Quantum Mechanics Course

Total: 450+ pages covering mathematical foundations through advanced quantum mechanics

Part VI, Chapter 4 | Page 2 of 3

Fermi's Golden Rule

Transition rates, density of states, and the photoelectric effect

Fermi's Golden Rule is arguably the most widely used result in quantum mechanics. It converts the time-dependent perturbation theory formulas into a simple, practical expression for the transition rate when the final state lies in a continuum. Named by Fermi (who called it "Golden Rule No. 2"), it underpins all of quantum scattering theory, spectroscopy, and nuclear/particle physics.

From Probability to Rate

Recall the transition probability for a constant perturbation:

$$P_{i\to f}(t) = \frac{|V_{fi}|^2}{\hbar^2}\frac{4\sin^2(\omega_{fi}t/2)}{\omega_{fi}^2}$$

For large $t$, we use the distributional identity:

$$\lim_{t\to\infty} \frac{\sin^2(\omega t/2)}{\pi(\omega/2)^2 t} = \delta(\omega)$$

Therefore, in the long-time limit, summing over a continuum of final states with density $\rho(E_f)$:

$$\sum_f P_{i\to f}(t) \to \int P_{i\to f}(t)\,\rho(E_f)\,dE_f = \frac{2\pi}{\hbar}|V_{fi}|^2\rho(E_f)\,t$$

The total probability grows linearly with time, giving a well-defined transition rate (probability per unit time).

Fermi's Golden Rule

The transition rate from an initial state $|i\rangle$ to a continuum of final states is:

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

where:

  • $\Gamma_{i\to f}$ is the transition rate (probability per unit time, in units of $\text{s}^{-1}$)
  • $|\langle f|\hat{V}|i\rangle|^2$ is the squared matrix element of the perturbation
  • $\rho(E_f)$ is the density of final states at the energy $E_f = E_i$ (energy conservation)

Energy conservation is built in: the delta function $\delta(E_f - E_i)$ enforces that transitions only occur to states with the same energy as the initial state (for a constant perturbation) or to states with $E_f = E_i \pm \hbar\omega$ (for a sinusoidal perturbation).

Density of States

The density of states $\rho(E)$ counts the number of quantum states per unit energy interval. It depends on the physical situation:

Free particle in 3D (box of volume $V$):

$$\rho(E) = \frac{V}{2\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\sqrt{E} = \frac{4\pi V\, p^2}{(2\pi\hbar)^3}\frac{dp}{dE}$$

Photon modes (radiation field):

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

Solid angle differential (scattering):

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

The density of states converts the Fermi Golden Rule from a formula for transitions to a specific state into a formula for the total transition rate.

Fermi's Golden Rule for Sinusoidal Perturbations

For a harmonic perturbation $\hat{H}'(t) = \hat{V}\cos(\omega t)$, the Golden Rule becomes:

$$\Gamma_{i\to f} = \frac{\pi}{2\hbar}|V_{fi}|^2\,\rho(E_i + \hbar\omega) \quad \text{(absorption: } E_f = E_i + \hbar\omega\text{)}$$
$$\Gamma_{i\to f} = \frac{\pi}{2\hbar}|V_{fi}|^2\,\rho(E_i - \hbar\omega) \quad \text{(stimulated emission: } E_f = E_i - \hbar\omega\text{)}$$

The factor of $1/4$ difference from the constant perturbation case comes from the $1/2$ in $\cos(\omega t) = (e^{i\omega t} + e^{-i\omega t})/2$.

Application: Photoelectric Effect

A photon of frequency $\omega$ ejects an electron from a hydrogen atom (initially in the ground state). The perturbation is the electric dipole coupling:

$$\hat{H}'(t) = -e\hat{\vec{r}}\cdot\vec{\mathcal{E}}_0\cos(\omega t) = -e\mathcal{E}_0\hat{z}\cos(\omega t)$$

The initial state is the hydrogen 1s state; the final state is a free electron with momentum $\vec{p}$ (plane wave). The matrix element is:

$$V_{fi} = -e\mathcal{E}_0\langle\vec{p}|z|1s\rangle = -e\mathcal{E}_0\int \frac{e^{-i\vec{p}\cdot\vec{r}/\hbar}}{V^{1/2}}\, r\cos\theta\, \frac{e^{-r/a_0}}{\sqrt{\pi}a_0^{3/2}}\, d^3r$$

Evaluating the integral and applying the Golden Rule with the free-particle density of states:

$$\frac{d\Gamma}{d\Omega} = \frac{256\,e^2\mathcal{E}_0^2\, a_0^5\, p}{3\pi\hbar^4}\frac{\cos^2\theta}{(1 + p^2a_0^2/\hbar^2)^6}$$

Key features of this result:

  • The $\cos^2\theta$ angular dependence means electrons are preferentially ejected along the electric field direction
  • Threshold condition: $\hbar\omega > |E_{1s}| = 13.6$ eV (the photon must have enough energy to ionize the atom)
  • The cross section falls off rapidly with increasing photon energy (as $\omega^{-7/2}$ for high energies)
  • This explains Einstein's photoelectric effect quantum mechanically

Validity of Fermi's Golden Rule

The Golden Rule is valid when:

  • Perturbation is weak: $\Gamma t \ll 1$ (system hasn't significantly decayed from initial state)
  • Time is long: $t \gg \hbar/\Delta E$ where $\Delta E$ is the energy range of final states (delta function has formed)
  • Continuum of final states: Discrete final states give oscillatory behavior, not a constant rate

These conditions can be summarized as: $\hbar/\Delta E \ll t \ll 1/\Gamma$. The rule fails for very short times (Zeno effect) and very long times (depletion of initial state).

Looking Ahead

On the next page, we explore selection rules for electromagnetic transitions, the electric dipole approximation, Einstein's A and B coefficients, and Rabi oscillations in two-level systems -- the regime where perturbation theory breaks down and exact solutions become possible.

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