WEBVTT
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PROFESSOR: Let's do adiabatic
evolution really now.
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Evolution.
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We're going to say
lots of things,
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but the take away message is
going to be the following.
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We're going to get maybe
even confused as we do this,
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but the take away
message is the following.
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You sort of begin in
some quantum state,
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and you're going to remain
in that quantum state
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as it changes.
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All the states are
going to be changing.
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The quantum states are going
to be changing in time.
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And you're going to remain
on that quantum state
00:00:49.380 --> 00:00:52.380
with an extra
phase that is going
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to have important information.
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That's basically all
that's going to happen.
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There's a lot of subtleties
in what I've said,
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and we have to unmask
those subtleties.
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But you're going to remain
in that state up to a phase.
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That phase is going to be
called something, Berry's phase.
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And there's a
dynamical phase as well
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that is simple and
familiar, but Berry's phase
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is a little less familiar.
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So you're going to get the
same state up to a phase.
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You're not going to jump states.
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After we'll do that, we'll do
Landau-Zener transitions, which
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are an example
where you can jump,
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and you will calculate
and determine
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how big is the
suppression to jump,
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and so it will
reinforce [INAUDIBLE]..
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So let me begin with this thing.
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Suppose you have an H of t.
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And now, you come
across this states that
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satisfy the following thing.
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H of t, psi of t, is
equal to E of t psi of t.
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If this equation doesn't
look to you totally strange,
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you're not looking hard enough.
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It is a very strange equation.
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It looks familiar.
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It looks like everything
we've always been writing,
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but it's not.
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Look what this is saying.
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Suppose you look at the
Hamiltonian at time 0.
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Then the state at time
0 would be an eigenstate
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of the Hamiltonian at
time 0 with some energy
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at time 0 and some state here.
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So this is what's called an
instantaneous eigenstate.
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It's an eigenstate
at every time.
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It's almost as if you find the
eigenstate at time equals 0.
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You find the eigenstate
at time equals epsilon.
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You find the eigenstate
at time equal 2 epsilon.
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Each time, and
you piece together
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a time dependent
energy eigenstate
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with a time dependent energy.
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We never did that.
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Our energy eigenstates
were all time independent.
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So what kind of
crazy thing is this?
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Well, it has some intuition.
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You know how to do it.
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You know the Hamiltonian at
every time, and at any time,
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you can find eigenstates.
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Now, you've solved
at time equals 0,
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and you solve it at
time equals epsilon,
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and at time equals
epsilon, you're
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going to have
different eigenstate.
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But at time equals
0, you're going
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to have lots of eigenstate.
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At time equals
epsilon, you're going
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to have lots of eigenstate,
but presumably, things
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are not changing too fast.
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You will know which
one goes with which.
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Like at time equals 0, I
get all these eigenstate,
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and at time equal epsilon,
I'll get this eigenstates,
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and presumably, you think,
well, maybe I can join them.
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I'm not going to go this to
that, because it's a big jump,
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and you can track them.
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So this you could
find many of those.
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These are called are
instantaneous eigenstates.
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They are a little
strange, because suppose
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you find those eigenstates,
this is so far so good,
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but maybe this goes like that,
and this crosses that one.
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Oh-- then how do you know
which one, should you go here,
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or should you go here, which
one is your eigenstate?
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So let's just hope
that doesn't happen.
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It's going to be very
difficult if it happens.
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Moreover, there's going to be--
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these states are
not all that unique.
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I could multiply
this Hamiltonian,
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this state by phase e
to the i chi of t here,
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and a time dependent phase,
the Hamiltonian wouldn't care.
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It would cancel.
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So these states are
just not very unique.
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Now, the more important thing
I want to say about them,
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they are the beginning
of our explicit analysis,
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is that do these psi's of t's
solve the Schrodinger equation?
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Are these the solutions of
the Schrodinger equation?
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We've found the
instantaneous eigen-- so
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are these solutions of
the Schrodinger equation?
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Is that what it means to solve
the Schrodinger equation?
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I hear no.
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That's true.
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Not at all.
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These are auxiliary states.
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They don't quite solve
the Schrodinger equation.
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And we'll try to use them to
solve the Schrodinger equation.
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That's what we're
going to try to do.
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So let's try to appreciate that.
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This are psi's of t.
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Now, my notation is going
to be a little delicate.
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Here is your
Schrodinger equation.
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The only difference
is that thing here.
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Now, we're suppressing
all spatial dependent.
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The Hamiltonian might
depend on x and p,
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and the wave function may
depend on x and p, [INAUDIBLE]
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x and other things,
and spin, other things
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will just suppress them.
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So this is the equation
we're trying to solve.
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This is the real equation
that we're trying to solve.
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And if you just
plug the top thing
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and try to see if that
solves that equation,
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you will find it
very quickly doesn't
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solve this equation at all.
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The left hand side,
if you plugged
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in there will appear a psi dot.
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If you thought psi of
t solves this equation,
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you will have a psi dot, and
here you will have an energy,
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and what is supposed to be
a psi dot, it's not obvious.
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It just doesn't solve it.
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So on the other hand,
we can try to inspire
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ourselves to solve it this way.
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You will write in ansatz.
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So we'll put a psi of t.
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We'll try to build our
solution by putting
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maybe the kind of
thing that you usually
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put for an energy eigenstate.
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When you have an
energy eigenstate,
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you would put an e to
the minus i et over h bar
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to solve the
Schrodinger equation.
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So let's do the same thing here.
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Let's put on top of the psi
of t an e to the minus i
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over h bar energy.
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But the energy depends on time
so, actually, the clever thing
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to try to put here
is an integral
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of the energy of time,
dt prime, up to t,
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because the main
thing of that phase
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is that its derivative
should be the energy.
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So that should help.
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So maybe this is
almost a solution
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of the Schrodinger equation.
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But that may not be the case,
so let's put just in case
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here, a c of t that
maybe we will need it
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in order to solve the
Schrodinger equation.
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So our idea is OK, we're
given those instantaneous
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eigenstates, and let's
use them to get a solution
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of the Schrodinger equation.
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Of course, if we
found that this is
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a solution of the
Schrodinger equation,
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we would have found that
with some modification,
00:10:06.960 --> 00:10:11.240
the instantaneous eigenstates
produce solutions.
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And that would be very nice.
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We will find,
essentially, that that's
00:10:16.770 --> 00:10:19.470
true in the adiabatic
approximation.
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So let's do this calculation,
which is important and gives us
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our first sight of
the adiabatic result.
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So here is the psi of t.
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Now let's substitute into
the Schrodinger equation.
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So I have the left
hand side is left
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hand side is i h bar dt
t of this psi would be--
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first, I differentiate the c.
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So c dot e to the minus i
h bar integral to t E dt
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prime, psi of t.
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Plus-- now I differentiate
this exponent, i h bar.
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So I'm sorry, I
have i h bar here.
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When I differentiate
this exponent,
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the i's cancel with the signs.
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The h bar cancels.
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I get an E evaluated at t.
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I'm differentiating with
respect to time here.
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So I get here, nicely, E of
t times the [? Hall ?] wave
00:11:45.980 --> 00:11:50.540
function again, psi of t.
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And then, finally, I get
plus i h bar c of t--
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i h bar c of t times the
exponent and the time
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derivative of psi minus i
over h bar t E dt prime times
00:12:29.130 --> 00:12:33.600
psi of t dot.
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So that is the dot of the state.
00:12:40.200 --> 00:12:44.640
You can differentiate the state,
means evaluating the state at t
00:12:44.640 --> 00:12:49.060
plus epsilon minus [INAUDIBLE]
t divide by epsilon.
00:12:49.060 --> 00:12:53.890
So we'll write it as
this psi dot in there.
00:12:53.890 --> 00:12:55.790
So what do we get here.
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Let's see the right hand
side, right hand side
00:13:08.690 --> 00:13:17.430
is H on the state
and H on the state
00:13:17.430 --> 00:13:22.920
comes here and ignores this
factor, ignores these factors.
00:13:22.920 --> 00:13:26.490
Our time dependent factors
come here and produces
00:13:26.490 --> 00:13:28.920
a factor of e of t.
00:13:28.920 --> 00:13:39.270
So H on psi of t is just
E of t times psi of t.
00:13:44.520 --> 00:13:48.465
So what happens, left hand
side equal to right hand side.
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This term cancels with this.
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This is nice.
00:13:53.970 --> 00:13:58.200
That's what the energy and
the instantaneous states
00:13:58.200 --> 00:13:59.460
should have done.
00:13:59.460 --> 00:14:04.890
But we're left with two
more terms that then cancel.
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These two terms.
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c dot is related to psi dot.
00:14:10.110 --> 00:14:15.570
So indeed, there's no obvious
way of generating a solution,
00:14:15.570 --> 00:14:19.140
because there is a term in the
Schrodinger equation with psi
00:14:19.140 --> 00:14:21.960
dot that must be
canceled or properly
00:14:21.960 --> 00:14:26.430
taken care of by c dot here.
00:14:26.430 --> 00:14:28.870
So what is the
equation that we have?
00:14:28.870 --> 00:14:36.620
We have this first term, but
the second term should be 0.
00:14:36.620 --> 00:14:38.900
So it's a simple equation.
00:14:38.900 --> 00:14:44.350
You can cancel
everything basically.
00:14:48.110 --> 00:14:51.870
The phase can be canceled
the i H can be canceled.
00:14:51.870 --> 00:14:59.810
So we get c dot of
t times psi of t
00:14:59.810 --> 00:15:05.360
is equal to minus c
of t times psi dot.
00:15:11.220 --> 00:15:14.390
That's what we have to solve.
00:15:14.390 --> 00:15:16.960
OK, we have to solve that.
00:15:19.930 --> 00:15:21.280
Let's see how it goes.
00:15:38.500 --> 00:15:40.540
Let's see.
00:15:40.540 --> 00:15:43.560
OK, let's try to
solve this equation
00:15:43.560 --> 00:15:51.390
by sandwiching psi of t,
one of those instantaneous
00:15:51.390 --> 00:15:54.450
eigenstates from the left.
00:15:59.700 --> 00:16:01.490
So what do we get here?
00:16:01.490 --> 00:16:03.530
Well, this is just a function.
00:16:03.530 --> 00:16:06.380
It just doesn't care,
and psi is supposed
00:16:06.380 --> 00:16:08.450
to be normalized state.
00:16:08.450 --> 00:16:13.410
Maybe I should have said, these
are instantaneous eigenstates,
00:16:13.410 --> 00:16:19.850
and psi of t, psi
of t is equal to 1.
00:16:19.850 --> 00:16:24.620
They're normalized at
every instant of time.
00:16:24.620 --> 00:16:27.090
That should not be
difficult to implement.
00:16:27.090 --> 00:16:35.130
So we get here c dot
of t equals minus c
00:16:35.130 --> 00:16:43.710
of t psi of t, psi dot of t.
00:16:43.710 --> 00:16:46.270
That's the kind of
differential equation.
00:16:46.270 --> 00:16:50.260
It just doesn't look
that bad at all.
00:16:50.260 --> 00:16:53.790
In fact, it's one of those
differential equations
00:16:53.790 --> 00:16:55.830
you can solve.
00:16:55.830 --> 00:17:02.340
And the answer is c
of t is equal to e
00:17:02.340 --> 00:17:14.739
to the minus 0 to t psi of t
prime, psi dot of t prime, dt
00:17:14.739 --> 00:17:15.239
prime.
00:17:18.950 --> 00:17:20.150
And that's the answer.
00:17:20.150 --> 00:17:22.910
In fact, take the
derivative, and you
00:17:22.910 --> 00:17:26.540
see this is an
equation of the form f
00:17:26.540 --> 00:17:32.360
dot is equal to a
function of time times f.
00:17:32.360 --> 00:17:35.390
This is solved by integration.
00:17:35.390 --> 00:17:37.370
That's what it is.
00:17:37.370 --> 00:17:38.416
Yes.
00:17:38.416 --> 00:17:42.230
AUDIENCE: [INAUDIBLE]
is the base?
00:17:42.230 --> 00:17:44.780
PROFESSOR: We will see that.
00:17:44.780 --> 00:17:48.050
That's my next point.
00:17:48.050 --> 00:17:52.540
We have here a c of t,
and we have an integral.
00:17:52.540 --> 00:17:57.990
Now, it looks decaying,
but it's actually a phase.
00:17:57.990 --> 00:18:00.950
Let's see that.
00:18:00.950 --> 00:18:07.535
So I want to understand
what is psi, psi dot.
00:18:12.130 --> 00:18:18.710
I claim that this quantity,
in fact, is purely imaginary.
00:18:24.820 --> 00:18:26.860
Let's see why.
00:18:26.860 --> 00:18:27.850
What is this thing?
00:18:27.850 --> 00:18:37.470
This is an overlap intuitively
over x of psi of x and t star
00:18:37.470 --> 00:18:42.806
d dt of psi of x and t.
00:18:48.400 --> 00:18:54.700
And this, this is
a dv x integral.
00:18:54.700 --> 00:18:56.880
It's a vector
integral in general.
00:18:56.880 --> 00:19:00.030
I don't have to put
arrows, I think.
00:19:00.030 --> 00:19:16.930
I have d dt of psi star psi
minus the psi star dt psi.
00:19:16.930 --> 00:19:22.530
This is just a little bit
like pre-integration by parts,
00:19:22.530 --> 00:19:30.880
is just saying a db
is d of ab minus b da.
00:19:30.880 --> 00:19:32.890
That's an identity.
00:19:32.890 --> 00:19:39.450
Now the first term, it's a d
dt of an integral, so over x.
00:19:39.450 --> 00:19:42.870
So the d dt goes out.
00:19:42.870 --> 00:19:47.470
And you have the integral
over x of psi star psi.
00:19:47.470 --> 00:19:49.300
That's the first star.
00:19:49.300 --> 00:19:56.630
And the second term
is minus the integral
00:19:56.630 --> 00:20:02.890
of dx of d psi star psi.
00:20:02.890 --> 00:20:12.920
But I will write it as psi d
dt of psi star here with a star
00:20:12.920 --> 00:20:14.510
there.
00:20:14.510 --> 00:20:15.790
Lots of stars.
00:20:15.790 --> 00:20:17.480
Sorry.
00:20:17.480 --> 00:20:19.520
Can you see that?
00:20:19.520 --> 00:20:23.070
The first term they
took out the derivative.
00:20:23.070 --> 00:20:25.910
The second term,
the sign is out,
00:20:25.910 --> 00:20:29.540
and you have the
psi star times psi,
00:20:29.540 --> 00:20:34.590
but that's a complex conjugate
of that other integral.
00:20:34.590 --> 00:20:37.930
The integral over space is 1.
00:20:37.930 --> 00:20:40.790
So this is 0.
00:20:40.790 --> 00:20:50.680
So this is equal to
minus psi, psi dot star.
00:20:53.430 --> 00:20:56.970
That thing in
parentheses is psi dot.
00:20:56.970 --> 00:21:03.210
So what did you show that
this complex number is minus
00:21:03.210 --> 00:21:04.810
its complex conjugate?
00:21:04.810 --> 00:21:10.485
So that thing is indeed a
purely imaginary number.
00:21:13.510 --> 00:21:16.080
OK, so we're almost there.
00:21:19.280 --> 00:21:20.840
So let's write this nicely.
00:21:25.120 --> 00:21:28.200
What did we find?
00:21:28.200 --> 00:21:34.560
Psi of t is equal--
00:21:34.560 --> 00:21:38.520
the psi of t, it's c
of t times this phase.
00:21:38.520 --> 00:21:40.800
So I'll put first
this phase here.
00:21:43.480 --> 00:21:46.370
I'll call it c of 0 here.
00:21:46.370 --> 00:21:50.470
I ignored it before, but
I could have put it here.
00:21:50.470 --> 00:21:52.360
I don't have to put it--
00:21:52.360 --> 00:21:55.910
there's no need for it.
00:21:55.910 --> 00:22:03.110
e to the minus i over h bar, the
integral from 0 to t of e of t
00:22:03.110 --> 00:22:06.230
prime dt prime.
00:22:06.230 --> 00:22:08.700
That's it.
00:22:08.700 --> 00:22:13.980
And then, I have this
factor, this c of t
00:22:13.980 --> 00:22:15.040
that I have to include.
00:22:15.040 --> 00:22:18.620
So let's put that
phase here, too.
00:22:18.620 --> 00:22:25.030
It's e to the-- this minus is
coming in to replace it by an i
00:22:25.030 --> 00:22:28.060
with another i in here.
00:22:28.060 --> 00:22:29.620
Why would you do that?
00:22:29.620 --> 00:22:32.600
Well, it's good
notation actually,
00:22:32.600 --> 00:22:40.480
psi, psi dot of t
prime, dt prime.
00:22:40.480 --> 00:22:45.460
And all that multiplying,
the instantaneous eigenstate.
00:22:49.100 --> 00:22:52.760
You see, this thing,
this i in front
00:22:52.760 --> 00:22:56.090
is telling you that if
you're using good notation,
00:22:56.090 --> 00:22:59.540
that this quantity
is a pure phase.
00:22:59.540 --> 00:23:03.800
And indeed, it's a pure phase,
because this thing is already
00:23:03.800 --> 00:23:05.390
known to be imaginary.
00:23:05.390 --> 00:23:08.360
So with an i, this is
real, and with this i,
00:23:08.360 --> 00:23:09.580
this is a pure phase.
00:23:09.580 --> 00:23:13.380
So it's just notation.
00:23:13.380 --> 00:23:14.370
So here it is.
00:23:20.180 --> 00:23:23.300
We did it.
00:23:23.300 --> 00:23:31.190
But I must say, we made
a very serious mistake,
00:23:31.190 --> 00:23:39.790
and I want to know if you can
identify where was our mistake.
00:23:39.790 --> 00:23:42.490
Let's give a little
turn to somebody else
00:23:42.490 --> 00:23:46.110
to see where is the mistake,
and then you have your go.
00:23:46.110 --> 00:23:52.660
Anybody wants to say
what is the mistake.
00:23:52.660 --> 00:23:55.790
the mistake is so serious
that I don't really
00:23:55.790 --> 00:23:58.052
have the right to--
00:23:58.052 --> 00:24:00.420
look, if I didn't
make a mistake,
00:24:00.420 --> 00:24:02.730
I've done something
unbelievable.
00:24:02.730 --> 00:24:06.910
I found the solution of
the Schrodinger equation
00:24:06.910 --> 00:24:10.840
using the instantaneous
eigenstate.
00:24:10.840 --> 00:24:13.900
I took the instantaneous
eigenstate,
00:24:13.900 --> 00:24:18.250
and now I've built the solution
of the Schrodinger equation.
00:24:18.250 --> 00:24:21.730
That is an
unbelievable statement.
00:24:21.730 --> 00:24:24.280
It would show that
you will remain
00:24:24.280 --> 00:24:28.580
in the instantaneous
eigenstate forever,
00:24:28.580 --> 00:24:31.770
and I never used slow variation.
00:24:31.770 --> 00:24:34.400
So this would be an exact state.
00:24:34.400 --> 00:24:35.810
This better be wrong.
00:24:35.810 --> 00:24:37.100
This cannot be right.
00:24:37.100 --> 00:24:39.975
It cannot be that you always
remain the same eigenstate.
00:24:39.975 --> 00:24:40.475
Yes.
00:24:40.475 --> 00:24:43.910
AUDIENCE: [INAUDIBLE]
00:24:43.910 --> 00:24:45.110
PROFESSOR: That's right.
00:24:45.110 --> 00:24:48.200
There's going to be a
problem with that equation.
00:24:48.200 --> 00:24:53.870
We did a little mistake here.
00:24:53.870 --> 00:24:57.980
Well, we didn't do a mistake,
but we didn't do our full job.
00:24:57.980 --> 00:25:00.380
Remember in
perturbation theory when
00:25:00.380 --> 00:25:04.640
you had to find the first order
of correction to the state,
00:25:04.640 --> 00:25:10.370
you put from the left a state
in the original subspace.
00:25:10.370 --> 00:25:13.000
You put the state
outside sub space.
00:25:13.000 --> 00:25:16.400
Here we dotted with psi of t.
00:25:16.400 --> 00:25:20.270
But we have to dot with every
state in the Hilbert space
00:25:20.270 --> 00:25:22.380
to make sure we have a solution.
00:25:22.380 --> 00:25:26.300
If you have a vector equation,
you cannot just dot with
00:25:26.300 --> 00:25:28.820
something and say,
OK, I solved it.
00:25:28.820 --> 00:25:32.580
You might have solved the x
component of the equation.
00:25:32.580 --> 00:25:36.300
So we really did not
solve this equation.
00:25:36.300 --> 00:25:40.790
So we made a serious
mistake in doing this.
00:25:40.790 --> 00:25:46.640
But the good thing is that
this is not a bad mistake
00:25:46.640 --> 00:25:48.890
in the sense of learning.
00:25:48.890 --> 00:25:51.500
The only thing I
have to say here
00:25:51.500 --> 00:25:54.800
is that this is
approximately true
00:25:54.800 --> 00:26:01.890
when the changes are the
adiabatic, if the change is
00:26:01.890 --> 00:26:02.780
adiabatic.
00:26:05.400 --> 00:26:08.400
And that is what
we're going to justify
00:26:08.400 --> 00:26:12.430
next time with another
detailed analysis of this.
00:26:12.430 --> 00:26:16.860
So we did a good effort
to find an exact solution
00:26:16.860 --> 00:26:20.370
of the Schrodinger
equation, and we came close.
00:26:20.370 --> 00:26:22.770
And this is a pretty
good approximation.
00:26:22.770 --> 00:26:25.890
This is the statement of
the adiabatic theorem.
00:26:25.890 --> 00:26:30.450
You pretty much follow the
instantaneous eigenstate up
00:26:30.450 --> 00:26:36.160
to a dynamical phase
and up to a Berry phase.
00:26:36.160 --> 00:26:39.090
But this is not an exact
solution of the Schrodinger
00:26:39.090 --> 00:26:41.670
equation, and in
some cases, there
00:26:41.670 --> 00:26:45.630
will be transitions
between those instantaneous
00:26:45.630 --> 00:26:47.780
eigenstates.