S行列
⟨ Φ f | ψ I ( + ∞ ) ⟩ = ⟨ Φ f | U ^ ( + ∞ , − ∞ ) | Φ i ⟩ = ∑ j ⟨ Φ f | Φ j ⟩ S j , i = ∑ j δ f , j S j , i = S f , i {\displaystyle \langle \Phi _{f}|\psi _{I}(+\infty )\rangle =\langle \Phi _{f}|{\hat {U}}(+\infty ,-\infty )|\Phi _{i}\rangle =\sum _{j}\langle \Phi _{f}|\Phi _{j}\rangle S_{j,i}=\sum _{j}\delta _{f,j}S_{j,i}=S_{f,i}}
である。すなわち
S f , i = ⟨ Φ f | U ^ ( + ∞ , − ∞ ) | Φ i ⟩ {\displaystyle S_{f,i}=\langle \Phi _{f}|{\hat {U}}(+\infty ,-\infty )|\Phi _{i}\rangle }