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

Revised on July 19, 2011 00:57:32 by Toby Bartels (76.85.192.183)