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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

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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,

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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

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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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00:11:08,620 --> 00:11:18,470
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

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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

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psi of t dot.

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So that is the dot of the state.

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00:12:40,200 --> 00:12:44,640
You can differentiate the state,
means evaluating the state at t

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00:12:44,640 --> 00:12:49,060
plus epsilon minus [INAUDIBLE]
t divide by epsilon.

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00:12:49,060 --> 00:12:53,890
So we'll write it as
this psi dot in there.

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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

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00:13:08,690 --> 00:13:17,430
is H on the state
and H on the state

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00:13:17,430 --> 00:13:22,920
comes here and ignores this
factor, ignores these factors.

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00:13:22,920 --> 00:13:26,490
Our time dependent factors
come here and produces

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00:13:26,490 --> 00:13:28,920
a factor of e of t.

194
00:13:28,920 --> 00:13:39,270
So H on psi of t is just
E of t times psi of t.

195
00:13:44,520 --> 00:13:48,465
So what happens, left hand
side equal to right hand side.

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00:13:51,000 --> 00:13:52,980
This term cancels with this.

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00:13:52,980 --> 00:13:53,970
This is nice.

198
00:13:53,970 --> 00:13:58,200
That's what the energy and
the instantaneous states

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00:13:58,200 --> 00:13:59,460
should have done.

200
00:13:59,460 --> 00:14:04,890
But we're left with two
more terms that then cancel.

201
00:14:04,890 --> 00:14:06,240
These two terms.

202
00:14:06,240 --> 00:14:10,110
c dot is related to psi dot.

203
00:14:10,110 --> 00:14:15,570
So indeed, there's no obvious
way of generating a solution,

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00:14:15,570 --> 00:14:19,140
because there is a term in the
Schrodinger equation with psi

205
00:14:19,140 --> 00:14:21,960
dot that must be
canceled or properly

206
00:14:21,960 --> 00:14:26,430
taken care of by c dot here.

207
00:14:26,430 --> 00:14:28,870
So what is the
equation that we have?

208
00:14:28,870 --> 00:14:36,620
We have this first term, but
the second term should be 0.

209
00:14:36,620 --> 00:14:38,900
So it's a simple equation.

210
00:14:38,900 --> 00:14:44,350
You can cancel
everything basically.

211
00:14:48,110 --> 00:14:51,870
The phase can be canceled
the i H can be canceled.

212
00:14:51,870 --> 00:14:59,810
So we get c dot of
t times psi of t

213
00:14:59,810 --> 00:15:05,360
is equal to minus c
of t times psi dot.

214
00:15:11,220 --> 00:15:14,390
That's what we have to solve.

215
00:15:14,390 --> 00:15:16,960
OK, we have to solve that.

216
00:15:19,930 --> 00:15:21,280
Let's see how it goes.

217
00:15:38,500 --> 00:15:40,540
Let's see.

218
00:15:40,540 --> 00:15:43,560
OK, let's try to
solve this equation

219
00:15:43,560 --> 00:15:51,390
by sandwiching psi of t,
one of those instantaneous

220
00:15:51,390 --> 00:15:54,450
eigenstates from the left.

221
00:15:59,700 --> 00:16:01,490
So what do we get here?

222
00:16:01,490 --> 00:16:03,530
Well, this is just a function.

223
00:16:03,530 --> 00:16:06,380
It just doesn't care,
and psi is supposed

224
00:16:06,380 --> 00:16:08,450
to be normalized state.

225
00:16:08,450 --> 00:16:13,410
Maybe I should have said, these
are instantaneous eigenstates,

226
00:16:13,410 --> 00:16:19,850
and psi of t, psi
of t is equal to 1.

227
00:16:19,850 --> 00:16:24,620
They're normalized at
every instant of time.

228
00:16:24,620 --> 00:16:27,090
That should not be
difficult to implement.

229
00:16:27,090 --> 00:16:35,130
So we get here c dot
of t equals minus c

230
00:16:35,130 --> 00:16:43,710
of t psi of t, psi dot of t.

231
00:16:43,710 --> 00:16:46,270
That's the kind of
differential equation.

232
00:16:46,270 --> 00:16:50,260
It just doesn't look
that bad at all.

233
00:16:50,260 --> 00:16:53,790
In fact, it's one of those
differential equations

234
00:16:53,790 --> 00:16:55,830
you can solve.

235
00:16:55,830 --> 00:17:02,340
And the answer is c
of t is equal to e

236
00:17:02,340 --> 00:17:14,739
to the minus 0 to t psi of t
prime, psi dot of t prime, dt

237
00:17:14,739 --> 00:17:15,239
prime.

238
00:17:18,950 --> 00:17:20,150
And that's the answer.

239
00:17:20,150 --> 00:17:22,910
In fact, take the
derivative, and you

240
00:17:22,910 --> 00:17:26,540
see this is an
equation of the form f

241
00:17:26,540 --> 00:17:32,360
dot is equal to a
function of time times f.

242
00:17:32,360 --> 00:17:35,390
This is solved by integration.

243
00:17:35,390 --> 00:17:37,370
That's what it is.

244
00:17:37,370 --> 00:17:38,416
Yes.

245
00:17:38,416 --> 00:17:42,230
AUDIENCE: [INAUDIBLE]
is the base?

246
00:17:42,230 --> 00:17:44,780
PROFESSOR: We will see that.

247
00:17:44,780 --> 00:17:48,050
That's my next point.

248
00:17:48,050 --> 00:17:52,540
We have here a c of t,
and we have an integral.

249
00:17:52,540 --> 00:17:57,990
Now, it looks decaying,
but it's actually a phase.

250
00:17:57,990 --> 00:18:00,950
Let's see that.

251
00:18:00,950 --> 00:18:07,535
So I want to understand
what is psi, psi dot.

252
00:18:12,130 --> 00:18:18,710
I claim that this quantity,
in fact, is purely imaginary.

253
00:18:24,820 --> 00:18:26,860
Let's see why.

254
00:18:26,860 --> 00:18:27,850
What is this thing?

255
00:18:27,850 --> 00:18:37,470
This is an overlap intuitively
over x of psi of x and t star

256
00:18:37,470 --> 00:18:42,806
d dt of psi of x and t.

257
00:18:48,400 --> 00:18:54,700
And this, this is
a dv x integral.

258
00:18:54,700 --> 00:18:56,880
It's a vector
integral in general.

259
00:18:56,880 --> 00:19:00,030
I don't have to put
arrows, I think.

260
00:19:00,030 --> 00:19:16,930
I have d dt of psi star psi
minus the psi star dt psi.

261
00:19:16,930 --> 00:19:22,530
This is just a little bit
like pre-integration by parts,

262
00:19:22,530 --> 00:19:30,880
is just saying a db
is d of ab minus b da.

263
00:19:30,880 --> 00:19:32,890
That's an identity.

264
00:19:32,890 --> 00:19:39,450
Now the first term, it's a d
dt of an integral, so over x.

265
00:19:39,450 --> 00:19:42,870
So the d dt goes out.

266
00:19:42,870 --> 00:19:47,470
And you have the integral
over x of psi star psi.

267
00:19:47,470 --> 00:19:49,300
That's the first star.

268
00:19:49,300 --> 00:19:56,630
And the second term
is minus the integral

269
00:19:56,630 --> 00:20:02,890
of dx of d psi star psi.

270
00:20:02,890 --> 00:20:12,920
But I will write it as psi d
dt of psi star here with a star

271
00:20:12,920 --> 00:20:14,510
there.

272
00:20:14,510 --> 00:20:15,790
Lots of stars.

273
00:20:15,790 --> 00:20:17,480
Sorry.

274
00:20:17,480 --> 00:20:19,520
Can you see that?

275
00:20:19,520 --> 00:20:23,070
The first term they
took out the derivative.

276
00:20:23,070 --> 00:20:25,910
The second term,
the sign is out,

277
00:20:25,910 --> 00:20:29,540
and you have the
psi star times psi,

278
00:20:29,540 --> 00:20:34,590
but that's a complex conjugate
of that other integral.

279
00:20:34,590 --> 00:20:37,930
The integral over space is 1.

280
00:20:37,930 --> 00:20:40,790
So this is 0.

281
00:20:40,790 --> 00:20:50,680
So this is equal to
minus psi, psi dot star.

282
00:20:53,430 --> 00:20:56,970
That thing in
parentheses is psi dot.

283
00:20:56,970 --> 00:21:03,210
So what did you show that
this complex number is minus

284
00:21:03,210 --> 00:21:04,810
its complex conjugate?

285
00:21:04,810 --> 00:21:10,485
So that thing is indeed a
purely imaginary number.

286
00:21:13,510 --> 00:21:16,080
OK, so we're almost there.

287
00:21:19,280 --> 00:21:20,840
So let's write this nicely.

288
00:21:25,120 --> 00:21:28,200
What did we find?

289
00:21:28,200 --> 00:21:34,560
Psi of t is equal--

290
00:21:34,560 --> 00:21:38,520
the psi of t, it's c
of t times this phase.

291
00:21:38,520 --> 00:21:40,800
So I'll put first
this phase here.

292
00:21:43,480 --> 00:21:46,370
I'll call it c of 0 here.

293
00:21:46,370 --> 00:21:50,470
I ignored it before, but
I could have put it here.

294
00:21:50,470 --> 00:21:52,360
I don't have to put it--

295
00:21:52,360 --> 00:21:55,910
there's no need for it.

296
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

297
00:22:03,110 --> 00:22:06,230
prime dt prime.

298
00:22:06,230 --> 00:22:08,700
That's it.

299
00:22:08,700 --> 00:22:13,980
And then, I have this
factor, this c of t

300
00:22:13,980 --> 00:22:15,040
that I have to include.

301
00:22:15,040 --> 00:22:18,620
So let's put that
phase here, too.

302
00:22:18,620 --> 00:22:25,030
It's e to the-- this minus is
coming in to replace it by an i

303
00:22:25,030 --> 00:22:28,060
with another i in here.

304
00:22:28,060 --> 00:22:29,620
Why would you do that?

305
00:22:29,620 --> 00:22:32,600
Well, it's good
notation actually,

306
00:22:32,600 --> 00:22:40,480
psi, psi dot of t
prime, dt prime.

307
00:22:40,480 --> 00:22:45,460
And all that multiplying,
the instantaneous eigenstate.

308
00:22:49,100 --> 00:22:52,760
You see, this thing,
this i in front

309
00:22:52,760 --> 00:22:56,090
is telling you that if
you're using good notation,

310
00:22:56,090 --> 00:22:59,540
that this quantity
is a pure phase.

311
00:22:59,540 --> 00:23:03,800
And indeed, it's a pure phase,
because this thing is already

312
00:23:03,800 --> 00:23:05,390
known to be imaginary.

313
00:23:05,390 --> 00:23:08,360
So with an i, this is
real, and with this i,

314
00:23:08,360 --> 00:23:09,580
this is a pure phase.

315
00:23:09,580 --> 00:23:13,380
So it's just notation.

316
00:23:13,380 --> 00:23:14,370
So here it is.

317
00:23:20,180 --> 00:23:23,300
We did it.

318
00:23:23,300 --> 00:23:31,190
But I must say, we made
a very serious mistake,

319
00:23:31,190 --> 00:23:39,790
and I want to know if you can
identify where was our mistake.

320
00:23:39,790 --> 00:23:42,490
Let's give a little
turn to somebody else

321
00:23:42,490 --> 00:23:46,110
to see where is the mistake,
and then you have your go.

322
00:23:46,110 --> 00:23:52,660
Anybody wants to say
what is the mistake.

323
00:23:52,660 --> 00:23:55,790
the mistake is so serious
that I don't really

324
00:23:55,790 --> 00:23:58,052
have the right to--

325
00:23:58,052 --> 00:24:00,420
look, if I didn't
make a mistake,

326
00:24:00,420 --> 00:24:02,730
I've done something
unbelievable.

327
00:24:02,730 --> 00:24:06,910
I found the solution of
the Schrodinger equation

328
00:24:06,910 --> 00:24:10,840
using the instantaneous
eigenstate.

329
00:24:10,840 --> 00:24:13,900
I took the instantaneous
eigenstate,

330
00:24:13,900 --> 00:24:18,250
and now I've built the solution
of the Schrodinger equation.

331
00:24:18,250 --> 00:24:21,730
That is an
unbelievable statement.

332
00:24:21,730 --> 00:24:24,280
It would show that
you will remain

333
00:24:24,280 --> 00:24:28,580
in the instantaneous
eigenstate forever,

334
00:24:28,580 --> 00:24:31,770
and I never used slow variation.

335
00:24:31,770 --> 00:24:34,400
So this would be an exact state.

336
00:24:34,400 --> 00:24:35,810
This better be wrong.

337
00:24:35,810 --> 00:24:37,100
This cannot be right.

338
00:24:37,100 --> 00:24:39,975
It cannot be that you always
remain the same eigenstate.

339
00:24:39,975 --> 00:24:40,475
Yes.

340
00:24:40,475 --> 00:24:43,910
AUDIENCE: [INAUDIBLE]

341
00:24:43,910 --> 00:24:45,110
PROFESSOR: That's right.

342
00:24:45,110 --> 00:24:48,200
There's going to be a
problem with that equation.

343
00:24:48,200 --> 00:24:53,870
We did a little mistake here.

344
00:24:53,870 --> 00:24:57,980
Well, we didn't do a mistake,
but we didn't do our full job.

345
00:24:57,980 --> 00:25:00,380
Remember in
perturbation theory when

346
00:25:00,380 --> 00:25:04,640
you had to find the first order
of correction to the state,

347
00:25:04,640 --> 00:25:10,370
you put from the left a state
in the original subspace.

348
00:25:10,370 --> 00:25:13,000
You put the state
outside sub space.

349
00:25:13,000 --> 00:25:16,400
Here we dotted with psi of t.

350
00:25:16,400 --> 00:25:20,270
But we have to dot with every
state in the Hilbert space

351
00:25:20,270 --> 00:25:22,380
to make sure we have a solution.

352
00:25:22,380 --> 00:25:26,300
If you have a vector equation,
you cannot just dot with

353
00:25:26,300 --> 00:25:28,820
something and say,
OK, I solved it.

354
00:25:28,820 --> 00:25:32,580
You might have solved the x
component of the equation.

355
00:25:32,580 --> 00:25:36,300
So we really did not
solve this equation.

356
00:25:36,300 --> 00:25:40,790
So we made a serious
mistake in doing this.

357
00:25:40,790 --> 00:25:46,640
But the good thing is that
this is not a bad mistake

358
00:25:46,640 --> 00:25:48,890
in the sense of learning.

359
00:25:48,890 --> 00:25:51,500
The only thing I
have to say here

360
00:25:51,500 --> 00:25:54,800
is that this is
approximately true

361
00:25:54,800 --> 00:26:01,890
when the changes are the
adiabatic, if the change is

362
00:26:01,890 --> 00:26:02,780
adiabatic.

363
00:26:05,400 --> 00:26:08,400
And that is what
we're going to justify

364
00:26:08,400 --> 00:26:12,430
next time with another
detailed analysis of this.

365
00:26:12,430 --> 00:26:16,860
So we did a good effort
to find an exact solution

366
00:26:16,860 --> 00:26:20,370
of the Schrodinger
equation, and we came close.

367
00:26:20,370 --> 00:26:22,770
And this is a pretty
good approximation.

368
00:26:22,770 --> 00:26:25,890
This is the statement of
the adiabatic theorem.

369
00:26:25,890 --> 00:26:30,450
You pretty much follow the
instantaneous eigenstate up

370
00:26:30,450 --> 00:26:36,160
to a dynamical phase
and up to a Berry phase.

371
00:26:36,160 --> 00:26:39,090
But this is not an exact
solution of the Schrodinger

372
00:26:39,090 --> 00:26:41,670
equation, and in
some cases, there

373
00:26:41,670 --> 00:26:45,630
will be transitions
between those instantaneous

374
00:26:45,630 --> 00:26:47,780
eigenstates.