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PROFESSOR: So
atom-light interactions.

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So we will focus just on
the electric field, E field.

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Magnetic field effects are
suppressed by the velocity

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of the electrons divided by c.

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And that you know is the
fine-structure constant.

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So magnetic field
effects are suppressed.

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Ignore magnetic v
over c corrections.

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With v over c,
again, of the order

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of the fine-structure constant.

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We will also think of
typically optical frequencies.

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Lambda in the optical range.

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So from about 4,000
to 8,000 angstroms.

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And that's much,
much bigger than a 0,

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which is about 0.5 angstrom.

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So that's good for
our approximation.

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That the wave is
relatively constant,

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being the wavelength
so large it's

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constant over the
extent of the atom.

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So we will think of
the electric field

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of the atom, E at the atom.

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Our electric field, a bit of
notation, will depend on time

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and it will be a real function
of time times a unit vector

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to begin with.

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This will get more
interesting because we're

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going to be dealing
with thermal radiation.

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So eventually, this
vector, n, will be pointing

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and we will average
it over all directions

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because thermal radiation
comes with all polarizations

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and in all directions.

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So there will be a little bit
of averaging necessary for that.

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That will happen next time as
we wrap up this discussion.

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So this picture is of
an atom sitting here.

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And in particular, its
electron, which is the particle

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that reacts the most
in the electric field.

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And there is a unit vector, n.

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And there's E of t here.

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So the electric
field, E of t, is

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2 E not in our conventions
cosine omega 3 times

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the vector, n.

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So what is the vector on
the scalar potential r, t.

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It's minus r times E of t.

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This is the formula
that gives you

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E as minus the gradient of phi.

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This formula is not true in
the presence of magnetic fields

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in general.

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There is a time derivative
of the vector potential.

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But again, we're ignoring
magnetic effects.

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So this is good enough for us.

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If you take the gradient
of this formula,

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the only r dependence is here.

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There's no r dependence
in this electric field.

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And in particular, we consider
the wavelengths to be very big.

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And this is good enough.

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So what is the
perturbing Hamiltonian

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due to the coupling
of the electric field

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to the charged particle?

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And we'll say atom, and we'll
put an electron or something,

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and we'll say the charge is q.

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Eventually it will be
minus E for the electron.

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But let's keep it at q.

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Delta H is q times
phi of r and t.

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So this is minus q times
r times E of t vector.

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Or minus q times r
times n times E of t.

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So so far, simple things.

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We're just considering
the electric field and how

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it adds on a charged particle.

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This is, of course,
the simplest situation.

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We will be considering in
this course soon, in fact,

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starting next lecture,
the general interaction

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of charged particles with
electromagnetic fields.

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But for the time being
and in this approximation,

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this is enough.

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And we will define
now a dipole operator.

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It's something that you
should keep in mind.

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It's the usual thing
when you define

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dipoles is you sum or integrate
over charges times position

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vectors.

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So this is a dipole operator.

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And I emphasized the
operator because of the r.

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When you have matrix elements,
transitions between states,

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everything will have to
deal with those matrix

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elements of r.

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And that's why we'll
have a dipole term there.

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So delta H at this
moment has become what?

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Minus the dipole times
the electric field.

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So let's do this.

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Minus the dipole.

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I'll do it here.

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Minus the dipole dotted
with electric field.

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Vector.

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Or minus d dot n with
2 times the magnitude

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of the electric field.

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Or minus v dot n times 2
E not cosine of omega t.

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Factors of 2 keep
us busy always.

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And we have to get them right.

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Remember when we did our
definition of perturbed

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Hamiltonian we said that delta
H was going to be equal to 2H

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prime cosine of omega t.

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And our transition,
amplitudes, and everything

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were written in
terms of H prime.

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So this was our definition.

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So at this moment, we
can isolate H prime here.

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So H prime is everything
except for the 2

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and the cosine of omega t.

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So H prime, for our
problem of atoms

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interacting with
electromagnetic fields,

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is a dipole interaction.

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And it's given by this
nice simple formula.

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That's our kind of
important end result.

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So if we have this
Hamiltonian, we

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have calculated the probability
for transitions, for example,

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from b to a, as a
function of time.

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When we have a
harmonic perturbation

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coupled to a two level
system, we have a probability.

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We don't have yet a rate.

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We just have a probability.

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And this is equal to the
other one, to the reverse one.

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It's 4Hab prime squared
over h squared sine squared

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of omega ba minus omega over
2 t over omega ba minus omega

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squared.

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That was the formula we had.

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In our case now, H
prime ab is that.

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So we'll get the following.

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4 E0 squared.

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The E0 factor here goes out.

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And then we will have
the matrix elements

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of the dipole operator, d.n ab.

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So it will all be matter
of the dipole operator.

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H squared.

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And these same factors.

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Sine squared omega ba
minus omega over 2T

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over omega ba minus
omega squared.

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So this is the
transition probability.

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We don't have a rate.

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But the rate will come when we
integrate over all the photons

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that contribute to this process.

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And we'll get an exact
analog of Fermi golden rule.

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So that will be next time.