Commutator
For operators $\hat{A}$ and $\hat{B}$ acting on wavefunctions, the following operator is called the commutator of $\hat{A}$ and $\hat{B}$.
- $ \com{\hat{A}}{\hat{B}} \coloneqq \hat{A}\hat{B}-\hat{B}\hat{A} $
If $\com{\hat{A}}{\hat{B}}=0$, we say that, $\hat{A}$ and $\hat{B}$ commute. In other words, there is no difference between the result of acting $\hat{A}$ after acting $\hat{B}$, and the result of acting $\hat{B}$ after acting $\hat{A}$.
Properties
- Bilinearity
- $ \com{\hat{A}+\hat{B}}{\hat{C}} = \com{\hat{A}}{\hat{C}}+\com{\hat{B}}{\hat{C}} $
- $ \com{\hat{A}}{\hat{B}+\hat{C}} = \com{\hat{A}}{\hat{B}}+\com{\hat{A}}{\hat{C}} $
- Alternativity
- $ \com{\hat{A}}{\hat{B}} = -\com{\hat{B}}{\hat{A}} $
- Jacobi identity
- $ \com{\hat{A}}{\com{\hat{B}}{\hat{C}}} + \com{\hat{B}}{\com{\hat{C}}{\hat{A}}} + \com{\hat{C}}{\com{\hat{A}}{\hat{B}}} = 0 $
- Additional Property
- $ \com{\hat{A}}{\hat{B}\hat{C}} = \com{\hat{A}}{\hat{B}}\hat{C}+\hat{B}\com{\hat{A}}{\hat{C}} $
- $ \com{\hat{A}\hat{B}}{\hat{C}} = \hat{A}\com{\hat{B}}{\hat{C}}+\com{\hat{A}}{\hat{C}}\hat{B} $
Read Commutator (ring theory) for further details.
Condition under which two operators are commutative
- If a common eigenfunction of any two different operators exists, they commute.
\[ \begin{cases} A\psi=a\psi \nl B\psi=b\psi \end{cases} \Rightarrow \com{A}{B}=0 \]
It’s very easy to show.
Anticommutator
We can also define anticommutators:
- $ \acom{\hat{A}}{\hat{B}} \coloneqq \hat{A}\hat{B}+\hat{B}\hat{A} $