I'm given this commutator:
$$\left[PXP,P\right]$$
Being $P\psi=-i\hbar\partial_x\psi$, and $X\psi=x\psi$
I've solved it in two ways, the first one is just aplying the commutator to some function $\psi$ and see what I get. My final result is:
$$\left[PXP,P\right]=-i\hbar^3\partial_{xx}$$
The second one is using some commutator properties:
$$\left[PXP,P\right]=-\left[P,PXP\right]=-(P\left[P,XP\right]+\left[P,P\right]XP)$$
$\left[P,P\right]=0$, so the second term goes away. I again expand the first term:
$$-P\left[P,XP\right]=-P(X[P,P]+[P,X]P)=-P[P,X]P=i\hbar P^2=\boxed{-i\hbar^3\partial_{xx}}$$
I again get the same result. When the teacher solved it in class, the final result was:
$$\left[PXP,P\right]=2i\hbar P^2$$
I have no idea where that $2$ comes from. Am I missing something? Am I doing something wrong?
Best Answer
You teacher seems to have made a mistake. I imagine that he/she did something like this: \begin{align} [PXP, P] &= P[XP,P]+[PX,P]P \\ &= P(X[P,P]+[X,P]P)+(P[X,P]+[P,P]X)P \\ &= P[X,P]P+P[X,P]P \\ &= 2i\hbar P^2 \end{align} Notice that the first equality is wrong. You can't peel operators off to the left and right if there are three operators in the first slot of the commutator!