# 運動常數

## 辨認運動常數的方法

• 最簡單，但最無系統的方法是靠直覺。假設一個物理量是運動常數（或許是從分析實驗數據而得到的結論）。經過數學證明，可以論定，在物體的運動過程中，此量的值是保守的。
• 假若一個物理量${\displaystyle A}$，既不是顯性地含時間，又與哈密頓量帕松括號等於零，則此物理量是保守的：
${\displaystyle {\frac {dA}{dt}}={\frac {\partial A}{\partial t}}+[A,\ H]}$

## 量子力學

### 導引

 ${\displaystyle {\frac {d}{dt}}\langle Q\rangle }$ ${\displaystyle ={\frac {d}{dt}}\langle \psi |Q|\psi \rangle }$ ${\displaystyle =\left\langle {\frac {\partial \psi }{\partial t}}|Q|\psi \right\rangle +\left\langle \psi |{\frac {\partial Q}{\partial t}}|\psi \right\rangle +\left\langle \psi |Q|{\frac {\partial \psi }{\partial t}}\right\rangle }$ ${\displaystyle ={\frac {-1}{i\hbar }}\langle H\psi |Q|\psi \rangle +\left\langle \psi |{\frac {\partial Q}{\partial t}}|\psi \right\rangle +{\frac {1}{i\hbar }}\langle \psi |Q|H\psi \rangle }$ ${\displaystyle ={\frac {-1}{i\hbar }}\langle \psi |HQ|\psi \rangle +\left\langle \psi |{\frac {\partial Q}{\partial t}}|\psi \right\rangle +{\frac {1}{i\hbar }}\langle \psi |QH|\psi \rangle }$ ${\displaystyle ={\frac {-1}{i\hbar }}\langle \psi |\left[H,Q\right]|\psi \rangle +\left\langle \psi |{\frac {\partial Q}{\partial t}}|\psi \right\rangle }$；

${\displaystyle {\frac {d}{dt}}\langle Q\rangle =0}$

## 參考文獻

1. ^ Morin, David. Introduction to classical mechanics: with problems and solutions. Cambridge University Press. 2008: 138. ISBN 9780521876223.
• Griffiths, David J. Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. 2004. ISBN 0-13-805326-X.