The commutator $[x,p]$

Position $x$ and momentum $p$ are operators in quantum mechanics. Acting on wave functions, the operator product $xp$ has the property

$$\hat{x}\hat{p}\Psi(x) = \frac{\hbar}{i}x\frac{\partial}{\partial x}\Psi(x) = \frac{\hbar}{i}x\Psi'(x)$$

$$\hat{p}\hat{x}\Psi(x) = \frac{\hbar}{i}\frac{\partial}{\partial x}x\Psi(x) = \frac{\hbar}{i}\left(\Psi(x)+ x\Psi'(x)\right)$$ (74)

The result depends on the order of $\hat{x}$ and $\hat{p}$: both operators do not commute. One has

$$(\hat{x}\hat{p}-\hat{p}\hat{x})\Psi(x) = i\hbar \Psi(x)$$ (75)

Comparing both sides, we have the commutation relation

$$[\hat{x},\hat{p}]:=\hat{x}\hat{p}-\hat{p}\hat{x} = i\hbar.$$ (76)

Here, we have defined the commutator $[A,B]:=AB-BA$ of two operators $A$ and $B$. Generalized to three dimensions with the three components $\hat{x}_k$ of ${\bf\hat{x}}$ and $\hat{p}_k$ of ${\bf\hat{p}}$, $k=1,2,3$, one has the canonical commutation relations

$$[\hat{x}_k,\hat{p}_l] = i\hbar\delta_{kl}$$

$$\delta_{kl} := 1,\quad k=l, .$$ (77)