Derivation
We want to derive the Hamiltonian H^=4Ec(a^1+a^1+a^1a^1+a^2+a^2+a^2a^2)−NEJ(a^1+a^2+a^2+a^1) from H^=∫Φ†[−2mℏ∇2+Vext+2gΦ†Φ]Φdr.
Assumptions:
- Our Wave function can be approximated as a combination of creation operators, i.e. Ψ≈Φ1(r)a1+Φ2(r)a2.
- ⟨Φ1,Φ2⟩=0, i.e. ∫RΦ1∗(r)Φ2(r)dr=0
- ∥Φi∥2=1
- Terms proportional to ∫RΦ1(r)2Φ2(r)2dr are neglected.
- ∫R∣Φ1(r)∣4dr=∫R∣Φ2(r)∣4dr
We start from
H^=∫Ψ†[−2mℏ∇2+Vext+2gΨ†Ψ]Ψdr=∫(Φ1(r)a1+Φ2(r)a2)†[−2mℏ∇2+Vext+2g(Φ1(r)a1+Φ2(r)a2)†(Φ1(r)a1+Φ2(r)a2)](Φ1(r)a1+Φ2(r)a2)dr=∫(Φ1∗(r)a1†+Φ2∗(r)a2†)[−2mℏ∇2+Vext+2g(Φ1∗(r)a1†+Φ2∗(r)a2†)(Φ1(r)a1+Φ2(r)a2)](Φ1(r)a1+Φ2(r)a2)dr=a1†∫Φ1∗(r)[−2mℏ∇2+Vext+2g(Φ1∗(r)a1†+Φ2∗(r)a2†)(Φ1(r)a1+Φ2(r)a2)]Φ1(r)dra1+a1†∫Φ1∗(r)[−2mℏ∇2+Vext+2g(Φ1∗(r)a1†+Φ2∗(r)a2†)(Φ1(r)a1+Φ2(r)a2)]Φ2(r)dra2+a2†∫Φ2∗(r)[−2mℏ∇2+Vext+2g(Φ1∗(r)a1†+Φ2∗(r)a2†)(Φ1(r)a1+Φ2(r)a2)]Φ1(r)dra1+a2†∫Φ2∗(r)[−2mℏ∇2+Vext+2g(Φ1∗(r)a1†+Φ2∗(r)a2†)(Φ1(r)a1+Φ2(r)a2)]Φ2(r)dra2=a1†∫Φ1∗(r)[−2mℏ∇2+Vext]Φ1(r)dra1+a1†2g∫Φ1∗(r)[(Φ1∗(r)a1†+Φ2∗(r)a2†)(Φ1(r)a1+Φ2(r)a2)]Φ1(r)dra1+a1†∫Φ1∗(r)[−2mℏ∇2+Vext]Φ2(r)dra2+a1†2g∫Φ1∗(r)[(Φ1∗(r)a1†+Φ2∗(r)a2†)(Φ1(r)a1+Φ2(r)a2)]Φ2(r)dra2+a2†∫Φ2∗(r)[−2mℏ∇2+Vext]Φ1(r)dra1+a2†2g∫Φ2∗(r)[(Φ1∗(r)a1†+Φ2∗(r)a2†)(Φ1(r)a1+Φ2(r)a2)]Φ1(r)dra1+a2†∫Φ2∗(r)[−2mℏ∇2+Vext]Φ2(r)dra2+a2†2g∫Φ2∗(r)[(Φ1∗(r)a1†+Φ2∗(r)a2†)(Φ1(r)a1+Φ2(r)a2)]Φ2(r)dra2=∫Φ1∗(r)[−2mℏ∇2+Vext]Φ1(r)dra1†a1+2g[∫Φ1∗(r)2Φ1(r)2dra1†2a12+∫Φ1∗(r)2Φ1(r)2dra1†2a12]+∫Φ1∗(r)[−2mℏ∇2+Vext]Φ2(r)dra1†a2+a1†2g∫Φ1∗(r)[(Φ1∗(r)a1†+Φ2∗(r)a2†)(Φ1(r)a1+Φ2(r)a2)]Φ2(r)dra2+∫Φ2∗(r)[−2mℏ∇2+Vext]Φ1(r)dra2†a1+a2†2g∫Φ2∗(r)[(Φ1∗(r)a1†+Φ2∗(r)a2†)(Φ1(r)a1+Φ2(r)a2)]Φ1(r)dra1+∫Φ2∗(r)[−2mℏ∇2+Vext]Φ2(r)dra2†a2+a2†2g∫Φ2∗(r)[(Φ1∗(r)a1†+Φ2∗(r)a2†)(Φ1(r)a1+Φ2(r)a2)]Φ2(r)dra2Ψ≈Φ1(r)a1†+Φ2(r)a2†Splitting the integralsSplitting the sum Pulling ai and ai† outPulling ai and ai† out