Fermi's Golden Rule - Inelastic collision probability

[Master J]

Fermi's Golden Rule for an inelastic transition states gives the probability of a transition, due to a perturbation (which is constant in a given time interval).

For 2 energy states A and B, it is proportional, among others quantities, to what I'll call E. This E is the small range of energy states around the final state B ---> ie. the uncertainty, since we can't know 100% the final energy state (uncertainty principle).

Obviously, for an inelastic collision, B - A > E, otherwise we would have conservation of energy.

So, I think I have all that right! But here is my question...the above mentioned probability, is it time INdependent? I have been told it is. However, it doesn't seem so to me:

B - A > E homework > E therefore, 2pih/t > E where h is h-bar and w the Bohr frequency between the 2 states.
So, as time goes on, it would seem to that the probability of finding, for example, an electron in a higher energy state would decrease, ie. it would have most likely relaxed by say photon or phonon emission.
Any insight in this? Am I right?

Perhaps...it is not explicitly time dependent??

 

[Bill_K]

What's contained in Fermi's Golden Rule is not a "E, a range of energies" Rather it's ρ(E), the density of states (number of final states per energy interval.) And the reason it's in there has nothing to do with the Heisenberg uncertainty principle, or because energy is being slightly unconserved.

It's in there because energy *is* precisely conserved, and the transition probability contains a delta function, δ(E_A - E_B). In order to get something finite, one sums over all final states having an energy in the immediate vicinity of E_B and defines a transition probability to this entire group of states.

The answer is time-independent because it's a first-order approximation. Does not take into account the fact that the occupation of the initial state will slowly decrease due to the interaction.

 

[Master J]

Ah, I have just realized that what I am looking at is in fact a precursor to FGR.

Sorry, I am not talking about FGR, which is a probability per unit time, i am talking about the probability itself. FGR is derived from the elastic consideration of this, and the equation I am talking about is from the inelastic. It is:

P = 2E. |V|^2. D / (B-A)^2 where D is the DOS, V is the matrix element.