Commutator of position and momentum

[Kara386]

How would $[p_x, r]$ be expanded? Where $r=(x,y,z)$, the position operators. Do you do the commutators of $p_x$ with $x, y,z$ individually? So $[p_x,x]+[p_x,y]+[p_x,z]$ for example?

 

[PeroK]

How would $[p_x, r]$ be expanded? Where $r=(x,y,z)$, the position operators. Do you do the commutators of $p_x$ with $x, y,z$ individually? So $[p_x,x]+[p_x,y+p_x,z]$ for example?

More generally, a vector operator such as $\mathbf{\hat{r}}$ represents three operators $(\hat{x}, \hat{y}, \hat{z})$, related in the same way as the components of a vector.

In this case, essentially by definition:

$[\hat{p}_x, \mathbf{\hat{r}}] = ([\hat{p}_x, \hat{x}], [\hat{p}_x, \hat{y}], [\hat{p}_x, \hat{z}])$

 

[Kara386]

More generally, a vector operator such as $\mathbf{\hat{r}}$ represents three operators $(\hat{x}, \hat{y}, \hat{z})$, related in the same way as the components of a vector.

In this case, essentially by definition:

$[\hat{p}_x, \mathbf{\hat{r}}] = ([\hat{p}_x, \hat{x}], [\hat{p}_x, \hat{y}], [\hat{p}_x, \hat{z}])$

Ah, thank you. :)