On the oddly pervasive notion of the conservation of energypic.twitter.com/j0iWw1sbTY
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Noether's theorem is a consequence of Lagrangian mechanics, so it's sorta begging the question. In some sense, though, you can get energy conservation in both GR and NM this way, which I guess is a good point!
Somewhat embarrassingly, I don't actually know what the quantum version of N's theorem is. I presume it's likely some obvious statement about operators commuting with the Hamiltonian or something...
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Yep, by definition a symmetry (e.g. x or t translations) commutes with the evolution operator e^{-iHt}, which means the generator of the symmetry commutes with the Hamiltonian so is conserved. For time translation symmetry the Hamiltonian commutes with itself so is conserved.
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To clarify: I know the standard results about conserved quantities in quantum mechanics. But prior to this thread I couldn't have put all the quantities in Noether's theorem in one-to-one correspondence with their quantum analogues.
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