M1L2g.txt

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Let me now spent the next 10 or 15 minutes
on what is called the dipole approximation
and the dipole Hamiltonian.
So we are talking now about one further expansion
of our expressions and this is the multipole expansion.
Let me first sort of give you this simple derivation
which you may have seen, but then you can appreciate
the rigorous derivation.
Either directly or through the vector potential,
we have formulated our theory in terms
of the electric and magnetic field
as a function of position.
But what we now want to exploit is that atoms are tiny
and, usually, the electric or magnetic field is not
changing over the extent of an atom or molecule.
In other words, what I'm telling you
is the relevant frequencies, the relevant components
of the electric and magnetic fields,
will, of course, be the components
which are in resonance with the atoms and molecules.
And what I'm telling you is that in many, many situations,
the wave lengths of the relevant modes
of the electromagnetic field are much, much longer
than the size of an atom.
So therefore, if we pin point our atom at the origin,
we may be able to neglect the spatial dependence
of the electromagnetic fields and replace it
by electric and magnetic fields at the origin
or if you want to go higher in a Taylor expansion,
we use gradients of it.
We've talked more about that, actually,
in 8, 4, 21 when we talk about multiple transitions,
but here I want to only focus on the lowest order which
is called the electric dipole approximation
because this is the most important case which
is used in atomic physics.
So for the electric dipole approximation,
we make actually several assumptions.
One is we neglect the quadratic term, the a square term.
And now if you look at the a dot p term,
and we are looking for matrix elements between two
atomic levels, we expand the vector potential
into plane waves.
Well, it's called the electric dipole approximation.
So maybe you want to express things
by the electric field, which is the derivative of the vector
potential.
So with that, we have now described things
by the electric field.
And what is needed now is if we assume that k r equal 0 or r
over lambda is very, very small, the exponential the plane wave
factor is approximated by 1.
And we are simply left in the electric dipole approximation
with a matrix element of the momentum operator.
We can simplify it further by writing the momentum operator
as the commutator between the position
operator and the Hamiltonian.
If you take the commutator between r or x and p square,
you simply get p and b vectors so that's what we are using.
And now, the matrix element becomes--
so what we have here now is r h minus h r, but h acting on 1
is because 1 is an eigenstate, gives the energy e 1.
And for the flipped part, we have
h r and now h acting on state 2 give us the energy e 2.
So therefore, by taking care of the Hamiltonian which gives us,
depending on which side h appears, e1 or e2.
We have used it to now the position matrix element.
And what we have here is, of course,
the transition frequency between the levels 1 and 2.
And all you've seen it, but if I can derive it in two minutes,
it's maybe [INAUDIBLE] exercise.
What I've shown you is that the momentum matrix element
can be replaced by the position matrix element, multiplied
by the resonance frequency.
But now we want to put things together
and I want to point out for you is that we have an omega here,
which is the frequency of the electromagnetic field,
and we have an omega 1, 2 here, which is the transition energy.
So therefore, if I put it together,
I started with a p dot a Hamiltonian,
but the p matrix element became an r matrix element.
The a was expressed by the electric field,
so therefore, we have now the electric field
times the position matrix element
or the dipole matrix element if you
multiply the position of the electron with a charge, too.
But in addition, we have this prefecter
so this becomes the dipole Hamiltonian e dot d.
When we replace that by 1, assuming
that we are interested in any way,
only in interactions with atoms where the radiation
field is near resonance.
So it seems that we have actually
made three assumptions to derive the electric dipole
approximation.
One was the long wave lengths approximation.
The second one was that we are near resonance.
And the third one was that we neglected the quadratic term
in the light atom interaction.
Well, there are often people who are wondering about it
when I said, OK, we're only interested
when the atoms in the electric magnetic radiation
near resonance.
That's OK, but atomic physics is the area of physics
with the highest precision.
We can make measurements with 10 to the minus 16
and time to the minus 17 position.
So therefore, if you have an electromagnetic field which
resonates with atoms, the typical frequency
of visible light is 10 to the 14 Hertz.
And you're just 1 megahertz away from the resonance.
It's 10 to the minus 8.
So question is would we observe correction
to the dipole approximation if omega 1, 2 is not
exactly omega.

Well the answer is no because, as I want to show you now
in the next few minutes and this is also
the end of our flight over the appendix of atom photon
interaction, the last two assumptions
are not really necessary.
So in other words, we go back to atom photon interaction
to this long appendix on the derivation of the QED
Hamiltonian.
And we go back to our fundamental Hamiltonian
and we want to do one approximation
and this is the only approximation which is needed.
It is the dipole approximation and that
means that the vector potential-- of course, we
only need the transverse part of it-- is replaced by its value
at the origin.
That's the only approximation.
So here is our Hamiltonian in its full beauty
and the only thing we have done is we've put in the origin
here.
For later convenience, we introduce the dipole operator
which is the position operator of particle
alpha times the charge of particle alpha.
And then we do simply a transformation.
So this is a unitary transformation
and we transform from the original Hamiltonian, which
includes the p dot a term and the a square term,
to a transformed Hamiltonian h prime.
And this transformed Hamiltonian has now
the electric dipole approximation
sort of getting in polarization Fourier space,
but this is the electric field and a
a dagger gives position times-- sorry,
these are normal modes-- it's a dipole operator
times the electric field.
And the a square term has disappeared.
So after the transformation, we have no quadratic term
and all we are left is this is nothing else than d dot
e, the dipole interaction.
And there is no prefecter omega 1, 2 over omega.
This is a rigorous transformation,
unitary transformation, to another basis.
In full disclosure, it involves a dipolar self energy which
is sort of a constant energy.
And if you look at the transformed velocities,
the transform velocities v prime are now
identical to the original canonical momentum.
So after that, the original what was the canonical momentum
in the original Hamiltonian h, becomes now velocity times mass
so p becomes now the mechanical momentum
and, therefore, p square is the kinetic energy.
Anyway, I think this is important
that we have this much more rigorous derivation
of the dipole approximation.
I don't want to go through that, I just
want to make sure you have seen it.
That if you want to do it really rigorously,
then you have to distinguish between the electric field d
and e.
So you have an electric field here and here
you have the polarization.
It's pretty much the same as in classical e and m.
And at the end of the day, we have
now for the primed operator for the Hamiltonian
after the canonical transformation,
after the unitary transformation,
we have now the only interaction term which
remains is the electric dipole interaction between the dipole
and this field, d.
But the field d prime corresponds
to the original transverse electric field.
I'm not going to explain the difference between e,
e prime, d, and d prime.
It can be done rigorously, but the take home message
is-- and with that I want to conclude--
that for most of this course, the interaction
of the electromagnetic field with our atoms
is described by this term.
We take the transverse electric field,
this is our radiation field, and it
couples to the operator of the atomic dipole moment.

I hope you got a take home message
that whenever you have any doubts about any part
of this derivation, or if you have any doubts whether it's
rigorous or not, you know now we are to look it up and learn
everything about it.
Any questions?