【SEOテキスト】宇田雄一,06.2.11,§1-3.時間発展と描像,[3]相互作用描像,(0)ハミルトニアン,H(t)=H0+H1(t),H0=H0†,H1(t)=H1(t)†,①基底と表示,B(I)(t)≡{|e(I)(f;t)>|f∈I},|e(I)(f;t)>≡U0(t)|e(s)(f)>,[ih(d/dt)U0(t)≡H0U0(t),U0(0)≡1,∴U0(t)=exp[-(i/h)tH0],U0(t1,t2)≡U0(t1)U0(t2)-1=exp[-(i/h)(t1-t2)H0]],|Ψ>≡∫dnfψ(I)(f;t)|e(I)(f;t)>,Ω|e(I)(f;t)>≡∫dngω(I)(g;f;t)|e(I)(g;t)>,②ベクトルと演算子,|ΨI(t)>≡∫dnfψ(I)(f;t)|e(s)(f)>,ΩI(t)|e(s)(f)>≡∫dngω(I)(g;f;t)|e(s)(g)>,|ΨI(t)>=U0(t)-1|Ψ>,ΩI(t)=U0(t)-1ΩU0(t),③時間発展,(ⅰ)|ΨI(t)>≡|Ψs(t)I(t)>,U1(t)≡U0(t)-1U(t),ΩI(t)≡Ω(t)I(t)=U0(t)-1Ω(t)U0(t),(ⅱ)ih(d/dt)U1(t)=H1I(t)U1(t),U1(0)=1,|ΨI(t)>=U1(t)|ΨH>,U1(t)†=U1(t)-1,ih(d/dt)|ΨI(t)>=H1I(t)|ΨI(t)>,(d/dt)ΩI(t)=(i/h)[H0I(t),ΩI(t)]-+[(d/dt)Ω(t)]I(t),(ⅲ)U1(t1,t2)≡U1(t1)U1(t2)-1,|ΨI(t1)>=U1(t1,t2)|ΨI(t2)>,U1(t1,t2)†=U1(t1,t2)-1=U1(t2,t1),U1(t,t)=1,U1(t1,t2)U1(t2,t3)=U1(t1,t3),ih(∂/∂t1)U1(t1,t2)=H1I(t1)U1(t1,t2),(ⅳ)ΩH(t)=U1(t)-1ΩI(t)U1(t)
|
