Time-Dependent Perturbation Theory

11.1 Fermi”s Golden Rule

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 $\rho(E_f)$:

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

11.2 Selection Rules for Electric Dipole Transitions

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.

11.3 Spontaneous Emission and Einstein Coefficients

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.