# 埃倫費斯特定理

${\displaystyle {\frac {d}{dt}}\langle A\rangle ={\frac {1}{i\hbar }}\langle [A,\ H]\rangle +\left\langle {\frac {\partial A}{\partial t}}\right\rangle }$

## 導引

{\displaystyle {\begin{aligned}{\frac {d}{dt}}\langle A\rangle &={\frac {d}{dt}}\int \Phi ^{*}A\Phi ~dx\\&=\int \left({\frac {\partial \Phi ^{*}}{\partial t}}\right)A\Phi ~dx+\int \Phi ^{*}\left({\frac {\partial A}{\partial t}}\right)\Phi ~dx+\int \Phi ^{*}A\left({\frac {\partial \Phi }{\partial t}}\right)~dx\\&=\int \left({\frac {\partial \Phi ^{*}}{\partial t}}\right)A\Phi ~dx+\left\langle {\frac {\partial A}{\partial t}}\right\rangle +\int \Phi ^{*}A\left({\frac {\partial \Phi }{\partial t}}\right)~dx\\\end{aligned}}}

${\displaystyle H\Phi =i\hbar {\frac {\partial \Phi }{\partial t}}}$

${\displaystyle (H\Phi )^{*}=-i\hbar {\frac {\partial \Phi ^{*}}{\partial t}}}$

${\displaystyle (H\Phi )^{*}=\Phi ^{*}H^{*}=\Phi ^{*}H}$

${\displaystyle {\frac {d}{dt}}\langle A\rangle ={\frac {1}{i\hbar }}\int \Phi ^{*}(AH-HA)\Phi ~dx+\left\langle {\frac {\partial A}{\partial t}}\right\rangle }$

${\displaystyle {\frac {d}{dt}}\langle A\rangle ={\frac {1}{i\hbar }}\langle [A,\ H]\rangle +\left\langle {\frac {\partial A}{\partial t}}\right\rangle }$

## 實例

### 守恆的哈密頓量

${\displaystyle {\frac {d}{dt}}\langle H\rangle ={\frac {1}{i\hbar }}\langle [H,\ H]\rangle +\left\langle {\frac {\partial H}{\partial t}}\right\rangle =\left\langle {\frac {\partial H}{\partial t}}\right\rangle }$

${\displaystyle \langle H\rangle =H_{0}}$

### 位置的期望值對於時間的導數

${\displaystyle H(x,\ p,\ t)={\frac {p^{2}}{2m}}+V(x,\ t)}$ ;

${\displaystyle {\frac {d}{dt}}\langle x\rangle ={\frac {1}{i\hbar }}\langle [x,\ H]\rangle +\left\langle {\frac {\partial x}{\partial t}}\right\rangle ={\frac {1}{i\hbar }}\langle [x,\ H]\rangle ={\frac {1}{i2m\hbar }}\langle [x,\ p^{2}]\rangle ={\frac {1}{i2m\hbar }}\langle xpp-ppx\rangle }$

${\displaystyle {\frac {d}{dt}}\langle x\rangle ={\frac {1}{m}}\langle p\rangle =\langle v\rangle }$

### 動量的期望值對於時間的導數

${\displaystyle {\frac {d}{dt}}\langle p\rangle ={\frac {1}{i\hbar }}\langle [p,\ H]\rangle +\left\langle {\frac {\partial p}{\partial t}}\right\rangle }$

${\displaystyle {\frac {d}{dt}}\langle p\rangle ={\frac {1}{i\hbar }}\langle [p,\ V]\rangle }$

${\displaystyle {\frac {d}{dt}}\langle p\rangle =\int \Phi ^{*}V{\frac {\partial }{\partial x}}\Phi ~dx-\int \Phi ^{*}{\frac {\partial }{\partial x}}\left(V\Phi \right)~dx}$

${\displaystyle {\frac {d}{dt}}\langle p\rangle =\left\langle -\ {\frac {\partial }{\partial x}}V\right\rangle =\langle F\rangle }$

## 經典極限

${\displaystyle {\frac {d}{dt}}\langle x\rangle =\langle v\rangle }$
${\displaystyle {\frac {d}{dt}}\langle p\rangle =-\ {\frac {\partial V(\langle x\rangle )}{\partial \langle x\rangle }}}$

${\displaystyle {\frac {dx}{dt}}=v}$
${\displaystyle {\frac {dp}{dt}}=-\ {\frac {\partial V(x)}{\partial x}}}$

${\displaystyle V\,'(x)=V\,'(x_{0})+(x-x_{0})V\,''(x_{0})+{\frac {1}{2}}(x-x_{0})^{2}V\,'''(x_{0})+\ \dots }$

${\displaystyle \left\langle {\frac {\partial V(x)}{\partial x}}\right\rangle \approx V\,'(x_{0})+{\frac {1}{2}}\ \sigma _{x}^{2}\ V\,''(x_{0})}$

1. 一個是量子態對於位置的不可確定性。
2. 另一個則是位勢隨著位置而變化的快緩。

## 參考文獻

1. ^ Smith, Henrik. Introduction to Quantum Mechanics. World Scientific Pub Co Inc. 1991: pp. 108–109. ISBN 978-9810204754.
2. ^ Tannor, David J. Introduction to Quantum Mechanics: A Time-Dependent Perspective. University Science Books. 2006: pp. 35–38. ISBN 978-1891389238.