Chap 1 Manybody wave function and Second quantization

i=1 H(ri); (5) HΨ(r1, r2, · · · ) = EΨ(r1, r2, · · · ). (6) (Note: For simplicity, unless necessary, we often write r as r.) Since there is no interaction between particles, the Schrodinger equation is separable. Assume Ψ(r1, r2, · · · , rN ) = φ(r1)φ(r2) · · ·φ(rN ), (7) then Hφαi(ri) = εαiφαi(ri), i = 1, 2, · · ·N. (8) Manybody eigenstate is a product of 1-particle states, Ψα1,α2,···(r1, r2, · · · , rN ) = φα1(r1)φα2(r2) · · ·φαN (rN ) (9) ≡ (r1, r2, · · · , rN |Ψα1,α2,··· ,αN ), with eigenvalues Eα1,α2,··· = εα1 + εα2 + · · ·+ εαN . (10) Notice that we have used a round bracket | · · · ) to represent a product state. The angular bracket | · · · 〉 is reserved for later use. Orthogonality: ( Ψα′1,α′2,··· ,αN |Ψα1,α2,··· ,αN ) = δα1α1δα2α2 · · · δαN αN (11) Completeness: ∑ α1α2···αN |Ψα1α2···αN )(Ψα1α2···αN | = 1. (12)