# Fock space and the Brillouin condition

1. May 5, 2013

### beans73

Hi there! in a recent lecture on fock space, i was given the brillouin condition for two-particle operators:-

<$\Phi_{0}|a^{†}_{a}a_{r}h|\Phi_{0}$>$= \frac{1}{2}\sum\sum$<$\Phi_{0}|a^{†}_{a}a_{r}a^{†}_{\lambda}a^{†}_{\mu}a_{\lambda'}a_{\mu'}$|$\Phi_{0}$><$\lambda\mu|g|\mu'\lambda'$>$= \sum[$<$rb|g|ab$>$-$<$rb|g|ba$>$]$

where h is the single particle hamiltonian, and g is the two body operator.

i'm just not quite sure of the proof of this. the textbook for this subject is sakurai's modern quantum mechanics, and it doesn't really cover fock space, so any help pointing me in a direction to look into this a bit more would be greatly appreciated :)

2. May 6, 2013

### DrDu

Try to express the full hamiltonian in terms of h and g.

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