The commutator of position and momentum, specifically ##[p_x, r]## where ##r=(x,y,z)##, can be expanded by calculating the individual commutators of ##p_x## with each position operator: ##[p_x, x]##, ##[p_x, y]##, and ##[p_x, z]##. This results in the expression ##[\hat{p}_x, \mathbf{\hat{r}}] = ([\hat{p}_x, \hat{x}], [\hat{p}_x, \hat{y}], [\hat{p}_x, \hat{z}])##. The vector operator ##\mathbf{\hat{r}}## comprises the three position operators, and their relationships follow the same principles as vector components. Thus, the expansion of the commutator reflects the inherent structure of these operators. Understanding this relationship is crucial for further analysis in quantum mechanics.
