# 埃伦费斯特定理

## 维基百科，自由的百科全书

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

## 导引

{\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))}

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

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

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

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

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

## 实例

### 保守的哈密顿量

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

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

### 位置的期望值对于时间的导数

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

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

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

### 动量的期望值对于时间的导数

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

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

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

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

## 经典极限

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

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

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

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

1. 一个是量子态对于位置的不可确定性。
2. 另一个则是位势随着位置而变化的快缓。

## 参考文献

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.