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PROFESSOR: OK, Let's
use the last 10 minutes

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to discuss an application.

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So that's our Fermi
golden rule there.

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Let's leave it in
the blackboard.

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So the example I will
discuss qualitatively

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will not compute the
rate for this example is

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auto ionization, or also
called Auger transitions.

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So we imagine of--

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I think the reason this
example is interesting

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is that there is continuum
states, sometimes in cases

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that you would not
think about it,

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or you wouldn't have
thought about them.

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So let's assume we
have a helium atom.

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So you have two electrons,
z equals 2, two protons

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and two electrons.

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And I will assume that we
have a hydrogenic state.

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So you see the Hamiltonian
of this whole ,

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atom there's a P1 squared over
2n plus a P2 squared over 2n,

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roughly for each electron,
plus or minus e squared over r1

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minus e squared over r2, and
then plus e squared over r1

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minus r2, which is
the Coulomb repulsion.

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So that's roughly
the Hamiltonian

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you need to consider,
at least to 0-th order.

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And let's assume this is
H0, and let's consider

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states of this Hamiltonian, H0.

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So these are hydrogenic states.

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We treat each electron
independently.

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And therefore, the
energy levels, E

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are defined by two principle
quantum numbers, n1 and n2,

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and they are minus 13.6
eV times z squared.

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The energy of an electron
in a nucleus with z protons

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gets multiplied by z
squared there times 1

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over n squared, and 1 squared
plus 1 over n2 squared.

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Those are the principle
quantum numbers.

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And z squared is equal to
4, because z is equal to 2.

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So this number
comes out at 54.4.

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So here is e of n1 n2 is minus
54.4 eV, 1 over n1 squared

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plus 1 over n2 squared.

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OK, let's look at the
spectrum of this atom.

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I'll erase here.

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So let's draw a line
here, put 0 here.

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Ground state.

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n1 is equal to n2 is equal to 1.

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The ground state
is at minus 108.

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That's a state that we call
state 1, 1 for the two quantum

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numbers being 1 and 1.

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OK, so then we go on,
and look what happens.

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Then we have, for
example, the state 2, 1.

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And the state 2, 1 is
going to be less bound,

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because instead of having 1
and 1, you have 1/4 plus 1,

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so it's less bound.

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3, 1, 4, 1.

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How about infinity 1?

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Infinity 1.

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That's n1, the first electron
being super Rydberg atom.

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It has its quantum
principle quantum

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number so high that it's 0,
and you have the other electron

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here with a 1.

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So this is at 54.4.

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The infinity 1 is here.

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One electron is bound
in the one state.

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The other electron
is out very far.

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It's almost free.

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You could have it free.

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If it goes free, we'll
have some more energy.

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So actually, after
you go here, there's

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a continuum of states, a
continuum of states over here

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in which every electron here
is an infinity and an electron

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with some momentum, k.

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And these are still
states that are

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here can be represented
by energies in this range.

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And then, eventually, you
have even this goes on here.

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So there's a lot of
bound states there,

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and then we can continue.

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How about the state--

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so I've put here infinity.

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I'm sorry, not infinity, k.

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I will put it here
like k times y,

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because the other state
remains in the 1 s.

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So one electron-- here both
electrons were in the 1s state.

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And here 2s, 3s, infinity.

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So one electron
remains in the 1s.

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One electron is free with
some momentum over there.

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Then, for example,
consider the state E 2, 2.

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This happens to be a 2 and a 2
is a quarter, a quarter, 1/2.

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So this is at minus 27 eV.

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It's right here in the middle.

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This is the state 2, 2.

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After the state 2, 2 sets in,
there is a state 3, 2, 4, 2, 5,

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2, infinity 2.

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So one term is infinity.

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The other is 2.

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So it's one quarter.

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And infinity 2 is that
the hydrogen level, 13.6.

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And after that infinity 2,
there is a new continuum

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of states that has some
momentum and are still in 2.

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And there's going to be an
infinite number of continuums

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arising here when you consider
one electron and the other one.

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So what is our other transition?

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This was at minus 27.2.

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The other transition we're
typically interested in

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is this one in which you have
this discrete state overlapped

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by the continuum of states in
which one electron is gone out,

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and one electron has
remained in the--

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has gone to the 1s state.

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So I'll write it down here.

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So what is the other transition?

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We have precisely this
situation with indicate

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that here in which
you have one state

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surrounded by a continuum.

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So the other transition, so
the line 2, 2, at minus 27 eV

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is in the middle of the
infinity 1 continuum.

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So we'll have a transition
into 2s squared--

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that's this state,
that's the name of it--

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goes to 1s times 3.

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The e 2, 2 is going to
be E1 infinity plus E3.

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And the e 2, 2 is minus 27.2.

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This is minus 54.4,
and this is E3.

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So E3 goes to 27.2 eV
for the free electron.

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So how do we visualize this?

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If you wanted to compute it,
and it's kind of a fun thing,

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but we will not do it, you
think of the initial 2s state.

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Have the atom.

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The one is the two
electrons here.

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And you think of them as having
an initial wave function,

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hydrogenic, and then suddenly,
because maybe something

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happened.

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This 1s electron got
excited, and suddenly,

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for an instant of time, the
two are there in the 2s.

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And then this
interaction turns on,

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and this is the v that makes
that transition from this 2s

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squared state to 1s 3.

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That has good matrix elements,
and this is going to happen.

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And this is the
dominant transition,

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because a radiative transition
in which one goes down here

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is fairly suppressed.

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It doesn't happen.

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So this is our example
of an application

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of Fermi's golden rule, and
we'll explore more next time.