Crucial step Quantum mechanics

[Gavroy]

hi

i am just doing some quantum mechanics. unfortunately, i have spent the last 2 hours on one single step, which i do not understand at all.

i don't get it, how you get from (19) to (20) in the attached file?

could somebody explain this to me?

 

[jtbell]

It might help people in deciding whether to download the PDF, to know that your question is about a fairly advanced topic: calculating the van der Waals forces between two hydrogen atoms in the ground state.

(Unfortunately I can't help.)

 

[Bill_K]

Isn't it just straightforward? If there's something subtle, I'm missing it too.

Σ_i |<B_i|Q|A>|² = Σ_i <A|Q| B_i><B_i|Q|A>

Completeness says Σ_i |B_i><B_i| = I

so the result is <A|Q²|A>

 

[Gavroy]

oh ya...thanks

 

[Gavroy]

sorry, I overlooked something:

Σ_i |<B_i|Q|A>|² = Σ_i <A|Q| B_i><B_i|Q|A>

could you explain this relation to me?

 

[dextercioby]

Well, you know that in algebra $z z^{*} = |z|^2$, right ? Now pick $z= \langle A|Q|B_i\rangle$.

 

[Gavroy]

okay...i got it...thanks