# 交換子

## 群論

G中两个元素gh交换子为元素

[g, h] = g−1h−1gh

## 環論

${\displaystyle [a,b]=ab-ba.}$

## 量子力學

${\displaystyle [{\hat {A}},{\hat {B}}]={\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}}}$

${\displaystyle [{\hat {A}},{\hat {B}}]=-[{\hat {B}},{\hat {A}}]}$
${\displaystyle [{\hat {A}},{\hat {B}}+{\hat {C}}]=[{\hat {A}},{\hat {B}}]+[{\hat {A}},{\hat {C}}],\quad [{\hat {A}}+{\hat {B}},{\hat {C}}]=[{\hat {A}},{\hat {C}}]+[{\hat {B}},{\hat {C}}]}$
${\displaystyle [{\hat {A}},{\hat {B}}{\hat {C}}]=[{\hat {A}},{\hat {B}}]{\hat {C}}+{\hat {B}}[{\hat {A}},{\hat {C}}],\quad [{\hat {A}}{\hat {B}},{\hat {C}}]=[{\hat {A}},{\hat {C}}]{\hat {B}}+{\hat {A}}[{\hat {B}},{\hat {C}}]}$
${\displaystyle [{\hat {A}},{\hat {A}}^{n}]=0,\quad n=1,2,3...}$
${\displaystyle [k{\hat {A}},{\hat {B}}]=[{\hat {A}},k{\hat {B}}]=k[{\hat {A}},{\hat {B}}]}$
${\displaystyle [{\hat {A}},[{\hat {B}},{\hat {C}}]]+[{\hat {C}},[{\hat {A}},{\hat {B}}]]+[{\hat {B}},[{\hat {C}},{\hat {A}}]]=0}$

${\displaystyle [{\hat {x}}_{i},{\hat {x}}_{j}]=0}$ ${\displaystyle [{\hat {x}},{\hat {x}}]=0}$${\displaystyle [{\hat {x}},{\hat {y}}]=0}$
${\displaystyle [{\hat {p}}_{i},{\hat {p}}_{j}]=0}$ ${\displaystyle [{\hat {p}}_{x},{\hat {p}}_{x}]=0}$${\displaystyle [{\hat {p}}_{x},{\hat {p}}_{y}]=0}$
${\displaystyle [{\hat {x}}_{i},{\hat {p}}_{j}]=i\hbar \delta _{ij}}$ ${\displaystyle [{\hat {x}},{\hat {p}}_{x}]=i\hbar }$${\displaystyle [{\hat {x}},{\hat {p}}_{y}]=0}$${\displaystyle [{\hat {y}},{\hat {p}}_{x}]=0}$${\displaystyle [{\hat {y}},{\hat {p}}_{y}]=i\hbar }$
${\displaystyle [{\hat {L}}_{i},{\hat {L}}_{j}]=i\hbar \epsilon _{ijk}{\hat {L}}_{k}}$ ${\displaystyle [{\hat {L}}_{x},{\hat {L}}_{y}]=i\hbar {\hat {L}}_{z}}$${\displaystyle [{\hat {L}}_{y},{\hat {L}}_{z}]=i\hbar {\hat {L}}_{x}}$${\displaystyle [{\hat {L}}_{z},{\hat {L}}_{x}]=i\hbar {\hat {L}}_{y}}$

### 正則對易關係

${\displaystyle [x,p]=i\hbar }$

### 與古典力學的關係

${\displaystyle \{x,p\}=1\,\!}$

${\displaystyle [{\hat {f}},{\hat {g}}]=i\hbar {\widehat {\{f,g\}}}\,}$