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.
1 Answer
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 $$S_xS_y\ne S_yS_x.$$
