OK for a system with spin 1/2 if one measures S_{x} the information on S_{y} is lost and measuring S_{y} after an S_{x} measurement one gets a 50% probability for S_{y} up. My question is if [S_{x},S_{y}]=i\hbar S_{z} has any other experimental implication. For example, if one measures S_{y} then S_{x} is there a meaning for -S_{x}S_{y}? If one measures S_{y} then S_{x} and then S_{x} followed by S_{y} do we get the "commutator"? After all we never get a projection of S_{z} no matter how ofter we measure S_{x} and S_{y}. And what is actually the meaning of i \hbar S_{z}? Does it make sense to multiply an operator with a complex valued constant? I am aware the question may come across silly but I wonder if these non commutation relations mean something else than simply that one measurement "kills the information" for a following type of measurement.

OK for a system with spin $1/2$, if one measures $S_{x}$ the information on $S_{y}$ is lost and measuring $S_{y}$ after an $S_{x}$ measurement one gets a $50\%$ probability for $S_{y}$ up. My question is if $[S_{x},S_{y}]=i\hbar S_{z}$ has any other experimental implication. For example, if one measures $S_{y}$ then $S_{x}$ is there a meaning for $-S_{x}S_{y}$? If one measures $S_{y}$ then $S_{x}$ and then $S_{x}$ followed by $S_{y}$ do we get the "commutator"? After all we never get a projection of $S_{z}$ no matter how often we measure $S_{x}$ and $S_{y}$. And what is actually the meaning of $i \hbar S_{z}$? Does it make sense to multiply an operator with a complex-valued constant? I am aware the question may come across silly but I wonder if these non commutation relations mean something else than simply that one measurement "kills the information" for a following type of measurement.