[QUE/QM-02002]

\(\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\)

1.       Use position representation        \[ \hat{q} \to q ; \qquad p\to -i\hbar \pp{q} \]      for the operators  \(\hat{p},\hat{q}\)  and prove the following relations    \begin{equation}\label{qm-que-02001;1}      [ \hat{q},\hat{p}^N] = i N\hbar \hat{p}^{N-1}; \qquad\qquad       [\hat{p},\hat{q}^N]  =-i N\hbar \hat{q}^{N-1}.      \end{equation}       Note you could have equally well used the momentum representation  \[ \hat{q} \to  i\hbar \pp{p}\qquad \hat{p} \to p \]   
2.       Prove the above commutators  \eqref{qm-que-02001;1} using canonical commutation relations,  \([q,p]=i\hbar\), only. Do  not use any  representation  for position or momentum operator.