Ladder Operator Technique
I have 2 small queries as attached. Not sure how they came about.
Thanks
Query 1: it's a bit hard to read your scan, but it looks as though you're asking why
$\displaystyle \hbar(\pm\hat{L}_{x}+i\hat{L}_{y})=\pm\hbar(\hat{L }_{x}\pm i\hat{L}_{y}).$
Is that correct? If so, I would just say this: the plus's and the minus's there are coupled. Take the "both positive" case on the RHS. You get the LHS when it's in the positive case. Take the "both negative" case on the RHS. You get the LHS when it's in the negative case. So those "plus-minus" symbols are coupled.
Query 2: You're asking how
$\displaystyle (\hat{L}_{z}\hat{L}_{\pm}-\hat{L}_{\pm}\hat{L}_{z})=\left[\hat{L}_{z},\hat{L}_{\pm}\right].$
Is that correct? Isn't that just the definition of the commutator?