Past exam of the mathematics course of the University of Cambridge / 2001 / ib / Paper 2 / 18F / a / Solution

The orbital angular momentum operators areOn smooth wavefunctions, $[x_i,p_j]=i\hslash {\delta }_{ij}$, while position operators commute with each other and momentum operators commute with each other. Applying the product rule for commutators givesUse it on the four terms in $[L_1,L_2]$: only $[x_2{p}_3,x_3{p}_1]=-i\hslash x_2{p}_1$ and $[x_3{p}_2,x_1{p}_3]=i\hslash x_1{p}_2$ survive, with the latter entering positively from the two minus signs. ThereforeThe cyclic versions similarly give $[L_3,L_1]=i\hslash L_2$ and $[L_3,L_2]=-i\hslash L_1$. Thus, for $L_{\pm }=L_1\pm i{L}_2$,Now define $L^2=L_1^2+L_2^2+L_3^2$. For each $i$,because the Levi-Civita symbol is antisymmetric in $j,k$ and the operator pair is symmetric. Linearity then gives $\boxed{[L^2,L_{\pm }]=0}$. The PDF superscript here is the squared total angular momentum, not the component $L_2$.