Selection Rules and Applications

When an atom interacts with electromagnetic radiation, the vector potential produces the perturbation $\hat{H}' = -(e/m_e c)\hat{\vec{A}}\cdot\hat{\vec{p}}$. In the long-wavelength limit ($\lambda \gg a_0$), the spatial variation of the field over the atom is negligible, and the perturbation simplifies to:

where $\hat{\vec{d}} = -e\hat{\vec{r}}$ is the electric dipole operator. The transition matrix element is:

The vector $\langle f|\hat{\vec{r}}|i\rangle$ is the transition dipole moment. If it vanishes, the transition is "forbidden" at the electric dipole level (but may still occur via higher-order multipoles).

The transition dipole moment $\langle n'l'm'|\hat{\vec{r}}|nlm\rangle$ is non-zero only when specific conditions are met. Using the Wigner-Eckart theorem and properties of spherical harmonics:

The physical origins of these rules:

• $\Delta l = \pm 1$ (parity rule): The photon carries one unit of angular momentum. Since $\hat{\vec{r}}$ is a vector operator (rank 1 tensor), it can only change $l$ by one. Equivalently, $\hat{\vec{r}}$ is odd under parity, so the initial and final states must have opposite parity.
• $\Delta m_l = 0, \pm 1$ (magnetic quantum number): Determined by the polarization of the radiation. $\Delta m = 0$ for linearly polarized light along $z$; $\Delta m = \pm 1$ for circularly polarized light.
• No restriction on $\Delta n$: The radial integral $\int R_{n'l'} r R_{nl} r^2 dr$ is generally non-zero for any $n, n'$, but favors small $|n'-n|$.

Examples: $2p \to 1s$ (allowed), $3d \to 2p$ (allowed), $2s \to 1s$ (forbidden: $\Delta l = 0$), $3d \to 1s$ (forbidden: $\Delta l = 2$).

Stimulated emission requires an external field, but atoms in excited states also decay spontaneously. A semi-classical treatment (treating the radiation field classically but including vacuum fluctuations) gives the spontaneous emission rate:

Key features:

• The rate scales as $\omega^3$: higher-frequency (shorter wavelength) transitions decay much faster
• This explains why optical transitions ($\sim 10^{15}$ Hz) are fast (~ns) while radio-frequency transitions ($\sim 10^9$ Hz) are extremely slow (~years)
• The 2p state of hydrogen has a lifetime $\tau = 1/A \approx 1.6$ ns

Einstein's 1917 analysis of thermal equilibrium between atoms and radiation established three fundamental processes and their rate coefficients:

Spontaneous emission (rate $A_{21}$):

Stimulated emission (rate $B_{21}\, u(\omega)$):

Absorption (rate $B_{12}\, u(\omega)$):

where $u(\omega)$ is the radiation energy density. The Einstein relations connect these coefficients:

These relations follow from requiring consistency with the Planck blackbody spectrum at thermal equilibrium. They show that spontaneous and stimulated processes are fundamentally linked.

When the perturbation is not weak, first-order perturbation theory breaks down. For a two-level system driven by a resonant sinusoidal field, the problem can be solved exactly. The Hamiltonian in the rotating-wave approximation is:

where $\delta = \omega - \omega_{21}$ is the detuning and $\Omega_R = |V_{21}|/\hbar$ is the Rabi frequency. The transition probability oscillates:

Key features of Rabi oscillations:

• On resonance ($\delta = 0$): $P_{1\to 2}(t) = \sin^2(\Omega_R t/2)$. Complete population transfer at $t = \pi/\Omega_R$ (a "$\pi$-pulse").
• Off resonance ($\delta \neq 0$): Oscillation frequency increases to $\sqrt{\Omega_R^2 + \delta^2}$, but maximum probability decreases to $\Omega_R^2/(\Omega_R^2 + \delta^2) < 1$.
• Perturbative limit ($\Omega_R \ll \delta$): Reduces to the perturbation theory result $P \propto \Omega_R^2/\delta^2$ with small oscillations.