Solving Density of States Problems in Molecular Quantum Electro Dynamics

[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.

Thank you all in advance
Scott

 

[eljose79]

Dear Scott,

Thank you for sharing your question with us. As a scientist in the field of Molecular Quantum Electro Dynamics, I can understand the challenges you are facing in determining the density of final states for your research. This is indeed a common problem in our field and there are a few ways to approach it.

Firstly, I would suggest looking into the concept of phase space density. This is a useful tool in situations where there is no well-defined quantization volume. It takes into account the momentum and position variables of the system, and can be used to calculate the density of final states.

Secondly, for situations where you want to consider only one direction, you can use the concept of partial density of states. This allows you to integrate over a specific region of momentum space, rather than the entire space. This can be useful in cases where there is a preferred direction for radiation emission.

Lastly, another approach could be to use the Wigner distribution function, which takes into account both the position and momentum variables and can be used to calculate the density of final states in a more rigorous manner.

I hope these suggestions are helpful to you in your research. Best of luck in finding a suitable solution for your specific situation.