Deriving Fermi's Golden Rule

Fermi's Golden Rule is the bedrock of spectroscopic transitions. It dictates the probability rate of a system transitioning from an initial state $ |i\rangle $ to a final state $ |f\rangle $ under a weak perturbation. To understand where this equation comes from, we must turn to Time-Dependent Perturbation Theory (TDPT).

1. The Unperturbed System

Assume we have a quantum system (like an atom) governed by an unperturbed Hamiltonian $ H_0 $. Its eigenstates $ |n\rangle $ satisfy $ H_0 |n\rangle = E_n |n\rangle $. The time evolution of these states is simply: $$ \Psi_n(t) = |n\rangle e^{-i E_n t / \hbar} $$

2. The Electromagnetic Perturbation

At time $ t = 0 $, we turn on an X-ray beam. The X-ray is an oscillating electromagnetic field, which we treat as a time-dependent perturbation $ H'(t) $. In the electric dipole approximation, this is: $$ H'(t) = -e \mathbf{E}_0 \cdot \mathbf{r} \cos(\omega t) = V \left( e^{i\omega t} + e^{-i\omega t} \right) $$ where $ V = -\frac{1}{2} e \mathbf{E}_0 \cdot \mathbf{r} $, $ \mathbf{E}_0 $ is the electric field amplitude, and $ \omega $ is the frequency of the X-ray.

3. First-Order Transition Amplitude

We want to find the probability amplitude $ c_f(t) $ of finding the electron in a final state $ |f\rangle $ at time $ t $, given it started in state $ |i\rangle $ at $ t=0 $. First-order TDPT gives the differential equation: $$ i\hbar \frac{dc_f}{dt} = \langle f | H'(t) | i \rangle e^{i \omega_{fi} t} $$ where $ \omega_{fi} = (E_f - E_i)/\hbar $ is the Bohr frequency between the states.

Integrating this from $ t = 0 $ to $ t $, and inserting our harmonic perturbation, we get: $$ c_f(t) = \frac{\langle f | V | i \rangle}{i\hbar} \int_0^t \left( e^{i(\omega_{fi} + \omega)t'} + e^{i(\omega_{fi} - \omega)t'} \right) dt' $$

4. Resonance and the Sinc Function

For X-ray absorption, the photon energy $ \hbar\omega $ must roughly match the energy difference $ E_f - E_i $ (so $ \omega \approx \omega_{fi} $). This makes the $ e^{i(\omega_{fi} - \omega)t'} $ term (the absorption term) dominate over the rapidly oscillating $ e^{i(\omega_{fi} + \omega)t'} $ term (which corresponds to stimulated emission). Dropping the emission term (the Rotating Wave Approximation) and integrating yields: $$ c_f(t) \approx \frac{\langle f | V | i \rangle}{i\hbar} \left[ \frac{e^{i(\omega_{fi} - \omega)t} - 1}{i(\omega_{fi} - \omega)} \right] $$

The probability of transition is $ P_{i \to f}(t) = |c_f(t)|^2 $. Multiplying $ c_f(t) $ by its complex conjugate gives: $$ P_{i \to f}(t) = \frac{4 |\langle f | V | i \rangle|^2}{\hbar^2} \left( \frac{\sin^2[(\omega_{fi} - \omega)t / 2]}{(\omega_{fi} - \omega)^2} \right) $$

5. The Long-Time Limit (Dirac Delta)

As time $ t $ grows, the function $ \frac{\sin^2(xt)}{x^2} $ becomes sharply peaked at $ x = 0 $ and its area approaches $ \pi t \delta(x) $. Replacing the sinc-squared term with the Dirac delta function gives: $$ \lim_{t \to \infty} \frac{\sin^2[(\omega_{fi} - \omega)t / 2]}{(\omega_{fi} - \omega)^2} = \frac{\pi t}{2} \delta(\omega_{fi} - \omega) $$

Because $ \delta(\omega_{fi} - \omega) = \hbar \delta(E_f - E_i - \hbar\omega) $, the probability becomes linearly proportional to time: $$ P_{i \to f}(t) = \frac{2\pi t}{\hbar} |\langle f | V | i \rangle|^2 \delta(E_f - E_i - \hbar\omega) $$

The Result: The Golden Rule

Finally, the transition rate $ W_{fi} $ is the derivative of the probability with respect to time ($ W_{fi} = dP/dt $). The $ t $ drops out, leaving the monumental equation: $$ W_{fi} = \frac{2\pi}{\hbar} |\langle f | V | i \rangle|^2 \delta(E_f - E_i - \hbar\omega) $$ This is Fermi's Golden Rule. It states that the transition rate is driven entirely by the square of the transition matrix element (the dipole operator) and strictly enforces energy conservation via the delta function.