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In a course I'm following I've read several time statements like:

In quantum mechanics a vector observable $\hat{\boldsymbol{V}}$ is conserved if both the module squared $\hat{V}^2$ and the projection along an axis $\hat{V}_z$ are conserved.

Obviously this is true for angular momentum $\hat{\boldsymbol{L}}$, but I'm not convinced about a generic vector operator.

If I understand correctly this statement, I guess there is the underlying hypothesis that every component of the vector doesn't commute with the others. This is not true in general, right?

Is this statement not correct? Does it refer only to angular momentum-like vector operators?

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  • $\begingroup$ I think it is not true. For example if the Hamiltonian is $H=\alpha\sigma_z$ (spin in magnetic field along $z$-direction) then the $z$-component $\hat{s}_z=\sigma_z/2$ and the module squared $\mathbf{\hat{s}}^2$ of the spin are both conserved, but $\hat{s}_x$ and $\hat{s}_y$ are not conserved (they change during Larmor precession). $\endgroup$ – Alexey Sokolik Jun 4 '17 at 13:54
  • $\begingroup$ Conserved under what? under time-evolution or under rotation? Are you restricted (implicitly or explicitly) to rotationally invariant Hamiltonians (The Hamiltonian if @AlexeySokolik is not...)? $\endgroup$ – ZeroTheHero Jun 4 '17 at 17:41
  • $\begingroup$ @ZeroTheHero I mean conservation during time evolution, because I never met the term "conservation" in application to rotations, in such cases the term "invariance" is usually used... $\endgroup$ – Alexey Sokolik Jun 4 '17 at 18:47

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