canonical commutation relation

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

(1)[p,x]x n = (iddxxx n)(xiddxx n) = i[(n+1)x nnx n] = ix n \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 (