Commutator of $p$ and $x^n$

We can use the commutator $[p,x]$ to help us. Remember that $px=xp+[p,x]$.

$$\begin{eqnarray*}[p,x^n]&=&px^n-x^np\\ &=&(px)x^{n-1}-x^np\\ &=&xpx^{n-1}+[p... ...p,x]x^{n-1}-x^np\\ &=&n[p,x]x^{n-1}=n{\hbar\over i}x^{n-1} \\ \end{eqnarray*}$$

It is usually not wise to use the differential operators and a wave function crutch to compute commutators like this one. Use the known basic commutators when you can.Nevertheless, we can compute it that way.

$$\bgroup\color{black}[p,x^n]\psi={\hbar\over i}{\partial\over\... ...{\partial\over\partial x}\psi={\hbar\over i}nx^{n-1}\psi\egroup$$

$$\bgroup\color{black}[p,x^n]={\hbar\over i}nx^{n-1}\egroup$$

It works pretty well for this particular case, but not if I have $p$ to some power...