量子力學中,費曼–海爾曼定理描述的是一個體系的能量對某個參量的導數與哈密頓量算符對同一參量的導數的期望值之間的關係。根據這一定理,通過求解薛定諤方程得到電子密度的空間分佈後,體系中的所有力都能通過經典靜電學求出。
該理論分別被不同的物理學家獨立地證明過,包括Paul Güttinger(1932)[1]、泡利(1933)[2]、海爾曼 (1937)[3]以及費曼(1939)。[4]
該定理的表達式如下:
式中
表示依賴於連續變化的參變量
的哈密頓量;
是該哈密頓量的本徵函數,通過哈密頓量間接依賴於
;
為能量,即哈密頓量的本徵值;
為積分微元。上述積分在全空間進行。
隨時間變化的波函數的費曼–海爾曼定理
因為一個一般的隨時間變化的波函數滿足含時薛定諤方程,所以費曼–海爾曼定理不再適用。
但是,該波函數滿足以下關係:
![{\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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ecdf7d998ea383a438ee88e36c8412b8f4a99ff7)
其中ψ滿足:
![{\displaystyle i\hbar {\frac {\partial \Psi _{\lambda }(t)}{\partial t}}=H_{\lambda }\Psi _{\lambda }(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26ff72eb5ba71575bf0bbdac5310b63a147cdd0e)
證明
以下證明只依賴於薛定諤方程,以及對於λ和t求偏導時,可以互換順序的假設。
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea16bf3f5a858d9db808b692b390d6a54b77e0f9)
參考