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PROFESSOR: Our subject today
then is Fermi's golden rule.

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So that's what we're
going to develop.

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Fermi's golden rule.

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Fermi's golden rule.

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And this has to do with
the study of transitions.

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And typically, the interesting
and sophisticated thing

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about this subject
is that you have

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a transition from
some initial state

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to a state that is part
of a continuum of states.

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That's what makes
it complicated.

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The transition from
one discrete state

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to another discrete
state with a perturbation

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is kind of a simple
matter to do.

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But when you can go
into a continuum,

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you have to integrate over
the set of final states,

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and that makes it a
lot more interesting.

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So we go from a discrete
state into a continuum.

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And that makes it
somewhat challenging.

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So we will consider
this in two forms.

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It's worth considering
the case of what

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we call constant perturbations.

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And you might say,
well, aren't we

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doing time dependent
perturbation theory?

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Yes, we are.

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But this kind of
perturbation, you

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will think of it as saying
that H is H0 plus a V that

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is time independent.

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But the way we think of
it is that here is time,

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here is time 0, and here is V.
V turns on at time equals 0.

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So there's a little
bit of time dependence.

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There was no V
before time equals 0,

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and there is a V
after time equals 0.

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So it's almost like the
Hamiltonian changes.

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And we want to see what
transitions we get.

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Because always the subject is
the subject of transitions.

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And we will see an application
of this later today.

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The other case is what is
called a harmonic perturbation,

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in which H is equal
to H0 plus delta H.

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And delta H is going
to be harmonic.

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2H prime.

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That's conventions
to put a 2 in there.

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It's convenient.

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Times cosine omega t.

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And this will be for
t between 0 and t0.

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Sorry if some people
can't see this.

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Too far to the right.

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2H prime cosine of omega t
when t is between 0 and t0.

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And it's 0 otherwise.

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And H prime, of course, is
time independent as well.

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Time independent.

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So these are the
two main cases when

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we will consider transitions.

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And we'll do the first one
in a lot of detail today.

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The subject is trying to get
the transitions understood

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for this case.

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And after that,
next time, we will

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do the case of the
harmonic perturbation that

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brings in a few new issues.

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But then, all of the
rest is the same.

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So once you've understood how
to do the constant transition,

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the harmonic perturbation
is going to be easier.

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So constant
perturbations will be

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useful to understand, for
example, the phenomenon

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of auto-ionization.

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Some atoms sometimes
ionize spontaneously,

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and this has to do
with this subject.

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Harmonic perturbations has to
do with even a more popular

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subject, which is atoms
interacting with radiation.

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You send in an
electromagnetic field.

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That is a harmonically
varying perturbation,

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and that's going to allow
you to calculate transitions.

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And by the time
we're done with this,

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you will be experts in
calculating atomic transitions.

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So one subject we
need since we're

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going to be doing transitions
between a discrete state

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or continuum is to
describe the continuum.

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And for that, for
the continuum, we

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use the concept of a density
of states in the continuum.

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Now, we will be considering
momentum eigenstates.

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And the momentum,
you know very well

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the momentum of a free
particle is a continuum.

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Takes absolutely
continuous values.

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So there is no way
you can count them

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or you can tell how many
there are per momentum range.

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It's like saying how many
numbers are there from 0 to 1.

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The way you have
to do that always

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is by adding an
extra parameter you

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wish you didn't have to add.

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And that's kind of
things that we have

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to do in physics sometimes.

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We have to add parameters.

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You consider when we discussed
last time the delta function

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perturbation, we broadened it,
and we were able to calculate.

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And then we saw that
the broadening size

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didn't matter for the
result. Therefore, good.

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So here, we'll do the same.

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You know that the
momentum eigenstates

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of a particle in an interval
or in a circle are quantized.

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And then you can count
quantized states.

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So we will put the
whole world in a box.

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A big box.

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Size of the galaxy, size of the
Earth, size of the laboratory

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maybe is big enough.

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And we'll put this parameter l
there for the size of the box.

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And we will get the
density of states

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because now the
states can be counted.

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Afterwards, by
the time we're all

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done with the transition rates
of the Fermi's golden rule,

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we hope that length
is going to disappear.

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So that's something
we will see that it

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happens in calculations.

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And there will be good
reasons why it happens.

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So we'll put the
world in a big box.

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It's not that big
the way I draw it,

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but it's supposed
to be very big.

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Length l.

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And you think of the
quantization of momentum

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by considering a wave function
that would be normalized

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and has--

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it's a momentum eigenstate.

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It has momentum in the x
direction, y direction,

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and z direction.

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And that wave function,
if all the sides

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of this world, this cubical
world with length l,

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that wave function is
properly normalized.

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You square it, and the norm is
equal to 1 times this factor

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squared.

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You integrate it.

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It's all good.

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So this is a nice
wave function in which

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the integral psi squared V
cubed x over the box is 1.

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And then what else do we do?

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We decide that we have to
consider periodic boundary

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conditions.

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Well, think of it as a
torus, properly speaking,

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in which each
direction is a circle.

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So the wave function
repeats itself

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after x increases by l,
after y increases by l,

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or after z increases by l.

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You could have chosen a box with
a finite big wall at the end.

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It would make no
difference for the counting

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at the end of the day.

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So the conditions are
that kx multiplied by l

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should be a multiple
of nx 2 pi times nx,

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ky times l, 2 pi times
ny, kz times l, 2 pi nz.

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And therefore, the
total number of states

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can be calculated by taking
a little differential

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of this thing.

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We say, well, if we let
kx vary by a little bit,

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dkx from this equation times
l is 2 pi dnx dky times l,

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2 pi times dny, and dkz
times l is 2 pi times dnz.

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So if I were to consider a
little interval of momentum

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defined by dkx, dky, dkz, a
little cube in momentum space,

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the number of
states in this cube

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would be dn, which
is dnx, dny, dnz.

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And this is equal to l over 2 pi
cubed d cubed k, which is dkx,

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dky, dkz.

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I say, if in this momentum range
the quantum number nx contains

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this number of values,
dnx, the quantum number ny

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can take a set of values, and
the same for the quantum number

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in the z direction.

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The total set of quantum
numbers is the product of them.

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So there is that many
states in this little cube

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of momentum space--

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that is, momentum space
that is between some

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k and a little more, called dk.

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Well, that's a famous formula.

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And the n is equal to l
over 2 pi cubed d cubed k.

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Some people know this by heart.

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But now we want to
write this in the terms

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of a density of states
as a function of energy.

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So if this is the
number of states

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in some interval
in momentum space,

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I can try to convert
this into saying,

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well, all these states,
because the momentum is varying

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within some little
bounds, the energy

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is varying within
some little bounds.

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So let's figure out
how much the dE is.

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And then the total
number of states

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will be given in this
range by the number

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of states per unit energy
multiplied by this thing.

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So this is the density
of states, which

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is states per unit energy.

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And many times, this is the
quantity we really want.

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We have here an
energy E as well.

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So we think of this V
cubed k as a little cube

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in momentum space.

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If momentum space
has an origin here,

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you can imagine states with
some momentum, and then

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a little cube in here saying
how much the momentum varies.

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So all these states
have some momentum

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and vary a little
bit in the momentum.

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Therefore, they all
basically have some energy

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up to some little variation.

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So how do we connect
these two things?

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Well, you remember, E is equal
to h squared k squared over 2m.

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And how about d cubed k?

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We'll try to think of d cubed
k physically as all the states

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that have momentum k.

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And now, this is a
space diagram now.

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This is x, y, and z.

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So here are the states
with momentum k.

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They point in this direction.

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So this is the direction
the states are pointing.

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And now we think of them as
having some possible angle

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here.

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d omega.

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So this is the direction.

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So let me not say
they're x, y, and z.

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Let me use angles here.

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Theta and phi, representing
the azimuthal and polar angle

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of this direction.

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So we have a little range
here and a little range

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of magnitude of momentum.

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So let me see if I can
draw it kind of nicer.

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You can imagine a
little cone as it grows,

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and the last part is a little
thick piece of the cone here

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of thickness dk.

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So d cubed k is the volume
of that little pillbox here.

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And it's k square
d omega times dk.

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So that's d cubed k.

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And now to relate
it to the energy

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we have here, this equation.

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So we take a differential.

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And it's d energy is
h squared kdk over m.

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So d cubed k here is k
dk, k d omega times k vk,

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and this is the same k dk here.

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So we can write it as k d omega.

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And for k dk we have m
over h squared d energy.

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So we're almost done
with our computation.

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We go back to this equation.

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And we have rho of E dE,
or dn, is equal to l over

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2 pi cubed d cubed k.

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But now we have
what d cubed k is.

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It's m over h
squared k d omega dE.

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So this is our formula.

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And this is rho of E dE.

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And here, for example,
you have rho of E.

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Actually, if it were rho of E,
I should have here k expressed

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in terms of E. Just makes the
formula a little more messy.

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But if you think of it as
a function of the energy,

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you have to say
the words properly.

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What is rho of E?

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Rho of E is the density
of states per unit energy,

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but when you're
only counting states

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that have a momentum
within an angle d omega.

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So this is the density of
states per unit energy,

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because this gives you states.

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00:19:17,930 --> 00:19:20,720
So this must be states
per unit energy.

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00:19:20,720 --> 00:19:23,390
But you're only looking
at the states that

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00:19:23,390 --> 00:19:25,910
are within an angle d omega.

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00:19:25,910 --> 00:19:29,510
If you were to look at the
density of all states that

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00:19:29,510 --> 00:19:34,100
have energy E, you would
have to integrate over omega.

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00:19:34,100 --> 00:19:38,210
But many times, we want
to make a transition.

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00:19:38,210 --> 00:19:40,030
You have an atom
and you're sending

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00:19:40,030 --> 00:19:41,350
an electromagnetic wave.

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00:19:41,350 --> 00:19:42,890
And you want to
make a transition,

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00:19:42,890 --> 00:19:46,700
and you want to see how
many electrons, for example,

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00:19:46,700 --> 00:19:49,190
are kicked out in
some direction.

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00:19:49,190 --> 00:19:54,830
So you need the density of
states within some solid angle.

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00:19:54,830 --> 00:19:58,130
So this is a
relatively useful thing

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00:19:58,130 --> 00:20:02,600
to have, not to integrate,
so that you can keep control

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00:20:02,600 --> 00:20:06,230
over your states and orient
them at a given angle.

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00:20:09,230 --> 00:20:16,310
So we've done a little
bit of basic preparation.

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00:20:16,310 --> 00:20:20,570
We've said what the Fermi
golden rule aims to do,

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00:20:20,570 --> 00:20:24,740
and it requires transition
to the continuum where we'll

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00:20:24,740 --> 00:20:28,070
have to use density of states.

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00:20:28,070 --> 00:20:30,590
And here is an example
of how you calculate

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00:20:30,590 --> 00:20:32,830
the density of states.