C. Cohen-Tannoudji, B. Diu, F. Laloe, Quanten mechanics 1 & 2

Generally, operators are not commutative, AB = BA
Commutator [A, B] =AB − BA
Examples:
f (x),
⇒
f (x),
∂
ψ = f
∂x j
∂
∂x j
∂
∂x j
ψ −
= − ∂
∂x j
f (x)
∂
∂x j
f
ψ − f
∂ ∂
ψ = − f
∂x j ∂x j
ψ
special case: f (x) =x i ⇒
x i ,
∂
∂x j
= −δ ij
[x i , x j ]=0
∂ ∂
,
∂x i ∂x j
=0
therefore: basic commutators of position and momentum operators:
[x i , x j ]=0 ; [p i , p j ]=0 ; [x i , p j ]=iδ ij