Time-Dependent Perturbation Theory

Static perturbation theory corrects stationary states. But most of what we observe — absorption and emission of light, decay of excited states, scattering rates — involves transitions between states driven by a time-dependent interaction. Time-dependent perturbation theory computes the probability that a system prepared in one eigenstate is found in another after a perturbation acts. Its central result, Fermi's golden rule, is the bridge from the postulates to measurable rates, and its structure reappears in scattering and QFT.

The Hamiltonian is $\hat{H}(t)={\hat{H}}_0+\hat{V}(t)$, with ${\hat{H}}_0$ solvable and $\hat{V}(t)$ small and (generally) time-dependent.

1. The Interaction Picture

The natural setting is the interaction picture (see heisenberg-picture.md §1), which factors out the trivial ${\hat{H}}_0$ evolution so that all motion of the state is due to $\hat{V}$. Writing $|{\psi }_I(t)⟩=e^{i{\hat{H}}_{0}t/\hslash }|{\psi }_S(t)⟩$, the state obeys

$$i\hslash \frac{d}{dt}|{\psi }_I(t)⟩={\hat{V}}_I(t)|{\psi }_I(t)⟩,{\hat{V}}_I(t)=e^{i{\hat{H}}_{0}t/\hslash }\hat{V}(t)e^{-i{\hat{H}}_{0}t/\hslash }.$$

Expanding $|{\psi }_I(t)⟩={\sum }_n{c}_n(t)|n^{(0)}⟩$, the amplitudes evolve as

$$i\hslash {\dot{c}}_k(t)=\sum _n⟨k|{\hat{V}}_I(t)|n⟩c_n(t)=\sum _n{V}_{kn}(t)e^{i{\omega }_{kn}t}{c}_n(t),{\omega }_{kn}=\frac{E_k^{(0)}-E_n^{(0)}}{\hslash }.$$

This is exact; perturbation theory solves it iteratively.

2. First-Order Transition Amplitude

Start in state $|i⟩$ ($c_n(0)={\delta }_{ni}$). To first order, integrate the right-hand side with the initial values:

$$c_f^{(1)}(t)=-\frac{i}{\hslash }{\int }_0^{t}d{t}^{\prime }{V}_{fi}(t^{\prime })e^{i{\omega }_{fi}{t}^{\prime }},f\ne i.$$

The transition probability is $P_{i\to f}(t)=|c_f^{(1)}(t){|}^2$. Two canonical time profiles:

• Sudden constant perturbation switched on at $t=0$: the integral gives $P_{i\to f}(t)=|V_{fi}{|}^2\frac{{\sin }^2({\omega }_{fi}t/2)}{(\hslash {\omega }_{fi}/2{)}^2}$, sharply peaked at energy conservation $E_f\approx E_i$ as $t$ grows.
• Harmonic perturbation $\hat{V}(t)=\hat{W}{e}^{-i\omega t}+{\hat{W}}^{†}{e}^{i\omega t}$ (e.g. an oscillating field): resonant transitions occur when $\hslash \omega \approx \pm (E_f-E_i)$ — absorption ($E_f=E_i+\hslash \omega$) and stimulated emission ($E_f=E_i-\hslash \omega$).

3. Fermi's Golden Rule

When the final states form a continuum (density of states $\rho (E_f)$), sum the probability over final states. The peaked function above integrates to a term linear in $t$, giving a constant rate:

$$\boxed{{\Gamma }_{i\to f}=\frac{2\pi }{\hslash }|⟨f|\hat{V}|i⟩{|}^2\rho (E_f){|}_{E_f=E_i}.}$$

This is Fermi's golden rule: the transition rate is set by the squared matrix element times the density of available final states, evaluated on the energy-conserving shell. It governs spontaneous and stimulated emission, photo- ionization, and scattering rates, and is the non-relativistic ancestor of the QFT decay-rate and cross-section formulas.

Validity requires that $\Gamma t\ll 1$ (little depletion of the initial state) yet $t$ long enough that the energy peak is sharp — an intermediate-time window. Over long times, exponential decay $P_i(t)=e^{-\Gamma t}$ emerges (Weisskopf–Wigner), and the finite lifetime $\tau =1/\Gamma$ gives the state a natural energy width $\Delta E=\hslash \Gamma$.

4. Selection Rules and Radiation

Matrix elements $⟨f|\hat{V}|i⟩$ usually vanish unless symmetry allows the transition. For electric-dipole radiation, $\hat{V}\propto \hat{\mathbf{r}}\cdot \mathbf{E}$, so $⟨f|\hat{\mathbf{r}}|i⟩$ must be nonzero, giving the atomic selection rules $\Delta \ell =\pm 1$, $\Delta m=0,\pm 1$ (from the angular integrals). Forbidden transitions proceed only via higher multipoles and are correspondingly slow — the reason for metastable states.

5. Adiabatic vs. Sudden Approximations

Two opposite limits are exactly tractable without the series:

• Adiabatic theorem: if $\hat{H}(t)$ changes slowly compared to $\hslash /\Delta E$ (the inverse gap), a system stays in the instantaneous eigenstate, acquiring a dynamical phase and a geometric (Berry) phase. Basis of adiabatic quantum computation and topological phases.
• Sudden approximation: if $\hat{H}$ changes abruptly, the state has no time to evolve and is simply re-expanded in the new eigenbasis; transition probabilities are overlaps $|⟨f_{\text{new}}|i_{\text{old}}⟩{|}^2$ (e.g. $\beta$-decay shake-up).

See also

• The Heisenberg / interaction picture — the formalism used here.
• Time-independent perturbation theory — the stationary counterpart.
• Uncertainty relations — lifetime and linewidth.
• QFT: Decay Rates — the relativistic sequel.