Transition Rates / Squared Dirac Delta

[MisterX]

I am not understanding something from my textbook. This is related to Fermi's Golden rule. It's about what happens when the matrix element of the perturbation $H'$ ends up being a Dirac delta for chosen normalization. Here is Fermi's Golden rule.
$\Gamma_{ba} = 2\pi \left|\langle b \mid H'\mid a \rangle \right|^2 \delta\left(\omega_a - \omega_b \right)$
I don't understand what is meant by "...is the decay rate into all the particles in the volume $V$. The number of particles in $V$ is $V/(2\pi)^3$."

 

[ChrisVer]

You can check for why that V/2pi^3 appears here:
http://www2.ph.ed.ac.uk/~gja/qp/qp12.pdf
in the 2nd page.
In fact since in the momentum space the volume is such as he proves it, the number of particles will be the inverse.

 

[Meir Achuz]

$$\delta({\bf 0})=V/(2\pi)^3$$.