A Hamiltonian that varies in time can drive transitions between stationary states. Write $$H(t)=H_0+V(t),$$ where $H_0|n\rangle=E_n|n\rangle$ and $V(t)$ is a weak time-dependent perturbation.

To first order, the transition amplitude from an initial state $|i\rangle$ to a different final state $|f\rangle$ is $$c_f^{(1)}(t)=\frac{1}{i\hbar}\int_0^t \langle f|V(t')|i\rangle e^{i\omega_{fi}t'}dt',$$ where $$\omega_{fi}=\frac{E_f-E_i}{\hbar}.$$ The exponential phase makes the response strongest when the perturbation contains frequency components near the energy-level spacing.

For a sinusoidal perturbation at angular frequency $\omega$, transition probability develops a resonant peak near $$\hbar\omega=|E_f-E_i|.$$ For transitions into a dense continuum of final states, the long-time result becomes Fermi's golden rule: $$\Gamma_{i\to f}=\frac{2\pi}{\hbar}|V_{fi}|^2\rho(E_f),$$ where $\rho(E_f)$ is the density of available final states at the energy selected by conservation.

The formula separates coupling strength from state availability: a transition can be weak because the matrix element is small or because few final states exist. Time-dependent perturbation theory underlies spectroscopy, spontaneous and stimulated transitions, scattering and decay rates across atomic, condensed-matter and nuclear physics.