How to deduce the expression of Sz from the raising and lowering operators?

Question

I know that we can define the $S_x$ and $S_y$ operators as follows: $$ S_x=\frac{1}{2}(S^++S^-),\\ S_y=\frac{1}{2i}(S^+-S^-). $$ The question is, how to deduce the expression of $S_z$ from the raising $S^+$ and lowering $S^-$ operators?

Answer 1

Write $$S^+=S_x+iS_y\\S^-=S_x-iS_y$$ Then $$S^+S^-=S_x^2+iS_yS_x-iS_xS_y+S_y^2=(S_x^2+S_y^2)-i(S_xS_y-S_yS_x)$$ Using $$S^2=S_x^2+S_y^2+S_z^2$$and $$[S_x,S_y]=i\hbar S_z$$ you get $$S^+S^-=S^2-S_z^2+\hbar S_z$$ Similarly $$S^-S^+=S^2-S_z^2-\hbar S_z$$ Subtracting the two, you get the commutator for $S^+$ and $S^-$: $$[S^+,S^-]=2\hbar S_z$$