Understanding Time-Dependent Perturbation Theory's 2nd Order Term

[actionintegral]

Time dependent perturbation theory...

second order term...

For some reason they replace

<E_{n}|H_{0}^2|E_{m}>

with

\Sigma<E_{n}|H_{0}|E_{i}><E_{i}|H_{0}|E_{m}>

I know why they are allowed to do this, what I don't understand is how it makes my life better?

 

[Meir Achuz]

The fact that \sum|E_i><E_i|=1 is called the completeness property of the states E_i>. It means that you have enough states to expand a function in.
Look at the Fourier sin series. There sin(2pi x/L) form a complete set.
It becomes useful when the |E_i> are eigenstates of H_0, while the |E_m> are not.