nLab
canonical commutation relation

In quantum mechanics, canonical quantization? replaces the momentum $p$ with the operator $-\mathrm{i}\hslash \frac{\mathrm{d}}{\mathrm{d}x}$. After this substitution, position and momentum fail to commute:

(1)$$\begin{array}{ccc}[p,x]x^n& =& (-\mathrm{i}\hslash \frac{\mathrm{d}}{\mathrm{d}x}x{x}^n)-(x\cdot -\mathrm{i}\hslash \frac{\mathrm{d}}{\mathrm{d}x}{x}^n)\\ & =& -\mathrm{i}\hslash [(n+1)x^n-n{x}^n]\\ & =& -\mathrm{i}\hslash x^n\end{array}$$\array{[p,x] x^n &=& \left(-\mathrm{i}\hbar \frac{\mathrm{d}}{\mathrm{d}x} x x^n\right) - \left(x \cdot -\mathrm{i}\hbar \frac{\mathrm{d}}{\mathrm{d}x} x^n\right) \\ &=& -\mathrm{i}\hbar [(n+1)x^n - n x^n] \\ &=& -\mathrm{i}\hbar x^n}

This is related to the combinatorics of placing a ball into a box and removing a ball from a box.