ヘルマン–ファインマンの定理
d E ( λ ) d λ = d d λ ⟨ ψ λ | H ^ ( λ ) | ψ λ ⟩ = ( d d λ ⟨ ψ λ | ) H ^ ( λ ) | ψ λ ⟩ + ⟨ ψ λ | d H ^ ( λ ) d λ | ψ λ ⟩ + ⟨ ψ λ | H ^ ( λ ) ( d d λ | ψ λ ⟩ ) = E ( λ ) ( d d λ ⟨ ψ λ | ) | ψ λ ⟩ + ⟨ ψ λ | d H ^ ( λ ) d λ | ψ λ ⟩ + E ( λ ) ⟨ ψ λ | ( d d λ | ψ λ ⟩ ) = E ( λ ) d d λ ⟨ ψ λ | ψ λ ⟩ + ⟨ ψ λ | d H ^ ( λ ) d λ | ψ λ ⟩ {\displaystyle {\begin{aligned}{\frac {\mathrm {d} E(\lambda )}{\mathrm {d} \lambda }}&={\frac {\mathrm {d} }{\mathrm {d} \lambda }}\left\langle \psi _{\lambda }\left|{\hat {H}}(\lambda )\right|\psi _{\lambda }\right\rangle \\&=\left({\frac {\mathrm {d} }{\mathrm {d} \lambda }}\langle \psi _{\lambda }|\right){\hat {H}}(\lambda )|\psi _{\lambda }\rangle +\left\langle \psi _{\lambda }\left|{\frac {\mathrm {d} {\hat {H}}(\lambda )}{\mathrm {d} \lambda }}\right|\psi _{\lambda }\right\rangle +\langle \psi _{\lambda }|{\hat {H}}(\lambda )\left({\frac {\mathrm {d} }{\mathrm {d} \lambda }}|\psi _{\lambda }\rangle \right)\\&=E(\lambda )\left({\frac {\mathrm {d} }{\mathrm {d} \lambda }}\langle \psi _{\lambda }|\right)|\psi _{\lambda }\rangle +\left\langle \psi _{\lambda }\left|{\frac {\mathrm {d} {\hat {H}}(\lambda )}{\mathrm {d} \lambda }}\right|\psi _{\lambda }\right\rangle +E(\lambda )\langle \psi _{\lambda }|\left({\frac {\mathrm {d} }{\mathrm {d} \lambda }}|\psi _{\lambda }\rangle \right)\\&=E(\lambda ){\frac {\mathrm {d} }{\mathrm {d} \lambda }}\langle \psi _{\lambda }|\psi _{\lambda }\rangle +\left\langle \psi _{\lambda }\left|{\frac {\mathrm {d} {\hat {H}}(\lambda )}{\mathrm {d} \lambda }}\right|\psi _{\lambda }\right\rangle \end{aligned}}}