What coordinates systems can the canonical commutation relation be generalised to? I also ask specifically for spherical coordinates. This is because I want to prove $\hat{\bf p}\cdot{\bf e}_r-{\bf e}_r\cdot\hat{\bf p}=-2i\hbar/r$. I have seen in places that $[\hat{r},\hat{p}_r]=i\hbar$. The equation I want to prove seems like a commutation relation (divided by $r$), i.e. $\hat{\bf p}\cdot{\bf e}_r-{\bf e}_r\cdot\hat{\bf p}=(1/r)[\hat{\bf p},\hat{\bf r}]$. Where did the factor of 2 go?
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$\begingroup$ The two commutation relations you've given aren't equivalent $\endgroup$– InertialObserverCommented Mar 16, 2019 at 3:55
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1$\begingroup$ If you like this question you may also enjoy reading this Phys.SE post. $\endgroup$– Qmechanic ♦Commented Mar 16, 2019 at 8:07
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$\begingroup$ @InertialObserver what do both represent/why are they different? $\endgroup$– user107224Commented Mar 16, 2019 at 9:21
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1 Answer
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I presume you're trying to construct a symmetric (or hermitian) $\hat p_r$ operator from the following identity
$$p_r=\frac{1}{2}(\vec p\cdot\hat e_r+ \hat e_r\cdot \vec p)$$
Hence $$\hat p_r=-i\hbar (\frac {\partial} {\partial r} + \frac {1}{r})=-i\hbar\frac {1}{r} (\frac {\partial} {\partial r})r $$
which yields the following communator $$[\hat p,\hat e_r]=-i\hbar (\frac {\partial} {\partial r} \frac {\vec r} {r}-\frac {\vec r} {r} \frac {\partial} {\partial r}) =-i\hbar (\frac {\partial} {\partial r} \hat e_r - \hat e_r \frac {\partial} {\partial r}) $$
$$[\hat p,\hat e_r] =-i\hbar (\frac {\partial} {\partial r} \hat e_r - \hat e_r \frac {\partial} {\partial r}) $$
Define $\hat p^{'}=-i\hbar\frac {\partial} {\partial r}$
$$[\hat p,\hat e_r] =[\hat p^{'},\hat e_r] $$
