\(\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\)
- 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 \[ \hat{q} \to i\hbar \pp{p}\qquad \hat{p} \to p \]
- 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.
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