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PROFESSOR: Today,
we're going to continue

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with the adiabatic subject.

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And our main topic is
going to be Berry's Phase.

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It's interesting
part of the phase

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that goes in adiabatic process.

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And we want to
understand what it is

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and why people care about it.

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And then, we'll turn
to another subject

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in which the adiabatic
approximation is of interest.

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And it's a subject of molecules.

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So I don't think I'll manage to
get through all of that today,

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but will we'll make an effort.

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So let me remind you
of what we had so far.

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So we imagine we have
a Hamiltonian that

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depends on time and
maybe had no dependents

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before time equals 0 turns on.

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And it has no further
variation after some time t.

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So the Hamiltonian
changes like that.

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And the adiabatic theorem
states that if you

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have a state at
time equals 0, which

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is a particular instantaneous
eigenstate, that

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is the instantaneous
eigenstate, then

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that's the full wave function at
time equals 0 and it coincides.

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Then, at time at any time in
this process, if it's slow,

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the process, the
state of the system,

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the full wave
function, psi of t,

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will tend to remain in that
instantaneous eigenstate.

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And the way it's
stated precisely

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is that psi of t minus--

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I'll write it like this--

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psi prime n of t, the
norm of this state

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is of order 1/T for any
t in between 0 and T.

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So I'm trying to state the
adiabatic theorem in a way that

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is mathematically precise.

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And let me remind you the
norm of a wave function

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is you integrate the
wave function square

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and take the square root.

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It's sort of the
usual definition

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of the norm of a vector is the
inner product of the vector

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with itself square root.

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So that's the norm
for wave function.

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And here, what this means is
that with some suitable choice

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of phase, the
instantaneous eigenstate

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is very close to the true state.

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And the error is a Fourier
1/T. So if the process is slow,

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it means that the change
occurs over long t,

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this is a small number.

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And there is some
instantaneous eigenstate

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with some peculiar phase--

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that's why I put the prime--

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for which this
difference is very small.

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And we calculated
this phase, and we

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found the state, psi
of t is roughly equal

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to e to theta n of t, e to the
I gamma n of t, psi n of t.

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And in this statement,
this is what I would

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call the psi n prime of t.

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And that's why the real
state is just approximately

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equal to that one.

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And we have these phases in
which theta of t is minus 1

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over h bar integral from 0
to t E n of t prime dt prime.

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That is kind of
a familiar phase.

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If you had a normal energy
eigenstate, time independent 1,

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this would be 1 to the--

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well, would be minus e
times t over h bar with an I

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would be the familiar
phase that you

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put to an energy eigenstate.

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Then it comes the
gamma n of t, which

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is an integral from 0 to t of
some new n of t prime dt prime.

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And this new n of t is I
psi n of t psi n dot of t.

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I think I have it right.

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So the second part of
the phase is the integral

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of this new function.

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And this new function is
real, because this part

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we showed before is imaginary,
where with an I, this is real.

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And this second
part, this gamma n,

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is called the geometric phase.

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This is the phase that has
to do with Berry's phase.

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And it's a phase that
we want to understand.

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And it's geometrical
because of one reason

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that we're going
to show that makes

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it quite surprising and quite
different from the phase theta.

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The phase theta is a
little like a clock,

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because it runs with time.

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The more time you
wait on an energy

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eigenstate, the more
this phase changes.

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What will happen with
this geometric phase

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is that somehow
properly viewed is

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independent of the time it takes
the adiabatic process to occur.

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So whether it takes us
small time or a long time

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to produce this
change of the system,

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the geometric phase will
be essentially the same.

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That's very, very unusual.

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So that's the main thing
we want to understand

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about this geometric
phase, that it depends only

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on the evolution of the state
in that configuration space--

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we'll make that
clear, what it means--

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and not the time it takes
this evolution to occur.

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It's a little more subtle, this
phase, than the other phase.

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So I want to introduce this
idea of a configuration space.

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So basically, we have that--

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let me forget about time
dependence for one second

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and think of the
Hamiltonian as a function

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of a set of coordinates,
or parameters.

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So the Rs are some coordinates.

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R1, R2, maybe up to R capital
N are some coordinates

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inside some vector space RN.

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So its N components.

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And what does that mean?

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It means maybe that
your Hamiltonian

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has capital N parameters.

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And those are these things.

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So you buy this Hamiltonian.

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It comes with some parameters.

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You buy another one.

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It comes with another
set of parameters.

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Those parameters can be changed.

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Or you construct them in
the lab, your Hamiltonians

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with different parameters.

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Those are the parameters
of the Hamiltonian.

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And suppose you have
learned to solve

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this Hamiltonian for all
values of the parameters.

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That is whatever the
Rs are you know how

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to find the energy eigenstates.

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So H of R times--

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there are some eigenstates, psi
n of R with energies En of R,

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psi n of R. And n
maybe is 1, 2, 3.

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And these are
orthonormal states,

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those energy eigenstates.

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So this equation
says that you have

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been able to solve this
Hamiltonian whatever

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the values of the
parameters are.

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And you have found all
the states of the system,

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n equal 1, 2 3, 4, 5, 6.

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All of them are in now.

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So this is a general situation.

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And now, we imagine
that for some reason,

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these parameters start to
begin to depend on time.

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So they become time
dependent parameters--

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can become time dependent.

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So that you now
have R of t vector.

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These are 1 of t
up to our Rn of t.

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So how do we represent this?

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Well, this is a Cartesian
space of parameters.

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This is not our normal space.

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This is a space where one axis
could be the magnetic field.

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Another axis could be
the electric field.

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Another axis could be
the spring constant.

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Those are abstract axis
of configuration space.

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Or this could be R1,
R2, the axis, R3.

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And those are your axes.

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And now, how do you represent
in this configuration space

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the evolution of the system?

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What is the evolution of the
system in this configuration

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space?

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How does it look?

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Is it a point?

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A line?

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A surface?

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What is it?

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Sorry?

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STUDENT: A path.

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PROFESSOR: It's a path.

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It's a line.

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Indeed, you look at your clock.

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And at time equals 0,
well, it takes some values.

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And you're fine, OK,
here it at time equals 0.

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At time equal 1,
the values change.

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There's one parameter,
which is time.

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So this traces a path.

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As time goes by, the core in
this changing in time and this

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is a line
parameterized by time--

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so a path gamma
parameterized by time.

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And that represents the
evolution of your system.

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At time equals 0, this point
could be R at t equals 0.

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And maybe this point
is R a t equal T final.

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And this system is
going like that.

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You should imagine the system as
traveling in that configuration

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space.

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That's what it does.

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That's why we put the
configuration space.

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And we now have a set--

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not the set-- a time
dependent Hamiltonian,

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because while H was a function
of R from the beginning, now

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R is a function of time.

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So this is your new Hamiltonian.

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And this is time dependent--

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dependent Hamiltonian.

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But now, the
interesting thing is

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that the work you did
before in finding the energy

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eigenstates for any position
in this configuration space

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is giving you the instantaneous
energy eigenstates,

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because if this equation here
holds for any value of R,

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it certainly holds for the
values of R corresponding

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to some particular time.

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So psi n of R of t is equal to
En of R of t psi n of R of t.

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So it's an interesting
interplay in which

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the act that you know your
energy eigenstates everywhere

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in your configuration
space allows

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you to find the
time evolved states,

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the time dependent energy
eigenstates, the instantaneous

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energy eigenstates
are found here.

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So what we want to do now
is evaluate in this language

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the geometric phase, this phase.

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I want to understand
what this phase is

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in this geometric language.