I've often seen spin 1/2 commutation rules as a principle valid for every angular momentum. In some text books there is a derivation from symmetries principles. My question is, if I have a spin $1/2$ particle, which measurements should I perform to know which are the commutation rules of $S_x, S_y, S_z$?


Commutation rules do not follow from measurements, but from comparing two sequence of rotations about different axes with inverse angles: \begin{align} 1-\lambda^2[L_x,L_y]&\approx e^{i \lambda L_x}e^{i\lambda L_y} e^{-i\lambda L_x} e^{-i\lambda L_y}\, ,\\ &= \left(R_x(\lambda)R_y(\lambda)\right)\left(R_x(-\lambda)R_y(-\lambda)\right) \end{align}

It may also be possible to obtain information about the average value of a commutator from the uncertainty relation \begin{align} (\Delta_{\vert\psi\rangle}L_x)(\Delta_{\vert\psi\rangle}L_y)=\frac{1}{2}\vert \langle \psi\vert [L_x,L_y]\vert \psi\rangle\vert \end{align} but this relation is state dependent so it’s not terribly useful.

  • $\begingroup$ I know there is this derivation from symmetry rules, I'd like to know if there is a way to derive commutation rules from experimental results. Is your opinion that: there aren't measurements that we can perform in order to understand which are the commutation rules? $\endgroup$ – SimoBartz Aug 16 '20 at 15:08
  • $\begingroup$ I do not know of a derivation using symmetry rules. The closest is in Sakurai or Townsend, and it’s only a symmetry argument inasmuch as one can argue that the measurement of spin along any axis should give the same outcomes, and this argument is IMO incomplete. $\endgroup$ – ZeroTheHero Aug 16 '20 at 15:14
  • $\begingroup$ ... and no I don’t think you can do measurement to infer commutation relations v.g. take an eigenstate of $L_x$ then $\langle L_y\rangle=\langle L_z\rangle=0$ so what can one infer from that? $\endgroup$ – ZeroTheHero Aug 16 '20 at 15:15
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    $\begingroup$ @SimoBartz This seems to be a special case of the question "How can we deduce a theory from experiments?" The answer is... we can't. Theories are tested, not deduced. We can never completely rule out the possibility that some other not-yet-discovered theory would work just as well and be just as efficient (but we shouldn't hold our breath waiting for that other theory, either). $\endgroup$ – Chiral Anomaly Aug 16 '20 at 17:45

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