Properties of Commutators
Definition
For two operators $A, B$, $AB - BA$ is defined as the commutator of $A, B$ and is denoted as follows.
$$ [A,B]=AB-BA $$
Properties
$$ \begin{align} [A, A] &= 0 \\[1em] [A, B] &= -[B, A] \\[1em] [A+B, C] &= [A, C] + [B, C] \\[1em] [AB, C] &= A[B, C]+[A, C]B \\[1em] [A,BC] &= B[A,C]+ [A,B]C \end{align} $$
Explanation
The main method of describing quantum mechanics is matrices. However, matrices do not satisfy the commutative property for multiplication. Therefore, when expanding an operator (matrix) $A,\ B$ as below, it is generally incorrect.
$$ (A+B)^{2} = A^{2} + 2AB + B^{2} $$
The correct expansion is as follows.
$$ (A+B)^{2} = A^{2} + AB + BA + B^{2} $$
Remembering that 'operator=matrix' is the way to reduce mistakes. The properties above are useful when calculating commutators.
Proof
(1)
$$ [A, A]=AA-AA=0 $$
■
(2)
$$ [A,B] = AB-BA = -(BA-AB) = -[B,A] $$
■
(3)
$$ \begin{align*} [A+B,C] &= (A+B)C-C(A+B) \\ &= AC+BC-CA-CB \\ &= (AC-CA) + (BC-CB) \\ &= [A,C]+[B,C] \end{align*} $$
■
(4)
$$ \begin{align*} [AB,C] &= (AB)C-C(AB) \\ &= ABC-CAB \\ &= (ABC {\color{blue}-CAB})+(ACB {\color{red}-ACB}) \\ &= (ABC {\color{red}-ACB}) + (ACB {\color{blue}-CAB}) \\ &= A(BC-CB) +(AC-CA)B \\ &= A[B,C] + [A,C]B \end{align*} $$
(5)
$$ \begin{align*} [A,BC] &= A(BC)-(BC)A \\ &= ABC-BCA \\ &= ({\color{blue}ABC} -BCA)+({\color{red}BAC} -BAC) \\ &= ( {\color{red}BAC}-BCA )+({\color{blue}ABC}-BAC) \\ &= B[A,C] + [A,B]C \end{align*} $$
■