# Canonical commutation relation for spherical coordinates?

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?

• The two commutation relations you've given aren't equivalent – InertialObserver Mar 16 '19 at 3:55
• If you like this question you may also enjoy reading this Phys.SE post. – Qmechanic Mar 16 '19 at 8:07
• @InertialObserver what do both represent/why are they different? – user107224 Mar 16 '19 at 9:21

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 $$\hat p^{'}=-i\hbar\frac {\partial} {\partial r}$$

$$[\hat p,\hat e_r] =[\hat p^{'},\hat e_r]$$