WEBVTT

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Good.

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So let's do, then,
our transitions.

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So we do the constant
perturbation.

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Constant perturbation.

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So as we said, delta
H is equal to V,

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and it's time independent.

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It just begins at time 0.

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And we'll examine what's
going on by time t0.

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So what are we going to do?

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We're going to
examine a transition

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to go from some initial
state i, initial state,

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to a final state f.

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So we don't have to say much
about what the Hamiltonian is

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or anything.

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For us, V is going
to have a constant.

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It's going to have
some matrix elements

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that once we do an
example you can calculate,

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but for the time being we
need not know too much.

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So I'm going to
use the key formula

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that was derived already
about perturbation theory

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and how you get the
transition amplitude.

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So we know that the coefficient
c associated to the m state

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to first order in
perturbation theory

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at that time t0 can be computed
as a sum over all n integral

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from 0 to t0 e to the i omega
mn t prime delta Hmn t prime

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over ih bar Cn 0 dt prime.

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So let me just say a few words.

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You may have seen this
yesterday in recitation.

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Here it is.

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The state has described
in this usual language

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of psi tilde of t was described
as a sum of c ms of ts ms.

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And we calculate those
in perturbation theory.

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So Cn's, we know them
at time equals 0,

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we imagine we know
the initial state

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and we want to
calculate them later.

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Well, the first order
in perturbation theory,

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the Cn's are called Cn's 1's.

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And they're given
by this formula.

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First order in
perturbation theory,

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because we have
a single delta H,

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this omega mn is the energy
of m minus the energy

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of n over h bar.

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And that's it.

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Those are all our symbols.

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So what do we have to do now?

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We have a transition from some
initial state to a final state.

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So what does that mean?

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It means that Cn 0,
it only exists when

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n represents the initial state.

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So I'll just write delta ni.

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At time equals zero,
system is in the state i.

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And at time t0, we're
asking for the probability

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to go into final state.

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So instead of using n and
m, we're just using f and i.

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And therefore, we'll
have m equal f.

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With these two
facts, the formula

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becomes cf the amplitude to
first order in perturbation

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theory to be in the state f
at time t0 is the sum is gone,

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sum over n just
applies for n equals i.

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So i will go here.

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We'll have one over
ih bar 0 to t0 e

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to the i omega final
state to initial state.

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That's m and n t prime Vfi,
because the delta H is V,

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and we're going from
initial to final.

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So fi here.

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And then the t prime.

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It's all gone.

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It's all become
very simple, though.

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You would say too simple.

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We have this is
time independent,

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so it goes out of the interval.

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So we just have the interval
of an exponential here.

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That's very, very easy.

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What is Vfi case vi?

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So

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This goes out, and we just have
to integrate this function.

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I'll write it in a
way that is simpler.

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Maybe you skip a line.

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I don't want to just
count the factors of is

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and doing the integral.

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When you integrate this,
you get another exponential.

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You're going to get
the exponential at t0

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minus the exponential at zero.

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All right, so we don't
want to do our integral.

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So I'll just write the answer.

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I was saying we get
an exponential here

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at t0 minus the
value at zero, then

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you take half of the
exponential out the form a sign.

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This are simple matters, so I
will not do the integral here.

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You get Vfi over Ef
minus Ei e to the i omega

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fi t0 two minus two i sine
of omega fi t0 over two.

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You can believe that.

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I think you can
believe the sine,

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and I have everything here.

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The h bar helped turn the
omega fi into Ef minus Ei.

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So we can now compute the
transition probability

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to go from the initial
to the final stage.

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So we'll write it like this.

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I to f-- it's a little funny.

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I don't know.

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You can write it
whichever way you want.

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Some people like it like that.

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I'm going to do it in the sense
that the initial state always

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appears as a cat, the
final state as a bra.

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So you draw the arrow
like that, more or less

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to keep the sense of
order in your brain.

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But if it doesn't
help you, write it

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whichever way you want.

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Pfi at t0 one is
the norm squared

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of this coefficient, cf one.

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The probability to be
found in the final state

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has to do with the norm
squared of this thing.

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So it's this that's part of what
was reviewed yesterday squared.

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So what do we get?

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That simplifies quite a bit.

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We get Vfi squared times
four sine squared omega fi

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t0 over two over ef
minus ei squared.

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And this is unit free.

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This has units of energy.

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V is a variation
of the Hamiltonian.

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It has units of energy.

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When you put states,
states are normalized

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so it doesn't change the units.

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And this has units of energy
squared, this has no units,

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and this is the answer.

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A little strange.

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There's a periodic variation
on the transition probability,

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and what does it mean to
have a weak perturbation we

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can ask already?

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And the answer in
general is quite simple.

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It's a pragmatic answer.

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A perturbation is
weak if this answer

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is very little, very small.

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Suppose this probability
comes out to be three,

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you know it's already too big.

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But if this is 10 to the minus
sixth times this function,

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that's reasonable.

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You're shining atoms
and one in a million

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goes and gets ionized.

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That's a reasonable thing.

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So the perturbation theory
is valid for whatever time

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you use this formula as long
as this number is small,

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and this could be arranged by
having Vfi sufficiently small.

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I want to understand
this function better,

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because this is a transition
from initial state

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to final state that
looks like that.

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So let's understand it better.

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Suppose one, Ef is different
from Ei, then how does it look?

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Well, it looks like this.

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I brought some other chalk
not that it helps too much.

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But it looks like this
as a function of time.

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The height here
is height four Vfi

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squared over Ef minus Ei
squared, and it's oscillatory.

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It goes to zero again at
time 2 pi over omega fi.

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OK, so this is the oscillation.

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Actually for a small time,
this grows quadratically,

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and then it starts blowing up.

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So here while the
initial behavior

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would be quadratic
for small time,

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this actually is quadratic
as we will see in a second,

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but you can more or less see
by the expansion of the sine,

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then the initial quadratic
growth gets tamed and becomes

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an oscillation here.

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This is valid for all times if
this number is relatively small

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so that we believe
perturbation theory,

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and that's that for that case.

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It's also interesting
that this gets suppressed

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as the energy of the final
state is different, more

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and more different, from the
energy of the initial state.

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So it always oscillates, but
if the state your transition

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is very far away, it is going
to be extremely suppressed

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by the quadratic factors.

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So this is an
important suppression.

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This is saying that transitions
that change the energy

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are not that favored.

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A constant perturbation
doesn't supply really

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energy to produce transitions
that change the energy much,

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and they are suppressed.

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So they produce them,
but they are suppressed.

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The other case
that is of interest

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is the case when
Ef is equal to Ei.

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I'm not saying that the state
f is the same as the state i.

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Not at all.

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It's a different
state but happens

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to have the same energy,
and in that case,

00:13:16.010 --> 00:13:19.590
we must take the
limit as Ef goes

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to Ei, remember omega fi is
Ef minus Ei so over h bar.

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So the limit as ef goes to
Ei of this Pif is how much.

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I'll kind of do it
in my head here.

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We have an h bar here that
is going to be left over.

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One over h bar,
so h bar squared.

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This is going to cancel.

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The four is going
to cancel with this,

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and we're going
to get vfi squared

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over h squared t0 squared.

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OK, so here it is,
the quadratic behavior

00:14:14.750 --> 00:14:17.480
when the energy
of the final state

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is the same as the energy
of your original state.

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Now, the transition probability
starts to grow quadratically.

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That cannot be valid
for too long time,

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because eventually that
number grows without bound,

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and that number could
become as big as 1,

00:14:37.530 --> 00:14:46.110
and that transition
is not reasonable.

00:14:46.110 --> 00:14:59.660
So this is valid
up to some max t0.

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And it's up to you,
depends on what Vfi is,

00:15:03.500 --> 00:15:05.150
how long you can trust this.

00:15:05.150 --> 00:15:06.900
So this is a growth.

00:15:06.900 --> 00:15:10.800
This is the same
growth we observe here.

00:15:10.800 --> 00:15:15.580
The limit as Ef goes
to Ei go to zero

00:15:15.580 --> 00:15:18.950
is actually the same as the
limit as t goes to zero,

00:15:18.950 --> 00:15:22.940
so it's the same
quadratic behavior.

00:15:22.940 --> 00:15:27.150
So finally, what's
going to happen?

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What are we aiming here?

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Well, we're aiming
to the case where

00:15:34.850 --> 00:15:41.870
we have in the energy line,
we have the initial energy Ei

00:15:41.870 --> 00:15:49.070
and then we're going to have
a continuum of final states

00:15:49.070 --> 00:15:50.880
that overlap with Ei.

00:15:54.480 --> 00:15:57.040
They're all over there.

00:15:57.040 --> 00:16:02.530
That's Ef all over there.

00:16:02.530 --> 00:16:06.160
Of course with our box, if
you come with your microscope,

00:16:06.160 --> 00:16:07.400
you see lines here.

00:16:10.080 --> 00:16:11.850
But they're all there.

00:16:11.850 --> 00:16:16.170
There is a continuum
overlapping with this,

00:16:16.170 --> 00:16:21.210
and now we're going to attempt
to sum over the continuum.

00:16:21.210 --> 00:16:22.755
And what should we observe?

00:16:25.590 --> 00:16:29.640
We should observe that
when we add the continuum

00:16:29.640 --> 00:16:33.310
physically, what do we need?

00:16:33.310 --> 00:16:41.430
We need to find what is called
a transition rate, in which you

00:16:41.430 --> 00:16:47.160
have the probability of
transition per unit time

00:16:47.160 --> 00:16:48.402
is a constant.

00:16:48.402 --> 00:16:49.860
You see, you have
a phenom-- you're

00:16:49.860 --> 00:16:52.800
shining light on an atom.

00:16:52.800 --> 00:16:55.080
OK, you shine
light and you wait.

00:16:55.080 --> 00:16:57.640
Eventually the atom ionizes--

00:16:57.640 --> 00:17:00.220
photoelectric effect.

00:17:00.220 --> 00:17:03.320
But if you have a billion
atoms, then you can shine light

00:17:03.320 --> 00:17:06.520
and you're going to have a
transition rate, basically how

00:17:06.520 --> 00:17:10.300
many atoms are going
to happen to ionize.

00:17:10.300 --> 00:17:13.480
So in order to have
a transition rate,

00:17:13.480 --> 00:17:18.579
the probability
that you transition

00:17:18.579 --> 00:17:22.089
has to be proportional
to the time

00:17:22.089 --> 00:17:25.550
that the perturbation
has been acting.

00:17:25.550 --> 00:17:33.110
So the probability of transition
must grow linear in t.

00:17:33.110 --> 00:17:35.660
Therefore, you have
a transition rate

00:17:35.660 --> 00:17:39.320
which is the probability of
transition per unit time,

00:17:39.320 --> 00:17:42.770
so you can grow linear
in time, so per unit time

00:17:42.770 --> 00:17:44.720
you have a transition rate.

00:17:44.720 --> 00:17:48.050
So somehow look
what's happening here.

00:17:48.050 --> 00:17:53.630
When Ef is different than Ei,
the transition probability

00:17:53.630 --> 00:17:55.460
is not linear in time.

00:17:55.460 --> 00:17:57.110
It does this.

00:17:57.110 --> 00:18:00.800
When E approaches--

00:18:00.800 --> 00:18:04.430
Ef approaches Ei, the
probability of transition

00:18:04.430 --> 00:18:07.700
goes quadratic in
time, and what we want

00:18:07.700 --> 00:18:12.080
is a probability of transition
that grows linear in time.

00:18:12.080 --> 00:18:15.560
That would define
a transition rate.

00:18:15.560 --> 00:18:19.530
So how is that going to happen?

00:18:19.530 --> 00:18:22.940
Well, we'll see it happen
in front of our eyes.

00:18:22.940 --> 00:18:26.700
The magic of integration
is going to do it.

00:18:26.700 --> 00:18:28.400
And moreover, we're
going to see that

00:18:28.400 --> 00:18:32.480
consistent with this intuition,
most of the transitions

00:18:32.480 --> 00:18:37.100
that are relevant are happening
within Heisenberg's uncertainty

00:18:37.100 --> 00:18:43.610
principle of a little energy
interval here around Ei.

00:18:43.610 --> 00:18:47.720
So this will be considered to
be at the end of the day energy

00:18:47.720 --> 00:18:50.030
conserving transitions.

00:18:50.030 --> 00:18:55.280
The Hamiltonian, the delta V
helps the transition happen

00:18:55.280 --> 00:18:59.070
but doesn't supply energy
at the end of the day.

00:18:59.070 --> 00:19:02.290
So this is what
we're getting to.