Photodetachment Cross Section Formulas: Explained

[Dahaka14]

I know the formulas for the length and velocity forms of the photodetachment cross section for an electron with energy $$\hbar\omega$$ are, respectively,

$$\sigma_{L}(\omega)=\frac{4\pi^{2}\alpha a_{0}^{2}\omega}{3}\sum_{f}|\langle\Psi_{f}|\sum_{j=1}^{n}z_{j}|\Psi_{i}\rangle|^{2}$$

and

$$\sigma_{V}(\omega)=\frac{4\pi^{2}\alpha a_{0}^{2}\omega}{3}\sum_{f}|\langle\Psi_{f}|\sum_{j=1}^{n}\frac{\nabla_{j}^{z}}{i\omega}|\Psi_{i}\rangle|^{2}.$$

Could someone please explain what these actually mean and how they are derived, or even better, where I can find a good reference on them? I asked my research professor about them and he was no help (these are where I got them from), and I can't find anything about them on the internet.

 

[olgranpappy]

I know the formulas for the length and velocity forms of the photodetachment cross section for an electron with energy $$\hbar\omega$$ are, respectively,

$$\sigma_{L}(\omega)=\frac{4\pi^{2}\alpha a_{0}^{2}\omega}{3}\sum_{f}|\langle\Psi_{f}|\sum_{j=1}^{n}z_{j}|\Psi_{i}\rangle|^{2}$$

and

$$\sigma_{V}(\omega)=\frac{4\pi^{2}\alpha a_{0}^{2}\omega}{3}\sum_{f}|\langle\Psi_{f}|\sum_{j=1}^{n}\frac{\nabla_{j}^{z}}{i\omega}|\Psi_{i}\rangle|^{2}.$$

Could someone please explain what these actually mean and how they are derived, or even better, where I can find a good reference on them? I asked my research professor about them and he was no help (these are where I got them from), and I can't find anything about them on the internet.

try Sakurai's quantum mechanics book. Look up Fermi's Golden Rule.