# Definition Fermi's golden rule (which was actually derived by Dirac and named so by Fermi) is a quantum mechanical rule for describing the transition rate between quantum states. $ \Gamma_{i \rightarrow f} = \frac{2 \pi}{\hbar} \rho(E_f) |\langle f | H^{\text{int}}|i\rangle|^2 $ where $\rho(E_f)$ is the [[density of states|density of states]] of the system at $E = E_f$ and $H^{\text{int}}$ is the interaction (perturbation) [[Hamiltonian]]. For a periodic perturbation with angular frequency $\omega$, we get $ \Gamma_{i \rightarrow f} = \frac{2 \pi}{\hbar} \delta(E_f - E_i - \hbar \omega) |\langle f | H^{\text{int}}|i\rangle|^2 $ # Derivation We start from [[Time-Dependent Perturbation Theory]]. We can write any state $\ket{\psi(t)}$ as: $ |\psi(t)\rangle = e^{-i E_i t/\hbar} |i\rangle + \sum_f c_f(t) e^{-i E_f t/\hbar}|f\rangle $ where $\ket{i}$ is the initial state we transition *from*, $\ket{f}$ are the possible final states we transition *to* and $c_f(t)$ is the probability amplitude of finding the system in the state $|f\rangle$ at time $t$. Clearly, $c_f(0) = 0 \, \forall \, f$, since we assume the system starts in the state $\ket{i}$ at $t = 0$. Solving the time-dependent [[Schrödinger equation]], for a periodic perturbation with angular frequency $\nu$ (angular frequency $\omega$), we get: $ c_f(t) = -\frac{i}{\hbar}e^{it \frac{\Delta E_f}{2\hbar}} \frac{\sin \frac{\Delta E_f}{2\hbar}}{\frac{\Delta E_f}{2\hbar}} \langle f | H^\text{int} | i \rangle $ where $\Delta E_f \equiv (E_f - E_i) - h\nu$. The probability amplitude is $P_f(t) = |c_f(t)|^2$. Then, we have: $ \Gamma_{i \rightarrow f} = \lim_{t\rightarrow \infty} P_f(t)/t = \frac{2 \pi}{\hbar} \delta(E_f - E_i - \hbar \omega) |\langle f | H^{\text{int}}|i\rangle|^2 $