= Solution
For a time-independent <Hamiltonian operator>, a <symmetry operator> $U$ satisfies $UHU^\dagger=H$, or $[U,H]=0$. Thus applying $U$ to an <energy eigenstate> preserves its <energy>. Commuting Hermitian <observables> can be diagonalized within each degenerate <energy eigenspace>, allowing states to be labelled by symmetry quantum numbers.
A continuous unitary symmetry $U(\varepsilon)=e^{-i\varepsilon Q/\hbar}$ has Hermitian generator $Q$. Differentiating the invariance condition gives $[H,Q]=0$. The <Heisenberg equation of motion> then yields
$$
\boxed{\frac{d\langle Q\rangle}{dt}=\frac{i}{\hbar}\langle[H,Q]\rangle=0}
$$
when $Q$ has no explicit time dependence. <spatial translations> connect to <momentum> conservation, <rotations> to angular-momentum conservation, and time-translation invariance to <energy> conservation. A central <potential energy> conserves all components of <angular momentum>, but they cannot all be assigned sharp simultaneous values because they do not commute; $H,L^2,L_z$ are a compatible choice. Discrete reflection symmetry gives <parity> labels without a continuously variable generator. A <quantum degeneracy> can be protected by a symmetry, but a <quantum degeneracy> alone does not identify its physical origin.
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