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PROFESSOR: So my comments.

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First one-- if the psi n of
t can be chosen to be real,

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the geometric face vanishes.

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So why is that?

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The geometric face is,
remember, the integral

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of this new factor, which was
i psi n of t psi n of t dot.

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You had to integrate that thing
to get the geometric face.

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We explained that this
quantity in general is--

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well, in general, it's always
imaginary, this quantity.

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And then with an i,
this is a real quantity

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which we've been using.

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But if this is
imaginary-- look at this.

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This is imaginary.

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How can it be imaginary
if psi is real?

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It can't be.

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If psi is real, imagine
doing that integral.

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You can kind of imagine.

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So it has to be
0 if psi is real.

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Yes?

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AUDIENCE: [INAUDIBLE].

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Why did we say
that was imaginary?

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PROFESSOR: We proved
this was imaginary.

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We did calculate the derivative
with respect to time of psi

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with psi, and we proved
that this was imaginary.

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And now I'm saying if this was
imaginary but if psi is real,

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the only possibility
is that this is 0.

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But you can prove, in fact,
that this is 0, if this is real.

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I can do it in a second.

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It's i integral psi n of
p-- because it's real--

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d dt of psi n of t.

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Output an x here,
and this is dx.

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That's what it is
if psi is real.

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Otherwise there
would be a star here.

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But if it's real,
there's no star.

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But this is just
the integral of 1/2

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of d dt of psi of
x and t squared dx.

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The d dt is that.

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And then the d dt goes
out of the integral--

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so this is i over 2
d dt of the integral

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of psi squared of x and t dx.

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But that integral is 1,
so the derivative is 0.

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So there's no geometric face.

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So if you have real
instantaneous eigenstates,

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don't even think
of Berry's phase.

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There's another case where
you don't get a Berry's phase.

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So when I'm speaking of
Berry's phase at this moment,

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I mean the Berry's phase from
a closed pathing configuration

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space.

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So if the configuration
space is one-dimensional,

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the Berry phase vanishes.

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Barry's phase vanishes for
1v configuration space.

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Why is that?

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Well, what do we have to do?

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The Berry's phase will be the
integral over the closed path

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in the configuration
space psi of r d, d r--

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because there's
just one side of r--

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d r.

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You have to
integrate that thing.

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That is the Berry phase--
is the integral of the Berry

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connection over--

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this is capital R--

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over the space.

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But now, if this configuration
space is one-dimensional--

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this is r--

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a closed path is a path that
goes like this and black.

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It retraces itself.

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So it integrates this
quantity with increasing r,

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and then it integrates
the same quantity

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with decreasing r,
and the two cancel,

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and this phase is equal to 0.

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So you cannot get a Berry's
phase if you have one

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dimension.

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You cannot get the Berry's phase
if you have real instantaneous

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eigenstates.

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But you can get a Berry's
phase in two dimensions,

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in three dimensions, and
there are several examples.

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I will mention one
example, and and then

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leave Barry's phase
for some exercises.

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So in 3D, for example--

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3D's nice.

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A 3D configuration space
is the perfect place

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to confuse yourself,
because you have

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three dimensions
of configuration

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and three dimensions of space.

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So nice interplay between them--

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r1, r2, r3.

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And then you have an
integral-- the Berry's phase

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is an integral-- over
a closed path here.

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So let's call gamma and
let's call this surface, s,

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whose boundary is gamma.

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So what is the Berry phase?

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The Berry phase is the
integral over gamma

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of the Berry connection d r.

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But what is that?

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That is, by Stokes'
theorem, the integral

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over the surface of
the curl of the Berry

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connection times the area.

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And so the area on that surface.

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Stokes' theorem--
remember in E&M?

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So here was a vector potential
gives you a magnetic field--

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so the integral
of a along a loop

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was equal to the flux
of the magnetic field

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through the surface.

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And now the berry
phase along the loop

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is equal to the
integral of the--

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we should call it
Berry magnetic field?

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No.

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When people think berries--

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curvature.

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As in the sense that the
magnetic field is the curvature

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of that connection.

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So this is called the
Berry's curvature,

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but you think about
magnetic field--

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Berry's curvature.

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So the Berry's
curvature-- people

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go with d is the curl sub r of
the Berry connection a of r.

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It's the magnetic field,
so it's the integral

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of d over that surface.

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So they're nice analogies.

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So one example you
will do in recitation--

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I hope-- you have
Max over there--

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is the classic example of
a spin in a magnetic field.

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So I will just say a couple
of words as an introduction,

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but it's a very
nice computation.

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It's a great exercise
to figure out

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if you understand
all these things.

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It's done in [INAUDIBLE]
explicitly in some ways,

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and it's a great
thing to practice.

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So if you want the
little challenge--

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I know you're busy with
papers and other things.

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but the idea is that you
have a magnetic field--

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the cogs of a
magnetic field, B0n--

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along some direction n, and
you have a spin 1/2 particle

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pointing in that direction n.

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But then you start
letting it vary in time--

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that unit vector.

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So this unit vector varies
in time and traces a path.

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And if it's idiomatic
process, the spin

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will remain in the instantaneous
energy eigenstate, which

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is spin up in this
direction, and it will track

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the magnetic field, basically.

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There will be some small
probability it flips,

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but it's small.

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And if this is an
idiomatic process,

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that error can be small.

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So as you move around here,
there will be a geometric face,

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and the geometric
face is quite nice.

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So for the spin up state--

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m+ state-- if it
follows this thing,

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the geometric phase for the spin
up state in this closed path

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in configuration space--

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this is the configuration
space of the magnetic field--

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will just be given by minus 1/2.

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This comes from spin
1/2 times an invariant

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of this loop, which is the
solid angle traced by this loop.

156
00:11:00,920 --> 00:11:03,940
So the geometric phase that's
acquired by this motion

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00:11:03,940 --> 00:11:07,570
is proportional just to the
solid angle of this loop.

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00:11:07,570 --> 00:11:13,380
It's a very nice result. Shows
how geometric everything is.