# 费曼-海尔曼定理

（重定向自费曼－海尔曼定理

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

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

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

$i\hbar\frac{\partial\Psi_\lambda(t)}{\partial t}=H_\lambda\Psi_\lambda(t)$

### 证明

\begin{align} \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{align}

## 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. Berlin: Springer. 1933: 162.
3. ^ Hellmann, H. Einführung in die Quantenchemie. Leipzig: Franz Deuticke. 1937: 285. OL OL21481721M.
4. ^ Feynman, R. P. Forces in Molecules. Phys. Rev. 1939, 56 (4): 340. Bibcode:1939PhRv...56..340F. doi:10.1103/PhysRev.56.340.