In derivation of the BdG mean field Hamiltonian as follows, I have a confusion here in the second step:
HMF−eff=∫d3rψ†↑(r)HE(r)ψ↑(r)+∫d3rψ†↓(r)HE(r)ψ↓(r)+∫d3r△⋆(r)ψ↓(r)ψ↑(r)+∫d3rψ†↑(r)ψ†↓(r)△(r)−∫d3r|△(r)|2U
=∫d3rψ†↑(r)HE(r)ψ↑(r)−∫d3rψ↓(r)H⋆E(r)ψ†↓(r)+∫d3r△⋆(r)ψ↓(r)ψ↑(r)+∫d3rψ†↑(r)ψ†↓(r)△(r)−∫d3r|△(r)|2U
=∫d3r(ψ†↑(r)ψ↓(r))(HE(r)△(r)△⋆(r)−H⋆E(r))(ψ↑(r)ψ†↓(r))+const.
with
HE(r)=−ℏ22m∇2
In the second step, we have taken
∫d3rψ†↓(r)∇2ψ↓(r)=−∫d3rψ↓(r)∇2ψ†↓(r)............(1).
I can prove (by integration by parts and putting the surface terms to 0) that ∫d3rψ†↓(r)∇2ψ↓(r)=∫d3r∇2ψ†↓(r)ψ↓(r)
but how is it justified to now take
∫d3r∇2ψ†↓(r)ψ↓(r)=−∫d3rψ↓(r)∇2ψ†↓(r)
in order to prove (1) ?
This post imported from StackExchange Physics at 2014-03-23 12:23 (UCT), posted by SE-user user38579