# 费曼-海尔曼定理

${\displaystyle {\frac {{\rm {d}}E}{{\rm {d}}{\lambda }}}=\int {\psi ^{*}(\lambda ){\frac {{\rm {d}}{{\hat {H}}_{\lambda }}}{{\rm {d}}{\lambda }}}\psi (\lambda )\ {\rm {d}}\tau },}$

• ${\displaystyle {\hat {H}}_{\lambda }}$ 表示依赖于连续变化的参变量${\displaystyle \lambda }$的哈密顿量；
• ${\displaystyle \psi (\lambda )\,}$ 是该哈密顿量的本征函数，通过哈密顿量间接依赖于${\displaystyle \lambda }$
• ${\displaystyle E\,}$ 为能量，即哈密顿量的本征值；
• ${\displaystyle {\rm {d}}\tau }$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

${\displaystyle {\bigg \langle }\Psi _{\lambda }(t){\bigg |}{\frac {\partial H_{\lambda }}{\partial \lambda }}{\bigg |}\Psi _{\lambda }(t){\bigg \rangle }=i\hbar {\frac {\partial }{\partial t}}{\bigg \langle }\Psi _{\lambda }(t){\bigg |}{\frac {\partial \Psi _{\lambda }(t)}{\partial \lambda }}{\bigg \rangle }}$

${\displaystyle i\hbar {\frac {\partial \Psi _{\lambda }(t)}{\partial t}}=H_{\lambda }\Psi _{\lambda }(t)}$

### 证明

{\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

1. ^ Güttinger, P. Das Verhalten von Atomen im magnetischen Drehfeld. Z. Phys. 1932, 73 (3–4): 169. Bibcode:1932ZPhy...73..169G. doi:10.1007/BF01351211.
2. ^ Pauli, W. Principles of Wave Mechanics. Handbuch der Physik 24. Berlin: Springer. 1933: 162.
3. ^ Hellmann, H. Einführung in die Quantenchemie. Leipzig: Franz Deuticke. 1937: 285. OL 21481721M.
4. ^ Feynman, R. P. Forces in Molecules. Phys. Rev. 1939, 56 (4): 340. Bibcode:1939PhRv...56..340F. doi:10.1103/PhysRev.56.340.