# Density of States

1. Aug 16, 2010

### xGAME-OVERx

Dear all,

I am doing summer research in the field of Molecular Quantum Electro Dynamics, and a persistent problem that I (and occasionally the group I am working with) have/has is knowing what to use as the density of final states (radiation states).

It is required in the Fermi Golden Rule, when calculating a rate:

$$\Gamma={{2\pi}\over{\hbar}}|M_{fi}|^2\rho$$

The most accepted definition in all the textbooks I can find is:

$$\rho={{k^2d\Omega}\over{(2\pi)^3{\hbar}c}}V$$

Which is useful, except for in two circumstances:
1) When there is not a well defined quantisation volume (i.e. all of space)
2) When we wish to consider one single direction, so cannot integrate over solid angles

Does anyone here know how to deal with these situations, or of an alternative, more useful, definition?

I have tried solving 1) by choosing an appropriate quantisation volume, but the answer one gets is rather dependent upon the volume chosen.

I have seen a book solve 2) by forming an intensity of emitted radiation instead of a rate, but this is of no use for my purposes (an intensity includes units $$m^{-2}$$, making it impossible to recover a rate in $$s^{-1}$$ because a single direction has zero area).
I have tried to solve 2) by invoking uncertainty principles, but this is a bit hand waving and not very rigorous.