Commutator problem (position, momentum)

[Kentaxel]

I'm having some difficulties with a certain commutator producing inconsistent results. Specifically I'm referring to

$$[p_x,x^3]$$

Depending on how i expand this it seems i get different coefficients, i.e

$$[p_x,x^3]=[p_x,x]x^2+x^2[p_x,x]=-i\hbar x^2 -x^2i\hbar=-2i\hbar x^2$$

However

$$[p_x,x^3]=[p_x,x^2]x+x[p_x,x^2]=([p_x,x]x+x[p_x,x])x+x([p_x,x]x+x[p_x,x])=-i4\hbar x^2$$.

Clearly I'm missing something here, but i can't quite figure out what.

 

[rubi]

The rule is $[X,AB] = [X,A]B + A[X,B]$. You seem to be using $[X,AB] = [X,B]A + A[X,B]$ instead.

 

[dextercioby]

Put a wavefunction from the Schwartz space over the reals to the right of the commutator, use the position representation, don't use Leibniz, but only the definition and check which of the 2 results is right.

 

[Kentaxel]

Yes of course, it should be

$$[p_x,x^3]=[p_x,x]x^2+x[p_x,x^2]=-3i\hbar x^2.$$

Thank you!