Time-Dependent Perturbation Theory

For a time-dependent perturbation $\hat{V}(t) = \hat{V}\,e^{-i\omega t}$ applied to an initial state $|i\rangle$The transition rate to a continuum of final states $|f\rangle$ is:

$\Gamma_{i \to f} = \frac{2\pi}{\hbar}|\langle f|\hat{V}|i\rangle|^2\rho(E_f)$

Where $\rho(E_f)$ is the density of final states at energy $E_f = E_i + \hbar\omega$.

Derivation. Using first-order time-dependent perturbation theory, the transition amplitude to state $|f\rangle$ is:

$c_f(t) = -\frac{i}{\hbar}\int_0^t \langle f|\hat{V}|i\rangle\,e^{i\omega_{fi}t'}\, dt'$

For a sinusoidal perturbation at frequency $\omega$:

$|c_f|^2 = \frac{|\langle f|\hat{V}|i\rangle|^2}{\hbar^2}\frac{\sin^2[(\omega_{fi} - \omega)t/2]}{(\omega_{fi} - \omega)^2/4}$

In the long-time limit, $\sin^2(xt)/x^2 \to 2\pi t\,\delta(x)$Giving:

$\frac{|c_f|^2}{t} = \frac{2\pi}{\hbar^2}|\langle f|\hat{V}|i\rangle|^2\,\delta(E_f - E_i - \hbar\omega)$

Summing over all final states with density $\rho(E_f)$:

$\Gamma = \int \frac{d|c_f|^2}{dt}\,\rho(E_f)\,dE_f = \frac{2\pi}{\hbar}|\langle f|\hat{V}|i\rangle|^2\rho(E_f) \quad \blacksquare$

The electric dipole matrix element:

$\langle f|\hat{\mathbf{d}}|i\rangle = -e\langle f|\mathbf{r}|i\rangle$

For hydrogen-like atoms, the selection rules are:

• $\Delta l = \pm 1$ (parity change required)
• $\Delta m = 0, \pm 1$ (for $z$, $x \pm iy$ polarisation respectively)
• $\Delta n$ unrestricted

The transition rate for $2p \to 1s$ in hydrogen:

$A_{2p \to 1s} = \frac{\omega^3}{3\pi\varepsilon_0\hbar c^3}|\langle 1s|e\mathbf{r}|2p\rangle|^2$

With $|\langle 1s|z|2p, m=0\rangle| = \frac{2^7\sqrt{2}}{3^5}a_0$This gives $A_{2p \to 1s} \approx 6.3 \times 10^8$ s$^{-1}$Corresponding to a lifetime $\tau \approx 1.6$ ns.

The Einstein $A$ coefficient (spontaneous emission rate) is related to the $B$ coefficient (stimulated emission/absorption):

$A_{21} = \frac{\hbar\omega^3}{\pi^2 c^3}B_{21}$

This relation, derived by Einstein in 1917 using thermodynamic arguments (detailed balance in a blackbody radiation field), was one of the first indications that spontaneous emission requires quantum electrodynamics.

1. Fermi’s Golden Rule requires a continuum: For transitions between discrete states (e.g., two bound states in an atom), the Rabi formula must be used. The density of states $\rho(E_f)$ in FGR ensures energy conservation is satisfied with finite probability.
2. First-order perturbation fails for strong fields: When the Rabi frequency $\Omega = |\langle f|\hat{V}|i\rangle|/\hbar$ is comparable to the detuning, higher-order effects (AC Stark shift, Rabi oscillations) become important.
3. Selection rules are not absolute: Forbidden transitions can proceed via higher multipole orders (E2, M1) or multiphoton processes, albeit at slower rates. A transition is truly forbidden only when all contributing channels vanish.
4. The rotating wave approximation (RWA): Dropping counter-rotating terms requires $|\omega - \omega_{fi}| \ll \omega + \omega_{fi}$. For very strong fields or ultrastrong coupling, the RWA breaks down.

• Laser cooling: Doppler cooling relies on repeated absorption—spontaneous emission cycles, with each cycle reducing atomic momentum by $\hbar k$.
• Optical pumping: Polarised light selectively populates magnetic sublevels via dipole selection rules, preparing spin-polarised atomic samples.
• Quantum optics: Spontaneous emission is the fundamental source of decoherence in quantum optical systems, limiting qubit lifetimes in cavity QED.
• Spectroscopy: Fermi’s Golden Rule underlies the interpretation of absorption spectra, photoemission, and inelastic scattering cross sections.
• Photovoltaics: The solar cell efficiency limit (Shockley—Queisser) derives from detailed balance between absorption and spontaneous emission.

• Quantum electrodynamics: Spontaneous emission is not predicted by non-relativistic quantum mechanics alone — it requires coupling to the quantised electromagnetic field vacuum.
• Scattering theory: The $T$-matrix formalism generalises FGR to higher orders. The optical theorem ($\sigma_{\text{tot}} = (4\pi/k)\,\text{Im}\,f(0)$) relates the forward scattering amplitude to the total cross section.
• Solid-state physics: Fermi’s Golden Rule describes electron—phonon scattering, carrier relaxation, and exciton decay in semiconductors.

Problem. A hydrogen atom in the ground state is illuminated by monochromatic light at $\lambda = 50$ nm, well above the ionisation threshold (13.6 eV). Estimate the photoionisation rate using Fermi’s Golden Rule.

Solution. The dipole matrix element for $1s \to$ continuum is approximately $\langle \epsilon_p | e\mathbf{r} | 1s \rangle \sim ea_0$. The density of continuum states at photoelectron energy $\epsilon = \hbar\omega - 13.6$ eV:

$k = \sqrt{2m\epsilon}/\hbar, \quad \rho(\epsilon) = \frac{V m k}{(2\pi)^3 \hbar^2} d\Omega$

The photoionisation cross section near threshold is $\sigma \approx 6.3 \times 10^{-18}$ cm$^2$ for hydrogen. At intensity $I = 10^{12}$ W/m$^2$, the rate $\Gamma = \sigma I / (\hbar\omega) \approx 10^{14}$ s$^{-1}$ — complete ionisation occurs within femtoseconds for intense fields.