WEBVTT

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PROFESSOR: OK.

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So time to complicate
the model a little bit

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to get more interesting
physics from it.

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So what am I going to do?

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I'm going to add an extra term.

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So the system goes
now to an H of t

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that is going to have
the alpha t over 2.

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And now it's gonna have a
little term of diagonal.

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So there's several ways
of thinking about this.

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H12 is going to be
constant in time.

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So it's a number.

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H12 star is it's
complex conjugate.

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If you're looking at t equals
0, H becomes just H12, H12 star.

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And the energy eigenstate
or the energy eigenvalues

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of this matrix are plus
or minus the norm of H12,

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the absolute value of H12.

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Just put a couple of lambdas,
calculate the eigenvalues.

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It involves H12 times
H12 star, which is

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the square of the norm of H12.

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And the energies are those.

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So look at what's happening
in your energy diagram

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as a function of t.

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At t equals 0, there are two
energies, H12 and minus H12.

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Those are the energies
at time equals 0.

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We're trying to get the analog
of what was going on there.

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And now you could
say, OK, at time

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equals 0, that's what I get.

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What do I get at large times?

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Well, at large times,
these are dominant.

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And these are very small.

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So you must get something
similar to these arrows here.

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So what I'll draw
here is this and this.

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And I cannot trust this
here for small time.

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But presumably, this is about
right here and about right here

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and about right here
and about right here.

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And the states must be
the same one, 0, 1 here;

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0, 1; 1, 0; 1, 0.

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So that's what you know just
without doing any calculation.

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That's what your system does.

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And now, a
Hamiltonian in general

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doesn't get levels crossing.

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That requires a coincidence like
having no off-diagonal element.

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So actually, what this
will give you is this.

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So that's how the system will
look as a function of time.

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Those are the energy
levels as a function

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of time, the instantaneous
energy levels.

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At every instant of time,
you now have the energies.

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These are the energies
of this matrix,

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the energies, the eigenvalues.

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One line computation, our E
plus minus equal plus or minus

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square root of H12
squared plus alpha

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squared t squared over
4, so plus or minus.

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So this is E plus
and E minus of t.

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All right.

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So here we go.

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We have a real system.

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And we have something
quite interesting actually.

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When you let H12 go to 0,
you're back to that place.

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And you just zoom through
like here you did.

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You just go through
and go through.

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So when the levels
do that, you just

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continue through any
particular state you were in.

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You don't start here
and then go here.

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You just go through.

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That's what you prove here.

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And therefore, when you take
the limit of H12 going to 0,

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if H12 goes to 0,
these things collapse.

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And you're back there.

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And you zoom by through.

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So when H12 will be
very, very small,

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you will be likely,
in fact, more likely

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to make the transition
than not to make it.

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So if you have a system for--

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this is just a micron separated
here, a milli-electron volt--

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let's be more precise here--

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you're more likely to
zoom through and make

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the nonadiabatic transition
because this will not

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be an adiabatic process.

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They're getting to
close to each other.

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On the other hand,
if you are far away,

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you're going to be
very unlikely to make

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the nonadiabatic transition
because you are very separated.

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The nice thing
about this problem

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is that it can be
solved analytically.

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Not terribly easy.

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It involves a little bit
of hypergeometric functions

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and some differential equations.

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But it can be solved.

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You will solve it
numerically in the homework.

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I will say a few words about it.

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And we'll discuss the
transition in this case.

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And the answer is
known analytically.

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It's a very famous result.
In fact, many people

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have written papers trying
to give simple derivations

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of this answer.

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So in order to just
write the answer

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and to see how it looks--

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so this is the answer
for a transition--

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we try to discuss the notion
of adiabatic process here.

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When is this
transition adiabatic?

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And what we do is this.

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You can imagine taking a tangent
here and another tangent here.

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And when it hits the lines,
the linear lines that we

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plotted here, bring them down.

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So you get a rectangle here.

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So you hit the
alpha t over 2 line.

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And you call this 2 tau
d or 2 tau, no, tau d.

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OK.

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So let's figure out
what that is because I

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say that this time is
the time that going up

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on the line, alpha t over
2, this time 2 tau d,

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you get the height, H12.

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That means that H12 is alpha
times 2 tau d divided by 2.

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So tau d is H12 over alpha.

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So tau d is usually thought
as the timescale associated

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to the change in
the Hamiltonian.

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That is you have an
original Hamiltonian, that's

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where this linear [INAUDIBLE].

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And over a time 2d,
the shape is changed

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into the final Hamiltonian.

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This is the process in
which the original system

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is changed into the new
system within this timescale.

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The timescale tau d.

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But we have another time
that is interesting here.

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You see, while this
change is happening,

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the separation of the energy
levels is by this distance H12.

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So the Hamiltonian
at t equals 0,

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the Hamiltonian looks
like 0, H12, H12, here.

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And if you have two states
governed by such Hamiltonian

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that is valid near
time equals 0,

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there's going to be oscillations
between states here.

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You can go from the first
state to the second state

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with some frequency
governed by this number.

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That is called the Rabi
frequency, Rabi oscillation.

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It's something
you've done in 805.

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In two-state systems,
you oscillate.

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And the frequency of oscillation
between the states 1 and 2

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is H12 divided by h bar or 2
pi divided by the period T12.

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That's the definition
of the frequency.

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So what do we have here?

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The process is
adiabatic if the time

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tau d for the change to happen
is much larger than the period

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capital T12 of the oscillation.

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So you can think of this
system during the time

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it spends in this box like
a two-level system separated

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by H12 that oscillates
between the two states.

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And in that time, 2 tau d,
all the change is happening.

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So you should have that
tau d is much bigger

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than the period of oscillation.

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And this corresponds
too because T12

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is the inverse of
omega 12 to omega 12

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tau d much greater
than 1, so either one.

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And that would be
adiabatic condition.

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Let's put the numbers here.

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Omega 12 is H12 norm over h bar.

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And tau d-- we often
found it there--

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is H12 over alpha.

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So over alpha.

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So this is much greater than 1.

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I don't know.

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This 2 here has nothing
to do with that formula.

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So this is the
adiabatic condition.

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And this is reasonable.

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You see, what is going
to make it adiabatic?

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The more these branches are
separated, the more difficult

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the transition, the better
the adiabatic approximation.

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And it's this thing here.

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The bigger this number,
the better you are.

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The slower alpha is,
the lower the slope

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is, the more time
it's going to take,

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the better the
adiabatic approximation.

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So this is really the thing.

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And the final formula
that I will write here

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is the probability for a
nonadiabatic transition

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is exponential of minus
2 pi omega 12 tau d.

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So the probability that you
cross the thing and jump

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from this down to here is this.

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It's suppressed by
the adiabatic factor.

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And that's what you will
check in the homework.