What is the meaning of $i\hbar S_{z}$? Does the commutation relation between $S_{x}$ and $S_{y}$ mean anything? 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.
 A: 
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 $−S_xS_y$?

You are right. This is highly abstract and hard to grasp in the
context of the observables $S_x$, $S_y$ and $S_z$. But there is
another way because these operators have two different roles:

*

*They represent measurable physical observables
(the components of spin angular momentum).

*They generate rotations around the coordinate axes.
For example: $R(\hat x,\alpha)=e^{-i\alpha S_x/\hbar}$
is a rotation around the $x$-axis by an angle $\alpha$.

These two roles are closely related to each other.
Actually these are two aspects of the same thing.
From the commutator relations between the spin operators, like
$$[S_x,S_y]=i\hbar S_z$$
you can derive corresponding commutator relations between rotations, like
$$R(\hat x,\alpha)R(\hat y,\beta)R(\hat x,-\alpha)R(\hat y,-\beta)
\approx R(\hat z,-\alpha\beta)
\quad \text{ if }\alpha,\beta\ll 1$$
Now this relation has a direct intuitive meaning
(and there is no $i\hbar$ involved anymore).
You can actually check this physically by doing the sequence of the
4 rotations ($R(\hat x,\alpha)$, $R(\hat y,\beta)$,
$R(\hat x,-\alpha)$, $R(\hat y,-\beta)$) on a shoe-box.
Then you will see, the rotations don't quite cancel,
and the box does not exactly come back to its original
orientation, but get a small rotation around the $z$-axis.

I wonder if these non commutation relations mean something else

The non-commutativity of rotations
$$R(\hat x,\alpha)R(\hat y,\beta)\ne R(\hat y,\beta)R(\hat x,\alpha)$$
is a direct consequence of the
non-commutativity of the spin components
$$S_xS_y\ne S_yS_x.$$
