フェルミの黄金律
⟨ ψ f | ψ ( t ) ⟩ = − i ħ ∫ 0 t d t ′ ⟨ ψ f | e − i H ^ 0 ( t − t ′ ) / ħ H ′ ^ ( t ′ ) e − i H 0 ^ t ′ / ħ | ψ i ⟩ = − i ħ e − i E f t / ħ ⟨ ψ f | H ′ ^ | ψ i ⟩ ∫ 0 t d t ′ e i ( E f − E i − ħ ω ) t ′ / ħ = − 2 i e i ( E f − E i − ħ ω ) t / 2 ħ ⟨ ψ f | H ′ ^ | ψ i ⟩ E f − E i − ħ ω sin  ( E f − E i − ħ ω ) t 2 ħ {\displaystyle {\begin{aligned}\langle \psi _{\mathrm {f} }|\psi (t)\rangle &=-{\frac {i}{\hbar }}\int _{0}^{t}dt'\langle \psi _{\mathrm {f} }|e^{-i{\hat {H}}_{0}(t-t')/\hbar }{\hat {H'}}(t')e^{-i{\hat {H_{0}}}t'/\hbar }|\psi _{\mathrm {i} }\rangle \\&=-{\frac {i}{\hbar }}e^{-iE_{\mathrm {f} }t/\hbar }\langle \psi _{\mathrm {f} }|{\hat {H'}}|\psi _{\mathrm {i} }\rangle \int _{0}^{t}dt'e^{i(E_{\mathrm {f} }-E_{\mathrm {i} }-\hbar \omega )t'/\hbar }\\&=-2ie^{i(E_{\mathrm {f} }-E_{\mathrm {i} }-\hbar \omega )t/2\hbar }{\frac {\langle \psi _{\mathrm {f} }|{\hat {H'}}|\psi _{\mathrm {i} }\rangle }{E_{\mathrm {f} }-E_{\mathrm {i} }-\hbar \omega }}\sin {\frac {(E_{\mathrm {f} }-E_{\mathrm {i} }-\hbar \omega )t}{2\hbar }}\end{aligned}}}