We are in a position to fairly easily calculate the spectrum of Black Body radiation.
Assume there is a cavity with a radiation field on the inside
and that the field interacts with the atoms of the cavity.
Assume thermal equilibrium is reached.
Let's take two atomic states that can make transitions to each other:
From statistical mechanics, we have
and for equilibrium we must have
We have previously calculated the emission and absorption rates.
We can calculate the ratio between the emission and absorption rates per atom:
where the sum is over atomic electrons.
The matrix elements are closely related.
We have used the fact that
The two matrix elements are simple complex conjugates of each other
so that when we take the absolute square, they are the same.
Therefore, we may cancel them.
Now suppose the walls of the cavity are black so that they emit and absorb photons at any energy.
Then the result for the number of photons above is true for all the radiation modes of the cavity.
The energy in the frequency interval
per unit volume can be calculated by
multiplying the number of photons by the energy per photon times the number of modes in that frequency interval
and dividing by the volume of the cavity.
This was the formula Plank used to start the revolution.