交換子

群G中两个元素g和h的交换子为元素

[g, h] = g⁻¹h⁻¹gh

它等于群的幺元当且仅当g和h可交换（即gh = hg）。

环或结合代数上两个元素a和b的交换子定义为：

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

量子力学中，经常用到对易关系（commutation relation），即

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

其中；${\displaystyle {\hat {A}}}$、${\displaystyle {\hat {B}}}$均为量子力学的算符，${\displaystyle [{\hat {A}},{\hat {B}}]}$是其对易算符，也称交换子。

如果上式等于零，则称${\displaystyle {\hat {A}}}$、${\displaystyle {\hat {B}}}$是对易的，即意味着${\displaystyle {\hat {A}}}$和${\displaystyle {\hat {B}}}$两个算符的运算顺序可以调换。反之则称非对易的，运算顺序不可以调换。

量子力學中，交換子有以下特性:

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

${\displaystyle [{\hat {A}},{\hat {B}}+{\hat {C}}]=[{\hat {A}},{\hat {B}}]+[{\hat {A}},{\hat {C}}],\quad [{\hat {A}}+{\hat {B}},{\hat {C}}]=[{\hat {A}},{\hat {C}}]+[{\hat {B}},{\hat {C}}]}$

${\displaystyle [{\hat {A}},{\hat {B}}{\hat {C}}]=[{\hat {A}},{\hat {B}}]{\hat {C}}+{\hat {B}}[{\hat {A}},{\hat {C}}],\quad [{\hat {A}}{\hat {B}},{\hat {C}}]=[{\hat {A}},{\hat {C}}]{\hat {B}}+{\hat {A}}[{\hat {B}},{\hat {C}}]}$

${\displaystyle [{\hat {A}},{\hat {A}}^{n}]=0,\quad n=1,2,3...}$

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

${\displaystyle [{\hat {A}},[{\hat {B}},{\hat {C}}]]+[{\hat {C}},[{\hat {A}},{\hat {B}}]]+[{\hat {B}},[{\hat {C}},{\hat {A}}]]=0}$

量子力学中的各个力学量之间，常用的对易关系有：

以下，${\displaystyle {\hat {x}}}$是位置算符、${\displaystyle {\hat {p}}}$是动量算符、${\displaystyle {\hat {L}}}$是角动量算符（包括轨道角动量、自旋角动量等），而${\displaystyle \delta _{ij}}$是克罗内克δ、${\displaystyle \epsilon _{ijk}}$是列維-奇維塔符號。其中i、j、k均可以指代x、y、z三个方向中的任意一个。

对易关系	更具体的形式
${\displaystyle [{\hat {x}}_{i},{\hat {x}}_{j}]=0}$	${\displaystyle [{\hat {x}},{\hat {x}}]=0}$、${\displaystyle [{\hat {x}},{\hat {y}}]=0}$
${\displaystyle [{\hat {p}}_{i},{\hat {p}}_{j}]=0}$	${\displaystyle [{\hat {p}}_{x},{\hat {p}}_{x}]=0}$、${\displaystyle [{\hat {p}}_{x},{\hat {p}}_{y}]=0}$
${\displaystyle [{\hat {x}}_{i},{\hat {p}}_{j}]=i\hbar \delta _{ij}}$	${\displaystyle [{\hat {x}},{\hat {p}}_{x}]=i\hbar }$、${\displaystyle [{\hat {x}},{\hat {p}}_{y}]=0}$、${\displaystyle [{\hat {y}},{\hat {p}}_{x}]=0}$、${\displaystyle [{\hat {y}},{\hat {p}}_{y}]=i\hbar }$
${\displaystyle [{\hat {L}}_{i},{\hat {L}}_{j}]=i\hbar \epsilon _{ijk}{\hat {L}}_{k}}$	${\displaystyle [{\hat {L}}_{x},{\hat {L}}_{y}]=i\hbar {\hat {L}}_{z}}$、${\displaystyle [{\hat {L}}_{y},{\hat {L}}_{z}]=i\hbar {\hat {L}}_{x}}$、${\displaystyle [{\hat {L}}_{z},{\hat {L}}_{x}]=i\hbar {\hat {L}}_{y}}$

• Fraleigh, John B., 2nd, Reading: Addison-Wesley, 1976 [2020-04-07], ISBN 0-201-01984-1, （原始内容存档于2021-07-09）
• Griffiths, David J., 2nd, Prentice Hall, 2004, ISBN 0-13-805326-X 含有內容需登入查看的頁面 (link)
• Herstein, I. N., 2nd, John Wiley & Sons, 1975
• Liboff, Richard L., 4th, Addison-Wesley, 2003, ISBN 0-8053-8714-5
• McKay, Susan, , Queen Mary Maths Notes 18, University of London, 2000, ISBN 978-0-902480-17-9, MR 1802994
• McMahon, D., , USA: McGraw Hill, 2008, ISBN 978-0-07-154382-8

• 正則量子化
• 正則變換
• 李導數
• 群
• 李代數
• 泊松括號
• 雅可比恆等式