Why the photoelectric absorption section finite at threshold

[zweiling]

I mean the photoelectric effect of the hydrogen atom.

It is weird. By the Fermi golden rule, the transition or absorption rate is proportional to the density of the final states. At threshold, the electron has zero momentum and thus zero density of state. Therefore, the absorption coefficient should be zero at the threshold. Where does this argument go wrong?

 

[mfb]

Exactly zero momentum is not a proper state. Every finite transition time gives a small momentum range.

 

[zweiling]

Exactly zero momentum is not a proper state. Every finite transition time gives a small momentum range.

anyway, close to the threshold, the absorption section should be close to zero. But in reality, it is finite.