time-dependent perturbation theory

Suppose a quantum system is sitting happily in a known state and you give it a gentle, time-varying nudge — a passing light wave, a switched-on field. Will it jump to a different state, and with what probability? Time-dependent perturbation theory is the systematic machinery for answering that, valid whenever the nudge is weak enough to treat as a small correction rather than a total upheaval.

The method splits the Hamiltonian into a solved part and a small time-dependent perturbation, H = H_0 + V(t). Any state is expanded in the known eigenstates of H_0 with time-dependent coefficients, |psi(t)> = sum_n c_n(t) e^(-i E_n t/hbar) |n>, and one solves for how the c_n evolve. To first order, starting in state i, the amplitude to be found in state f is c_f(t) = (-i/hbar) integral_0^t <f|V(t')|i> e^(i omega_fi t') dt', where omega_fi = (E_f - E_i)/hbar. The transition probability is |c_f(t)|^2. When V oscillates at frequency omega, this amplitude builds up resonantly when omega matches omega_fi — the mathematical origin of resonant absorption and emission of light.

This is the workhorse behind almost all of light-matter interaction: absorption, stimulated and spontaneous emission, and the derivation of selection rules and Fermi's golden rule all start here. Be honest about its limits: it is a perturbative expansion, trustworthy only while V is weak and the transition probability stays much less than one. Push it too hard, or wait too long exactly on resonance, and the first-order result grows past unity — a signal that the approximation has broken and you must resum the series, which for a driven two-level system gives the non-perturbative Rabi oscillations instead.