D.41 Derivation of the Einstein B coefficients

The purpose of this note is to derive the Einstein

Unlike what you may find elsewhere, it will not be assumed that the
atoms are either fully in the high or fully in the low energy state.
That is a highly unsatisfactory assumption for many reasons. For one
thing it assumes that the atoms know what you have selected as

Since both the electromagnetic field and the collisions are random, a
statistical rather than a determinate treatment is needed. In it, the
probability that a randomly chosen atom can be found in the lower
energy state

It is assumed that the collisions are globally elastic in the sense
that they do not change the average energy picture of the atoms. In
other words, they do not affect the average probabilities of the
eigenfunctions

The evolution equations of the coefficients

Further, because the equations are linear, the solution for the
coefficients

Here

Now consider what happens to the probability of an atom to be in the
excited state in the time interval between collisions:

Here

Because the typical time between collisions

Therefore, if the change in probability

Now if this is averaged over all atoms and time intervals between collisions, the first term in the right hand side will average away. The reason is that it has a random phase angle, for one since those of

Summing the changes in the probabilities therefore means summing the changes in the square magnitudes of

If the above expression for the average change in the probability of
the high energy state is compared to (7.46), it is seen
that the Einstein coefficient

Now the needed

What is left is

(This really assumes that the particles are in a very large periodic box so that the electromagnetic field is given by a Fourier series; in free space you would need to integrate over the wave numbers instead of sum over them.) The square magnitude is then

where the final equality comes from the assumption that the radiation is incoherent, so that the phases of different waves are uncorrelated and the corresponding products average to zero.

The bottom line is that square magnitudes must be summed together to
find the total contribution of all waves. And the square magnitude of
the contribution of a single wave is, according to (D.25)
above,

Now broadband radiation is described in terms of an electromagnetic
energy density

If a change of integration variable is made to

Recall that a starting assumption underlying these derivations was that

Note that this is essentially the same analysis as the one for
Fermi’s golden rule, except for the presence of the given field
strength

Consider for a second the limiting process that the field strength

In any case, while the term

This must still be averaged over all directions of wave propagation
and polarization. That gives:

where

To see why, consider the electromagnetic waves propagating along any axis, not just the

The Einstein coefficient