Proof of Bloch's theorem Ψ(x+a)=Ψ(x)=ekaΨ(x) letΨ(x)=eKxΦk(x) Ψ(x+a=eKaΨ(x)=eikaeikx④k(x) =eKx+aΦ(x+a) ∴.Φk(x+a)=Φk(x) Attention:E≠h2K2/2uProof of Bloch’s theorem (x a) (x) e (x) iKa       let (x) e (x) k iKx      ( ) ( ) ( ) ( ) e x a x a e x e e x k iK x a k iKa iKa iKx           (x a) (x)  k   k : / 2 2 2 Attention E   K