Scattering of Photons

The scattering process clearly requires terms in
that **annihilate one photon and create another**.
The order does not matter.
The
is the square of the Fourier decomposition of the
radiation field so it contains terms like
and
which are just what we want.
The
term has both creation and annihilation operators in it but not
products of them.
It changes the number of photons by plus or minus one, not by zero as required for the scattering process.
Nevertheless this part of the interaction could contribute in second order perturbation theory,
by absorbing one photon in a transition from the initial atomic state to an intermediate state,
then emitting another photon and making a transition to the final atomic state.
While this is higher order in perturbation theory, it is the same order in the electromagnetic coupling constant
,
which is what really counts when expanding in powers of
.
Therefore, we will need to **consider the
term in first order and
the
term in second order** perturbation theory
to get an order
calculation of the matrix element.

Start with the first order perturbation theory term. All the terms in the sum that do not annihilate the initial state photon and create the final state photon give zero. We will assume that the wavelength of the photon's is long compared to the size of the atom so that .

This is the matrix element
.
The **amplitude to be in the final state
is given by first order
time dependent perturbation theory**.

Recall that the absolute square of the time integral will turn into . We will carry along the integral for now, since we are not yet ready to square it.

Now we very carefully put the interaction term
into the formula for **second order time dependent perturbation theory**, again using
.
Our notation is that the
**intermediate state of atom and field is called**
where
represents the state of the atom and we may have zero or two photons, as indicated in the diagram.

We can understand this formula as a second order transition from state to state through all possible intermediate states. The transition from the initial state to the intermediate state takes place at time . The transition from the intermediate state to the final state takes place at time .

The **space-time diagram** below shows the three terms in
Time is assumed to run upward in the diagrams.

Looking again at the formula for the second order scattering amplitude, note that we integrate over the times and and that . For diagram (a), the annihilation operator is active at time and the creation operator is active at time . For diagram (b) its just the opposite. The second order formula above contains four terms as written. The and terms are the ones described by the diagram. The and terms will clearly give zero. Note that we are just picking the terms that will survive the calculation, not changing any formulas.

Now, reduce to the two nonzero terms.
The operators just give a factor of
and make the photon states work out.
If
is the intermediate atomic state, the **second order term reduces to**.

The terms coming from the integration over can be dropped. We can anticipate that the integral over will eventually give us a delta function of energy conservation, going to infinity when energy is conserved and going to zero when it is not. Those terms can never go to infinity and can therefore be neglected. When the energy conservation is satisfied, those terms are negligible and when it is not, the whole thing goes to zero.

We have calculated all the amplitudes.
The **first order and second order amplitudes should be combined, then squared**.

Note that the delta function has enforced energy conservation requiring that , but

The final step to a **differential cross section** is to divide the transition rate by the
**incident flux of particles**.
This is a surprisingly easy step because we are using plane waves of photons.
The initial state is **one particle in the volume moving with a velocity of **,
so the flux is simply
.

The

This is called the

Also note that the formula yields an infinite result if
.
This is not a physical result.
In fact the cross section will be large but not infinite when energy is conserved in the intermediate state.
This condition is often refereed to as ``the intermediate state being **on the mass shell**'' because
of the relation between energy and mass in four dimensions.