:11,040
is going to be the momentum.

315
00:22:11,040 --> 00:22:17,100
Interestingly, the derivative
of S with respect to time

316
00:22:17,100 --> 00:22:18,390
is minus the energy.

317
00:22:23,480 --> 00:22:26,750
What will happen in the
semiclassical approximation

318
00:22:26,750 --> 00:22:29,900
is that this S over
there will depend on x,

319
00:22:29,900 --> 00:22:34,400
and this p will depend
on x, and it will be

320
00:22:34,400 --> 00:22:36,800
this p that depends on x here.

321
00:22:36,800 --> 00:22:39,890
And we will see how
to solve this equation

322
00:22:39,890 --> 00:22:45,560
in an approximation scheme where
the changes are a little slow.

323
00:22:45,560 --> 00:22:50,630
And the notation S
here is also motivated

324
00:22:50,630 --> 00:22:54,020
because actions in
classical mechanics

325
00:22:54,020 --> 00:22:57,620
actually have this property.

326
00:22:57,620 --> 00:22:59,300
The gradient of
the action-- if you

327
00:22:59,300 --> 00:23:02,340
think of the action as a
function of coordinates,

328
00:23:02,340 --> 00:23:07,060
which is something you don't
usually do, but if you do,

329
00:23:07,060 --> 00:23:09,560
in somewhat advanced
classical mechanics,

330
00:23:09,560 --> 00:23:12,380
you see that the derivatives
of the action-- spatial

331
00:23:12,380 --> 00:23:16,120
derivatives are the momentum,
and the time derivatives

332
00:23:16,120 --> 00:23:16,745
are the energy.

333
00:23:16,745 --> 00:23:20,900
It's a nice relation
between classical mechanics,

334
00:23:20,900 --> 00:23:25,400
and justifies once more the name
of semiclassical approximation,

335
00:23:25,400 --> 00:23:28,290
which we will continue
to develop next time.

336
00:23:28,290 --> 00:23:30,140
See you then.