= Solution
The <Time-dependent Schrödinger equation> $i\hbar\partial_t|\psi\rangle=H(t)|\psi\rangle$ governs unitary evolution of a closed system. If $H$ is independent of time, expand in its <energy eigenstates>:
$$
\boxed{|\psi(t)\rangle=\sum_n c_ne^{-iE_nt/\hbar}|n\rangle}.
$$
An individual <energy eigenstate> changes only by a <phase>, so its <probability density> and time-independent-observable expectations are stationary. Superpositions can have interference terms oscillating at $(E_n-E_m)/\hbar$; fixed <energy> populations do not imply that every <observable> is constant.
For general $H(t)$, $U(t,t_0)=\mathcal T\exp[-(i/\hbar)\int H(t')dt']$ requires <time ordering> unless <Quantum Hamiltonians> at different times commute. The equivalent Heisenberg description evolves <observables> rather than the state, giving $d\langle A\rangle/dt=(i/\hbar)\langle[H,A]\rangle+\langle\partial_tA\rangle$. Explicit time dependence can transfer <energy> from an applied perturbation.
For a weak perturbation $V(t)$ to a known $H_0$, the first-order transition amplitude from $n$ to $m\ne n$ is
$$
c_m^{(1)}(t)=-\frac i\hbar\int_{t_0}^t\langle m|V(t')|n\rangle e^{i(E_m-E_n)t'/\hbar}\,dt'.
$$
Its squared modulus gives the leading transition <probability> while depletion remains small. A nearly monochromatic perturbation emphasizes <energy> differences near its quantum of <frequency>; a continuum and suitable long-time limit yield <Fermi's golden rule>. These approximations have stated weak-coupling and time-resolution limits; they do not replace exact unitary dynamics in every regime.
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