Commutator Practice

Classical Mechanics Level 2

Compute the commutator of operators $[\hat{x}^2,\hat{p}]$ , where $\hat{p} = -i\hbar \partial_x$ is the momentum operator and $\hat{x}$ is the position operator.

i\hbar p

i\hbar

2i\hbar x

i\hbar x

1 solution

Matt DeCross
Mar 11, 2016

One can rewrite this commutator in terms of the canonical commutator:

$[x^2,p]=xxp-pxx=xxp-xpx+xpx-pxx=x[x,p]+[x,p]x= 2i\hbar x.$

Hats have been omitted, but all $x$ and $p$ above are operators. In the last equality the canonical commutation relation was used.

Commutator Practice

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