Rate expression for emission
Photon density of states
We consider cavity modes (i.e. electromagnetic radiation) in a volume V. For simplicity we consider this to be a cubic box of length L. This gives us a "wave in a box" analogous to the "particle in a box" in the sense that there are boundary conditions on a plane wave solution. In particular we require that
for x with similar expressions for the y and z components. This is also equivalent to requiring that an integral number of wavelengths fit into each dimension of the box L = n
l. Since k = 2p/l we also have k = n2p/L.The three components of the allowed k-vectors are:
To get the total number of modes N = n x m x l we use the above expression
The differential number of modes in a differential volume of k-vector space is
We substitute V = L3 and convert to spherical polar coordinates
We can express the angular terms as the solid angle d
W = sinqdqdf. We also substitute w = kc.
Since
we can also write this as
Therefore the number of states per unit energy is
