On the oddly pervasive notion of the conservation of energypic.twitter.com/j0iWw1sbTY
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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.
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.
