费曼-海尔曼定理量子力学中，费曼–海尔曼定理描述的是一个体系的能量对某个参量的导数与哈密顿量算符对同一参量的导数的期望值之间的关系。根据这一定理，通过求解薛定谔方程得到电子密度的空间分布后，体系中的所有力都能通过经典静电学求出。 该理论分别被不同的物理学家独立地证明过，包括Paul Güttinger（1932）[1]、泡利（1933）[2]、海尔曼 （1937）[3]以及费曼（1939）。[4]该定理的表达式如下： d E d λ = ∫ ψ ∗ ( λ ) d H ^ λ d λ ψ ( λ ) d τ , {\displaystyle {\frac {{\rm {d}}E}{{\rm {d}}{\lambda }}}=\int {\psi ^{*}(\lambda ){\frac {{\rm {d}}{{\hat {H}}_{\lambda }}}{{\rm {d}}{\lambda }}}\psi (\lambda )\ {\rm {d}}\tau },} 式中 H ^ λ {\displaystyle {\hat {H}}_{\lambda }} 表示依赖于连续变化的参变量 λ {\displaystyle \lambda } 的哈密顿量； ψ ( λ ) {\displaystyle \psi (\lambda )\,} 是该哈密顿量的本征函数，通过哈密顿量间接依赖于 λ {\displaystyle \lambda } ； E {\displaystyle E\,} 为能量，即哈密顿量的本征值； d τ {\displaystyle {\rm {d}}\tau } 为积分微元。上述积分在全空间进行。随时间变化的波函数的费曼–海尔曼定理 因为一个一般的随时间变化的波函数满足含时薛定谔方程，所以费曼–海尔曼定理不再适用。 但是，该波函数满足以下关系： ⟨ Ψ λ ( t ) | ∂ H λ ∂ λ | Ψ λ ( t ) ⟩ = i ℏ ∂ ∂ t ⟨ Ψ λ ( t ) | ∂ Ψ λ ( t ) ∂ λ ⟩ {\displaystyle {\bigg \langle }\Psi _{\lambda }(t){\bigg |}{\frac {\partial H_{\lambda }}{\partial \lambda }}{\bigg |}\Psi _{\lambda }(t){\bigg \rangle }=i\hbar {\frac {\partial }{\partial t}}{\bigg \langle }\Psi _{\lambda }(t){\bigg |}{\frac {\partial \Psi _{\lambda }(t)}{\partial \lambda }}{\bigg \rangle }} 其中ψ满足： i ℏ ∂ Ψ λ ( t ) ∂ t = H λ Ψ λ ( t ) {\displaystyle i\hbar {\frac {\partial \Psi _{\lambda }(t)}{\partial t}}=H_{\lambda }\Psi _{\lambda }(t)} 证明 以下证明只依赖于薛定谔方程，以及对于λ和t求偏导时，可以互换顺序的假设。 ⟨ Ψ λ ( t ) | ∂ H λ ∂ λ | Ψ λ ( t ) ⟩ = ∂ ∂ λ ⟨ Ψ λ ( t ) | H λ | Ψ λ ( t ) ⟩ − ⟨ ∂ Ψ λ ( t ) ∂ λ | H λ | Ψ λ ( t ) ⟩ − ⟨ Ψ λ ( t ) | H λ | ∂ Ψ λ ( t ) ∂ λ ⟩ = i ℏ ∂ ∂ λ ⟨ Ψ λ ( t ) | ∂ Ψ λ ( t ) ∂ t ⟩ − i ℏ ⟨ ∂ Ψ λ ( t ) ∂ λ | ∂ Ψ λ ( t ) ∂ t ⟩ + i ℏ ⟨ ∂ Ψ λ ( t ) ∂ t | ∂ Ψ λ ( t ) ∂ λ ⟩ = i ℏ ⟨ Ψ λ ( t ) | ∂ 2 Ψ λ ( t ) ∂ λ ∂ t ⟩ + i ℏ ⟨ ∂ Ψ λ ( t ) ∂ t | ∂ Ψ λ ( t ) ∂ λ ⟩ = i ℏ ∂ ∂ t ⟨ Ψ λ ( t ) | ∂ Ψ λ ( t ) ∂ λ ⟩ {\displaystyle {\begin{aligned}{\bigg \langle }\Psi _{\lambda }(t){\bigg |}{\frac {\partial H_{\lambda }}{\partial \lambda }}{\bigg |}\Psi _{\lambda }(t){\bigg \rangle }&={\frac {\partial }{\partial \lambda }}\langle \Psi _{\lambda }(t)|H_{\lambda }|\Psi _{\lambda }(t)\rangle -{\bigg \langle }{\frac {\partial \Psi _{\lambda }(t)}{\partial \lambda }}{\bigg |}H_{\lambda }{\bigg |}\Psi _{\lambda }(t){\bigg \rangle }-{\bigg \langle }\Psi _{\lambda }(t){\bigg |}H_{\lambda }{\bigg |}{\frac {\partial \Psi _{\lambda }(t)}{\partial \lambda }}{\bigg \rangle }\\&=i\hbar {\frac {\partial }{\partial \lambda }}{\bigg \langle }\Psi _{\lambda }(t){\bigg |}{\frac {\partial \Psi _{\lambda }(t)}{\partial t}}{\bigg \rangle }-i\hbar {\bigg \langle }{\frac {\partial \Psi _{\lambda }(t)}{\partial \lambda }}{\bigg |}{\frac {\partial \Psi _{\lambda }(t)}{\partial t}}{\bigg \rangle }+i\hbar {\bigg \langle }{\frac {\partial \Psi _{\lambda }(t)}{\partial t}}{\bigg |}{\frac {\partial \Psi _{\lambda }(t)}{\partial \lambda }}{\bigg \rangle }\\&=i\hbar {\bigg \langle }\Psi _{\lambda }(t){\bigg |}{\frac {\partial ^{2}\Psi _{\lambda }(t)}{\partial \lambda \partial t}}{\bigg \rangle }+i\hbar {\bigg \langle }{\frac {\partial \Psi _{\lambda }(t)}{\partial t}}{\bigg |}{\frac {\partial \Psi _{\lambda }(t)}{\partial \lambda }}{\bigg \rangle }\\&=i\hbar {\frac {\partial }{\partial t}}{\bigg \langle }\Psi _{\lambda }(t){\bigg |}{\frac {\partial \Psi _{\lambda }(t)}{\partial \lambda }}{\bigg \rangle }\end{aligned}}} 参考 ^ Güttinger, P. Das Verhalten von Atomen im magnetischen Drehfeld. Z. Phys. 1932, 73 (3–4): 169. Bibcode:1932ZPhy...73..169G. doi:10.1007/BF01351211. ^ Pauli, W. Principles of Wave Mechanics. Handbuch der Physik 24. Berlin: Springer. 1933: 162. ^ Hellmann, H. Einführung in die Quantenchemie. Leipzig: Franz Deuticke. 1937: 285. OL 21481721M. ^ Feynman, R. P. Forces in Molecules. Phys. Rev. 1939, 56 (4): 340. Bibcode:1939PhRv...56..340F. doi:10.1103/PhysRev.56.340.
