# 埃倫費斯特定理

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

## 導引

\begin{align} \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{align}

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

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

$(H\Phi)^*=\Phi^*H^*=\Phi^*H$

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

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

## 實例

### 保守的哈密頓量

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

$\langle H\rangle=H_0$

### 位置的期望值對於時間的導數

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

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

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

### 動量的期望值對於時間的導數

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

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

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

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

## 經典極限

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

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

$V\,'(x)=V\,'(x_0)+(x-x_0)V\,''(x_0)+\frac{1}{2}(x-x_0)^2 V\,'''(x_0)+\ \dots$

$\left\langle\frac{\partial V(x)}{\partial x}\right\rangle\approx V\,'(x_0)+\frac{1}{2}\ \sigma_x^2\ V\,''(x_0)$

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.