← Part VI/Fermi's Golden Rule

5. Fermi's Golden Rule

Reading time: ~28 minutes | Pages: 7

Transition rate to continuum of final states - foundation of decay and scattering theory.

The Problem

Previous formalism gave transition to discrete state $|f\rangle$

Many physical processes involve continuum:

Atomic decay (photon emitted into free space)
Ionization (electron ejected into continuum)
Scattering (final momentum not quantized)
Particle decay

Density of States

Number of states per unit energy:

$$\rho(E) = \frac{dN}{dE}$$

For particle in box with volume $V$:

$$\rho(E) = \frac{V}{(2\pi)^3}\frac{d^3p}{dE} = \frac{Vp^2}{2\pi^2\hbar^3}\frac{dp}{dE}$$

For non-relativistic particle $(E = p^2/2m)$:

$$\rho(E) = \frac{V}{2\pi^2\hbar^3}(2m)^{3/2}\sqrt{E}$$

Fermi's Golden Rule

Transition rate from initial state $|i\rangle$ to group of final states:

$$\Gamma_{i\to f} = \frac{2\pi}{\hbar}|V_{fi}|^2\rho(E_f)$$

$\Gamma$ has units of inverse time (decay rate)
$V_{fi} = \langle f|\hat{H}'|i\rangle$ is matrix element
$\rho(E_f)$ evaluated at $E_f = E_i$ (energy conservation)
Valid for weak, constant perturbations

Derivation Sketch

Step 1: From time-dependent perturbation theory:

$$P_{i\to f}(t) = \frac{|V_{fi}|^2}{\hbar^2}\frac{\sin^2[(\omega_{fi})t/2]}{(\omega_{fi})^2}$$

Step 2: Sum over final states in energy range $[E, E+dE]$:

$$P_{total}(t) = \int |V_{fi}|^2\frac{\sin^2[(\omega_{fi})t/2]}{(\omega_{fi})^2}\rho(E_f)dE_f$$

Step 3: For large $t$, $\frac{\sin^2(x)}{x^2} \to \pi t\delta(x)$

Step 4: Transition rate = $\frac{dP}{dt} = \frac{2\pi}{\hbar}|V_{fi}|^2\rho(E_f)$

Physical Interpretation

Transition rate proportional to:

Coupling strength: $|V_{fi}|^2$ - how strongly states interact
Phase space: $\rho(E_f)$ - how many final states available

More available final states → faster transition

Lifetime and Decay

If state can decay to multiple channels:

$$\Gamma_{total} = \sum_{\text{channels}}\Gamma_i$$

Lifetime:

$$\tau = \frac{1}{\Gamma_{total}}$$

Population decays exponentially:

$$N(t) = N_0 e^{-t/\tau} = N_0 e^{-\Gamma t}$$

Example: Spontaneous Emission

Atom in excited state $|e\rangle$ decays to $|g\rangle$ + photon

Interaction: $\hat{H}' = -\vec{d}\cdot\vec{E}$ (electric dipole)

Photon density of states:

$$\rho(\omega) = \frac{V\omega^2}{\pi^2 c^3}$$

Decay rate (Einstein A coefficient):

$$\Gamma = \frac{\omega^3|\langle g|\vec{d}|e\rangle|^2}{3\pi\epsilon_0\hbar c^3}$$

Example: Photoionization

Photon ionizes atom: bound state → continuum

$$\Gamma = \frac{2\pi}{\hbar}|V_{fi}|^2\rho(E_{\text{electron}})$$

where $E_{\text{electron}} = \hbar\omega - |E_i|$

Cross section:

$$\sigma(\omega) = \frac{4\pi^2\alpha\hbar\omega}{c}|M_{fi}|^2\rho(E_f)$$

Selection Rules

If $V_{fi} = 0$ by symmetry, transition forbidden (or highly suppressed)

Electric dipole selection rules for hydrogen:

$\Delta \ell = \pm 1$
$\Delta m = 0, \pm 1$

Forbidden transitions can occur via:

Magnetic dipole ($\sim \alpha^2$ weaker)
Electric quadrupole ($\sim (ka)^2$ suppressed)
Two-photon processes

Validity Conditions

Fermi's Golden Rule valid when:

Weak coupling: $|V_{fi}| \ll E_i$
Continuum: Closely spaced final states
Irreversible: No return to initial state
Long times: $t \gg \hbar/|V_{fi}|$

Applications

Atomic physics: Spontaneous emission lifetimes, photoionization cross sections
Nuclear physics: Beta decay, alpha decay rates
Particle physics: Particle decay widths, scattering cross sections
Solid state: Electron-phonon scattering, optical absorption
Quantum computing: Decoherence rates, gate errors

← Time-Dependent Perturbation Variational Method →

Quantum Mechanics Course